 Welcome to this lecture number 14 on unsteady radial flow in confined and unconfined aquifers. Here in this lecture, in the previous lecture we discussed on unsteady flow into the wells and in this lecture we will be moving on to unconfined as well as unconfined aquifers and of course, the flow is radial and it is unsteady. And here in this 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 unsteady radial flow in confined aquifers. So, in the previous lecture we had this, we discussed the solution of Thay's equation by two graphical methods namely one by type curves and two by Cooper Jacob approximation. And in this lecture we will discuss another method for the solution of the Thay's equation which is used in the unsteady radial flow in confined aquifers and that method is the another method for solution of Thay's equation by this developed by Chow. So, which is this method, that is why this method is known as the Chow's method, Chow's method of solution. So, here what is done is, so in a pumping test, so the a curve is plotted and a semi-logarithmic plot, the observation data is plotted on a semi-log plot that means the drawdown axis is linear this one scale and the time axis is on logarithmic scale. And here, so this is on the plotted curve, so arbitrary points are chosen and their coordinates that is the time coordinate t as well as the drawdown s, so they are determined just like say here. So, this is the, this is drawdown s and then this is log of time t and here, so this is a tangent to the curve at a chosen point and determine the 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 the log t axis and here say this is maybe this is 1, this is 0, this is 1, this is 2, 3 and 4 and so on. So, here say 2 points are chosen on this plot such that the difference in log t is equal to 1 and so that the corresponding difference in this say for example, these are the 2 points here. So, this is, so this difference is 1 where and this is, so this difference is delta s that is the change in drawdown per that is 1 scale difference of the log value of time and here, so this f u is a parameter which is a function of this well function u, well function parameter. So, this is equal to s the drawdown divided by delta s, so drawdown divided by delta s, this is delta s and then the original value of this is s here. So, this is the value s the drawdown the typical value s and then, so this is delta s. So, this s by delta s is denoted as f u, then what is done is the corresponding values of w u and u are obtained from the figure. So, this figure is, so here we have this f u which is s by delta s and then here we have this w u that is the well function. So, here the corresponding values of w u and u are obtained from this figure. So, here this in this, so this f u, so this is also on logarithmic scale, so this is log scale and w is also on log scale. So, this is a log log plot of f u versus w u 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 the curve and so these along this the curve, so the values of the well function u are plotted are denoted and here typically, so this is say this f u axis may start with 10 to the power minus 1 and then this is 1, this is 0.1 and then this is 1 and then so this is 10 and similarly here this is, so this is 10 to the power minus 2 and then this is 10 to the power minus 1 and then this is 10 to the power 0 or this is 1 and then so here this is 10 and then so on. So, basically both these scales are, this is a log log plot and here typically the values of u, so they start with say here at the top it may start with say point this u is equal to point triple 0 1 that is 10 to the power minus 4 and so here we will get this is 10 to the u is equal to here, u is equal to 0.001 and somewhere here we will get u is equal to 0.01 and here this is u is equal to 0.1 and then here where the slope changes, so rough when this u is 1 and here this all the way it is close to this one, so this u is equal to say 3. So, typically this is how the value of the well function u changes and then so using this, so here what is done is, so using this equation that is f u is equal to s by delta s, so this one, so this f u is equal to s by delta s, so let us denote this equation as 1 and then so here the corresponding values of w u and u are obtained from the figure that is this figure. So, this is the relationship between f u and w u, so this is the Chou's relationship between f u, w u and u the well function u, so this u is the well function and so here what is done is the formation constants that is the transmissivity, the aquifer formation constant with transmissivity t is obtained by, so the equation that is the typical drawdown is equal to q divided by 4 pi t and into this w u, so here this w u is known, s is known, q is known and so therefore, t can be determined, so that is t the transmissivity t is equal to q divided by 4 pi s into w u and so this is the first this one and then next the the storativity, the storativity s is obtained by the equation s is obtained by, so this is a storativity s obtained by the equation that is r square by t is equal to 4 t by s that is the well function equation, so we know that well function is equal to r square s divided by 4 t into t, so by this expression we get that is, so this is the well function equation, from this the storativity s is equal to that is u into 4 pi and u into 4 transmissivity t into time since pumping divided by r square, so like this using this Chou's relationship between this f u and w u, so we can determine the formation constants of the aquifer and so now we will move on to the unsteady radial flow in unconfined aquifers, so this Chou's method is a third method by which we can solve the Thay's equation and now we will move on to unsteady radial flow in unconfined aquifers. So far we have seen the confined aquifer wherein it is under pressure and wherein the things are somewhat more straight forward I should say as compared to the unconfined condition and here in this case, so this there will be it represents 3 types of behaviors say 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 3 different kinds of nature, so the first one is this segment 1 wherein the slope is quite steep, so this here we can denote, so this as the segment 1 wherein the slope is you can say it is relatively steep, then so this is what happens is, so the as time increases the as time further increases the drawdown will just marginally increase and so this one here we can denote this as segment 2 or stage 2, segment 1 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 3, so here what happens is, so the gravity drainage and obviously in this unconfined aquifers, so the 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 flow towards well in unconfined aquifers is characterized by this slow drainage of interstices, so basically and water flow or rather ground water flow towards a well is towards a well in an unconfined aquifer shows a slow drainage through the interstices or the pores that is why initially what happens is when the pumping starts, so then so this drawdown increases and this is drawdown increases relatively steeply in this first segment and then once so this it has increased, so then what happens is, so the here, so the that is the cone of depression, so here there will be compaction of the aquifer as well as expansion of the water as pressure is reduced from pumping, so initially there is compaction of aquifer, compaction of water as pressure is as there is a reduction in pressure due to pumping, so this the first segment that is, so the first segment having the steep slope or steep drawdown, so here what happens is, so this will continue for a very short while and here so the drawdown reacts similar to a unconfined aquifer, so that means, so here this is the gravity here in this region it more or less behaves like a confined or artesian aquifer or pressure aquifer and afterwards what happens is, so this gravity drainage starts, so this is basically here you can say this is segment 1, here you can say this is analogous 2, so here this segment 1 is analogous 2 say confined aquifer or artesian aquifer or pressure aquifer, confined flow, next here the segment 2, so here so this is slow gravity flow in the segment 2 and here so this is because of the expansion of the cone of depression, so in the in segment 2 there is expansion of the cone of depression, here you can say there is a gradual expansion of the cone of depression and hence slow gravity drainage and so this continues and next is in the third segment in segment 3, 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 segments, segment 1 having analogous to confined flow, segment 2 having slow gravity flow and then in segment 3, so there is this is unsteady flow again which is may be again somewhat like segment 1 and then again the slow 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 developed for a fully penetrating well in an unconfined aquifer by Newman in 1975 as S is equal to the drawdown S is equal to Q divided by 4 pi T the discharge divided by 4 pi into transmissivity and here so this is a so well function of three parameters that is UA, UY, eta and here where this UA, so each of them represent one segment UA is equal to R square into storativity divided by TT, so in case of the unsteady flow in a confined aquifer it was the well function U is equal to R square S by 4 TT 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 so this is this W of UA, UY and so this is the note this is denoted as the unconfined well function UA is given by R square S by 4 TT and then UY is given by R square SY divided by TT and this is applicable for higher T values higher values of time. So, it represents segment 3 and this eta, so eta is given by R square KZ divided by B square KR, so here this KZ is the vertical hydraulic conductivity and KR is the horizontal hydraulic conductivity and obviously R is the radius and then B is the that is the unconfined aquifer thickness. So, using these 3 parameters that is this the unconfined well function is more complicated as compared to the well function in case of a confined aquifer wherein there is only one parameter that is well function parameter that is U is equal to 4 square that is R square S by 4 TT whereas, in this case it is a function of 3 parameters that is UA, UY as well as eta and here the theoretical curves for this UA and UY as well as eta are given by this Newman's curves. So, the Newman's curves for unconfined well function that is W UA, UY, eta, so here in this Newman's curves we have along the vertical axis of course, here also this is a log-log plot and so here we have, so this is W UA, UY and eta and here it starts with say 0.01 then 0.1, this is 1, 10, 100, so this is the unconfined well function which is plotted along the that is the vertical axis and then here we have that is 1 by UY along the horizontal axis. So, it starts with say 10 to the power minus 5 then 10 to the power minus 4, 10 to the power minus 3, 10 to the power minus 2, 10 to the power minus 1 then this is 1 then it further extends. So, this is 10, then this is 10 square or 100 and then this is 10 cube or 1000 and here what happens is, so somewhere between this 10 to the power minus 1 and 1, so here this is the curve it goes up to, so this is 1 and then similarly somewhere between 10 to the power minus 5 and 10 to the power minus 4 and so this curve goes up to say little over 10. Here, so these are the, so this is the, here actually let me, so this is the phase curve for 1 by UA and this is the phase curve for 1 by UY and in between we have, let me use the different color, this one that is, so this is 7, so this is 2, this is 2 and so here this is 0.2, next here this is 0.001, so this is 0.2 then this is 0.001, so these are all the, so this is the eta values, so this is eta is equal to 7, eta is equal to 2, eta is equal to 0.2 and then this eta, so this is eta is equal to 0.01, so this is how the theoretical curves for the, you can say this is the 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 confined, I am sorry, unsteady radial flow in a confined aquifer wherein, so it is, there is, it is only there is a well function parameter that is U is equal to r square s divided by 4t into t whereas, in this case, so there are 3 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, so which is a ratio of r square by b square that is the distance from the well axis, the radial distance from the well axis, square of that divided by the square of the unconfined aquifer thickness of course, that is a, itself is a variable and multiplied by the ratio of the vertical hydraulic conductivity and the horizontal hydraulic conductivity. So, like this in this, the unsteady radial flow in an unconfined aquifer is, even though it is the unconfined aquifer is the one which is much, which is the first aquifer as we encounter when we go from the ground surface, but here, so because it is at the top and then so there it is more, it is subjected to more fluctuations because of the natural as well as natural ground water recharge and it is the reason and many times, if it is a, this one in some cases where the, even the operation may also play a predominant role at if the, if there is a tropical desert kind of situations like Oasis or anything. So, therefore, so it represents more intensive, so unlike the unsteady radial flow in confined aquifers, so this unsteady flow in unconfined aquifers represents more this one. So, now with this, we will go to, we will just, I will briefly, that is start with, that is leaky aquifers, so that is the unsteady flow and here, so this leaky aquifers, so they may have either a bottom confining layer which may be leaky or they may have a top confining layer which may be leaky or even both the top layer as well as bottom layer may be leaky and of course, one, if the bottom layer is having more perforations or in that case, so what happens is there will be, the 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 then in that case, the, this leaky aquifer may gain in terms of ground water, so therefore, so it represents an entirely different this one and here, so this is the, the Walton presented the theoretical curves for leaky aquifer, so this is, so this leaky aquifer may have say with single leaky confining or rather confining, say semi confining layer, say leaky that means semi confining layer 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 1960 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 W U, R by B and here, we have this is 1 by U and of course, both are on this one and here, this access the W U, R by B, so that is, this is denoted as leaky well function, so this is the, so in case of confined aquifer, it is W U, there is a well function that is W U, whereas in case of unconfined aquifer, we have unconfined well function, so that is W U A U Y, eta whereas in this case, the leaky aquifer, so this is 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, R by B and it is the theoretical Walton's theoretical so it is here, the W U access will start at say 0.01 and then, this is 0.1, so this is 1 and then, 10 and then, similarly here, the 1 by U access will start at 0.1, 1, 10, 100, 1000 and 10000 or say 10 to the power 4 and here, so this Thais curve is the one which starts somewhere in between that is 0.1, that is 0.1 and 1 and so it starts here, so this is the Thais curve, so here, this is R by B is equal to 0.05 and this is the Thais curve and here, from the same point, so this is R by B is equal to 2.5 and in between, so there are different of this one, so this is R by B is equal to 1 and here, this is R by B is equal to 0.1, so like that, so this in this case, the drawdown which is a function of R and T is given by Q divided by 4 pi T into the leaky well function that is W of U, R by B and again, so here, this U is the same as the confined well function parameter that is R square S divided by 4 T T and this and R by B is equal to R into under square root K dash divided by K B B dash and this U is the same as the one and here, this is here, this B dash is the equator thickness, so this is the K dash is the hydraulic conductivity of aquitard, 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. We will stop here and we will continue in the next lecture and we will move on to the further topics in this well hydraulics. Thank you.