Welcome to this lecture number fifteen in

which we will continue with the previous lecture that is the unsteady radial flow in leaky

aquifer and here ahhh let us start with the ahh sketch actually the diagrammatically sketch

in which so there is a well it is fully penetrating into the leaky aquifer and

so this is the water table and here for simplicity we are considering

this aquitard that is the semi confining layer only at the top although actually the aquitard

can be at the bottom also instead of the top or it can be both at the top as well as bottom

which was briefly listed in the previous lecture but for the simplicity we are considering

only at the top aquitard or the top semi confining layer so this is and the bottom confining

layer is fully impervious so this is the aquitard art that is semi confining layer

so it is a top or a aquitard at the top and this is the

impervious confining layer at the bottom and this well

is fully penetrated here let me so this is the ahh water table and here so

there is a radially inward flow into this well which is fully penetrating into the leaky

aquifer so here let me mention here so this is the well fully penetrating a leaky aquifer

so here this is the this is the unconfined aquifer

and this is the leaky aquifer and this unconfined aquifer and this aquitard ahh through this

aquitard so there is a some contribution of groundwater into the leaky aquifer here so

this aquitard at the top so it has so the ahhh hydraulic conductivity there is k dash

similarly this leaky aquifer so with the hydraulic conductivity k and this one the thickness

of the aquitard is b dash and similarly the thickness of the leaky aquifer is b so with

this that means k and b represents the hydraulic conductivity and the thickness of the leaky

aquifer while k dash and b dash represents the hydraulic conductivity and thickness of

aquitard which is situated at the top so with this definition sketch so this is the the

definition sketch and in the previous lecture so it is a the expression for the leaky aquifer

well function was written as a as follows so this is denoted by

w of u comma r by b where this b represents this equivalent aquifer thickness that means

it is a thickness aahh which represents the thickness of leaky aquifer at the bottom having

a top aquitard and here so this is a the the thies modified theis equation is a the equate

aquifer well function is a w u comma r by b and here obviously this u is a r square

into the storage coefficient divided by 4 transmissivity into the time since pumping

and then r by b so this r by b so this r and b is it is r into under square

root k dash which is the hydraulic conductivity of the top aquitard divided by k into b into

b dash so this can also be written as r divided by under square t that is the transmissivity

of the leaky aquifer divided k dash by b dash which is ahhh the ratio of hydraulic conductivity

of the top aquitard as well as thickness of top aquitard so this is this represents this

bottom the denominator that is square root of t divided by ahh k dash by b dash that

is equivalent to b here and with this definition ahh sketch as well as parameter so now the

the equation for the draw down is given this q divided by 4 pi into the transmissivity

of the leaky aquifer multiplied by leaky aquifer well function that is wu comma r by b b is

the capital it is capital b it represents the equivalent ahhh thickness of the leaky

aquifer as well as in it as well which is confined by a top aquitard or top confining

layer and here so ahh this one and here let us consider ahh in case of a fully confined

aquifer so what happens is this incase of fullu confined aquifer so this k dash that

is the hydraulic conductivity of the aquifer becomes 0 because it is fully confined at

the top as well as the bottom so here for a fully confined aquifer k dash=0 so this

implies that b tends to infinity and therefore this w u comma r by b becomes wu that is leaky

aquifer well function so this becomes this gets transformed

into this confined aquifer well function which is simply known as well function

so therefore so this is a so this relationship for the drawdown in terms of pumping discharge

as well as the transmissivity of the leaky aquifer and this u which is the well function

parameter which is r square into storage storativity divided by 4pi t into t 4 times transmissivity

into time since pumping comma r by b so this r is the radius of the well as well as b is

the equivalent thickness of the leaky aquifer as well as the top aquitard so this one and

here also we can force this is a graphical method similar to the type curve which is

ahhh followed in the aquifer confined aquifer so here also that similar method can be used

here also so what is done here is so there is a log-log plot of log of wu comma r by

b versus log of 1 by u then another so this is the type curve so this is the type curve

for leaky aquifer similarly there is a plot the drawdown that is log of s versus log of

t this is the field curve for the leaky aquifer and both this will have a similar shape of

the this one and here what is done is so in this case this may be here this ahh field

curve so where will be different points depending upon the

and here this is the leaky aquifer well function which we already we discussed in the previous

lecture and here so this will be something like this so if we match the field curve over

the type curve so that most of this data points on the field curve will overlap or will lie

more or less on the almost on the the type curve so in that case while at the same time

maintaining the parallel nature of the axis so this is a so here i am just and

next is this one the field curve and then so this is the

matching that is a matching point determination so this is the graphical method for for determining

leaky aquifer parameters formation constants so here what is done is so the firstly there

is the type curve having this log of wu comma r by b versus log of 1 by u so it will have

a this kind of nature and then here so it is matched with so this is log of t and this

is log of s so wherein we have number of data points like this so in this case ahh this

is the matching point so this is the and once we do that for any using any point we can

get the we can determine the ahhh the leaky aquifer formation function such as the transmissivity

and storativity so for any point so here let us say typical matching point

in this case so this represent ahhh log of t so this is log s for the matching point

similarly this represents log t for the matching point and this represents

that is log wu comma r by b and this represents log w and so therefore here what we can do

is using this matching technique we can determine the formation constants of

the leaky aquifer so by this matching technique so that is we know this s versus t so in that

case the corresponding this one is there so therefore this is u=r square s divided by

4tt so here this based on the matching point 1 by u is known so therefore u is known log

of 1 by u is known so therefore the ue this law 1 by is known and then this time is known

and then this r which is the radius of the leaky well fully penetrating through this

that is known so therefore we should be able to calculate the this one they so this is

one equation and then secondly we have the other equation this is s=q divided by4 pi

t into w u comma r by b so here this is known this is known based on this so this is known

so therefore t=that is q divided by 4 pi into s into wu comma r by b so this is a for

the so this is for the matching point once this t is known so then so this is here

it can call this as equation so this you can call equation one so

therefore using this equation one and the expression for the well function leaky aquifer

well function parameter that is u so we can determine this storativity of the well function

storativity of the leaky aquifer=u into 4 tt divided by r square so this is for the

matching point so this is also the matching point this also for the matching point so

this is equation two so like this once we match the both the curves similar to the one

which we did in the confined aquifer similar to the one we did in the confined

aquifer so here we are matching the individual points on this pumping test based on this

pumping test data where in the drawdown as well as the time since pumping that is plotted

in this curve and we match that with this the plot of that is the like is the the walton’s

theoretical curve for the leaky aquifer so this is the type curve so this is also the

walton’s theoretical curve so using this and matching maintaining the parallel nature

of the vertical as well as the horizontal axis so you get the matching point and then

for any point we will get four different values of log of the leaky well function parameter

as well as the log of 1 by u log of leaky well function as well as log of reciprocal

of the well function leaky well function parameter and the log of drawdown as well as the log

of time since pumping so then simply substitute all values and then obtain the the transmissivity

and ahh firstly the transmissivity using this expression that is the ahhh theis the modified

thies equation for the leaky aquifer as well as ahhh from that the the storativity ahh

of leaky aquifer using as per the equation two so like this we can solve leaky aquifer

and of course imagine and here we can ahh imagine how involved it is so here considering

things which is simply we have simplified it to a great extent so

that means although in case of a leaky theoretically or there can be it can have semi confining

layer at the top as well as bottom and more than that so the even between these two semi

confining top as well as bottom one may have less hydraulic conductivity one may have more

hydraulic conductivity so if a top aquitard is having less hydraulic conductivity then

the leaky aquifer may be losing its ground water volume whereas on the other hand if

the ahh ahhh that is the bottom ahhh confining semi confining layer is having a less hydraulic

conductivity then in that case it could be a gaining leaky aquifer generally even without

any pumping well so therefore we have even with the simplified one wherein the the bottom

confining layer is taken to be fully impervious and there is only the top that is the semi

confining layer or a aquitard layer for the leaky aquifer and that too we are considering

that to be having uniform hydraulic conductivity of k dash and a uniform thickness of b dash

with this the ahh we are we are in a position to determine the formation constants of the

leaky aquifer such as the transmissivity and storativity using this the equations so that

so this will complete ahh the unconfined radial flow in a leaky aquifer and now we will go

to that is the next component of this lecture that is the well flow near aquifer boundaries and here let us start with the well flow near

the stream and in this case say let us consider a stream and of course here i am ahhh exaggerating

the the depth of the stream and here there is a pumping well on one side of the stream

and of course we are considering only the so this is the stream this is the perennial

stream so this is the perennial stream

and then this is the pumping well fully penetrating well and we are considering

so there is a bottom confining layer which is common for the well as well the perennial

stream this is q okay let me follow the color code that is green for this representing water

and then so this is the fully penetrating well and here so we will once

so when there was no pumping so this is the so this is the original water table when there

is no pumping and once the pumping starts so there will be a cone of depression and this is the the shape of cone of depression

fully penetrating this is a pumping well pumping so here which is also can be denoted as a

discharging well and it is on one side of this perennial stream now here

so the this case the water table will have a flatter this cone of depression will have

a flatter slope away from the ahh stream and steeper slope towards the stream and this

one can be ahhh represented by that is to separate ahh this one that is say let me redraw

this one so here let me write here so this is a

flatter slope of cone of depression whereas this is a steeper slope of the cone of depression

steeper slope will be towards the perennial stream and the flatter slope is away from

the perennial stream and so these are the so this is the perennial stream so now this

can be thought of as a so let me redraw so this is q

and this is the the ground level and so this is the non-pumping or the original water table that is non-pumping water

table and here so the the cone of in this case suppose we replace this the upstream

bank i am sorry the near bank of the perennial stream by a barrier here the resultant cone of depression so this can be thought of as a symmetrical

that is a recharge well so this is so here this is the the imaginary this is the

this is the image recharging image well so which is imaginary then so this is symmetrical

with respect to the so this recharging image well so this is the line of symmetry so this

is the actual discharging well which is on one side of the perennial stream

and because of the perennial stream so it is getting more contribution more radial contribution

on the perennial stream side on the other side for this discharging well so therefore

this cone of depression shows this unsymmetrical slope the flatter slope on the side away from

the perennial stream and the steeper slope towards the perennial stream so this can be

explained by introducing so this is a an image well which is symmetrical with respect to

the line of symmetry so which line of symmetry represents the nearest bank of the perennial

stream to the discharging well and so here these distance are say if this is ahhh if

this distance is if this distance is –a – representing

because it is to the left of the line if symmetry and this will be + a so therefore here so

this is the resultant cone of depression and then this is the

cone of impression due to image well so this image well is analogous

in its effect it is analogous to contribution by this contribution to the discharge ahh

by the perennial stream so therefore this is the

cone of depression and this is the cone of impression so the

resultant of the cone of depression in the absence of perennial stream so therefore and

this perennial stream can be replaced by symmetrical image well when which is recharging so therefore

the contribution of this this one the perennial stream is a replaced by a symmetrical recharging

image well so finally we get this depression cone of depression as shown by this ahhh solid

green line which is resultant of cone of depression in the absence of perennial stream shown by

this ahhh dot and dash convention as well as the cone of impression due to the image

well which is a imaginary and which is having the same which is providing the same effect

as the perennials stream so here like this the because of the boundary so in this case

upstream that that is the near boundary of the perennial streamso the ahhh the cone of

depression will show an unsymmetrical slopes steeper slope towards the stream and

the flatters slope away from the stream now let us consider

the the next case where in there is a solid boundary

and so here in this case we can also draw the that is the for this figure we can also

draw the the flow net

so here we will get this is the pumping well and this is the imaginary recharge well the

recharge image well and here so the streamlines so there will be and then this is the boundary

here and here so this is the pumping well and then

this is the ahhh this is the discharging real well and this is the recharging

image well and as usual so this is a and here so like this we get so these are the streamlines for the recharging

image well and then similarly which are radially diverging radially outward as one having radially

outward shape were as the discharging real well the stream lines are converging and obviously

these are the streamlines and then the equipotential lines so will be ahhh circles then similarly

here the so these are the equipotential lines for the discharging real well okay anyway

it is obvious and so this is equipotential line and then this is the ahhh here is the

equipotential line this is the recharging image well so like this so this is the flow

net and next we will consider the case of a barrier

boundary so far we have studied we consider the case of a stream and now let us consider

the case of a barrier boundary and in this case so this is the barrier boundary and here we have a pumping well or discharging

well so this is a q from the discharging and then

the the static water table or the original water table so this is the

static water table that is original water table and let us consider so this height as

h0 from the impervious boundary and then so this is the and in this case the the final

the shape of this the water table so in this case it will be

reverse of the perennial stream so in case of perennial stream so the the cone of depression

was having steeper slope towards the well and the flatter slope away from the steeper

slope towards the stream and platter slope away from the stream in this case it is reverse

so because of the boundary so the contribution of ground water contribution towards the boundary

will be less therefore we will have a flatter slope whereas the ground water contribution

away from the boundary will be more so that it will have a steeper slope so in this case

so this is a flatter slope of cone of depression and this is a steeper slope of cone of depression

so this is just the barrier boundary so this is a just the reverse of a case having a perennial

stream so we will so this is a so we considered ahhh

in this first we considered the ahhh that is the well flow near ahh well flow near perennial

stream and next we will consider well we consider well flow near barrier boundary so we will

stop here and then i will continue the next lecture thank you

Sir…numerical problems bhi karwa dijiye