 Welcome to this lecture number 8 on permeability and its determination. So, in the previous lecture, we were discussing about the intrinsic permeability, which is essentially a function of flow medium only and it has the dimensions of square of length and its expression it is this is the intrinsic permeability is denoted by k 0 and it is linked with the permeability or hydraulic conductivity k by the formula k is equal to k 0 into g divided by nu, where nu is the kinematic viscosity and by this we also know that by Darcy's equation we have this hydraulic conductivity or permeability which can be expressed as velocity v of the ground water divided by the hydraulic gradient that is minus dh by dl. So, that will be the hydraulic conductivity or permeability and equating that to k 0 into g by nu. So, finally after simplifying we get an expression for this intrinsic permeability also known as specific permeability as this v the product of v the ground water velocity multiplied by nu the kinematic viscosity of ground water divided by the product of the acceleration due to gravity g and the hydraulic conductivity I am sorry the hydraulic gradient that is minus dh by dl. So, this is the expression for the intrinsic permeability in terms of the ground water flow velocity the ground water kinematic viscosity the acceleration due to gravity and the hydraulic gradient. So, this intrinsic permeability k 0 is measured in terms of as we have already discussed that it has the units of square of length it is measured in terms of that is micrometer square of micrometer that is square of 10 to the power 6 meter which is 10 to the power minus 12 meter square and in case of petroleum engineering. So, another unit known as Darcy is also under use and this Darcy is related to this micrometer square by this expression 1 Darcy is equal to 0.987 micrometer square. So, you can roughly say that 1 Darcy is almost equal to say 1 micrometer square. So, this is the intrinsic permeability which essentially is a function of flow medium only having a dimensions of square of the length. Now, let us come to the we have already seen in the previous class we have seen the the storativity or storage coefficient which is the volume of ground water that an aquifer can release or absorb per unit surface area per unit change in the head. And so this is denoted by the letter s and we have also seen the hydraulic conductivity which is given by the Darcy's expression as the ground water velocity is equal to the hydraulic conductivity k into the hydraulic gradient that is minus d h by d l. So, now these two storativity as well as permeability also known as hydraulic conductivity and storativity is also known as storage coefficient along with that there is a third parameter which forms what is known as the what are known as the three formation constants of the aquifer. And that is this transmissivity also known as transmissibility and it is defined as the ground water flow rate through a unit width of aquifer under a unit hydraulic gradient. So, here we are considering a unit width of aquifer and a unit the flow is taking place under a unit hydraulic gradient. So, we know that we have an expression based on the Darcy's law that the ground water flow rate q is equal to the ground water velocity which is given by k into i and the area of flow. So, therefore q is equal to k into i into a and by this definition of transmissibility or transmissibility. So, this q will be equal to t when this hydraulic gradient i is equal to 1 as well as the area of the ground water flow is equal to the unit width multiplied by the saturated thickness of the aquifer denoted by b. So, therefore when i is equal to 1 and a is equal to b the saturated aquifer thickness. So, then q is equal to t. So, therefore this transmissibility or transmissibility t is given by the product of the hydraulic conductivity k multiplied by the saturated thickness of aquifer that is b. So, in case of unconfined aquifer. So, this b will be varying because the unconfined aquifer is not confined at the top. So, therefore b will be varying as per the water table shape of the water table shape obviously the slope also. Whereas, in case of a confined aquifer. So, the saturated aquifer thickness is constant at a particular location and so therefore, so the there the transmissibility or transmissibility t is more constant as compared to that of an unconfined aquifer. So, the hydraulic conductivity k the transmissivity also known as permeability the transmissibility t also known as transmissibility and the storage coefficient that is s also known as storativity. So, these three form what are known as the formation constants of the aquifer and here this transmissibility t it has the expression given by k times b. k is the hydraulic conductivity b is the saturated aquifer thickness. So, therefore this transmissibility t its dimensions are l t to the power minus 1 into l. So, this is l square t to the power minus 1. So, this transmissibility is expressed in terms of meter square per second or meter square per day or it can be meter square per hour depending upon the time unit in which the ground water flow is described. Now let us come to the determination of hydraulic conductivity or permeability. So, this hydraulic conductivity or permeability can be determined by either field tests, laboratory tests and in the field tests we have pumping hole test that is pumping test of course, we also have say tracer test and we also have this auger hole test. So, these form what are known as the field test in which the hydraulic conductivity or permeability k is determined in the field and in the laboratory it can also be determined by what is known as the falling head permeability test or falling head permeability test also permeability test and the other one is the constant head permeability test or permeameter. So, the instrument which is used the experimental setup which is used for determining the hydraulic conductivity or permeability in the laboratory is known as permeameter and its specific purpose is the determination of the k or the hydraulic conductivity value. So, therefore, this is both is whether it is a constant head whether it is operating at constant head in that case it is known as constant head permeameter and whether it is operating at a falling head that means, when the head is falling continuously as the ground water flow takes place. So, in that case it is known as falling head permeameter. So, these two form the laboratory test whereas, the other is one that is the pumping test, tracer test, the auger hole test they form what are known as the field test. Now, let us come to firstly the constant head permeameter or say permeability test. So, here we have in the laboratory a setup in which the ground water is made to flow through a soil or a rock sample under constant head. So, let me draw the schematic diagram here. So, there will be a supply of water. So, this is the supply, this is the continuous supply. So, this is the overflow and here so this is the constant water level and here so the ground water or this water is made to flow through a setup in which this is the sample here soil or rock sample and here to retain this soil sample. So, there are two porous plates one at the top and one at the bottom of the soil sample and then there is this. So, there will be a collecting tank. So, this is a collector in which the water after flowing through this sample which is enclosed between two porous plates one at the top and one at the bottom. So, this sample so this has a so the cross sectional that is a sample cross sectional area is A. So, which is perpendicular to the flow direction in this case the flow is from bottom to top and this is known as the constant head H. So, this is the constant head H which is essentially the level difference between the constant level in the apparatus after the flow takes place through the soil or rock sample as well as the ground water so constant water level in the supply side. So, this is the sample cross sectional area A and in this case so collector say suppose let us say the collected volume in the collector is V. Let us show this by H V and this sample cross sectional area and then this sample length so this is L. So, this sample cross sectional area so this is a cross flow direction and this L is a long flow direction. So, in this case and the collected volume is V in say time t. So, in that case the hydraulic conductivity K is given by V the volume multiplied by so divided by time t. So, this will give the discharge divided by the area that will give the velocity and divided by the hydraulic conductivity so that is H by L. So, therefore, so the hydraulic conductivity K is given by the volume V multiplied by the sample length L divided by the cross sectional area A multiplied by the time of collection of volume that is t multiplied by the constant head H. So, this is the expression for hydraulic conductivity in constant head permeability test. So, simply this hydraulic conductivity is expressed in terms of say 4 linear parameters rather 2 linear parameters 1 area parameter 1 volume parameter and 1 time parameter. So, here this is the collected volume is the V the cross sectional area is the A and then the 2 linear parameters are the sample length along flow and the constant head between the supply the constant level supply tank and the level of ground constant ground water level after it has flown through the soil or rock sample and then t is the time of collection. So, this is the constant head permeometer or which is also known as the constant head permeability test. Now, let us go to another laboratory method which is the variable head permeability test here unlike the constant head permeometer or constant head permeability test the head in the supply side will be varying. So, obviously through due to gravity flow. So, this head in the supply side is continuously decreasing. So, the sample will the sample and that part is almost the same. So, there is the soil sample or soil or rock sample is enclosed between 2 porous plates. So, this is the soil or rock sample through which permeability variable head permeability test is conducted and. So, here this is the collector. So, this is the collector and here we have this water and the sample it is assuming this to have a circular cross section. Let us consider the diameter of the sample as r c and here. So, this is the supply side circular tube with diameter equal to r t or rather 2 r t. So, the diameter of the circular sample and 2 r t is the diameter of this pipe which supplies water for flow through the soil or rock sample and the initial head. So, this is let this be the head h 1 and this head is measured with respect to this one. So, this is h 1 is the initial head and then let us say this is h 2. So, this is the h 1 is at time is equal to 0 and h 2. So, this is the water level. So, this is at time is equal to t. So, here in this case this is let us say this is the h 1. So, the initially the head is at this level which is above the reference level which is the constant water level maintained after flow through the soil or rock sample. So, this is the datum with which the heads are measured. So, that is h 1 and then as the water flows through the sample. So, this head decreases and at an arbitrary time t. So, this head decreases from h 1 to h 2 h 1 is at time is equal to 0 h 2 is at time is equal to t. So, now and the collectors with the collected volume. So, now we can say here that the ground water flow rate or the flow rate through the soil or rock sample is given by the cross sectional area. It is pi into RT square multiplied by dh by dt which is the rate of change of head with time. So, that is represents the velocity in the vertical direction and this q is also equal to the pi into Rc square where Rc is the radius of the circular sample through which the water flows circular soil or rock sample multiplied by the hydraulic conductivity k into h by L and this L is the sample length along flow. So, now if we equate these two expressions and integrate. So, here this is pi into RT square dh by dt is equal to pi into Rc square into k into h by L. So, let us simplify this one. So, therefore, here we can write this as a dh by h with limits say from initially limit of h 1 to the final limit of h 2 is equal to. So, dh by h and of course, if you want you can keep this Rc square by RT square which is a constant you can take it outside into k by L. So, all this will be outside the integral sign and here. So, there will be definite integral when h is equal to h 1 time is equal to 0 when h is equal to h 2 the time is equal to t and here. So, this is dt. So, after simplifying this after integrating and then simplifying we get an expression for the hydraulic conductivity through this variable head permeability test as RT square by Rc square into L by t L is a sample length 3 is a time of collection into natural log of h 1 by h 2 where h 1 is the initial head h 2 is the final head after the flow which results in the continuous decreasing of the head through this variable head permeability. So, this is the expression for k in variable head permeability test or variable head permeameter. So, while in the constant head permeability test. So, the expression for the hydraulic conductivity is simply given by the product of the collected volume multiplied by the sample length divided by the product of the sample cross sectional area and the time of collection and the constant head. Whereas, in this case in case of variable head permeability test. So, this hydraulic conductivity k is given by RT square by Rc square of course you can also write this k as the area of the tube divided by the area of the column into L by t. So, this is the A t is the tube area and A c is the sample column area into L by t into natural log of h 1 by h 2. So, therefore, so this is the expression for the hydraulic conductivity and if both the tube as well as sample which are in circular shape generally that is the case. In that case simply this k is equal to RT square by Rc square into L by t into natural log of h 1 by h 2 where h 1 is the initial head at time is equal to 0 and the h 2 is the final head after the flow of water through the soil or rock sample at time is equal to t. So, these are the two laboratory test which are used to determine the hydraulic conductivity or permeability. And in this case the laboratory test many times they do not yield they do not give the proper results because in most of these cases the sample through of soil or rock through which the hydraulic conductivity laboratory test is conducted it will be a disturbed sample. So, therefore, they may not give the actual value. So, that case we go for the field test firstly let us consider the field permeability determination test or permeability test and in this firstly let us say the tracer test. So, in this tracer test what is done is suppose there is a ground water or soil or rock sample. So, here two holes are drilled one at the upstream and another one at the downstream. So, this is the upstream hole which is A and then this is the downstream hole which is B and so in the upstream hole. So, this is tracer is added. So, this is sampling for tracer is done at the downstream hole and here in this case obviously there will be hydraulic gradient. So, this is the and the level difference between the water level in the upstream and downstream is H and say suppose this. So, this is the flow direction and this is the length between these two holes L. So, in that case the velocity of flow is given by the flow velocity the ground water flow velocity ground water it is the interstitial flow velocity. So, this is V A. So, this is given by the hydraulic conductivity K divided by the porosity N multiplied by H by L which represents the hydraulic gradient and this is also given by L by T where. So, this is the T is the travel time N is the porosity and H is the difference in the water level between the upstream hole and the downstream hole upstream hole where in the tracer is added and downstream hole where the sampling of the tracer is done. That means the time measurement as well as its concentration and. So, here by equating these two expressions. So, from this expression we get this K is equal to H I am sorry L square N into L square divided by H into T that is the hydraulic conductivity K is given by the porosity of the soil or rock sample N multiplied by the square of the distance between the two holes the upstream the one at the upstream and downstream. So, let me show the central line here for these two. So, this is obviously L is the center to center distance and H is the level difference of ground water in the upstream hole A and the downstream hole B and T is the time of travel. So, sample is added here. So, the initially and then the stopwatch is started and as soon as the sample reaches this downstream hole B this stopwatch is closed. So, that it will give the travel time T. So, therefore, so this hydraulic conductivity is given by the porosity multiplied by the square of the length divided by the H which is the difference in the water level between upstream hole and downstream hole and T the travel time. So, this is the first of the field permeability test for determining the permeability or hydraulic conductivity. Of course, there is also another technique known as the point dilution method and let us not go into that and so there is also now let us come to another field test which is known as the auger hole test. Here what is done is an auger hole is drilled in the ground consisting of soil or rock where we need the permeability or the hydraulic conductivity value and so this is Lw which is the length of the well or the auger and this auger hole or well it has a diameter of 2 into Rw and in this case so let this be the so this is the impermeable or highly permeable layer and with respect to this so this is the water table denoted by Wt and the draw down in this auger hole let this be y the level difference between the water table and the water level in the auger hole and the height of this water table above this datum of impermeable or highly permeable layer let this be h. So, here so it can be shown that the hydraulic conductivity K is given by that is C a coefficient so this is a dimensionless constant in C divided by 864 and this K so this is it directly gives the hydraulic conductivity value in meters per day and 864 comes from the this is 24 into 3600 which is the number of seconds in a day that is 86400 into dy by dt which is the the rate of change of the y with time so this dy by dt so this is measured in centimeter per second measured in centimeter per second so initially this auger hole is drilled and then so some water is removed that is water is pumped out from this auger hole so therefore there will be a draw down of y created and then so because the water table is above the so there is a draw down so what happens is so this y goes on decreasing so therefore this dy by dt represents the rise of the rate of change of this draw down with time so this K is directly given by this expression C is a constant so this is the dimensionless constant so this is the expression for the hydraulic conductivity through this auger hole test now let us come to the last anyway of course we will go into this that is the pumping test and in this pumping test so through a well in the field so the time versus draw down that means the time and is continuous measurement is done and then this consists of continuous measurement of time and draw down in a well subjected to pumping and by using the appropriate expressions so the hydraulic conductivity K value is determined so these are some of the methods for the determination of the hydraulic conductivity initially we discussed about the laboratory test wherein we started with the constant head permeability permeameter or constant head permeability test and we also discussed the variable head permeability test or variable head permeameter then we moved on to the field test which will give a better result for simple reason that the sample is undisturbed in case of field test as compared to the laboratory sample of soil or rock which is generally disturbed so which consisted of there are three types of that one is the tracer test the auger hole test and lastly the pumping test and of course the actual pumping the one we will see the subsequent chapters. Now let us come to the heterogeneity and anisotropy so here in all these we made an assumption that the soil or rock layer through which the groundwater flows so is a homogeneous and isotropic that means it is the hydraulic conductivity is same in all directions in all the three directions that is Kx, Ky, Kz whereas in actuality it is not the case now let us consider the some of the anisotropic aquifers wherein this let us consider say separate layers wherein the let us say this is the wherein the flow is taking place firstly let us consider that is the groundwater flow along the layers or strata and in this case let us consider so this is the so these are the three layers the first layer let us consider the permeability value as K1 and let us consider the width as the thickness of this aquifer as B1 and let us consider the second as a K2 the second layer having hydraulic conductivity K2 and a thickness B2 and then let us also consider let us consider an aquifer having a permeability Ki having a thickness of Bi and let this be the last the lowermost aquifer having a thick nth aquifer having a permeability or hydraulic conductivity Kn and thickness Bn and in all these let the flow be along the layers or along the strata so in this case so this is case of heterogeneous aquifer of course here in this case we are not considering the anisotropy and we are considering only flow in one direction so in this case the equivalent permeability that is Ke so this is expressed as summation from 1 to n Ki into Bi divided by summation Bi from 1 to n so and this is the expression for the equivalent this permeability and the expression for transmissibility or transmissivity is simply given by Ke into summation Bi that is 1 to n which is simply summation 1 to n that is Ki into Bi so this is the expression for transmissivity and this is the expression for the equivalent permeability so the ground water flow is along the layers or strata sometimes it may so happen that in a heterogeneous aquifer the ground water flow is taking place across the stratum or layers so this is the ground water flow across the heterogeneous strata so here let us consider say this is the impervious strata and this is the other impervious strata may be top or bottom you can say and then suppose this is the ground water flow and let us consider the strata the flow are perpendicular to the flow direction each having length of say L1 and hydraulic conductivity K1, L2 hydraulic conductivity K2 and so on say suppose and here let us say this is Li the hydraulic conductivity Ki and lastly let us say this is Ln with hydraulic conductivity Kn so in this case the ground water flow is perpendicular or across the heterogeneous layers so in this case so the total flow length is given by this L is equal to summation Li 1 to n and the equivalent permeability that is Ke is given by summation Li 1 to n divided by summation 1 to n of Li by Ki so basically this is the length and Li by Ki represents the travel time so this is the expression for the equivalent permeability in case of a heterogeneous aquifer where in the ground water flow is across the heterogeneous flow stratum so we will stop here and we will continue in the next class in the next lecture on the and move on to this further articles in this. Thank you.