 This is the best way to study what is the diffusive contaminant transport. So if you add a small amount of dye in a fluid, dyes could be any color and what happens after some time, this dye diffuses in water, when you are kids we were asked to put a small amount of potassium per magnet in a water column. The moment you put a small drop of potassium per magnet, you come after some time and you find the holes water in the glass has become pinkish reddish. So this is a diffusive process solid. So diffusion is molecular diffusion, you are not shaking it, you are not stirring it, you are just putting a drop of potassium per magnet and because of the molecular activity, the entire system is becoming uniformly colored or contaminated. So this is a case if I take a solution where the plug remains intact, alright. But there could be a situation where after certain time the concentration might drop. So what I am showing over here is, if you look at this band of the color, this shows the concentration. So starting from the initial value, this front is decreasing as far as its width is concerned and that is because of the diffusive process, alright. So a better way to understand this would be, in this case there is no diffusion taking place. However, if you really want to show the diffusion, you know what is going to happen, the intensity of the diffusive contaminant has decreased and it has spread over in the direction of the flow. Now, there are different types of diffusive contaminant transport processes and we talk about steady state diffusion and this is the diffusion flux constant with time and this is where the fixed first law is applicable, I am sure you must have studied fixed law. Do you find this equation somewhere in geomechanics? So JD is the diffusive flux, if you remember, minus corresponds that the concentration of the chemicals is going to decrease with respect to X, so you use the sign of minus, theta is the porosity of the porous media and del C by del X is the concentration gradient. Do you find this somewhere in geomechanics? Have you used it somewhere? If I replace C by the pore pressures U, what is del U by del X, pore pressure gradient not with respect to time with respect to length of the sample, where would this be used? This concept, del C by del X equal to some coefficient multiplied by del square U upon del T square, yes, so truly speaking, consolidation phenomena is nothing but a diffusive transport of water through the porous media, alright? So this equation is quite known, well known, so this D is nothing but del square upon del T, what are the dimensions of CV, coefficient consolidation, that is also L square by T, clear? So when you talk about the non-steady state diffusion, so we use this law and this is what is very well known to you, so del C by del T equal to del by del X multiplied by diffusion coefficient del C upon del X. So this becomes rate of change of concentration with time and this is the rate of change of concentration with distance. So when diffusion is occurring, it is a process which is going to happen over a length of the sample and over a period of time, it is a coupled phenomenon, alright? This is correct? So there you did lot of tests to get the value of CV in construction process. Now again the question is how would you obtain D value? So you will have to perform some test where D can be obtained very easily. So you have to measure the concentrations over a period of time in the sample and along its linear dimensions also. Now coming to the chemical energy field, we have been doing lot of things in this context and I have cited 3, 4 PAD theses, you should go through the PAD theses of Dr. Guru Murthy, Dr. Rakesh who are the scientists from Atomic Energy and they were interested in finding out the contaminant transport through intact and fractured dogmas, particularly of radioisotopes and Dr. Rakesh was more interested in finding out the contaminant transport through saturated and unsaturated soil mass. The reason is simple, this work is related to the deep disposal of atomic waste where if you remember I think we have discussed earlier, we dispose the atomic waste of intermediate and high concentrations in geological repositories, in rocks which are unfragmented, non-weathered. However, when you are talking about the low level waste, this could be disposed in the trenches and the depth of disposal will be hardly 10 to 15 meter of the trenches of the vaults. Vaults are the ones which are specially maintained system by RCC lining and in which the atomic waste is deposited. So these are the 2 situations which we have studied that is the rocks and the soils. Stations are saturated and unsaturated and rocks are intact and fractured. I would say this is the first time somebody had studied the radio-nuclide migration through fractured and intact rock mass in 2002 as well as that and Dr. Rakesh studies this in saturated and saturated soils way back in 2005. So we studied chloride, iodide, cesium, strontium in their active as well as inactive forms. So active form when I write for these anions and cations, these are radio isotopes particularly of cesium and strontium. And so we have completed the studies on all 4 possibilities that is intact rock, fractured rock, saturated soils, unsaturated and saturated soils, reactive contaminants and non-reactive contaminants. And this is where the question was how will you obtain the diffusion coefficient. So we developed diffusion cells. So these are the simple diffusion cells which we created, they would not cost to even 1000 rupees. So what I have done is I have taken a small glass tube and partitioned it in 3 compartments by using the rock samples. So this system I have utilized for finding out the diffusion coefficient through intact rock mass. So those of you who play carambord, you have the strikers. So what you have to do is you have to collect the rock samples from the field and make thin samples out of it, 1 mm, 2 mm, 5 mm, 10 mm thick samples. And those thick samples have been fixed in the diffusion cells, clear? So these are the compartments which get created. I can create a groove on the top of this through which the samples can be taken over a period of time. This is what is known as sampling. The central portion is filled up with the contaminant in the liquid form and then we keep the system as it is for several days. So as and when diffusion occurs from the central portion to the left and right portion, the CT can be obtained by sampling through the port. We call this as a sampling port and to avoid evaporation of the solution, we keep it closed most of the time. Now when you are dealing with a fracture rock mass, this fracture was done under the laboratory situation. There are sampling sample splitters which have been designed and done. So you keep a rock sample or a rock core and then by using a splitter you can create a cleavage or a fracture in the sample. And then just like as if I am doing a falling head test, I will fit this rock core in a standing tube, fill up the upstream side with the contaminant solution. So this becomes C0, fill up the bottom portion with the pure water and let the contaminant migrate through this. And I can keep a port over here through which I can sample out over a period of time. So this becomes CT the way it is. Now I think I am going to answer your question what you are asking, the setup is ready and then I can record these results alright. So what you will observe here is these are known as diffusion characteristics to save our time we kept both the diffusion cells in the centrifuge and we did study that elevated G value. Though we later on realize that the G is not going to influence the diffusion coefficient because this is a molecular diffusion. So this was the paper which was read by several reviewers and this is a history now anyway. So what you get is you get CT by C0 with respect to time and normally these tests were done for months together 3 to 4 or 5 months alright because for getting one sample each point on this curve through a rock of let us say granite where the permeability would be 10 power minus 21 meter per second would take you several days or months. So this is a game of patience like you have to do these tests and be very patient. What centrifugation did is it accelerated the seepage which is taking place through the rock samples, advective flow. So once you have got these diffusive characteristics we normally consider the initial portion take the slope of this and we have shown that this is how the diffusion gets modeled with respect to n, n is the centrifugation effort which you are putting in and then you can solve these equations you can use del c by del t equal to d and del c del square c by del x square to solve this and what we have shown is that the diffusion coefficient in the model is same as the diffusion coefficient in the prototype which has to be true. By using this modeling what we could do is for fragmented or the fractured rock mass and the intact rock mass we have done in 7 minutes we have simulated 50 days of response of 6 meter thick fragment rock mass. And in 75 minutes of centrifugation it is about more than one and a half year of simulation for a 0.3 meter thick intact rock mass. So these techniques were utilized for the first time I would say and we could get some results which were used by people who are into the mathematical modeling and they have used these coefficients and the parameters which we have suggested and they have gone for the environmental impact analysis. Now it all looks very simple but I am sure when this was done we had no clues we started from the scratch and we were lucky to be successful in whatever we touched I hope you can realize anything this comes to your mind you got the answer to your question how the slope is computed how the sorption and other things are done. Sir the question is like this diffusion will always occur or not because it is a if it is a molecular phenomena like then you will always be having like there is no concept of conservative or non conservative in this like if it is a molecular phenomenon then you always will be having some sort of go to the concepts diffusion is caused because of the concentration gradient. So as long as delta C is not 0 the diffusive contaminant will occur clear that is fine that is what your question is anything else yes. And this diffusion like it is not related to how do I say it like the chemical sits on the molecular particle or like what happens see when we say molecular diffusion the molecular the soil. The diffusion is occurring through the liquid phase water clear I gave you an example that if you put the drop of potassium permagnet in this standing water still water even then the entire water becomes pinkish reddish and that is because of the molecular diffusion in the geomaterial like I was trying to think what it happens. Pores are important we will come to that so the liquid phase which is being which is present in the pores of the geomaterials the molecular diffusion is that going to occur through that. So imagine all the pores are filled up with water and then there is a concentration gradient. So what is going to happen all this concentration will migrate through the pore solutions and then it will get equilibrated once C1 is equal to C2. As long as C1 is greater than C2 this whole thing will keep on migrating. Geomaterial has nothing to do with this thing that is what I am. Geomaterials are only going to have the pores but pores cannot be in the air. Individual activity of the I think I have given you the best possible technical answer. So the pores which are saturated in the geomaterials are alloying they are acting like the pipes which are connected with the point number 1 to point number 2 and the entire system is fully saturated. So the pores are completely filled up with water is it not there could be a partial saturation also which we are going to talk about is this okay. So the pores are within the porous media but yes you can create a diffusive contaminant transport in this room also do you have any answer how would you create this gas or let us say you bring a small bottle of scent perfume open it at that end and what happens slowly you can smell it over there. So this is the diffusive transport of chemicals in a fluid which is there this is also molecular by the way because this molecular diffusion is taking place in air but we are talking about the molecular diffusion through the water. So sometimes you also call this as a free molecular diffusion in water. This is something which was created by my student Dr. Siddip for his PhD thesis. We were trying to study how diffusion occurs in soils alright and this is where we have clubbed the effect of migration of concentration and electric field. So in today's discussion if you remember I started with four modules of advection you know diffusion thermal heat flux and electrical flow. So this is a beautiful example of how the migration of contaminants can be studied by using impedance spectroscopy. So impedance spectroscopy is the electromagnetic field which you are using for detecting the movement of contaminant front in soils. So again we took a tube and this tube one third of this tube was filled up with soil which was mixed with some contaminant. So this becomes a so this is the tube which has been taken glass tube or this could be a prospect tube in which one third of the portion has been filled up with the soil which is spiked with contaminants alright and then rest two thirds is the uncontaminated soils. So you just keep the system over a period of time on a horizontal platform and then what you will observe is that the contamination species from C will migrate into U and I wanted to see how this pattern of contaminant migration would be. Unfortunately you cannot take XRA or XRD not XRD XRA you cannot do all these type of systems. So a better way is to utilize the electromagnetic field in the form of electrical properties. So what we have done is we have embedded different electrodes and this is how the electrodes have been embedded into sample. A prime forms one pair, BB prime forms another pair. So across the you know diameter of the sample we are measuring the electrical properties and we are showing that how diffusion migrates. So the objective is to see how much contaminant migrates from right to left and for that impedance spectroscopy has been done. So this cell itself became a very interesting tool which was created by Dr. Siddique and by using this type of systems we could study the diffusion coefficient of the saturated and unsaturated soils for different contaminants. If you are interested in knowing how this was done please read the papers written by him. Now this is another interpretation of the breakthrough curve. So if you have CT by C0 versus time you know we take the slope of this curve and this is what is defined as the diffusion coefficient. And there is a law which is defined as Archie's law. This is D equal to some constant multiplied by porosity power some other constant. So if you see the papers which have been published by my student Dr. Paresha we have tried to come out with a classification scheme for soils where we tested several soils and we have obtained the diffusion coefficient for different type of contaminants passing through these soils. Can you tell me where could this equation be utilized if I know the values of the two parameters at eta and small n. If I say D is known and if these coefficients are known I can utilize this equation for obtaining the porosity of the geometry. I am sure you must have realized that obtaining porosity is a very difficult task that to precise. Conventional way of taking the sample and putting it in water and taking out and weighing it is not going to give you the real porosity of the system why because the pores might be smaller than the water molecule. So under no circumstances this water molecule is going to enter into the pores of the geomaterial and hence we have to change these techniques of determination of porosity. So one of the ways to determine the porosity in the most precise manner would be do good diffusion tests and if you know the diffusion coefficient of the contaminant and the porous media because diffusion coefficient is constant for a given system of contaminant and the porous media is interaction between porous media and the contaminants then I can obtain the porosity alright. So people who are working in atomic industry and those who are interested in disposal of the waste in the best possible manner so that the waste does not contaminate the near field and the far field remember we have talked about all this in the beginning of the course, far fields would be in few kilometers with time changing ranging from let us say tens of years to hundreds of years however if I want to see what is happening immediately within few hours within few days few meters away from the disposal site this becomes near field. So those who are working in these type of studies they want to not take any risk and they want the best possible characterization of the materials. So this is interesting way of looking at the things that if I am doing a contaminant transport test diffusive contaminant transport test I can obtain the porosity also. So we did several tests by taking the soils of different types with different saturations with 180 60 corresponds to saturation levels the moisture content and the volumetric moisture content and different densities these are two different type of soils. And what we have measured is we have measured the impedance. So I will show you quickly how this impedance can be of some help in establishing the migration of diffusive contaminant front. So we have created one molar sodium chloride solution and 0.01 molar strontium chloride solution because these contaminants are normally present in most of the sludges which are coming out of the industrial sludge and then you can analyze the sodium concentration and strontium concentration by using ICP or atomic absorption. So this is the classical way of doing this. Now what you are observing over here is if you plot impedance with respect to length of the sample you remember that setup where I have shown you uncontaminated soil up to one third and sorry uncontaminated soil up to two thirds and contaminated soil up to one third of the sample two compartments. So this is the contaminated soil and this is the uncontaminated soil. And when you leave this over a period of time what you will observe is the contaminants migrate into uncontaminated soil region and because of migration of the contaminants the impedance what happens to the impedance? Impedance should increase or impedance should decrease impedance decreases alright why? Because the soil becomes more conducting. So this concept we have utilized that impedance is inversely proportional to the concentration of chemicals and by doing some simulation and modeling we have obtained the diffusion coefficient of the soils. So this is how the influence of saturation has been plotted when you have the you know at a given time for different states of the material how the diffusive contaminant transport will occur. As I say if you really want to understand and learn about these processes please follow the papers which have been written by Sri Deep and myself and you can come across the trivialities and the difficulties associated in doing this test. Fortunately all these things are well established now and you can go through and you can follow it better. What I wanted to demonstrate is that simple equipment, simple techniques can be developed in modern day you know persuasion of science and technology and you know to depend upon high-fi gadgets to execute your research ideas. The contaminated soil why is it remaining constant? The concentration is decreasing. Yeah it is a good question. So concentration of the contaminant could be so high that there is a minor change and for all practical purposes I can assume the concentration of contaminant soil to remain constant. Clear? Suppose 30,000 ppm and what is diffusing is hardly 1 or 2 ppm. So this is sort of a constant loading which I am maintaining on the system and similarly in this case this is the case of constant concentration being exposed to the soil sample. So delta C is insignificant and that is the reason the C remains constant. Good observation. Then this is the equation you know diffusion contaminant transport where if you look at this equation this is one of the forms of the solutions which we have discussed earlier second fixed law del C by del t equal to d into del square C by del x square. So Ct by C0 is nothing but the normalized concentration, l is the length of the sample and rd is what is appearing over here is the retardation coefficient. I think we discussed about this sometime back and I had asked you to write down on the on your notebook which I am going to discuss. So this rd coefficient indicates that what is the retardation coefficient of the geomaterial which is related to the sorption of the geomaterial. Now dE is the diffusion coefficient not the free. This is through the pores we call it as effective diffusion coefficient. What people are trying to do is they are trying to solve this expression. The moment I get dE term, the moment I get rd term, rd term is dependent upon the density of the soil mass and it is volumetric moisture content. What it indicates is that Ct by C0 over a period of time and length is going to be an exponential decay, is this okay? dL minus dE time and lc, lc could be the total length of the sample and x is the distance. So those terms are appearing in cos and sin terms in the form of x and x0 at different t values. So if you solve this expression, you can get how Ct by C0 is changing with respect to x. It is a time series. So this is how you have got the time factor from consolidation equation also.