Welcome to this lecture number fourteen on

unsteady radial flow in confined and unconfined aquifers here in this lecture in the previous

lecture ahh discussed on unsteady flow into the wells and in this lecture we will ahh

moving on to unconfined as well as unconfined aquifers and of course the flow is radial

and the it is unsteady and here in the ahh in the previous lecture so there was this

unsteady flow equation which was solved by two methods that is two graphical methods

one is the time drawdown method as well as the am sorry this the

unsteady radial flow and confined aquifers so

in the previous we had this we discussed the solution of theis equation

by two graphical methods namely 1 by type curves and two b cooper jacob

approximation and in this lecture we will discuss another

method for the solution of the theis equation which is used in the unsteady radial flow

in unconfined aquifers and that method is the

another method for solution of theis equation by this develop by chew so which is ahhh that

is why this method is known as the chew’s method chew’s method of solution so here

what is done is so in a pumping test so the curve is plotted and semi logarithmic plot

ahh observable observation observed data is plotted on a semi-log plot that means the

ahhh drawdown axis is ahhh linear this one scale and the time axis is on logarithmic

scale and here so this is ahhh on the plotted curve so the arbitrary points have chosen

ahh and the coordinates that is the time coordinate t as well as the draw down s so they are determined

just like say here so this is the this is ahhh drawdown s and then this is log of time

t and here so this is a tangent to the curve at a chosen and determine to drawdown difference

delta s in feet per log cycle of time so here actually say that is say suppose this is the

say these are the points suppose we are getting so what should be done is so this is a log

at axis and here say this is may be this is 1 ahh this is 0 this is 1 this is 2 3 4 and

so on so here say two points are chosen on this plot such that the difference in log

t=1 and so that the corresponding difference in this say for examples these are the two

points and here and this difference is 1 where and this is so this difference is delta s

that is the change in drawdown that is one scale difference of the log value in time

and here so this fu is a parameter which is function of this well function u well function

parameter so this function=s the drawdown divided by delta s so drawdown divided by

delta s so this is delta s and then the original value of this is s here so this is a the value

s the drawdown the typical value of s and then so this is delta s so this f by delta

s by delta s is a ahhh denoted as ahhh fu then what is done is the corresponding value

of w and u are obtained from the figure so this figure is

so here we have this fu which is s by delta s and here we have the w that is we function

so here so the corresponding value of w and u are obtained from this figure so here this

in this so this fu so this is also one logarithmic scale so this is a log scale and w is also

log scale so this is a log-log plot of fu versus wu and here so this curve will have

a shape of like this so initially the slope will be flat and eventually

the slope increases so this will be the type of curve and so these along this the curve

so the value of well function u are plotted are denoted and here say typically so this

is say this fu axis may start with 10 the power – 1 and then

this is 1 this is 1 and then this is 1 and this so this is 10 and similarly so this is

a this is the 10 to the power -2 and then this is 10 to the power -1 and then this a

10 to the power 0 and this is 1 and then so here this is the 10 and then so on so basically

both these scales are on this is log-log plot and here typically the values of u so they

start with say here it at the top it may start with say point this u=0001 that is 10 to

the power –4 and so here 10 to the power u=to here u=001 and somewhere here you

will get ahh u=001 and here this is a u=1 then this is here where the slope changes

so rough this u is 1 and here this all the way it is close to this one so this u=say

3 so typically this is how the value of the well function u changes and then using this

so here what is done is so they ahhh using this equation that is fu=s by delta s so

this one so this fu=s by delta n let us denote this

equation as 1 and then so here the corresponding w and u are obtain from the figure that is

this figure okay so this is the ahhh relationship between fu and w so this is chew’s relationship

between fu wu and u the well function u so this u is the well function

and so here what is done the is the formation constants that is the transmissivity

the aquifer formation constants with transmissivity t is obtained by

so the equation that is the typical drawdown=q divided by

4 pi t and into this w and here this w is known s is known q is known and so this t

can be determined so that is t the transmissivity t=q divided by 4 pi s into wu okay and so

this is the the first ahh this one the and then next the ths storativity the storativity

s is obtained by the equation s is obtained by so this is the storativity s

obtained by the equation that is r square by t=4t by s that is the well function equation

so we know that well function=r square s divided by 4t into t by this expression we

get that is ahh so this is the well function equation from this the storativity s=u into 4pi and

u into 4 transmissivity t into time since pumping divided by r square

so like this using this chew’s relationship between this fu and wu so we can determine

the formation constants of the aquifer okay so now we will move on to the unsteady radial

flow in unconfined aquifers so this chew’s method is a third method

by which we can solve the thies equation and now we will move on to unsteady radial flow

in unconfined aquifers so far we have the the confined aquifer where in the it is under

pressure the wherein the things are somewhat ahh more straight forward i should say as

compared to the unconfined condition and here in this case so this there will be it represents

three types of behavior suppose we plot the drawdown s versus time t on log scale so this

is log of s the drawdown and then the log of t the time since the pumping so here so

it indicates three different kinds of nature so the first one is this segment wherein the

slope is a quite steep so this here we can denote so this as the

segment one wherein the slope is you can say it is relatively steep then so this is what

happens is as a time in increases the as time further increases the drawdown just marginally

increase and so this one here you can you can denote this as segment 2 or stage 2 segment

having a relatively steep slope then segment 2 having a relatively flat slope then again

in the third this one so this is here you can say so this is segment three so here what

happens is so the the gravity drainage and obviously in an in this unconfined aquifer

so this drainage is by gravity and this gravity drainage

is not immediate and obviously that is why it constitutes an

unsteady flow condition and so the water float ah water flow towards well ahh in unconfined

aquifers is characterized by slow drainage of interspaces so basically ahhh and water

flow are at the ground water flow towards well is a towards a well in an unconfined

aquifer shows a slow drainage through the interspaces or the pores

that is why initially what happens is when the pumping starts so then so this drawdown

increase and the this is drawdown increase relatively steeply in this first segment and

then once this it has increased so when what happen is so the ahhh here so the

that is the cone of depression so here the compaction of the aquifer as well as expansion

of the water as pressure reduced from pumping so initially

is there a compaction of aquifer and expansion of water as pressure is as there is a is a reduction

in pressure due to pumping so this the first segment that is so the first

segment having a steep ahhh slope our steep draw down so here what happens is so this

ahhh ahhh will continue for a very short while and here so the drawdown reacts similar to

an unconfined aquifer so that means so here this is the gravity ah here in this region

it more or less behaves like a confined or artesian aquifer or pressure aquifer and after

wards what happens is so this gravity drainage so this is basically here you can say this

is segment one here you can say this is analogous to so here this segment one is analogous to

say confined aquifer or artesian aquifer or pressure aquifer confined flow next here the

segment two so here so this is slow gravity flow in this segment two and here so this

is because of the expansion of the cone of depression so in the in segment two there

is expansion of the cone of depression here we can say is a gradual expansion of

the cone of depression and hence slow gravity drainage and so next continues and next is in the third

segment in segment three so the time drawdown curve almost resembles

non equilibrium type curves that is unsteady ground water flow curves

so and therefore so there are three distinct segment segment one having analogous to confine

confined flow segment two having slow gravity flow and then in segment three so there is

this is a ahhh unsteady flow again which is maybe again somewhat like segment one and

then again this slope flattens like that so here so therefore in such case so this

the relationship between the drawdown and the discharge was drawdown s

and discharge q was a developed for a fully penetrating well

in an unconfined aquifer by newman in nineteen seventy five as s=the drawdown s=q divided by 4 pi t the

discharge divided by 4 pi into transmissivity and here so this is a so well function as

a 3 parameters that is ua uy eta and here where this ua so each of them represent one

segment ua=r square into storativity divided by tt so in case of the unsteady flow in a

confined aquifer it was the well function u=r square s by 4tt whereas in this case

so this ua is r square s divided by simply t the transmissivity multiplied by the time

since the beginning of pumping and here so this is a so this is a this w of ua uy and

so this is the this is denoted as the unconfined well function ua is given by r square s by

4tt and then uy is given by r square sy divided by tt and this is applicable for for higher

t values higher values of time so it represent segment three and this eta so eta is given

by r square kz divided by b square kr so here this this kz is the vertical hydraulic conductivity

and kr is the horizontal hydraulic conductivity and obviously r is the radius and the b is

the that is the unconfined aquifer thickness so using these three parameters are that is

this the unconfined well function is more complicated as compared to a well function

in case of confined aquifer wherein there is only one ahh parameter that is well function

parameter that is u=4 square that is r square s by 4t where as in this case it is a function

of three parameters that is ua uy as well as eta and here the theoretical curve for

this ua and ui as well as eta are given by this newmens curves so the newman’s curve

for unconfined well function that is wua comma uy comma eta so here in this newman’s curve

we have along the the vertical axis of course here also this is log-log plot and so here

we have so this is a this is a w ua uy and eta and here it is starts

with say 01 and then 1 and this is 1 10 100 so this is the unconfined well function which

is plotted along the vertical axis and then here we have that is a 1 by uy along the horizontal

axis so it starts with say 10 the power – 5 then 10 to the power – 4 10 to the power

-3 10 to the power -2 10 the power -1 then this is 1 then further extends so this is

10 then this is a 10 square of hundred and then this is 10 cube or 1000 and here what

happens is somewhere between this 10 to the power -1 and 1 so here this is the the curve

goes up to so this is 1 and then similarly somewhere between 10 the power -5 and 10 to

the power -4 and so this curve goes upto say little over 10 and here so these are the so

this is the here actually let me so this is the theis curve for 1 by ua and this is the

face curve for 1 by uy and in between we have color this one that is

so this is a 7 and say this 2 this is 2 and so here this is 2 and next here this

is 001 so this is 2 and this 001 so these are the so this is eta values so this is eta

=7 eta=2 eta=2 and this this eta so this is a eta=01 so this is how theoretical curves

for a the can say this is newman’s theoretical curves for unconfined well function so like

this here we will get the it is more complicated as compared to the unconfined flow radial

flow in a in a confined i am sorry un steady radial flow in a confined aquifer where in

so it is a there is a the it is only there is a well function parameter that is u=r

square s divided by 4t into t whereas in this case so there are there are three parameters

one is ua which is given by r square s by tt representing the first segment uy which

is equal to r square sy divided by tt which is applicable for the higher values of t representing

the third segment and then eta which is a ratio of square by b square that is the the

distance from the ahh well axis the radial distance from the well axis ahh square of

the that divided by the square of the unconfined aquifer thickness of course that is the itself

is a variable multiplied by the the the ratio of vertical hydraulic conductivity and the

horizontal hydraulic conductivity so like this so the in this the unsteady radial flow

in an unconfined aquifer is even though it is a unconfined aquifer is the one which is

much which is the first aquifer as you as we encounter when we go from the ground surface

but there so because it is at the top and then sot here it is more ahh it is subjected

to ahhh more fluctuations because of the natural as well as naturally ground water recharge

and it is the reason and many times if it is a aaa this one ahhh in some cases where

the even evaporation may also predominant role at if the they okay if there is a tropical

desert kind of situation like oss or anythingso therefore it is represents more intensive

so unlike the ahh unsteady radial flow in confined aquifer so this unsteady flow in

unconfined aquifer represents more this one so now with this we will ahhh go to we will

just briefly start with this leaky aquifers so that is the unsteady flow and here so this leaky aquifer so they may

have either bottom confining layer which may be leaky and

of course one if the bottom layer is having is having more perforations or in that case

what happens is there will be aquifer will be losing water whereas on the other hand

if the top confining layer is more leaky as compared to the bottom confining layer so

in that case the this leaky aquifer may gain in terms of ground water so therefore so they

it represents the entirely different this one and here so this is a the ahhh ahh walton

presented the theoretical curves for leaky aquifer so this is a so this leaky aquifer

may have say with single leaky confining or other confining means says semi confining

layer say leaky that means semi confining layer or with double semi confining layers

so double means this confining layer may be at the top or as well as at the bottom so

in this case the theoretical curves were developed by walton in nineteen so theoretical curves

for leaky aquifer so this is the walton’s theoretical curves for leaky aquifer so they

were developed in the year nineteen sixty and here so similar to the well function for

the unconfined aquifer here we have a well function for the leaky aquifer and that is

denoted by wu comma r by b and here we have this is 1 by u and of course

both are on this one and here this axis to the wru comma r by b so that is ahhh this

is denoted as leaky well function so this is the

so in case of confined aquifer it is wu this is well function that is w whereas in case

of unconfined aquifer we have unconfined well function so that is w ua uy eta whereas in

this case the leaky aquifer so this is a somewhere in between a confined aquifer and an unconfined

aquifer so here there are two parameters the first parameter is u and the second parameter

is r by b and so this is the leaky well function that is w u comma r by b and it is the theoretical

walton theoretical curve so it is here the wu axis will start at 01 and then this is

1 so this is 1 and then 10 then similarly here the 1 by u axis will start at 1 1 10

100 1000 and 10000 or say 10 to the power 4 and here the theis curve is the one which

starts somewhere in between that is 1 okay that is 1 and 1 and it is starts here and

this is the the theis curve so here this is r by b=05 and this is the theis and here

for this form same point so this is r aby b=25 and in between so there are different

this one so this r by b=1 and here this is b=01 so like that so in this case the

drawdown which is function of r and t is given q divided by 4 pi t into the leaky well function

that is w of u comma r by b and again so here this u is the same as the ahh confined well

function parameter that is r square s divided by 4tt okay and this and r by b=r into under

square k dash divided by kb b dash okay and this u is same as this one and here this is

the here this b dash is the aquitard thickness so this is the k dash is the hydraulic conductivity

of aquitard or leaky aquifer so that is leaky aquifer and b dash is the thickness of aquitard

that means leaky aquifer and k and b are the for the regular aquifer okay we will stop

here and we will continue ahhh in the next lecture on we will move on to the further

ahh topics in this well hydraulics thank you

excellent sir,

thanku sir for clearing my doubts……

Sir please a numerical on this