 Let us talk about the third mechanism of contaminant transport which is dispersion. So I mean, yeah, there is a literal meaning is getting dispersed, is it not thinning out of something, spreading, scattering. So here you have a plug of certain concentration and now because of the flow of water what is happening, the concentration over a distance gets spread out. That means here the concentration is dropping now. So this is the concentration profile, you know, at this end you have higher concentration, at this end you have low concentration on the x axis. So Ct by C0 at this point is 1, Ct by C0 at this point is 0, is a step gradient. Now as this concentration front moves, what happens, there is a spread out of the concentration and Ct by C0 becomes substantial. Now what is the logic when you try to study the dispersion phenomenon, we use these type of models which tell you how dispersion phenomena is taking place. So this is a micro mechanism associated with the dispersion mechanism. The analogy is similar to the pipe flow. Now if I ask you to draw the velocity profile in the pipe flow, you can draw it maximum at the centre and at the two ends because of the wall friction or the friction of the pipe the velocities are less, the same analogy I can use over here. So if you consider the velocity contrast within the sample as far as the pore size is concerned, the pores which are bigger would allow you, allow the migration of water to be faster as compared to the pores which are quite close to each other. There could be a concept of the path length also, the difference between discharge velocity and the seepage velocity if you remember. So the short path is normally straight and the tortuous path is going to be the longish path and then of course we talk about the friction in the pores. So these are the three mechanisms which are normally used to define the dispersion process. So the velocity is going to be more in the central portion of the pore as compared to the place where the water is in contact with the soil grains because this is where the friction is getting mobilized and the velocity of the flow is going to be less. So what causes dispersion? Ultimately the dispersion is because of the contrast of the velocity in different layers of the sample. So imagine a situation where within the pores you have a velocity contrast. So this is going to result in the dispersion process. Similarly, the long and short path distances within the pores are also going to create dispersion phenomena and of course the pore size, if pore sizes are not uniformly distributed the dispersion is going to be prevalent, is this part okay? It so happens that in case of the clays and the sands normally we do not talk about the dispersion because the seepage velocities are going to be extremely small and dispersion gets ruled out. Sir, why at the bottom the dispersion is least? It is not the bottom dispersion is least, you are talking about this figure? It is getting reduced. No, it is not that, it is the just to show you that if the pore distance is too much the velocity is going to be more, discharge is going to be more as compared to the smaller pores. Then logic could be the path length, third logic could be the friction. So at the ends of the pipe there are, the friction is also there because of the pipe surface and that path length is also there. So both the factors are contributing, as we go above the friction is getting reduced only the path factor is there. So see friction is going to be constant in the pipe flow, is it not? So at bottom the friction will be the shear surface. Let us start the bottom, basically this is the contact between the fluid and the pipe where the friction is mobilizing, correct? Now within the fluid, the central portion of the fluid is migrating faster as compared to the portion which is in contact with the pipe, I think that part is understood. So this contrast in the velocity itself is going to create dispersion or the variation in the pore sizes or the variation in the flow path. So these are the three models which you create to study the dispersion phenomena. So in short the dispersion is because of, you know, change in or let us say gradient in the velocity which gets created in the pore system and the answers or the reasons could be 1, 2, 3. This is how you are modeling it. So from this point onwards, this is the poromechanics it starts. I am sure you must have heard the name of the poromechanics in geomechanics is becoming a very advanced subject where the pore size analysis is being done by using the most sophisticated instruments because now the realm of the discussion you must be realizing is intra-pore, inter-pore, all right. So what is happening inside the pore is becoming more interesting and more important. So I will be talking about how to do the pore structure modeling subsequently because as you said sometime back, you know, what is the importance of the porous media. So porous media is defined by its pore structure and then the question is how to quantify the pore structure, how to differentiate between the pore structures, clear. So in today's world everything is being done and I will discuss with you how it is done. This picture just shows you how the solid particle and the fluid interaction takes place, what is the tortuous path and what is the direction of the flow and truly speaking how the porous media is defined or how it is modeled, is this part okay. So when we talk about the dispersion, we use the coefficient md and this is equal to another coefficient, we call it as a dispersion coefficient in the lateral direction multiplied by the seepage velocity. So AL is the dynamic dispersivity, vs is the seepage velocity and AL is a term which depends upon the flow path, these are empirical relationships which are used for modeling the dispersion phenomena. But as I said because when the flow takes place through the compactor soils, the seepage velocity is going to be extremely small, is it not. And hence md term becomes insignificant, tortuous path I think you understand is a seepage path and the general direction path is a discharge path. The fourth process through which the contaminants might migrate in the porous media is hydrodynamic dispersion. So hydrodynamic dispersion is a process where you know some mechanical activity is required to trigger or to accelerate the molecular diffusion, a good example would be the potassium permagnet which I added to the water column in the glass, if I let it happen on its own, it will take a lot of time, a better way would be you stir it with a spoon. So what will happen? The entire thing will become uniform colored or uniformly concentration in no time. So what hydrodynamic dispersion does is, it gives you the mechanical effect as far as the distribution of the concentration is concerned in the system. I hope you can realize that in a porous media hydrodynamic dispersion cannot be valid, is this okay? But hydraulics people, when they are talking about some discharge coming and entering into another reservoir of water, let us say when you are disposing the sewage in the sea. So what is done is you take the pipe and flush out all the sewage into a big reservoir, clear? So in this case, hydrodynamic dispersion is valid. But as far as geomaterials are concerned, hydrodynamic dispersion is ruled out again. This is the governing equation which deals with hydrodynamic dispersion, DL equal to AL into VS plus DI. So DI is again the diffusion in the free state. So DL is the coefficient of hydrodynamic dispersion, the units remains in L square by T, L is the distance, T is the time and if you solve this equation, you will get CT by C0, error functions and there are exponential terms and if you substitute the values of time, at a given time what is the value of the concentration of the contaminants or at a distance. So if you see the L, L would be the distance of the point where I want to find out the concentration, this can be obtained by using this type of expressions. So these are simple models which are used for obtaining the concentration of contaminants in the porous media. Now if I club all these things together and as a geo-technical engineer, I will be more interested in advection diffusion equation because as I said molecular diffusion, molecular dispersion and hydrodynamic dispersions are ruled out in our context, you know, this is how this equation looks like. So this is a clubbed form of advection and diffusion process, we call this as one-dimensional ADE, one-dimensional Y because you are talking in terms of only Z axis, ADE corresponds to advection diffusion equation, the rate of change of concentration with respect to time is diffusion coefficient multiplied by del square C by del Z square, this is the second fixed law, I hope you understand minus seepage velocity into del C by del Z, what is this component? This is V into C mass flux, clear? So this is because of the seepage velocity, the rate of change of concentration along the length of the sample, this coefficient KD is the distribution coefficient which defines the sorption, desorption phenomena which we are talking about, eta is the porosity of the soil mass or the porous media, rho dry is the dry density of the porous media, del C by del T is rate of change of concentration with respect to time. So this equation is one-dimensional ADE, now if you do a bit of mathematical jugglery, you can obtain the KD parameter. So there are several ways of doing the analysis of this equation and this is where the practice of environmental geomechanics is important. Normally you will come across situation where you are doing monitoring let us say, so suppose if I dispose the waste at a given point and I would like to measure what is the concentration of the chemical species that they are certain X, Y, Z and T. So I can put sensors over there, I can drill the go logs, I can collect the water sample, I can take out the soil sample and I can establish what is the concentration at this point. In other words, I can know the value of C at a given time and Z by doing experiments or by doing field studies, clear? And I can substitute it there, I have to compute Vs, seepage velocity which is not difficult to do. If I know the discharge velocity, I can divide by the porosity and I can get the seepage velocity. Now what is going to be difficult is how to obtain KD parameter because that is a new subject in itself. So when you ask this question to yourself, how to obtain KD which is sorption desorption phenomena then we have to enter into sorption desorption which is a sort of chemical characterization of soil contaminant system. Another way of looking at this equation would be if I can measure KD parameter, if I know the porosity, if I know the dry density of the porous media, if I know the seepage velocity, if I know the diffusion coefficient, I can substitute these terms in this expression and what I can obtain is del C by del T, so rate of change of concentration with respect to time and what I have to do is I have to obtain rate of change of concentration with respect to distance. So again you have to follow the time series, is that okay? Now one more interesting thing from this equation which can be observed is if I take this term on the left hand side, what happens to this expression, this expression becomes del C by del T, 1 plus KD into rho dry upon porosity. So 1 plus rho dry into KD upon porosity is known as retardation parameter which I had shown you at this place. If you remember, we were talking about the RD parameter over here, is it not? So this RD is equivalent to 1 plus rho dry multiplied by distribution coefficient KD upon porosity. All the countries which are into nuclear activities, I hope you understand, nuclear activities you understand, what do they do? They have to have their researchers who are going to solve these expressions for various reasons. I hope you got the point. It is easy to run a reactor by using nuclear source of power but the question is where you are going to dispose the atomic waste, nuclear waste, you got it? So suppose you have an atomic power plant, you have to solve this equation to make sure that this is going to be the zone in which no rehabilitation should occur number one. If at all this zone is becoming too limited because of the encroachment of the you know land by the people and suppose in a country like Japan where the space is extremely less, how do restrict this zone is a challenge which a geotechnical engineer has to take and this is where I would like to use barrier systems. You remember, we were talking about containment and if I cannot contain it, what I should do? I should remediate it, decontamination. So everything is getting interlinked now and that becomes interesting problem, modeling, giving a solution to the cause of something. This is part clear but I am sure you must be realizing the difficulties also which I have told you because the parameters are not known. Now the question is if I am dumping the waste over here and if I have to find out what is happening over here, I have to keep on taking the samples from different places of the soils at what radial distance, at what depth, how many samples we will take out? Sampling is not going to be easy and imagine if the disposal depths are in few hundreds of meters, 200 meters, 300 meters, 400 meters, so what is going to happen? You will be having a zone of influence which might be running in few kilometers. So a big challenge is how many samples should be taken and hence what type of subject we are going to now use to answer this question, reliability analysis. So what is the reliability of the sampling? What is the reliability of the data which you are generating by lab test or the field test? The reliability of the data is which is going to be corresponding to real-life situation. Now this is the practice of environmental geomechanics, I hope this point is clear to all of you. What are the factors which influence the contaminant transport mechanisms? Grain size of the porous media, yes. In my college again they were using some lagmere model like equations to find this desorption, sorption parameters and then they can put it into the. I am very happy that you are much ahead of time, so that time will come I will discuss. So these are all known as sorption, desorption, isotherms, we will talk about that. So the isotherms which I was talking about are the sort of breakthrough curves which will give you the KD parameter, we will discuss about this. So let me start with the factors which are responsible for contaminant transport in porous media. First is grain size, seepage velocity is a direct function of the grain size distribution, texture is a function of grain size, pores, structure, pore distribution is a function of grain size, density, seepage velocity, concentration of chemicals, viscosity of the contaminants, hydraulic conductivity. So these are as per the porous media is concerned. If you include all these terms the porous media is defined. Now what are the parameters which will be affecting the contaminants themselves, so type of contaminant and its concentration clear and its activity, so whether it is active, non-reactive, radioactive, non-radioactive, so the type of contaminants, type of the soil condition, so this is a big matrix which people are trying to solve. Another thing is contaminant transport is going to depend upon the mechanism, so just now we have studied four mechanisms, radioactive, diffusive, dispersive and hydrodynamic dispersion. So you have to see what mechanism is controlling the contaminant transport and there are few regulatory numbers which I will be discussing can be utilized to establish what type of contaminant transport is occurring in the porous media. So I am sure from this discussion you can realize that I can write a series of parameters like this that the concentration of a contaminant in the porous media would be a function of so many properties, characteristics and I hope you can guess now I am heading towards what pi Buckingham theorem is you have done it. So C is the concentration of contaminants in the pore water, mu is the dynamical viscosity of the fluid, D is the diffusion coefficient, S is the mass of the adsorbed contaminants per unit volume, V s is the seepage velocity, T f is the surface tension, rho f is the density of the fluid, then G is the accession due to gravity, then L is the characteristic microscopic length, characteristic microscopic length and then you have time which is the physical time at which you want to find out the concentration migration and soil properties, what properties have been left out chemical characteristics, mineralogical characteristics, surface area, cation exchange capacity, its mineralogy and composition. So once you put all these things together this becomes a real contaminant transport through geometries. Then we have different similitudes, we call them as non-dimensional numbers. So what are the coefficients or the numbers which control the transport mechanisms? We have concentration number, advection number, diffuser number, capillary effect number, adsorption number, dynamic effects number alright. So what we have done is we could manage a similitude between the numbers for some cases and I will show you how that was done, unfortunately there are few numbers which cannot be simulated like Reynolds number and Peclet numbers. So Reynolds number I hope you understand is nothing but the inertial forces divided by the viscous forces and Peclet number talks about the diffusion coefficient, C pi velocity multiplied by the characteristic length upon diffusion coefficient of the contaminant alright. Now what I will show you is how this discrepancy was resolved. So if I say that there is a relationship between Reynolds number and Peclet number, this can be proved and how this was proved I will show you. So mu is the viscosity of the contaminant in the solution form, rho f is the density of the contaminant, diffusion coefficient of the contaminant, density of the fluid, C pi velocity and L u is the microscopic length such as the particle size and normally we take it as d10 or d50 or the mean particle size of the soil. So if you follow all this, this is the work which some of my students did. What we wanted to do is we wanted to see what type of contaminant transport occurs in compacted soils. I think this is what you are asking sometime back. So if I plot Peclet number against Reynolds number, this is how the situation looks like. So I can divide the whole area of the graph in 4 parts. As far as the Reynolds number is less than 0.4, 0.5, 1, 2, 3, 4, 5, 0.5 and the moment it becomes greater than 0.5, there is a difference between the diffusive and advective diffusive contaminant transport. When the Peclet number is 0.1, 2, 3, 4, 0.4, a combination of Reynolds number less than 0.5 and Peclet number greater than 0.4 will create dispersion. This is proven by previous studies, Reynolds number more than 0.5 and Peclet number more than 0.4 will create advection and the fourth one is advective diffusive contaminant transport. We did some experiments in centrifuge and numerical modelling and what we could show is that the type of soils normally we get, they follow advective dispersive phenomena. It was very difficult for us to create samples where diffusive and advective diffusive contaminant transport could be created in the laboratory situation, why? Because reconstituting the samples for a given soil where the Reynolds number, sorry Peclet number would be less than 0.4 is going to be difficult. So I am just trying to show you this type of study to give you a feel that simple experiments can be done to simulate a type of mechanism of contaminant transport. All these type of the topics are open topics, anybody is free to work on them and create either a porous media or a contaminant, the combination of which will give you a sort of a mechanism and you can get an answer that why this mechanism prevails.