 Now, once you have done all these mechanisms how you are going to use them. So, what I have shown you was the simplest possible models which were being used, but nature is not so simple our profession is not so simple and that is where we have to take help for few models which define advective, diffusive, contaminant transport in tandem, in tandem means together coupled phenomena. So, this is the form of the equation. If you do lot of mathematical jugglery you will end up getting this equation. Can you recognize few terms appearing here? Del c by del t is nothing, but concentration gradient with respect to time, d i is diffusion coefficient, del square c by del x del z square is variation of concentration with respect to distance. What is V s? C page velocity del c by del z is nothing, but again concentration gradient in the distance domain, rho dry is the density of the media. What is K d? K d is the distribution coefficient divided by porosity of the porous media multiplied by del c by del t. I have not intentionally shown the derivation of this equation which will take lot of time, but I just wanted to show you how these type of equations can be used for simple modeling purpose. So, if I ask you a question in broad sense what this parameter corresponds to d i into del square c by del z square sin u s. What would be this term? This is diffusive contaminant transport. What about this term? Del c by del z into V s advective contaminant transport. What about this thing which I have encircled? This is the retardation of the porous media. See the flux is moving from one point to another point. What you are trying to find out is the rate of change of concentration. So, rate of see how concentration is moving from one point to another point it is nothing, but primarily because of diffusion. If there is a seepage velocity the concentration gets reduced. So, you have reduced this much value from this term. So, diffusive contaminant transport minus advective contaminant transport minus retardation capacity of the porous system. So, when you say retardation capacity of the porous system these two terms are very well known to you. It is nothing, but the porous media characteristic which you have been talking about. Now, the term which is appearing for the first time or which is quite new for you is k d. Now, this is what is known as distribution coefficient. What is the meaning of term distribution? See this is what I have shown you earlier that if you allow enough time for a contaminant to come in contact with the porous system because of very low seepage velocities. What is going to happen? It starts communicating with the porous media. Now, how much communication is going on? It is nothing, but distribution that means what percentage of contaminant gets or is getting interacted with the porous media. So, that means this is where we say what is the fraction of the contaminant which is getting distributed from a liquid phase to the solid phase of the soil mass. So, you should appreciate that these are all philosophies and philosophies have been put in mathematical forms clear. So, now let me ask you a question if k d is much more what happens to the concentration at a given point and whether it whether this type of situation is useful helpful to you or not? That is right. That is right. What is the meaning of this? If you have a clay barrier whether clay minerals are very active what is that they are doing? That is right correct. So, this is the whole philosophy of designing a clay barrier system. You have to design clay barriers using a sieve. I am using the word sieve which not only allows water to move out, but it stops cations also from the solute or contaminant. So, the name of this type of sieve is molecular sieve. That means your porous media is acting now like a molecular sieve. It will not allow certain molecules to even pass through itself and that becomes the best possible barrier system which should be utilized at a waste containment facility. Is this part clear? Now, d c by d t you can obtain very easily. You can measure the concentration by conducting either the boreholes or in the laboratory you can measure repeatedly after certain time by taking a sample or dosing and then you can measure by using atomic absorption AIS or ICP or whatever. So, this term is very well known d c by d t. d i you can obtain by conducting different experiments. I have shown you one experiment in the previous lecture by using diffusion cells. Del square c by del z square can also be obtained rate of change of concentration with respect to time. You can monitor at two three different places how concentration is changing and you can know the concentration profile change. V s is easily measurable. The most challenging task is K D and this is a big headache. Government of India spends almost not less than 100 crore rupees in the research where K D determination should be done very precisely. Most of the mines where uranium is been taken out. If your K D parameter is not determined properly, the life becomes hell clear. So, one side too much of industrialization too much of mining activity very good for your nation to become a high profile nation. But, then second side what is going to happen your own populace is going to die because of all this contamination. So, this becomes a very tricky issue. So, there is a very big exercise going on right now where K D determination has to be done in different labs. About 22 labs have been selected in the country our lab is one of them and we are now trying to formulate a methodology to streamline the methodologies and methods which are adopted for determination of K D. So, the first step in modeling would be after you have got these parameters you should get an answer from this equation in the form C as a function of T and Z. So, what is the concentration at a given time at a given distance that is what you are interested in as a user. So, D i is the diffusion coefficient and K D is the distribution coefficient. Now, with this information I would be in a position to take you into the world of sorption desorption mechanisms. I hope you will appreciate this unless this was given to you there is no point talking about sorption and desorption process. Most of the mathematical models are available these days in the form of finite element codes and finite difference codes and there are several commercially available packages which are available which can give you a solution to A D E this is also known as A D E advection diffusion equation. Now, let me ask you a question whether this is a one dimensional equation or a two dimensional equation or three dimensional equation is a one dimensional equation. So, you should appreciate the point that one dimensional equation will have its own limitations when you solve the real life problems. So, that is where you have to talk about del C by del t as a function of del C by del z del C by del x del C by del y. So, keeping the time domain constant how the distance domain is varying in terms of concentration becomes a real challenge alright. So, I hope this part is clear that there is a new term defined as K D and we have to obtain K D by somehow. So, this we will talk about a bit later. Let us understand what are the parameters or the factors which decide the type of contaminant transport mechanism. So, the first is grain size I have given you some very raw examples like sand and clay and what is the difference between the mechanism of contaminant transport in sands and clay advection diffusion alright. So, grain size is very important density is very important C by velocity is extremely important concentration of contaminant is very important viscosity of the fluid or this solute is very important hydraulic conductivity of the media is very important then comes factors affecting the behavior of contaminant. So, these are two different things these parameters influence the mechanism you agree. So, mechanisms are four advection diffusion dispersion and hydrodynamic dispersion. So, these are the parameters which are going to define what type of mechanism going to prevail in a system. Now, the factors affecting the behavior of contaminant or the fate of contaminant. So, the first one is contaminant itself whether it is reactive or non reactive clear or radioactive or decaying type there could be a contaminant which does not come out of the porous system it gets decayed completely within it alright. So, there could be a situation the second term would be soil condition what type of soil conditions you are talking about and of course, the mechanism which is going to govern. So, the whole idea of showing you these these points is what is the point I think this is what I wanted to demonstrate to you ultimately it boils down to the mechanism which is governing the contaminant transport. So, it is not a very easy and simple way of modeling though it appears to be because this term itself will take care of all these parameters plus many more parameters into account ok. Let me quickly run through if you if you want to do modeling what are the parameters which would be required. So, concentration C of a contaminant in the porous media can be defined by these factors. So, C is a function of viscosity diffusion coefficient sorption coefficient seepage velocity surface tension density of the fluid g value gravity 2 type of dimensions I have used here you are doing a course on centrifugal modeling. So, I am sure that you must you would have been using this type of equation which are nothing but pi Buckingham theorems. So, multiplied by the time and the soil properties soil properties happen to be lumped parameter it is very difficult to distinguish between one property and the property just like that. So, concentration of contaminant in the pore solution of the pore water is defined by these parameters where mu is the dynamic viscosity of the fluid d is the diffusion coefficient s is the mass adsorbed of the contaminant per unit volume b s corresponds to the interstitial flow velocity is nothing but seepage velocity t f is the surface tension of the fluid particle interface rho f is the density of the fluid g is the acceleration due to gravity l is the characteristic macroscopic length what is macroscopic length physical dimension. So, my question to you is if I am using or let me not ask you let me let me show you how it will vary you should try to catch the point that what is the difference between macroscopic dimensions and microscopic dimension. So, l mu is the characteristic macroscopic length which is nothing but the particle size and of course, your t is nothing but the time. So, all these parameters are required to form a pi Buckingham theorem how many parameters you have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So, out of 11 how many equations can be formed 10 clear the most vulnerable set of the equation would be with 9 unknowns if I make 2 parameters dependent on each other. So, that means, any contaminant transport problem will have at least 9 parameters which either you have to know beforehand or you have to determine or you have to compute it by some mechanism. So, this is just to show you the complexity of the problem when you talk about contaminant transport in porous media. Now, your colleagues from hydraulics cannot deal with such a complicated situation because what they deal with is all these things will have disappear there. So, l properties microscopic lens macroscopic lens will not come to the picture your seepage velocity nothing but discharge velocity diffusion coefficient also becomes a free molecular diffusion in water. So, it is very easy to obtain it clear. So, there it is a subset of the problem which we are talking about. So, contaminant transport in porous media happens to be a very very broad problem though contaminant transport in free water supply happens to be a subset of it clear. So, just to give you an idea about how modeling is to be done. Now, when you do modeling you must have learned a lot in your centrifugate modeling course that you require some dimensionless numbers. So, these are the coefficients of contaminant transport mechanisms which you have to generate out of those 9 parameters which govern the contaminant transport. So, the first parameter is known as concentration number. So, concentration number is nothing but concentration divided by densities of fluid and this ensures similarity of concentration of you know homologous points in the sample and the prototype advection number. Advection number is nothing but seepage velocity multiplied by time divided by macroscopic or microscopic macroscopic that is right the physical length of the sample. So, this will ensure kinematic similarity of motion in the model and the prototype diffusion number diffusion number is nothing but D multiplied by T upon L square. So, L square upon T is nothing but inverse of D in dimension. So, this becomes a non dimensional number. So, this ensures similarity of diffusion process in the model and prototype. Capillary effects number most of my students are working in capillary effects on the soil. In fact, you may also be working in the same so, how do you define capillary effect number? The surface tension coming to the picture macroscopic length macroscopic length multiplied by the gravity effect and the fluid density. So, truly speaking this is nothing but the ratio of the surface tension force and the inertia forces. Inertia is nothing but because of density of the fluid. So, inertial forces divided by the surface tension forces are nothing but the capillary effects. So, this talks about similarity of the capillary effect in the model and prototype adsorption number how much is getting sobbed on to the system divided by density of the fluid. So, a very very dense fluid may have more sorption or less sorption. It is a difficult question to answer because again it will depend upon the activity of the fluid, but a good way of defining adsorption number would be a dense fluid like honey whether it will diffuse easily in the porous system or water will get diffuse into the porous system with some contaminants correct. So, that means, a denser material should have lesser sorption into the system. Similarly, your dynamic effect number. So, dynamic effect numbers are nothing but macroscopic length or microscopic length macroscopic length physical dimension of the system because gravity is coming to the picture. So, l upon t square is nothing but inverse of g. So, this becomes a non-dimensional number. Now the way I look at these numbers is these are nothing but the controlling valves. What is the meaning of this? If I want to study what type of phenomena is going to occur in a porous system when a sudden contaminant comes in contact with it. If I use these numbers I get a very clear cut mathematical image of the mechanism which is going to prevail in the system. Otherwise you cannot perceive and visualize it, but if you are playing with these numbers it gives you a feeling that if number is between a and b this type of mechanism is going to prevail. If this number is between c and d this type of situation may prevail or may not prevail. That means, what you are doing is now you are one step very close to quantifying mathematically the mechanism which is going to govern in a porous media. So, remember what we have done? We have talked about different mechanisms. We have talked about the models. We have emphasized on how to get the parameters which are involved in the models and now we are talking about how to quantify the whole process. This is particular. So, there are 4 steps in contaminant transport mechanisms when we talk about contaminant transport in porous media. Well, there are few anomalies. What are the anomalies? Discrepancies. The discrepancy number 1 is the Reynolds number. You have been dealing with Reynolds number always when you talk about seepage flow. How do you define this term? It is inertial forces divided by viscous forces. So, this is nothing, but the inertial force rho f is the density of the fluid multiplied by seepage velocity multiplied by why microscopic why not microscopic that is right because of this is the seepage which is taking place through porous that is right. So, please remember that this is not the macroscopic number and then we use another word or another number which is known as Peclet number. So, Peclet number talks about concentration migration, particularly dispersion phenomena. So, you have seepage velocity multiplied by again the microscopic length of the sample specimen divided by diffusion coefficient. Now discrepancies in the way that truly speaking there is a school of thought which says that Reynolds number should also get modeled when you do different modeling exercise. But then a great respite is even if it gets modeled the Reynolds number is so low that even if you multiply by 300 it is always going to be less than 1. This type of analysis and debate was presented by Dr. Ashok Gupta in lot of his papers worked on the centrifuge modeling of seepage analysis. So, this is where we defended one analysis by saying that R e has to be always less than 1 otherwise this is not a flow through porous system. It becomes a open pipe flow where R e varies from 1 to 10 which is not the case with any soil or rock system. Now comes the second issue that is the Peclet number. So, Peclet number even if it is higher than certain n value n value is nothing, but acceleration or artificial environment condition. So, here also Pe is going to be less than 1 and order of magnitude of Pe and R e would be in 10 power minus 7, 10 power minus 6. So, any acceleration level which you may apply on the soil sample would not cross R e and Pe more than unity clear. So, I have defined these coefficients over here in the relationship between Pe and R e. So, if you just substitute these terms and you derive equation you will find that Pe equal to nu divided by rho f into d into R e. Now this becomes a coefficient which relates Reynolds number with Peclet number. Again I will repeat Reynolds number is nothing, but the number which controls seepage or the flow in porous system and Peclet number is the number which regulates transportation of contaminants from one point to another point. If there is no flow there will not be any contaminant transport except for the case when you are talking about diffusion. So, this is where R e becomes much less. So, mu is the viscosity of the contaminant, rho f is the density of the contaminant, d is the diffusion coefficient of the contaminant, rho f is the fluid density, b is the seepage velocity, l u is the characteristic microscopic lens such as particle size it is equal to d 10 or d 50 of the mean or the mean particle size of the system. How do you define mean particle size? If d 10, d 50 is known not average whereas, there are different ways of defining average particle mean particle size. The way I would like to define this is under root of d 10 into d 50. This is again depending upon different research groups and authors. So, as you say normally d 10 plus d 50 by 2 is never done. It is I know, but when you talk about the mean particle size it has to be a sort of a you know GP of the two terms. So, it is a very interesting world of researchers. Now, I will show you how these modeling can be done. Now, this paper was the presented by my students Sridhi Bhattan and Moran's at a conference. What we did is actually my dream is to develop this model further. If you plot Peclet number and Reynolds number on x and y axis, you have four types of contaminant transport mechanisms. So, depending upon in which zone the values are falling without doing much of you know analysis I can tell you immediately that what type of mechanism I am going to control. This is what actually I would have loved to do half way through. So, most of the points which we could do are falling in dispersion. It is very difficult to model soils to get some results in advection diffusion and diffusion zones though it looks very easy, but you make a sample and repeatedly you will end up because we did lot of analysis here 3, 3, 7 plus 10, 13, 15 about 45 samples we have tested here. Each point corresponds to almost 3 repetitions of 4 repetitions. So, advection is very easy to get. Now, in sense if I keep on adding clay, what is happening? I am traversing from right hand side to the left hand side. So, that means, Reynolds number is decreasing and so is Peclet number clear. So, I fall in this domain, but this domain itself is such a big domain where I can reduce Reynolds number, but then I should have expected increase in Peclet number. So, that becomes a modeling challenge. I am just trying to show here that two numbers can also be correlated with each other and if you have a sort of a electronic interface where automatically Pe and Re are recorded from a landfill and you have a database of this type where immediately you will come to know what type of mechanism is going to prevail in this landfill and you can take precautionary measures. Is this particular? Yes. In some of the situations, yes you can create. So, artificially you can induce Reynolds number which are much, much higher which are even Peclet numbers will also become very high because of that. So, these are situations which you can simulate, but the most trivial situation would be in this domain, but nature does lot of contaminant transportation only in this domain. So, all your salt water intrusion problems are somewhere here, a bit of them will be falling in advection diffusion. So, what is happening? Re is increasing at the cost of Pe. Here Pe is increasing at the cost of Re. So, these are perfect if this line goes in this direction it becomes a pure clay aquifer. If this line goes in this direction, it becomes a purely sanded aquifer. So, you can do modeling for a certain type of aquifer also if you have this type of a database, all right. I hope you must be clear by this time that how these parameters can be utilized in solving real life problems. So, let me quickly finish now sorption process. So, sorption is nothing, but you have any grain or a parking space in general where most of the cations come, molecules come, they get adhere to this, they get parked over here. So, in simplest possible form this is the sorption phenomena. So, it has two components absorption and adsorption. What is absorption? Atoms or molecules move into the bulk of a porous material, example the absorption of water by a sponge. Now, once this mechanism is over then the tendency of the system is to migrate into the molecules or atoms. Now, this is what is going to be adsorption. So, if you read the definition, atoms or molecules move from the bulk phase that is solid liquid or gas onto a solid or liquid surface. So, for the penetration. So, purification by adsorption where impurities are filtered from the liquid or gases by their adsorption onto the surface of a high surface area solid such as activated charcoal. So, these are basic difference between the two. Absorption still is a physical phenomena. However, adsorption has to be a physical, chemical, mineralogical phenomena. That means, this is where all the activities or the total activities of the system comes to the picture. This is particular. Stumps related to sorption are adsorbates that is the molecules that have been adsorbed onto the solid surface. Substrates or adsorbent, the surface to which adsorbates are adsorbed. In case of adsorbed cations tightly held on the surfaces of negatively charged dry clay particles, clay particle is substrate and cations are adsorbates. I am sorry it should be adsorbates. So, it is a good schematic diagram. These slides have been prepared by Sochit. So, you have a clay particle which is negatively charged and then the cations are adhering to this clay platelet. So, it is a electrical balance is there first of all that nullification of charge is there. So, this type of mechanism where the cations are getting pugged onto the clay particles is a sort of a sorption phenomena alright. So, clay particles are providing enough space for the cations to come and get pugged over there. The reverse mechanism is desorption that means, whatever gets out freely. So, the phenomena whereby a substance is released from or through a surface becomes a desorption phenomena. Process is the opposite of sorption and occurs in a system being in the state of sorption equilibrium between bulk faces alright. A good example of this would be you have these water filters which are normally used in the home. So, dirty water you have a sludge formation on the candle of the water filter alright. So, it is a simple absorption process first of all, but because of impurities which are quite active the tendency of this system would be to migrate into the candle or that calcium carbonate thing which is the candle. So, it migrates into it. So, suppose if you do not wash it for several months and then take fresh water supply and keep this candle over there what happened there is a reverse migration of all the sludge which is formed on the candle into the fresh water. So, this becomes a sort of a desorption process. Now, these mechanisms are normally used when you are dealing with remediation techniques in the real life. So, when the concentration of the pressure of substance in the bulk face is lowered some of the soft substance move to the bulk stage. So, these are the copybook style of defining the desorption process. Now, this is a slide which shows these processes in the simplest possible form you have the clay layers in which the cations are getting logged or bugged. So, they are strongly soft into system when the system is dry you add water to this with the hydration going on and this is detachment from the surface of the clay. And you keep on doing this process much more water if you add what happens all of these systems are free to come out. So, this is what I have written here that clay layers you have cations which are trapped into this these are nothing, but the adsorbed ions cations in the dry condition cations are strongly soft on clay particles you add a bit of water. So, water molecules make them quite hefty the tendency is to get detached from the clay platelet is a sort of a desorption process which has started. So, the water molecules wedged into the interlayer after adding water and this is the final stage this is the intermediate stage. So, the cations get fully hydrated which results in the adsorption from clay surface. So, this is how the sorption desorption mechanism keeps going in the nature alright. So, I will stop here today.