# Lecture 28: Well Hydraulics-1 Hi, ah this is lecture number 28 on Well Hydraulics.
So, the last class ah we discussed on ah you know a Aquifers Properties. So, and this class
we are going to ah see some of the well hydraulics mostly the ah ah water flow to well in confined
as well as in unconfined aquifers . So, before going to the details let us see
the draw down pattern in both confined as well as unconfined aquifers . So, in case
of confined aquifer, if you see clearly here ah so, there is a well which is penetrating
vertically down and and and into the you know the the confined aquifer. So, this is a confined
aquifer. So here if you ah observe so, this is a confined
aquifer. So, it has a thickness b and so, this is initial ah water level if you see
this this is initial water level so, before pumping. And so, if you take this as a datum
ok this is the datum . So, from here to here this is h naught. So, this is initial ah static
ah water level. So, in case of confined aquifer since it is. So, since the water is under
pressure between you know 2 layers right 2 ah you know ah impervious layers so, then
due to this pressure so, water level rises in ah the well.
So, the water level ah is at static piezometric level ok. So, that is h naught. So, and if
you see you know if you pump water continuously. So, at a particular time interval so, the
water level is going to be you know deplete down so, if you think that this is the water
level. So, ah the patterns this is called the drawdown curve. So, that any point if
you take. So, this is a 2 coordinates. So, one is the radial distance. So, that is R
and then the other is the head ok. So, the other is head. So, the radial distance from
here to here. So, from centre well to so, from centre of well to this point and then
your h ok So, so, this has 2 components R and h ok.
So, this is the drawdown curve. So, this varies with time ok and also with the distance look
at this. So, at at end of these end of this drawdown
so, there is a change in the the pressure with respected to ah distance will be 0, because
at at this point at this point there the the curve is tending to the flat like 0 slope.
So, d h by d l d r ah will be 0 at this point ok . Then so, this drawdown denoted with ah
denoted with S right so, and the drawdown which is the difference between the static
water level and the head ok. So, ah so, here the drawdown can be. So, this is h and the
drawdown S is equal to so, h naught minus h ok. So, the initial water level minus the
change in water level here h ,so, this difference will give the drawdown .
. So, ah and then ah the cone of depression here.
So, this is the pattern the drawdown pattern is like a cone, whereas this is the kind of
a cone. So, just like you you have you know the spongy material right, this is the spongy
material ah which is initially completely saturated, if you put a straw and take water
from here. So, it so, the water which is adjacent to this opening will be first ah removed,
then after that it will be influenced to the extreme points. So, if you see this. So, this
acts like a cone ok . So, so, the same thing is going to happen
here and similarly here unconfined aquifer. So, the aquifer is lying on a bed rock or
impervious surface impervious layer here. So, there is no other impervious layer here
and top and this is open to the atmosphere. And, the water level here we are seeing. So,
this is called the static water level and also water table this is called water table.
So, in confined case this is water table in ah. So, sorry in in unconfined case it is
water table where as confined case it is piezometric surface ok. So, ah the similarly if you take
water from unconfined aquifer so, same kind of drawdown pattern you can see ok. So, this
is. So, the distance between static ah water level and the the pumping water level. So,
this is called a pumping water level. So, so that is called drawdown ok .
So, and the next is so, some of the definitions are important in order to understand hydraulics
of the well. So, first one is static ah water level. So, some of the things we have already
discussed just in the previous slide. So, the water level at which water stands ah in
a well before pumping starts. So, that is what so, if you have a well right ah which
is penetrated into the aquifer ok . So, this is ah what you call ground level and this
is a impervious surface. And so, initially the water level is this. So, this is called
static water level. So, if you see ah so, before pumping whatever
water which is standing in inside ah the well. So, that level is called the static water
level. So, generally this is water table in phreatic wells whereas, ah ah piezometric
surface in case of confined aquifer or artesian wells ok .
So, the pressure at static water level is atmosphere. So, static water level generally
is atmosphere even in case of piezometric surface. So, the static water at the pressure
at ah water level I mean these water surface is the atmosphere. So, because then that that
really equals the pressure ok equals the pressure the atmosphere pressure. Suppose you have
pressurized ah force ok just like if you see the fire fighters. So, fire fighting force
if you if you see. So, this is initially under pressure and based on the particular pressure.
So, suppose the building is on fire right. So, the water is going to water is going to
ah I mean eject or eject to the building. So, the point where the water is reaching
at the end it cannot you know further eject ok.
So, this point at this point so, it will be equal to atmosphere ok this is a pressure
and this point equal to the atmosphere. So, the same thing here the water level rises
to the atmosphere equal to atmosphere. So, and the ah express as the vertical distance
from the ground surface to the water level in the well. So, this is known and static
water level for artesian wells is above water table because of the the the pressure due
to confined ah layers ok. And then piezometric surface so, this is a
artesian well. So, the height at which a water will stand in piezometric pipe open to the
atmosphere extend to the confine aquifers. So, the piezometric surface we are talking
about the water level in the well in confined aquifer ok. So, or on the water artesian well
or the water level in flowing well. So, flowing well the water level in the sense suppose
so, this is a well. So, the is going to flow like this the flowing well right . So, so
this is your ah piezometric surface. So, the rise in water in a pipe in general
h is equal to P by W where the pressure at the bottom and W unit weight of the ah water.
So, suppose this is the water and if you have a pipe in it right. So, the water level definitely
rises here. So, this is h ok. So, the water level h so, definitely this is the pressure
at this point that this is a pressure, pressure this point. So, ah. So, at that pressure and
then and then divided by unit weight of water will give the the rise in the water level
in the pipe ok. See the pumping water level here ah so, once
you pump water from the well. So, the time at which the water level ah present in the
ah well is called the pumping water level. So, the level of water in the well when pump
suppose you have ah this is the so, this is the well. So, initially this is called static
water level. So, then you started pumping it. So, then after that you have seen the
water level ah is depleted down. So, this is called at time t right. So, this is this
is called the pumping water level ok. So, this is a static water level h naught let
us say, this is pumping water level h . So, then drawdown so, drawdown is difference
between static water level and pumping water level. So, the same you see . So, ah this
is the bedrock and then have a well. So, static water level and this is the pumping water
level right. So, the difference between these 2 will give the sorry ah so, the difference
between these 2 will give the a drawdown. So, that is denoted with S.
So tube casing in the boundary point of drawdown ah here, so the limit of drawdown is the tube
casing. So, your drawdown should not go beyond the ah in the tube casing otherwise what happens.
So, the casing which has perforations right just for say perforations suppose this is
the thing right. So, the thing is the pumping water level should not go beyond the ah the
casing. So, what happened in that case? So, this will be free right and your yield will
be less the first of all and the second there is a chance of sediments enter into the system
ok. And, area of influence the area which gets
affected by the pumping of well is called area of influence suppose. So, here so, you
have a this is a drawdown curve right. So, the area so, the area so, if you take the
the top loop right . So, this is the well ok. So, the area which is influenced by this
well right is called the the pumping of the well is called area of influence. So, this
is the area of influence and the boundary is called the circle of influence this is
the circle of influence ok . And, then so, next is well yield. So, the
well yield is the volume of water discharge from it per unit time. So, in a unit time
how much volume of water which is discharged from the well is called ah well yield.
And, specific capacity of well so, this is yield per unit drawdown. So, for unit drawdown
ah the yield is ah denoted with specific ah capacity of the well and they are open wells.
So, open wells are also called dug wells and this to the water bearing formation or aquifer.
So, this ah drive water from formation close to the surfaces and large diameter of open
well the permits ah storage of large quantity. So, generally the diameter of open well varies
from 1.2 to 1.5 meter ok . So, generally these open wells are seen in irrigated fields nearby
irrigated fields where mostly the water being used for irrigations open wells ok .
And, then so, the next is so, I in the lasts ah you know ah presentation. So, we mentioned
2 properties of leakage aquifers one is the ah or semi confined aquifer ok. So, one is
hydraulic resistance and the leakage factor. So, here in case of ah so, I want to repeat
those to see some of the properties. So, if you see the hydraulic resistance, which
is a property of semi confined aquifer. So, hydraulic resistance denoted with C, which
is equal to b square by K b. So, b is leakage factor here and C hydraulic conductive resistance
that is in days and K is hydraulic conductivity of an of the aquifer, b is the thickness thickness
of the aquifer. So, here suppose you have a leaky ah leaky layer here and this is aquifer
let us say this is bedrock this is aquifer, aquifer. In the sense the soil which is completely
saturated with water right. So, and there is a leakage. So, water can go up or even
during recharge water can go enter in to the aquifer ok .
So, ah here K is the aquifer ah aquifer conductivity and b is aquifer thickness. And so, remaining
C and B are properties of your ah semi confined ah layer. So, in this equation, we are not
seeing any property related to semi confined ah layer. So, here generally ah so, we are
going to see that right ah. So, B so, C hydraulic resistance that is in days ok.
So, C is the ah generally so, C is equal to ok . So, b ah this is in day’s right. So,
b dash by k dash ok. So, where b dash is thickness of your ah I mean semi confined layer this
is b dash and k dash is hydraulic conductivity of the semi confined layer. So, that will
give the ah C ah there is hydraulic resistance. So, days, because this is in metres divided
by this is in metre per day you get ah days ok .
So, then ah so, the property here is when C is infinity so; that means, the hydraulic
resistance is infinity this is too much hydraulic resistance. So, then the K value is this is
K is impervious ok. So, the K is 0 and that is impervious. So, when this will be infinity
; that means K is 0. So, b let us say b is not equal to 0. So, in that case K equal to
0 whereas, when C equal to 0 ah what happen K should be equal I mean tends to infinity.
So; that means, this is an aquifers the perfect aquifer ok .
So, and then next property is leakage factor. So, in case of leakage factor as I said the
determines the distribution of leakage into or from the semi impervious layer. So, are
the same equation if we use C is equal to right ah B square by k b right. So, then B
is equal to ah C k b square root right K b c square root, where c is equal to D dash
or b dash by K dash, ah D dash is saturated thickness of aquifer and K dash is hydraulic
conductivity of the aquifer. So, these this aquifer is the semi confined
confined layer basically ok and B generally determined with pumping test.
So, high value of B indicates the greater resistance because C is proportional to B
. So, just we have seen increasing C the hydraulic resistance ah. So, definitely in a ah proportional
to B. So, B increases means c also increases ok .
So, then the next is tube wells. So, what is that tube wells? So, tube wells are constructed
by pushing a pipe below the ground surface and pushing through the different geological
formation, which consists a water bearing and non-water bearing strata ok. So, this
is a tube well is a pipe, which is being pushed down into the ah into the ground.
So, and also through the different the geological formations sometimes you know this is an aquifer
and this is not an aquifer right and again there is another aquifer ok. So, so that that
is the definition of tube well. So, it can connect to the single aquifer or multiple
aquifers ok . So, here in in tube well there are 2 parts
one part one is blind pipe and perforated pipe or well screen. So, this is a pipe if
you see ok ground surface and then there is a aquifer ok . So, then so, this the from
top to up to this this is called blind pipe, which does not contain any perforation ok.
Otherwise, the the next part the below there is a thin pipe. So, which is required because
this is the this is ah what is that aquifer? So, this collects the water through the perforations
ok . All right so, the filter points this is popular
in deltaic region. So, deltaic region if you see the water table is very shallow.
So, so, water table is shallow you do not require ah any blind by here. So, directly
you can ah use a screen pipe right the whole screen pipe can be used right. So, because
the water table is up here ok so, all maybe even will be less, but but since the water
table is shallow you do not require any ah blind pipe ok in case of the so, those are
filter points. So, then what is the cavity well. So, the
cavity well is a tube well. So, opposite to the filter point so, this consists of blind
pipe in does not contains any ah contain any screen pipe. So, here the cavity wells are
generally used in ah ah clay soils. So, like if if you have a clay layer if you have a
clay layer. So, this is a ground surface right and this is a clay layer. So, that is aquiclude
right aqui clude. So, then the only thing for this so, the clay
layer will act as and the support right. So, only the blind pipe is required up to these
and since this is an aquifer it does not require any screen point right. So, this really ah
have a stability right, it creates stability to the pipe and also does not allow to sinking
of well ok. So, then if you see if you take water from
here so, it forms a nice you know you know ah spherical bulb ok, spherical bulb through
3 D view if you see. So, it is like a like a bulb ok. So, water which is nearby is extracted
by this one ok. So, cavity well is a tube well penetrates
through confined aquifer and does not consist of a screen ok and the water penetrates only
through bottom opening. So, here so, this is a blind pipe
and is the ground surface this is a course sand is here and this is clay layer and also
sandy layer. So, we are looking for the clay layer. So, once it hits the clay layer and
that is it. So, water enters ah the critical velocity just like it clears a bulb ok here
. And, the next is partial penetrating well.
So, here if you see ah in previous case the the well was here up to this is a fully penetrating
well right. So, in case of partially penetrating well, so the water is ah flowing through well
in 3 dimensional basically. So, 3 dimensional why because the bottom since
the bottom is open partially I mean it is kept at partially, it is not fully pass through
the ah confined aquifer ok. So, there is a possibility of vertical flow down. So, this
is a horizontal flow ok, horizontal vertical flow. So, this all combines your 3 dimensional
ah flow, ah a well having screen length less than the aquifer thickness. So, this screen
length is not ah equal to the the aquifer thickness. So, then then that is called the
partially penetrating well. So, so, then ah what is the steady state ah
flow to well ok. So, in this case of studies steady state so, the pumping water level will
be equal to the recharge Q right Q 2. So, recharge Q 2. So, the pumping if the pumping
water level Q 1 is equal to Q 2, which is recharge ah or or the well water flow to well.
So, then the d v by d t is. So, the change in velocity will be 0. So, then this will
be to Q 1 is equal to Q 2 and we say the ah the flow is steady state in in the well ok
. So, then how to find out the ah the study
state flow to well in confined ah unconfined aquifer? So, in case of unconfined aquifer
so, this is this is the bed rock right. And, this is the ah ground surface and this is
the static water level or static this is ah water table. And, this is the well and of
course, ah after sometime after pumping for for you know at time t. So, the water level
reached to this point is called pumping water level, and the corresponding ah drawdown curve
is like this and this is d h by d r represented with and at any point here this is r right.
This is a distance from the centre of well and then the corresponding head at this point.
And, H is the head at the extreme point or it is called static water level table and
R R is the the radius of influence R ok. So, these with these parameters so, let us derive
an equation to find out a flow into the well ok .
So, here in case of steady state so, there are assumptions that ah flow is horizontal
and penetrating the well radially ok. So, the flow which is taking place this is horizontal
right and also this is horizontal, but radially it is doing it ok. And, then well is pumped
at a constant rate this Q is constant here right so, then the Q 2, which is also equal
to Q 1. And, Dupuit ah Forchheimer assumptions are
valid in this case. So, using the Darcy’s law because the Q which is ah going in is
equal to going out so, Darcy’s law is well valid. So, Q is equal to K i a right. So,
then the K is hydraulic conductivity. So, if you have so, if you have this is a
ah a well right ah . So, this is a this is a bedrock and this is a surface water and
salt surface and this is a well going Q right. So, then at this point if you so, see water
is flowing horizontally into the system so; that means, if you take any any you know point
here right. So, any any so, well point here ok so radially going .
So, this is the well. So, water is going horizontally. So, then ah so, d h by d r will give the your
your ah what you call this drawdown curve d h by d r and 2 pi r h 2 pi r h because the
what this is h for example, right. So, water is really flowing to the surface surface right.
So, we have to take the surface area. So, that is 2 pi r. So, this is 2 pi r into h.
So, that is surface area of cross section and ah and k is hydraulic conductivity. So,
then if you just re shuffle that h d h is equal to Q by 2 pi r 2 pi k into d r by d
r ok. So, then the next we are going to integrate that with the boundary condition.
So, then you can get the solution. So, h d h is equal to integration Q by 2 pi k into
d r by d r. So, this and boundary conditions are when r equal to r w so; that means, exactly
at the radius, radius of the well. So, r is equal to r w r w is the well radius. So, h
is equal to n h w. So, h at this point h is h w ok. Then, h is equal to h 8 suppose this
is a ah drawdown curve. So, when h is equal to h. So, when h is equal to h . So, that
will be possible only when r is equal to capital R. So; that means, that extreme point so,
in extreme point r equal to capital R. So, put these ah ah limits right r w R and
if you can integrate this finally, get the equation Q where Q to pi K H plus h w h w
into H minus h w ok . Q is equal to pi k into H square minus h w
square divided by l n of R by r w ok . Yeah. So, and the this is the same equation
ok, the same equation will are talking and these are the the definitions.
And, in the next so, how to estimate ah the aquifer properties like hydraulic conductivity
and transmissivity ok. So, hydraulic conductivity transmissivity so, K once you have one parameter
the other parameter can be estimated because T equal to k into aquifer thickness b ok.
So, here um in order to do that so, install a 2 observation wells observation well, one
and observation well 2 nearby the pumping well. So, this is a pumping well.
And, then so, in each observation well we are going to find out the draw down S 1 and
draw down S 2 ok. So, that is the ah use of this using 2 data ah draw down points. So,
you can find out the K value and T value. So, let us see how we can do that? Ok.
So, here ah in this ah in this. So, assume that there are 2 observations wells ok. So,
then in if you have an observation well nearby knowing so, draw down is equal to s is equal
to capital H, which is static water level minus small h which is the pumping water level
ok . So, then just ah substitute for a substitute for you know H, small h which is equal to
capital H minus s ok . In this in the previous equation Q is equal
to 5 h ah then h 2 square minus h 1 square because if you remember 5 k into H square
minus h w square right. So, let us this is also known as pi k into h ah h 1 ah h 2 square
minus h 1 square. So, the same thing so, this is h 2 ah the second observation point h 1,
the first observation point ok . So, now, substitute these values ah the h
2 and h 1 right. So, pi k into H minus H 2 square minus ah H minus S 1 square by l n
by r 2 by r 1 ok . So, the next so, and once that is done.
So, the pi K ah H minus S 2 square, H minus S 1 square into just multiply by 2 H and divide
by 2 H. So, take one here and see how it how we can transform? So, H minus H 2 square by
2 H right by 2 H is equal to H square plus H 2 square minus 2 H S 2 by 2 H.
So, the finally, if you simplify that ah this will be equal to H by 2 into S 2 dash S 2
dash ok. So, where S 2 dash is equal to S 2 plus S 2 dash square by 2 H ok. Now, substitute
this ah into here right into here. So, 2 pi 2 h pi K H by 2 ok so, minus S S 2 dash minus
H by 2 minus S 1 dash. So ah because this will be H minus S 2 2 H
is equal to so, H by 2 right ah minus S 1 S 2 dash similarly for S 1 ok. So, then simplifying
that you get 2 pi T by S 1 dash minus S 2 dash l n by r 2 by r 1 ok. So, where so, here
T is equal to so, pi not pi H into K. So, T is equal to K into H H is the thickness.
So, that is the static water level. So, ah substituting T in place of K H you get Q is
equal to 2 pi T into S 1 dash minus S 2 dash by l n by r 2 by r 1 .
So, this way so, the final ah so, finally, you get a T is equal to Q into l n r 2 by
r 1 2 pi into S 1 dash by S 2 dash. . So, let us see an example here.
So, the example so, you have a you have ah a 25 centimetre diameter well in an unconfined
aquifer pumped at uniform rate Q is equal to given. So, the drawdown observed at 1 meter
and hundred metre distance are 8 meter and 0.4 meter respectively, assume the thickness
of the saturated aquifer is 25 meter. So, the values given Q is given r 1 r 2 and
the corresponding S 1 S 2 also given. So, at r 1 1 meter ah this is 8 metre S 1 and
100 metre it is a 0.4 metre ok. So, then h 1 is equal to H minus r 1 ah sorry, draw down
observed at one metre hundred metre distance from the centre of the well are 8 metre and
0.4 metre ok . So, then um so, h 1 is equal to H minus r
1 h 2 is equal to H minus r 2. So, this will be 17 metre and this will be 24.6 meter and
substitute in in the unconfined aquifer r 2 by r 1 right. So, then so, substitute in
this equation. So, you will get ah the K value, which is
Q by ah pi h 1 square by h 2 square l n r 2 by r 1. And, substitute the values and K
value you get 0.0 1 3 9 and multiplying with ah H you get T is equal to 0.3475 metre square
per minute ok. So, ah this lecture, what we focussed is some
of the important definitions in case of well hydraulics. And, then how to ah determine
the flow to well during study state and unconfined aquifer ok and we also solved the problem
on unconfined case. So, and using knowing the ah knowing the draw down ah at 2 observation
points. So, we would be able to estimate ah the aquifer parameters such as K and T. So,
that is hydraulic conductivity and transmissivity. Thank you.