 Welcome to this lecture number 15 in which we will continue with the previous lecture that is the unsteady radial flow in leaky aquifers and here let us start with the sketch actually the diagrammatical 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 simplicity we are considering only 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 that is semi-confining layer so it is top or rather aquitard at the top and this is the impervious confining layer at the bottom and this well is fully penetrating and here let me so this is the water table and here so there is 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 and through this aquitard so there is some contribution of ground water into the leaky aquifer and here so this aquitard at the top so it has so the hydraulic conductivity that is K dash and 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 represent the hydraulic conductivity and the thickness of the leaky aquifer while K dash and B dash represent the hydraulic conductivity and the thickness of the aquitard which is situated at the top so with this definition sketch so this is the definition sketch and yesterday I am sorry the in the previous lecture so it is the expression for the leaky aquifer well function was written as follows so this is denoted by W of U, R by B where this B represents an equivalent aquifer thickness that means it is a thickness which represents the equivalent of leaky aquifer at the bottom having a top aquitard and here so this is the phase the modified phase equation is the leaky aquifer well function is that is W U, R by B and here obviously this U is R square into the storage coefficient divided by 4 transmissivity into the time since pumping and then this R by B so this R by B so this R by 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 root T that is the transmissivity of the leaky aquifer divided by K dash by B dash which is the ratio of the hydraulic conductivity of the top aquitard as well as the thickness of the top aquitard so this is this represents so this bottom the denominator that is square root of T divided by K dash by B dash that is equivalent to this B here and with this definition sketch as well as this parameter so now the equation for the drawdown is given by this Q divided by 4 pi into the transmissivity of the leaky aquifer multiplied by the leaky aquifer well function that is W U, R by B, B is the it is capital B which represents the equivalent thickness of the leaky aquifer as well as which is confined by top aquitard or top semi-confining layer and here so this one and here let us consider in case of a fully confined aquifer so what happens is this in case of fully confined aquifer so this K dash that is the hydraulic conductivity of the aquitard becomes 0 because it is fully confined at the top as well as bottom so here for a fully confined aquifer K dash is equal to 0 so this implies that B tends to infinity and therefore this W that is the U, R by B becomes W U 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 relationship for the drawdown in terms of the 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 4 T into T 4 times transmissivity into the times in pumping, 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 there is a graphical method similar to the type curve which is followed in the aquifer in the 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 W U, R by B versus log of 1 by U and then another so this is the type curve so this is the type curve for leaky aquifer similarly there is if we plot the drawdown that is log of S versus log of T so this is the field curve for the leaky aquifer and both this will have a similar shape of the so on and here what is done is so in this case say this may be here this in the field curve so there will be different points depending upon the and here this is the leaky aquifer well function which 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 these data points on the field curve will overlap or will lie more or less on the almost on the type curve so in that case while at the same time maintaining the parallel nature of the axis so this is so here I am just and the next is this one the field curve and then so this is the matching that is matching point determination so this is the graphical method 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 W U, R by B versus log of 1 by U so it will have this kind of a nature and then so here so it is matched with so this is log of T and this is a log of S so wherein we have a number of data points like this so in this case so this is the matching point so this is the and once we do that so for any using any point we can get the we can determine the leaky aquifer formation constants such as the transmissivity and storativity so for any point so here let us say typical matching point in this case so this represents log of S and similarly this represents log of T so this is a log S for matching point similarly this represents log T for matching point and this represents that is log W U, R by B and this represents log W U and so therefore here what you 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 is equal to R square S divided by 4 T T 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 U this 1 by U is known and U 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 the so this is one equation and then secondly we have the other equation that is S is equal to Q divided by 4 pi T into W of U, R by B so here this is known based on the matching point so this is known so therefore T is equal to it is Q divided by 4 pi into S into W U, R by B so this is for the so this is for the matching point once this T is known so then so this is here we can call this as equation so this you can call as equation 1 so therefore using this equation 1 in the expression for the well function leaky aquifer well function parameter that is U so we can determine the the storativity of the well function storativity of the leaky aquifer is equal to U into 4 T T divided by R square so this is for the matching point so this is also for the matching point this is also for the matching point so this is equation 2 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 the pumping test data where in the drawdown as well as the time since pumping that is plotted in this curve which is the field known as a field curve and we match that with this the plot of that is the it 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 so maintaining the parallel nature of the vertical as well as the horizontal axis so you get the matching point and then for any point so you will get 4 different values of log of the leaky well function parameter as well as a log of 1 by U log of leaky well function as well as log of reciprocal of the leaky well function parameter and the log of drawdown as well as the log of the time since pumping so then simply substitute all these values and then obtain the transmissivity and firstly the transmissivity using this expression that is the phase the modified phase equation for the leaky aquifer as well as from that the the storativity of the leaky aquifer using as per this equation 2 so like this we can solve leaky aquifer and of course imagine and here you can imagine how involved it is so here we are considering things which are simply we have simplified it to a great extent so that means although in case of a leaky aquifer 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 layers at the top as well as bottom one may have less hydraulic conductivity one may have more hydraulic conductivity so if the top aquitard is having less hydraulic conductivity then the leaky aquifer may be this one may be losing its ground water volume whereas on the other hand if the that is the bottom confining semi-confining layer is having 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 this one wherein the the bottom confining layer is taken to be fully impervious and there is only the top that is semi-confining layer or aquitard layer for the leaky aquifer and that too we are considering that to be having uniform hydraulic conductivity of k dash and the uniform thickness of b dash with this we are in a position to determine the formation constants of the leaky aquifer such as the transmissivity and storativity using this the equations given. So that so this will complete the 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. Here let us start with the well flow near a stream and in this case say let us consider a stream and of course here I am exaggerating the the depth of the stream and here there is a pumping well on one side of the stream and of course in this case we are considering only the so this is the stream so 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 as the perennial stream this is q let me follow the 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 the cone of depression fully penetrating and 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 in this case the water table will have a flatter this cone of depression will have a flatter slope away from the stream and a steeper slope towards the stream and this one can be represented by that is two separate this one that is so 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 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 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 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 recharge well so this is here this is the the imaginary this is the this is the the image well recharging image well so which is the imaginary and 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 than the other side for this discharging well so therefore this cone of depression shows this unsymmetrical slopes with the flatter slope on the side away from the perennial stream and a steeper slope towards the perennial stream so this can be explained by introducing so this is 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 distances are so if this is if this distance is if this distance is minus a minus representing because it is to the left of the line of symmetry and this will be plus 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 the contribution by this contribution to the discharge by the perennial stream so therefore the 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 I am sorry this is so therefore and this perennial stream can be replaced by a symmetrical image well when which is recharging so therefore the contribution of this this one the perennial stream is replaced by a symmetrical recharging image well so finally we get this resultant cone of depression as shown by this solid green line which is the resultant of the cone of depression in the absence of perennial stream shown by this dot and dash convention as well as the cone of impression due to the image well which is imaginary and which is having the same which is providing the same effect as the perennial stream so here so like this the because of the boundary so in this case it is the upstream that is the near boundary of the perennial stream so the the cone of depression will show a non-symmetrical slopes steeper slope towards the stream and a flatter slope away from the stream now let us consider the the next case wherein 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 then this is the imaginary recharge well recharge image well and here so the stream lines so they will be and then this is the boundary here and here so this is the pumping well and then this is the this is the discharging real well and this is the recharging image well and as usual so this is here so like this we get so these are the stream lines for the recharging image well and then similarly which are radially diverging radially outward as one having a radially outward shape whereas for the the discharging real well the stream lines are converging and obviously so they these are the stream lines and then the equipotential lines so will be circles then similarly here they so these are the equipotential lines for the discharging real well animates obvious and so this is equipotential line and then so this is the here this is the equipotential line for the recharging image well so like this so this is the flow net and next we will consider the the case of a barrier boundary so so far we have studied the we considered 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 a discharging well so this is q from this discharging well 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 the reverse of the perennial stream so in case of perennial stream so the the cone of depression was having a steeper slope towards the well and a flatter slope away from the steeper slope towards the stream and flatter slope away from the stream in this case it is reverse so because of the boundary so the contribution ground water contribution towards the boundary will be less so therefore it will have a flatter slope whereas the ground water contribution away from the boundary will be more so therefore 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 just the reverse of a case having a perennial stream so we will so this is so we considered in this firstly we considered the that is the well flow near well flow near a perennial stream and next we will consider well flow we considered well flow near a barrier boundary so we will stop here and then we will continue in the next lecture thank you.