Mod-01 Lec-14 Unsteady Radial Flow in Confined and Unconfined Aquifers

Mod-01 Lec-14 Unsteady Radial Flow in Confined and Unconfined Aquifers


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

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