 We have been talking about geomaterial characterization in general and in particular the chemical characterization and I will continue with this. In today's discussion I will give you a brief idea about the contaminant transport in porous media and this is important to be studied at this juncture before we start dwelling into desorption and desorption studies because the pretext of sorption and desorption is contaminant transport in porous media. When the interaction of contaminants occurs with the porous media then only we talk about this sorption and desorption process. Now this is a multidisciplinary subject, many technologists would utilize the concept of sorption and desorption and very evolving area particularly in the realm of environmental geomechanics and hope you will realize that this becomes very, very important due to the fact that is the quantification of the interaction which we have been studying since long. So if you want to study how to quantify the interaction between contaminants and the porous media the best way would be to study the sorption, desorption mechanisms and from there you quantify the whole process. And of course we will continue with our discussions on thermal, electrical and magnetic characterization subsequently. So to begin with the contaminant transport in porous media hope all of you understand we have cited several situations, examples where the contaminants come in contact with the porous media or the soil mass or the geomaterials and this also has the pretext to the fact that flow through porous media is very well understood when we deal with water that is you know you have already talked about the seepage consolidation and stability of the structures where we have discussed the influence of water seeping through the porous system. Now if this water contains contaminants and these contaminants are migrating through the porous media then the entire mechanics or the interaction process becomes difficult and intricate to understand and that is what actually we are trying to attempt to study. So the concept of hydraulic conductivity is quite well established you have done constant head test and falling head test the simple analogy which comes to my mind is if I adulterate the water with some contaminants and if I allow this water to migrate through the porous media and if I ask you how much the hydraulic conductivity has changed this is one question. The question would be what is the concentration gradient of the contaminants which have been retained by the soil mass you know when the contamination is passing through this third could be what is the fate of the contaminants in the soil mass. So these are the three issues which we are bothered about and we would like to study. So when chemicals flow through the porous media this becomes very interesting important and contemporary situation very practical situation because under no circumstances the water which is flowing in the porous media in real life is going to be you know demineralized water or saline water or it could be contaminated water. So we have cited several examples where the liquid form of contaminants starts migrating through the porous media waste storage facilities landfills mostly and when the soils become contaminated and how would you remediate them and particularly when the leaching is occurring in the liquid phase. Contaminants have to be defined here before we enter into this discussion of contaminant transport through porous media. So we have to understand what is the basic characteristics of contaminants these are the inorganic species or these could be organic species also this could be reactive or non-reactive also you know this could be radioactive and non-reactive also. So sometimes we call the non-reactive contaminants as conservative contaminants alright and conservative contaminants are normally used by us every day we consume lot of salt through our food or through different types of you know eatables and medical scientists would be using different type of salts for tracers suppose if you want to find out what is the problem with your digestive system so they do biopsy they do endoscopy and all those things. So that is also a good example you know you drink some tracers barium chloride mostly those of you who might be aware of this and then they trace how much of the barium chloride has been sobbed by the intestine and that are the places where there is some damage done to the intestine so you can do endoscopy and you can find out you know what is the extent of damage to the colon or intestine. So in this context we use some units to define what is the amount of contaminants which is present in the solvent and the best way to define the concentration of contaminants in the solution would be milligrams of contaminants in 1 liter of water so we call this as milligrams per liters. Another unit could be parts per million so parts per million is 1 gram of the solution which is dissolved in million grams of the another solution I am sorry this should be the grams of the solute divided by millions of grams of the solution. So please correct this is not correct it should be the grams of the solute which is dissolved in the solvent millions of grams of solvent. Now coming to the types of flow through the porous media the basic equation which you have used in the form of the Darcy's law if you remember velocity is proportional to hydraulic gradient. Now this equation can be extended to study how the contaminants are going to migrate through the porous media provided the fabric and the state of stress of the soil sample is not changing or the porous media is not changing. So in this circumstances we can assume that j is equal to phi into x where j is the flow rate alright and phi is a constant we call this as conductivity coefficient or for flow and x is the driving force. So if you use this equation what it shows is that there is a flow of the flux which is occurring and this is equivalent to or equal to the coefficient of the flow multiplied by the driving force. So one of the ways to correlate this equation with the Darcy's law would be q equal to k into i into a. So a is the area of cross section and capital Q upon area of cross section will be small q discharged per unit area and that will be equal to k into i. So there k is nothing but phi gets replaced by k and hydraulic conductivity which is known as hydraulic conductivity and x is nothing but i the driving force. So this part you have already mastered is it not Darcy's law take a sample and apply heads h1 and h2 across the two ends and the delta h which is h1 minus h2 acting over the length of l of the sample is nothing but the hydraulic gradient and small q is k into i. Now I can extend this concept to the flow of current also through the samples is it not. So in that case what I have to do is the sample length remains l and if I know what is the amount of current which is passing through the sample what I have to do is I have to apply v1 and v2 voltage across the sample and then v1 minus v2 gives delta v and the phi term becomes the conductivity term and j is nothing but the flow rate. So flow rate is i current this is equal to conductivity multiplied by delta v upon l. So delta v upon l is also defined as electric field if you remember is it not delta v upon l is the electric field. Now here in case of electricity we use Ohm's law and this Ohm's law is nothing but i equal to specific conductivity multiplied by voltage drop. This can also be extended to heat flow through the geomaterial provided the length of the sample is l and I am applying temperatures t1 and t2 across the two ends. If t1 is greater than t2 the way it was same as v1 greater than v2 and h1 greater than h2. So delta t gives me the temperature gradient and l is the length. So temperature gradient upon l is the temperature gradient multiplied by k. Now this k is the thermal conductivity and q is the heat which is passing through the sample and this is what we define as 4 years law. Now what is remaining is the chemicals alright. So if I apply the concentration of chemicals c1 and c2 across the two ends of the sample this type of situation may occur in the coastal area where you have aquifer of certain length and one side the aquifer is exposed to the seawater saline water and on the other side it is connected to the land towards the land. So c1 is greater than c2 and hence there is a flow of concentration in this direction. So d becomes here the diffusion coefficient multiplied by concentration gradient delta c upon l fixed law. Now I hope you can understand that what we have done is we have you know discretized all the fluxes in their components and then we are saying that if any type of flow has to occur through the porous media it can be modeled by using Darcy's law case of advection flow of water if the water is having species of chemicals then we can use the fixed law and if this water which is having the species of chemicals is at elevated temperature and if it is migrating through the porous media I can utilize Fourier's law to solve this and there could be a situation where the voltage is also acting across the sample electrical voltage or the sample is exposed to the electromagnetic field. So this is where I can use the Ohm's law. So the point is any type of flow which is occurring through the porous media can be discretized into different flux conditions and then I can superimpose them depending upon the nature of the problem and I can solve it alright. So this is the general philosophy of the concept when we talk about how the flow or contaminant transport takes place through porous media. One thing which I would like to highlight is I hope you understand when you apply this law or this equation what is the principle unknown? The principle unknown would be the phi term. So small k hydraulic coefficient or hydraulic conductivity or coefficient of permeability you have obtained by doing different tests in the laboratory or in the field. So you did falling head test, you did constant head test, you did you know triaxial test, you did consolidation test also to get small k value and in field if you have to obtain the hydraulic conductivity then you can do Packer's test or pumping in, pumping out test and so on. So that means the big question is how would you obtain the coefficient of flux transport. So this was quite easy in case of conductivity, electrical conductivity I have to pass current of some value measure the voltage across the two ends and obtain the sigma value alright. In case of heat we will have to pass heat through the soil sample and making sure that heat does not alter the state of the material including the moisture content because I hope you understand that if you are passing the DC current through the sample DC is a heating current. So when you pass DC current through the soil sample the chances are the moisture content may get altered and I am not interested in that. So I will be using AC current most of the time so that AC is a non heating current alright and then I can get the value of thermal conductivity of the soil. Similarly, I will have to do some test to obtain the diffusion coefficient of the contaminants through a porous media. So I am sure you must be realizing that this is a complicated issue. First of all you should be having proper devices which you can utilize to obtain these parameters is this correct? So unfortunately or fortunately when we started working on all these problems there was nothing which was existing in the market. You could not order anything to get off the shelf and say that I will start doing my experiments. So what we have done mostly is we have devised different equipments, we have devised different types of methodologies to obtain all these coefficients and this is where our major emphasis has been. Once you have obtained these coefficients then the life is simple you can plug them in mathematical models and you can do the analysis the way you have done the seepage analysis by using flow nets this is part clear. So the basic challenge is how to get electrical conductivity, how to get thermal conductivity, how to get diffusion coefficients. So from this point onwards my emphasis would be to explain to you how these coefficients were obtained. Of course this is a very elaborate manner in which we have done this and most of these studies have been done by my PhD scholars and each thesis has been attributed to finding out these parameters and the setups which have been done. Sir when chemical flow is taking place like if there is some sorption along with Darcy flow the soil fabric will change. Yes let us start the things with very simple situations alright and then we will complicate it as much as you want do not worry. First to begin with start with the simple things so that life becomes easy this is correct this rule of life. I will answer your questions how everything has been done. Now I am sure you must have realized that what is coupling now if I couple these modules with each other this becomes a coupled flow this is okay. That is a simple mathematical you know superimposition. So let us start understanding the flow through the porous media. So the first thing which comes to mind is you are talking about the advection and all the flow of water because of the hydraulic gradients is advective flow clear. Now we also define this as convection process the you know the idealization would be something like this. If I consider a certain volume of a chemical we call this as a plug. So this is a solute which is getting transported from left to right and this is a situation at t equal to 0 time. Now when t1 is approached this plug simply gets displaced by certain distance and at t2 it gets further displaced by some distance without losing its integrity that is more important. So solute which is also contaminant gets transported through the seepage velocity along with the flowing fluid which is water in response to gradient hydraulic gradient. So we define this as seepage velocity equal to hydraulic gradient multiplied by hydraulic conductivity and divided by the porosity of the porous media. Now this is where I can answer your question that what we are assuming here is that the solute and the porous media both are conservative. So it is a simple situation where you are talking about flow through let us say perfect glass tube alright and contaminants or the solute is let us say conservative salt which is not going to react with the walls of the tube or the soil particles. So under these circumstances what is happening is the entire concentration simply moves from one point to another point without reacting with the porous media and without getting decayed or changed altered. Now if these conditions are valid this contaminant transport is known as advective contaminant transport alright and then I can find out what is the relationship between the time and the seepage velocity. So if I want to find out the time which is required for this plug of contaminant to travel from one point to another point I can use this relationship V s is known L is the distance travelled by the contaminant and I can get the relationship as porosity into L upon k into i. I am sure you must have utilized this equation in your undergraduate when we were talking about let us say in an open channel flow if I ask you to find out the velocity of the water how do it. So you will add some conservative contaminant at a given point and you will measure the concentration at another point you know the time you know the distance you can find out you know what is the velocity of the flowing water. Now again coming back to the point suppose if I want to find out what is the advective mass flux which is passing through the porous media. So advective mass flux is velocity multiplied by concentration and you must have done this analysis V1 into C1 equal to V2 into C2 conserving the mass flux in an open channel flow. So C happens to be the concentration of the solute which is the mass of the solute per unit volume of the mixture and we know other parameters hydraulic conductivity we know the hydraulic gradient this is nothing but the discharge velocity and then I can equate this alright. So this is a simple situation where I can find out what is the mass flux which is entering the soil mass and which is coming out of the soil mass and remember we are assuming that the both contaminants and the porous media are conservative they do not react with each other. The second mechanism of contaminant transport is diffusion what is happening here. Suppose if I take soil sample in a glass tube and if I connect it upstream as well as downstream side with the water path and if I eliminate the delta H how would you eliminate the delta H by keeping H1 and H2 constant but what I will do is I will use the water which is going to pass through the sample by adding some salt into it. So what I have done is by maintaining the heads I have eliminated the advection process is this okay. But now what you will observe is because of the concentration gradient between this and this this side is fresh water this side is contaminated water. Now if you do this experiment what you will observe is I have to take out the sample of the discharge water or even if it is not discharging let us say in the steady state condition I can take some sample from here and I can do ICPMS or atomic absorption spectroscopy to find out what is the concentration of the water on the downstream side. And then if I plot the CT value CT is the concentration of the contaminants present in the downstream side and this is normally normalized with the C0 value initial concentration. So this becomes a non-dimensional term CT upon C0. Now this is plotted on Y axis and time is plotted on T axis any idea what this graph is known as have you ever used this anywhere suppose if I change C by population of a nation. Yes so environmental scientist or the anthropologist will be very eager to find out you know what will be the population after a certain time in our case C is concentration. So this is the concentration which is coming out of the sample at a given time and C0 is the initial concentration. So this becomes a breakthrough curve BTC have you heard about it breakthrough curve in environmental engineering course I am sure you must have talked about the BTCs. It so happens that BTC cannot be equal to 1 why if soil mass and contaminants both are reactive. So this is what answers your question now the moment you take a reactive soil mass and reactive contaminants some sorption is going to occur and because of that CT is always going to be less than C0 value. In a totally conservative system we do not talk about the concentrations but there could be another situation that you have a reactive contaminant which is passing through a conservative soil mass. So there will not be any sorption so CT by C0 could be equal to 1. Now this graph the moment you get will give you the diffusion coefficient capital D alright. So this is a simple way now what I have to do is I have to create instrument in such a manner where I will expose the sample on one side to higher concentration another side to the lower concentration wait for some time let the steady state diffusion process set up into it draw this relationship get the slope and that is the diffusion coefficient alright. So by definition the way we define this is solute or contaminants this is migrating through the sample in the absence of any hydraulic head. In real life we would have a situation where advection also gets coupled with diffusive contaminant transport alright because you cannot stop it. So this becomes advective diffusive contaminant transport where the heads are also acting water is flowing to the porous media and there is a difference in concentration of the chemicals which are being carried by the water when it enters the sample and when it comes out of sample. So this becomes a typical advective diffusive contaminant transport. Again the flow will take place from higher concentration to the lower concentration and delta C is caused by C1-C2 and when this process is going to end when Ct tends to be equal to C0. So the moment the concentration gets equilibrated on both the sides the diffusive contaminant transport is going to cease. At what point we will be taking the slope for diffusion coefficient? Normally we take the linear portion and what I can do is I can manipulate the whole thing in such a manner that I can plot T because these diffusive times are extremely long times. So normally T can be plotted on a log scale. So this becomes a semi log scale I will take the central portion of the curve which is this almost a straight line and that slope is used alright.