 The third mechanism of contaminant transport is dispersion. It is also the dictionary meaning of word dispersion is something which disappears or thins out scattering or spreading. So, you can say spreading of contaminants is dispersion phenomena alright. Now, if you consider this figure what you notice in the first part of the figure is that there is a flow shown as the positive values or dotted lines and followed by a concentration dose over here ok. So, this is the plug of concentration which is flowing along with a flow stream. Now, what I have plotted here is on the y axis C T by C naught that is concentration of contaminants at a given time and normalize with initial concentration C naught as a function of spatial variation that is the distance. So, what you notice is that at x equal to 0 C T by C naught is 1 and then dispersion starts taking place after certain distance and then by the time you reach some value of x the C T dies down completely. That means, this is a complete dispersion of the contaminant in the flow stream. Similarly, as the front increases or moves further you can see now dispersion taking place in the much more control volume and then after certain time or after certain distance the dispersion predominates. So, the flow is taking place at the same time dispersion is also taking place. So, let us now consider the issues which cause dispersion to occur in the porous system. So, this diagram basically shows the contaminant or the pollute is spreading out from the flow path. So, this is the flow path and then there is a concentration which keeps on dispersing flowing out disappearing and so on. The mechanism which governs this process is mixing or spreading of the solute. So, in day to day life when you add sugar to a cup of tea you need not to stir it that could also be a way is it not, but then you have to wait for how many hours when the entire cup of tea becomes sweet. So, what do you do? Stir it. You stir it. What is the meaning of that? You are accelerating the dispersion phenomena. This becomes a sort of a mechanical mixing. So, that is the fourth mechanism of contaminant transport where mechanical dispersion comes in a picture. So, this is somewhere in between diffusion and mechanical dispersion alright. So, when you encounter a situation where mixing or spreading of solute takes place this is what is termed as simple dispersion phenomena. I am sure during your undergraduate courses when you were studying environmental engineering you must have come across a mechanism which can be classified as a dispersion mechanism. Can you name it? Dispersion is you have a chimney or a stack and from the stack the flue gases are going into the environment and then air is also flowing at a certain velocity. So, what happens to the plume? This plume gets dispersed from the point of origin to a certain point and there the concentration is almost negligible. So, it is a good example of dispersion in air, but because we are talking about geo environmental engineering our aim is to understand dispersion in geo materials that too in geo environment clear where you have porous media and the fluid in the form of water table. So, you think of a situation where pollutants are being discharged in the geo environment and dispersion phenomena takes over. So, in simplest possible form as we studied here if mixing or spreading of contaminant is taking place it is a simple dispersion process. The only difference between the stack emitting the gases in air and contaminants being disposed in geo environment would be did you follow the question? The basic difference between the stack emitting pollutants in the air and the pollutants getting discharged in the geo environment would be in the media itself first of all there you have a very rare media air here you have a media which has certain properties. So, first of all the difference in the media second thing is flow velocities. If flow velocities are much more in air as compared to the flow velocities even in wrong water clear. So, these are the two attributes which got changed and the mechanisms getting completely changed is this part clear. So, when you say mixing or spreading of the solute what becomes important? The flow velocity of the medium or the carrier becomes much more important clear. The attribute is you will not have any plug formation here what happens to the plug. If you remember in the previous lecture we were talking about the plug moving from one point to another point to another point without showing any significant physical distortion to itself and that was the mechanism of advection or convection clear. But here what is going to happen? If you consider this plug what has happened to the plug? The plug that got physically distorted. So, this is the difference between advection and dispersion phenomena. Another good example is most of your colleagues who are working in hydrogeology or maybe in hydraulic structures analysis they must be talking about you know jet flow under a in a reservoir. So, think of a situation like BMC Bombay Municipal Corporation where the entire sludge is going to be disposed of in the sea. So, you have a water column of the sea and then there is a pipe which is disposing of the entire waste into the sea clear. So, the pipe interface when it comes in contact with the water column of the sea this is where the dispersion is going to take place. If dispersion does not take place the intensity of the contaminants will always remain very very high. So, we have talked about three situations. One is dispersion in air which is very rare rare in the sense density wise. The second is dispersion in geo environment where the attributes of the material are totally different and the velocities are totally different and third is the situation dispersion taking place in a water body clear. So, ultimately what happens is there is a physical deformation or distortion to the contaminant plug which takes place in dispersion mechanism. Now, another attribute is the concentration is going to be negligible at flow rates which are very very low and short distance of transport dispersion phenomena, but when you talk about the micro mechanics that what is causing this to happen this is where this type of model become very important. So, what you will notice is if you want to model dispersion phenomena in a geo material we have to take into account three things. The first one is sizes of the pores and this is where I think you will appreciate the point that why pore size analysis becomes much more important than particle size analysis. If you remember when you had come to the my to my lab I had told you that nowadays we do not talk about particle size distribution much. We are more interested in knowing how pores are interconnected how pores are communicating with each other whether they are dead end pores whether they are well connected pores whether pores are saturated or partially saturated what type of capillary tubes are existing and so on. So, this is where pore size analysis becomes very important. The second attribute is path length what type of path lengths are being covered by this solute or the contaminants and third one is the friction within the pores itself all right. So, these three attributes become very important now let us understand what is the influence of the pore size. I hope you will appreciate this point if the inter spacing between the pores is too much the flow velocity is going to be fast clear. Now, you can imagine it like this a fast flow path getting mixed with a slow path what is going to happen there is a velocity contrast within the pores and hence dispersion is going to get actuated. If the velocities were same dispersion would not have occurred it would have been advection, but because there is a contrast in the velocity of the porous media because of different pore sizes which are available in the system the dispersion is going to get actuated. This is part clear the second model to consider would be path length. The chances are the contaminant or the solute may follow this path clear or it may follow this path. So, there could be a short path, there could be a long path again this is going to lead to the velocity of flow wherever there is a contrast in the velocity of the flow the dispersion phenomena will dominate friction in the pores. Do you find this type of situation somewhere in civil engineering which you might have analyzed much more in details where very good flow in a never in open channel it has to be in conduits and pipes no no no please this type of situation occurs as you rightly said this is the peculiarity of a conduit or a pipe flow clear. What is the property? Why do you define friction coefficient? What is the friction coefficient which you use in pipe flow? What is the name of the friction coefficient? Manning's C parameter. So, what is the significance of this? The significance is if you draw a velocity profile within the pores again what you notice is the central portion has a higher velocity as compared to the velocities on the edges. So, these type of models help you in understanding what type of this rugacity that is right correct Manning's is basically this is Manning's equation is used normally for open channel flow, but the equivalent is C parameter which is for chase is constant. So, that can be used for your pipe flows. So, here what happens is the friction between the grains is always less in the central portion as compared to the friction which is very close to the grain. So, this is the micro mechanics model which is normally used to understand dispersion and the modeling of the dispersion all right. I hope you will agree with the point that our job is more challenging as compared to the flow taking place through a conduits or even the open channel because here we have to deal with something which is quite stochastic and probabilistic and non deterministic. So, that is why this becomes a challenging task. Now, if you magnify this model let us say the solute has different opportunities to travel from one end to another end. One is solute may adopt a path A to A, it may adopt a path B to B and it may adopt a path C to C. Now, this is where the concept of tortuosity comes into the picture. If you draw a straight line from this point to this point that is a length of the sample physical length of the sample, but the tortuous structure of the soil or the pores allows the effective length of the flow path much more higher than the physical length. So, that means, what is the root cause of dispersion? The root cause of dispersion is flow path and which can be quantified in the form of tortuosity factor clear. That is one of the reasons when you talk about seepage analysis that tortuosity factor is not much taken into account. It is normally assumed to be some constant value, but when you are talking about the contaminant transport dispersion mechanisms all those parameters become quite significant. Just wait a minute here and if you try to understand one more mechanism which may be going on simultaneously. If flow path is more, the time taken is much more. What is the literal meaning of this? Contaminant get more time to interact with the porous media. So, this is where the interaction is starting. Is this part clear? So, the interaction is going to start when you talk about the flow of the contaminants in the porous structure and there is enough time for contaminants to interact with it alright. In today's lecture after two three slides I will introduce the concept of sorption and followed by desorption. Now, this point is the genesis of sorption. Is this clear? So, the biggest challenge is how to quantify all these mechanisms and that is where you require some parameters which are nothing, but quantification of a mechanism. The way you have defined consolidation by using CV. So, what is CV basically? CV is nothing, but coefficient of consolidation. How much consolidation is taking place can be defined mathematically if you use parameter CV. Now, if I ask you a question, how much dispersion is going to take place? You need a parameter to define either the inter pore structure or the connectivity of the pores or the frictions which is which is acting on the contaminant flow because of the edges of the pores or because of the micro mechanism which is going on between the contaminants and the pore structure. So, the basic idea is if you want to quantify all these terms processes you just require some mathematical terms. So, we will be talking about now mathematical terms which will define a mechanism of contaminant transport. So, let us go much more into the complexities of the pore structure. If there is no dissolved contaminants in water it is a simple case of seepage. And you always assume till now that a pore structure is inactive, water is also inactive. So, you talked about only seepage flow. When you complicate it a bit your pore structure is active, contaminants are also active. What is going to happen is going to get guided by all these mechanisms put together. This is where we use a word retardation factor. Now, what is the meaning of word retardation? Something wants to move, but porous media is trying to retard its motion. Why it is so? Because it has a affinity towards contaminants you got this point. Now, why is this affinity coming? Either the system is hyper or active or the contaminants are hyper or active. Any one of them being passive this type of situation will never arrive. That means, retardation will never take place in systems which are passive. So, then the question is whether retardation is a good mechanism for engineers or not? It is. So, nature is trying to retard the flow of contaminants from one point to another point. And this is the fundamental property of the porous media contaminant interaction. So, let us consider this bit more. This is how the porous media is being depicted. You have solid particles they could be semi solid also they could be porous they could be whatever. Then you have lot of tortuous flows which is taking place. And then there is some general flow direction of the this is nothing but the discharge. And this part is nothing but the seepage. So, you think of a system where seepage velocities are predominant as compared to the discharge velocities. What is going to happen in this type of systems? Dispersion is going to be much more. What is the significance of this? As compared to sands dispersion should be less or more in place. More in place. No less because you remember flow is going to take place. So, flow yes that is right. Then you grouped up. See the more flow is taking place the chances of flow contrast within the porous media are much more higher in sands as compared to clays. Clays are going to show you pure diffusive contaminant transport because there is no advection taking place. So, basically the way we define dispersion phenomena is nothing but variation in the velocity due to tortuous nature of flow path. On larger scale dispersion is caused by different flow rates resulting from heterogeneities which are encountered in the porous media. Now, let me ask you one more question. When you say heterogeneities encountered what is the significance of this? In what form these heterogeneities are getting implicated in your analysis? In other words these heterogeneities correspond to what? This is material heterogeneity yes. Is this the flow behavior heterogeneity? Yes, all sorts of heterogeneities are in built into the system. K x, K y, K z in three dimension is a sort of a flow phenomena heterogeneity. So, I hope now you can understand very easily as long as you talk about isotropic homogeneous material you will never be able to capture these mechanisms which are going to contaminate your occupiers much more and that is the reason why these type of studies are becoming much and more important. This process is repeated millions of time times by millions of water particles. So, this process keeps on going and this is the equation which is normally used as to define dispersion phenomena. Now, m d equal to a l into v s where a l happens to be dynamic dispersivity. If you see the units they are always going to be meters unit of length v s is the seepage velocity meter per second. Now, a l is normally defined as 0.0175 multiplied by length to the power 1.46 if length happens to be less than 3500 meters. So, 3.5 kilometers systems can be assumed to be working with this. So, what is m d? m d is nothing, but the coefficient of dispersion. So, which term is more important? Now, you can see I think my suggestion to you is that do not look it as a mathematical equation. It is not a mathematical equation truly speaking. It is a sort of a superimposition of two things one is a l what is a l? a l depends upon the physical dimension of the aquifer and what is v s? v s is nothing, but the flow. So, this is where the porous media is coming in contact with it is straight how seepage is going to take place. So, if v s is very less your dispersion coefficients are going to be extremely less. However, if v s is more where m d is going to be more clear that is the difference between sands and clays yeah everything that is right. So, a gross model is something like this if you consider a certain unit length of the porous media which is nothing, but capital L the value of a l would be some factor multiplied by that term. Of course, these models are being developed by several people and there is no clear clarity about which model to be used and how to be used. Now, let us talk about hydrodynamic dispersion. So, we have discussed about advection diffusion and now hydrodynamic dispersion. What is your guess? If you remove the word hydrodynamic we have already talked about dispersion. Now, if I add hydrodynamic what is the significance of this term? Hydrodynamic this is nothing, but stirring your cup of tea with mechanical mixing. So, mechanical mixing that means it should have two attributes one is no one is there should be a dynamics in the flow or in the fluid and there should be a fluid phase. So, fluid phase which gets agitated because of external disturbance the dispersion which is getting caused because of this is hydrodynamic dispersion. So, these are the attributes processes of molecular diffusion and mechanical dispersivity cannot be separated in flowing round water. Truly speaking hydrodynamic dispersion is a very fascinating field not to geotechnical engineers why, but more to hydraulics people why look at this. So, our seepage velocities are very very less. So, we rule out completely hydrodynamic dispersion in porous media for that matter even dispersion we do not talk about. So, for us the most important mechanisms are advection and diffusion clear. So, processes of molecular diffusion, molecular diffusion we have discussed already. So, diffusion is always because of molecular activity and mechanical dispersivity. So, when you are stirring the fluid with a spoon or with some some stirrer you are providing lot of energy into the system and then the entire sugar disappears and this and this T becomes sweet or saturated. So, introduction of a factor which takes into account mixing and diffusion will yield this type of a function where D L is the coefficient of hydrodynamic dispersion which is a function of simple dispersion, simple dispersion plus what is D I diffusion coefficient. So, D I correspond to molecular diffusion and A L V S corresponds to mechanical dispersivity clear. So, it is nothing but the combination of the two. So, if you put V S equal to 0 what is the significance of this seepage is very very less in per V S material. What is significance of this clays compacted clays intact rocks where truly speaking there is no flow of water what is going to govern your diffusion coefficient molecular diffusion coefficient is nothing but hydrodynamic dispersion. But then because V S term is not appearing we will never use the term D L we will say that this is the diffusion mechanism which is going to control the contaminant transport at a geological level how deltas are formed Sundarbans I think you can imagine the situation now. So, rivers are bringing tons of loads of sediments and they come and they mix on the interface of C. So, there is a density contrast there is a reduction in velocity or enhancement in the velocity reduction in the velocity clear. So, sediments deposit and deltas are formed and then your diffusion mechanism starts. So, from fresh water sorry from salt water to the fresh water. So, it is a back action of the C into the deltas and of catch is formed where you cannot drink the water because the entire land is contaminated because of diffusion clear. So, these type of situations become very important. So, D L is the coefficient of hydrodynamic dispersion again the units are same as what you have been using always L square by T meter square per second or whatever all right. Now, hydrodynamic dispersion modeling is very interesting and most of time people use if you remember the type of situation I have discussed earlier you are setting up an industry somewhere and the basic question is down the time line or may be down the distance line what is going to happen to the geo environment. So, the question is what is the concentration of a contaminant at a distance L from the source at a given time T. So, this is the equation which is used where C is nothing but C T that is the concentration at a given time equal to initial concentration half of this you must have done error function analysis no anyway. So, this is error function of L minus V S upon into T is nothing but the distance divided by 2 times diffusion coefficient of hydrodynamic dispersion multiplied by time to the power half is this ok. So, this is a simple model which you can use to obtain C at a given time if distance is known or vice versa. Now, most of the environmental agencies what they are doing any guess how they are using this equation what is known what is unknown in this expression you can obtain V S exactly. So, if you drill a bore log or bore hole you take out a sample from there of the ground water you know the value of C T. So, L is fixed C T is fixed now you have choice either you can find out after how much time exactly or you can verify your models by computing D L if time is also known. So, this is where the concept of observation wells comes into the picture. So, every unit where the disposal of contaminants or hazardous waste is being done it has to be surrounded by a series of observation wells. So, that whenever you dose the landfill immediately you should find out what is happening down the stream in terms of time and distance and whether your disposal facilities are working perfectly alright or not. This is a broad philosophy of how landfills can be designed in such a way. So, that the environment remains unaffected for that matter not only landfills any industry. So, this becomes a simple mathematical modeling exercise I am sure if you go and use the MATLAB or your excel sheet format there you will find error function. So, you just compute V S value multiplied by a given time distance is known D L you can obtain by doing some experiments easily. So, this term is known and you can find out the error function of this clear and then you can get the C T value. So, this is the simplest possible form of the dispersion modeling when you are designing projects of importance. This is particular the most challenging thing would be lot of people are using finite element codes for doing ground water modeling hydrogeological modeling. What is the biggest challenge in this whole study? What is easily predictable? What is easily not predictable? What is you know unreliably predictable and so on? Are you getting my point? So, mathematically this equation looks very simple and straightforward that is true yes porous media does not remain same. So, first thing is how you are going to develop an equivalent model for your entire domain of length L. How many boreholes I should do? How many samples I should take out on the ground? How much time I will be taking to analyze these samples and at the last what is the predictability and accountability and dependability of the parameters which I am going to get from these analysis. But the bottom line is we are all engineers. So, we should not ask these type of questions again and again why at least we are getting some answer to a real problem which is better than no answer that is it alright. So, this is how you can use the simple models in determining the environmental impact analysis. Now, nobody can now you should not cut a sorry figure in front of anybody that you do not know how to do environment impact analysis. So, this is the simplest form alright of environmental impact analysis for which when you become a consultant you can earn lakhs of rupees as your consolidation fee. So, most of the people are working right now in these areas any doubts. Now, let us complicate this further in your opinion what is the degree of uncertainty and degree of complexity if you are doing this type of analysis. See here it looks very easy to say that you can obtain d l how you are going to obtain d l either you are going to device as laboratory setup. So, what you are going to do you are going to cut out a sample which you are assuming to be representative of the entire soil mass and testing you know and getting the results dependent on the property of that sample which may not be a real representation of the entire domain. If you are talking about the field test life is easy or more cumbersome I think there have been lot of efforts where people have done in situ testing to obtain dispersion coefficients diffusion coefficients seepage coefficient and so on. But there you have to conduct the experiments for several years or few months to obtain the results and that is where the climatic conditions environmental conditions material properties have no control over all right. So, this becomes a very intricate exercise and you require lot of dedicated efforts to model the parameters all right.