Mod-01 Lec-15 Unsteady Radial Flow in Leaky Aquifers (Contd.); Well Flow Near Aquifer Boundaries

Mod-01 Lec-15 Unsteady Radial Flow in Leaky Aquifers (Contd.); Well Flow Near Aquifer Boundaries


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

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