 So, we will just go to the first method which I want to discuss in this session and then afternoon we will be discussing about the finite method and then packages. So, the finite difference methods as I mentioned this is one of the first methodology which was come as a numerical method say it is the even though the various things the underground work of finite difference method were existing at the end of the 19th century beginning was 20th century, but as a methods to solve the engineering problems it came into picture only by 1955 or 54 something like that. So, that the reason as I mentioned the number of say large set of equations to be solved there was no methodology to solve such a large system of equation for a large problems. So, that is why this finite difference method came into picture. So, the depending upon your familiarity or the complexity of the problem you may choose the which method you want. So, here we will be discussing the finite difference method. So, finite difference method water and environmental problems are concerned this is the most commonly used methods for various problems like ground water, surface water, watershed management, fluid mechanics etcetera. So, the basics of finite difference method is what as I mentioned we are trying to represent the continuous variation of the function concerned by a set of values at points on a grid dog intersecting lines. So, that is what we are trying to do here is that there will be a system say a domain we know what kind of domain we are having. So, that domain we are trying to represent in terms of a grid points. So, the and then we will be trying to see how the variation taking place with respect to given function. So, it is a for example, if H is what we want to find out then it is gradient that means del H by del x in x direction or del H by del y in y direction. So, the gradient of the function are then represented by differences in the values at neighboring points and finite different variation of the equations are formed. So, the first step is to carefully determine the physical concepts as we have already seen and then next step is to translate the concept of variation to the mathematical terms. So, that also we have seen and then appropriate boundary conditions initial conditions and the next step is after these steps only the finite difference method if we are going to use. So, what we are trying to do we are replacing the governing differential equation by a set of different different equations applicable to the system of nodes. So, these steps shows the what is will be generally doing in any kind of finite difference modeling. So, first one is we are having as the governing equations and the boundary conditions and then we will be dividing the domain into grids and then we transform the partial differential equations into difference equation. So, that is the essential step in finite difference method and so that the partial differential equation we will be having instead of partial differential equations we will be having a system difference equations and then with respect to the difference equations we will be applying the boundary conditions and then after application of the boundary conditions we just to solve a system of equations. It may be just like a linear system of equations and once we solve this linear system of equations, we will be getting the solution, if it is a key for problem it can be head variation or it can be concentration variations, if it is say hydrodynamics problems are considered it may be the variation of the depth variations or the velocity variations or whichever variable you are considering. So these are the essential steps while using the final difference method. So if you go through the literature the basic final difference, so we will be just discussing some of the basic aspects of final difference method before trying to discuss about some real field problems how we can solve. So there are three commonly used final difference approximations for the solution of partial differential equations. So these are so called a backward difference scheme, forward difference scheme and central difference schemes. So that is as I mentioned we will be putting a grid upon the domain which we are considering. So with respect to that grid points we are trying to represent how the variation is taking place for the gradient that means if the h is the variable which we are considering. So how h is will be varying with respect to one naught point another naught point. So that we will be described with respect to these three take rings backward forward and generally central difference. So in the backward difference scheme as I mentioned if you want to find out how the head is varying. So this del h by del x say the variation in head in x direction that can be represented with respect to the h i that means the head at the considered known which we are putting and then what is just the previous node that is h i minus 1. So h i minus h i minus 1 divided by delta x. So I will just show in this there is a figure here. So if you see this figure you can see that if this is the domain which of our interest. So here we will be discretizing so delta x if it is in x direction this and y direction here so delta x is the interval which we discretize. So this we can number it 1, 2, 3 etc. So that is why this i i minus 1 j i j etc are put with respect to the grid intersecting points. So with respect to this in the backward difference as I mentioned this del h by del x x variation x direction is put. So here what is the head at this location minus what is the head at this location divided by what is the distance delta x so that it will be giving the del h by del x. So similar way del h by del y also we can write in y direction so all these variations we can represent this way so that is the backward difference. So then if it is we are having say the second order equation so this is for the first order equation and if there is any second order equation we can again take one more difference of these. So del square h by del x square will be represented del by del x of this h i minus h i minus 1 divided by delta x so after doing the approximation the simplification we can write this is h i that means the head at ith node minus 2 times the head at the previous node h i minus 1 plus h i minus 2 divided by delta x square so that is the second order. So if you have third order equation then you can go take one more gradient of this so like that any kind of problems can be solved. So this is the essence of the backward difference. So very similar way as we have seen the forward difference also with respect to this grid you can see that if this is the point which we want to find out say we are i j the node which you are considering so then the forward difference what we backward difference with respect to this node we have checked what is with respect to the previous node. So here this forward difference is considered what is the with respect to the forward node that means here with respect to this node so i plus 1 h i plus 1 minus h i divided by delta x so that is what we are trying to do in the forward difference. So del h by del x at ith node will be represented as h i plus 1 minus h i divided by delta x and then if you have higher order tends then like del square h by del x square can be written as del by del x of del h by del x so it will be the del h by del x at i plus 1 minus i divided by delta x. So finally again we will be having an equations like this h i plus 2 minus 2 h i plus 1 plus h i divided by delta x square so this i plus 2 i plus 1 i everything indicates the nodal position with which we will be representing over which we will be discretizing the domain and then the next one is so called central difference. So in the central difference what we will be doing here if we are having this i plus ij node at this location then we will be checking what is happening with respect to the previous as well as the forward one. So here this i plus j is here so what is the value here what is the value here divided by 2 delta x that will be the value with respect to the central difference. So we can have the final difference with respect to the central difference backward difference and forward difference so this is central difference. So here say it can be i plus 1 or i plus half and i minus half so both are same with respect to how we represent the system and then the second order equations will be del square h by del x square again as we have seen we will be taking one more derivative with respect to this and so that finally the del square h by del x square we can write as h i plus 1 minus 2 h i plus h i minus 1 divided by delta x square. So we generally so if you know these three differences backward forward and central difference then we can have a combinations of systems say when we deal we can have a combinations of system and then of course a number of other techniques are also there for the final difference schemes. So finally you can see that this final difference approximations you can see all of you know what is Taylor series so actually it is coming the final difference schemes are coming with respect to Taylor series and this is so once you for the given system so for the each with respect to grid points we can write the difference equation like this whether using backward forward central or whatever it you want to do and then we can write a complete system of equations and then we can apply the boundary conditions and then we can try to solve the system. So the general principle is that unknown variable which are the functions of x y or time t are represented for every values of time and by a final number of values where i j indicate the counter variables x y directions respectively so that is what with respect to this if this if your real problem domain is rectangle or irregular also we can deal accordingly that I will show when we study the real field problem so accordingly we can choose the gridding and then if you want to have more accurate then you should have very fine mesh or fine grid so that the distance will be from one point to another point will be very small so you can discretize the domain like this. So now let us see how we can use this way of approach to solve some of the governing equations say for example if you consider the steady state flow problem so for one dimensional problems say if you consider an aquifer system then say a confined aquifer system for one dimension we can write t in steady state t del square h by del x square is equal to minus cube or q is the either recharge or pumping so this term del square h by del x square as we have already written you can use a central difference or backward or forward and then you can write the equations like this and very similar for two dimensions also we can write the equations correspond to del square h by del square del square plus del square by del y square this term and this q is the pumping rate t is the transmissivity of the aquifer system which we consider so this will be the final difference equation for the given grid points so like that if there are 100 say a number of nodes will be there in both directions you can see that x direction y direction number of nodes will be there so with respect to each grid points we will be writing the equations like this and then see this q is the pumping rate which we are which is one of the excitation for the given system so t is the transmissivity so the appropriate values will be putting that is the steady state then you can see that most of our problem there will be variation with respect to space as well as time so head varies with respect to space and time so flow domain can be discretized spatially and our difference equation can be derived say one dimensional two dimensional three dimensional problems so that is the spatial variation and then the time is considered also we have to discretize with respect to time so here say for example say if it is one dimensional problem delta x is in this direction that is a spatial variation and delta t is this direction so the time variation also we can represent so the if you say for example a forward difference scheme in time then we can write del x by del t that is the time variation that is n plus h ij n plus 1 minus h ij ni ij represent which node you consider with respect to space and then n represent the previous time step and n plus 1 is a current time step so divide by delta t so if you are doing a simulation for 24 hours and if you are having a time step for 1 hour then this delta t will be 1 hour so accordingly which way you are discretizing you can write the time variation also like this so as we have seen the time is also considered we can have backward forward or central difference so you can see that del h by del t with respect to the backward here n and n minus 1 so this is the current time and central difference is considered h ij n plus 1 minus h ij n minus 1 divided by 2 delta t so like this we can write for central difference forward difference or backward difference so that is also with respect to time and then another important aspect in final difference method is so called an explicit and implicit scheme so most of the time will be having the numerical methods when we write the code we have to see that whether which method you are using so what is explicit scheme what is implicit scheme so for example when we consider say an equation like this we have seen we can see that the flow equation so for round water flow is concerned we can represent this del h by del t as a function of h so this is the equation so this del h by del t that is we as we have seen we will be applying the forward backward or any of the time difference and then we can write like this so this is equal to a function so this function you can be you can see that this may be varying with respect to say it may be our pumping time like we have seen in the previous equations here so with respect to here you can see the one-dimension equation this q by t so with respect to this when we consider so this variation the pumping can be the previous time step since if the present time step if that is known or if it is that unknown values then it can be written like this with respect to time that is this right hand sadly we can write like this so in explicit final difference scheme if f is taken at time t at time t the values of variable h is known therefore the values of the h at t plus delta t can be found out explicitly and then the time steps required in explicit method is should be very small so we have to see that we are separately we are writing whatever that is the just the previous time step what is there accordingly this right hand side is approximated and then right away you can see that this s t is known we can directly find out s t plus delta t so that we are directly getting the solution but only the problem here is that this delta t should be very small you can see that if you put a very large time step then whatever you are getting the accuracy will be very very less so you have to be very careful when you use the explicit scheme and then no it is most of the time we do not have to write a complete system of equation to solve in explicit method we can even forward with respect to time and then wherever the the the non values are there directly with respect to that we can find out the solution so and then the other method is so called implicit scheme the implicit scheme what we do here the right hand side as I mentioned that the function it is not only a variant of the the the previous time step which is a non value but we consider the current times also how the variation is taken with respect to a waiting function so here we we we consider a waiting function so which is theta 0 plus theta 1 is equal to 1 so we put a certain weights may 0.5 0.5 or 0.4 0.6 like that so we can put a certain weights like this or it can be also this can be fully implicit in terms of theta 1 is equal to 1 and theta 0 is equal to 0 so that means we do not consider what is happening the previous time step and then we can write the equation so then it is called fully implicit or it is semi implicit it will be when we use this theta 0 is 0.5 and theta 1 is 0.5 that means we are taking what is happening the previous time step 50% effect under the current time step 50% effect so then when we write like this it is so called the semi implicit scheme so we can have a explicit scheme or semi implicit scheme or implicit scheme so accordingly we can write the equation now say for example if you consider the full system of equations in groundwater flow the equation can be written del square by del square this is a simplified form of the equation so the equation can be written del square h by del square plus del square h by del y square is equal to s by t del h by del t minus r by t where s is the sorry coefficient t is the transmissivity r is either recharge or pumping and so this equation if you want to write in explicit format then you can see that with respect to the n time which is the current time step where we write for a specific time n and firstly we write like this and then here you can see the say for all these terms del square h by del x square we will be writing the the final different scheme this term we will be writing and del h by del t also represented so finally we will be having a system like this so this is corresponding corresponding to x variation y variation and then the time variation and then here you can see this term this pumping or recharge time is only with respect to the previous time step we are considering so that is the way which we consider in explicit scheme so once this equation is written we can repeat for all the all the grid points so that we can form a system of equations then say for example with the time the gridding is considered when the final difference we can have equal grid in x y direction or we can use unequal also depending upon the problem and whatever way you want to solve so say for example if you use the equal gridding then you can directly what we use with respect to this equation you can explicitly write the solution so we are trying to find out what is the value at n plus 1 time step like this so explicitly you can see that all complete right hand side is now known only one value is here so that is unknown so you can you are explicitly getting the solution so that is why the for these problem it is called a explicit methods so here of course since it is unsteady problem then we have to put some initial conditions and then we can supply the boundary conditions and then we can proceed so when we consider the time and space variation we can have various schemes we can combine the forward backward central etc and we can have a number of schemes in final difference if you go through final difference literature we can see forward in time central in space forward in time backward in space forward in time forward in space so different types of schemes can be used depending upon the type of problem which we are solving and depending upon the scheme which you are choosing the solution accuracy may increase so depending upon that you can select and now if you consider the full equation for the the ground water flow equation in the implicit scheme we can write the equation like this so here as I mentioned this q by t or r by t which is pumping or each are term so when we consider the implicit scheme what is the variation with respect to the current time step is also considered for this so you can see that this right hand side is here this is unknown time here with respect to the current time step so we cannot directly solve the system of equation you have to write the the system of equation for entire grid and then you have to solve it separately so here for the given system of equation dash carriage by del x square we write the equation for this term coming and then dash carriage by del y square we are having this term and then this s by t and this that is the del s by del t is written like this and then this term is the pumping or each are term is considered we are having the current time step is considered so that is the implicit scheme so here the head values at node ij and surrounding four nodes are unknown for time step n plus 1 therefore all the equations obtained for the n plus one time step should be solved by say by by some schemes to get these values so the solution is implied and not explicitly as in the case of explicit scheme therefore it is called implicit final difference equation strictly q should be q n plus 1 that means the the current time step when we are using that it should be the correct way of approach so if you use the correct way of approach then you have to write the system of equation for entire grid points and then you have to solve it finally for the system so generally when we say depending upon the problem you can have you can write this system with respect to as I mentioned it can be fully explicit scheme fully implicit scheme or we can have a semi implicit scheme so here we can write to say for example this variation with respect to del square but del x square and the right hand side so with respect to that if you use a variant the coefficient alpha which is varying from 0 to 1 we can write alpha in the fully explicit plus 1 minus alpha into fully explicit so you can see that when alpha is equal to 1 then you can see that it will be fully implicit alpha is equal to 0 it will be fully explicit and then semi implicit or different kinds of schemes are possible as I mentioned so like this say for example when we consider del square by del x square we can write del square by del y square we can write so we can have either a fully explicit or fully implicit or a semi implicit scheme depending upon the problem you consider so say for example for the when you are considering a general equation like what we have done here in this slide so here you can see the right the general equation can be written either so that you can choose when you are writing a computer code if you write like this the advantage is that you can how you can solve it by explicit or implicit or semi implicit or any kind of combinations can be used so this equation we can write in a general format like this so finally this alpha the only what the input should be alpha is equal to either 0.5 either 1 or 0 accordingly the you can have an explicit scheme or an implicit scheme so finally as I mentioned if alpha is equal to 0 it will be a fully explicit scheme and alpha is equal to 1 it is a fully implicit scheme and then alpha is equal to 0.5 it is semi implicit or so called a Krang-Nickerson scheme such a scheme is called a Krang-Nickerson scheme so this the advantage is that this fully implicit and then even the semi implicit scheme it is unconditionally stable the mathematically there is certain procedure to see the stability of the numerical method so you can see that it is unconditionally stable and then the advantage is that we can go for larger time step but the disadvantage is that we have to write the complete system of equations together and then we have to solve it either through iterative methods or the the schemes like a Gauss elimination or Gauss serial iteration schemes so the final equations say for example two-dimensional ground to water flow equation is considered we can write a final system of equation like this and then we can say after the system of equations are written the next stage is that we have to apply the boundary conditions and then we have to go for the iterative or the these solutions so here it should be noted that right hand side is equal to initial categories for the first time step where for the time seen solution we will have to define the the initial conditions and then according to the initial conditions only it will be forwarding with respect to time and then if you use for say for example semi implicit scheme or Krang-Nickerson scheme we are having alpha is equal to 0.5 and then we can we can just rewrite the equation with respect to the these values so you can choose whatever scheme you want and then you can apply the boundary condition for the given system of equation and then you can solve the problem so just to show as a how we can so apply this parent difference method for a real problem so here one of the case studies which are done by some of my students I will be presenting a case study here so here the case study is so called in the Patanjiru industrial belt in near Hyderabad so this is an aquifer system is about the total area is about 500 square kilometer so a large area and so the question here is that say so we have already delineated the aquifer system like this and then here in this location a large number of industries were built in 1977 onwards you can see that now all of you have heard about this special economic zone so in 1970s also during Indragram this time or monarchy their size time they they also have these kinds of ideas special economic zone so you can understand what is in a special economic zone what happens one man gives lock of incentives and then what happens is that most of time the environmental routes and regulations are diluted that is very really what is happening is that all these environmental regulations will be diluted into for the favor of the industries so that is what is happened in this area so called a Patanjiru industrial belt in in near Hyderabad so here about 700 industries were established from starting from 1977 onwards so of course most of these industries are small scale and medium sized industries there are pharmaceutical industries then the paint industries the metal industries a large number of industries were in this location so free power were given there were absolutely there were no environmental regulations during that time and here you can see that there are some small streams and then one major river called Nakhavagu in Telangu Bagu means river and then another river which is so called Manjira river is coming so you can see that here this is the major stream and this is the starting of this the stream here and here is the ending of the river from this with respect to this aquifer system so what we why we undertook this study here the question was that say last 30 years or 35 years these industries were say operating and then the initially 77 onwards 10 years absolutely there were no treatment plans most of the industries what they do they just say put their all effluents to this nearby streams like in Nakhavagu or this kinds of small streams and then other solid waste they just put in the land fields nearby so all this you can see that in this area the rainfall is about 110 centimeter rainfall so with respect to that say the directly coming effluents as well as what is coming from this landfill all these things percolate to the underground system and also the surface water system. So in the after 1977 within five years all this area got polluted the surface water got polluted and then people start complained so finally the government the state government and pollution control board has to do something and finally they they put certain certain routes and regulations for this and finally I think after 10 years they have started some effluent plant and some treatment have been started in this area so the question what we want to analyze here is with respect to this effluent treatment plants and then with respect to various sources how the contamination is spreading how the the system is behaving be especially the ground water system of course surface water is also very much affected that also we have conducted a large number of field studies so our question here is with respect to numerical model we want to see how the system is behaving how the pollution is spreading and then if say the question was that I say if we stop the pollution how many years more we need to remediate the system so that also we have analyzed some of my students analyzed later so here I am just showing the how we can develop a final difference model and then how we can solve this problem so here even though this final difference method you can it is not so complicated method you can write your own code but very established models are available in literature which is so called a mod flow modular type flow which is actually three-dimensional package which is developed by USGS United States Geological Survey and this is with the another standard software used all over the world in many type of problems especially ground water flow problems and then of course transport problem also since transport problem we have to first solve the ground water flow equations and then we have to find out the head variations and then only we can find out see page velocity and using that see page velocity only we can solve the transport equations so here the package so called a mod flow is used so the various of course we have to study the various aquifer properties various the source of pollution we have to see the head variation all these things we have to study while doing a realistic field investigations using a computer model so here for this aquifer system the aquifer depth varies it is actually a two layer aquifer system and aquifer the depth varies from 10 to 25 meter and a number of pumping wells are there and then also the the porosity varies from 39 to 43 percent and then the hydraulic conductivity at various locations varying from 2.5 meter to 100 meter per day and as I mentioned the the annual rainfall is about varying from 802 1500 millimeter or 80 centimeter to 150 centimeter so if you take annual rainfall say here we are considering is 80 centimeter and then a recharge takes place from varying from 100 to 110 millimeter per year to the aquifer system that is the field study is indicated and then a number of pumping studies were done so that we can identify how much is the the hydraulic conductivity variation so as I mentioned here this is a two layer aquifer system so here the aquifers are shown here and to solve this problem say you have to see that when you go for modeling you especially when transport modeling is concerned condemn our final aim is to see how the condemnation spreading is taking place with respect to the various sources so we put over this 500 square kilometer area we use the very fine mesh we use the small rectangles of 160 meter by 150 meter size and then that is a discretization you can see in the this is actually the discretization for the area actually this is one by Mr. Shajil here who is here sitting here so this very fine mesh you can see that that will give the the flow variation and the the the head variation and the the concentration variation very accurately so that was our purpose so 160 meter by 150 meter cell has been used and then the of course we have to see the various aspect when you construct the model define the boundaries then define whether you are going for a layered model 3d model 2d model and then of course you have to see how the how the the pumping effects you can see that this is a larger area 500 square kilometer area so number of pumping wells will be there so that we have to find out how many pumping wells and then what are what is the pumping taking place at various wells and then permeability variation we have put into various zones how the the permeability variation is taking place and then the of course as I mentioned once the problem is defined we should give a boundary condition since the the variation is with respect to the boundary conditions actually the investigation in this location showed that there is not much variation with respect to the the flow is considered there is not much variation with respect to time so we actually simulated the problem as a steady state model for flow but transport is of course time dependent model we are done so here the head is actually for the given problem in this location you can see these two areas actually as I mentioned one river is entering here and the river is going here so at these two locations actually the the head the hydraulic the the head was available so these two boundary conditions were taken and the head the variation with respect to this is about 57 meter difference that has been taken in the you know 72 meter difference has been taken in this model so that have the boundary condition for flow and then as I mentioned the the aquifer since the there is a river at the middle a major river at the middle so the aquifer is coming like this so there is the other sides of the aquifer system we put in no no flow boundary conditions only the head is known at the exit of the river and the entry of the river other location we put as a no flow condition since the flow is taking place this direction so the the first stage of the model preparation is as I mentioned the pre-processing so pre-processing include we have to define the domain we have to put the grid as we have already seen here the grid then we have to define the boundary conditions and if it is unsteady flow simulation then we have to define the initial conditions also then as I mentioned we have to see the variation of the the hydraulic conductivity we have to see the variation of the if it is the porosity variation we have to find out so all this the the include the pre-processing of the model so this is the the the grid which we used for the simulation so this yellow things indicates the various well locations and and here this is the major industry pollution so we have to use very fine mesh in these regions actually it is not essential to have a uniform grid we can have also non-uniform mesh wherever our major interest of area is there there you can have very fine mesh and wherever the not much area of interest say for example this region since the condemnation the flow is this direction so here if you want you can have a very coarse mesh at this region and this location we can have fine mesh so then of course as I mentioned various processes we have already discussed so we have to calibrate with respect to the the various parameters like the hydraulic conductivity porosity etc so a rigorous calibration process has been done for this area so we have calibrated with respect to some of the observed values observed head values and then we have done as I mentioned we have gone for a steady state simulation here so once the the it is done then the next stage is we have to represent how the head is varying so you can see that head variation conduits are plotted here so here the the variation of head is produced by solving the flow equations by using the mod flow package then of course you can say as I mentioned the major flow is in this direction of the river so here you can see that the the flow variation is taking place like this so that is also we have plotted and then as I mentioned our major purpose here is to do a transport model since we want to see how the condemnation is spreading with respect to this industry pollution so of course then when we come back to so the essentially the procedure is same using finite difference method so you can just discretize the transport equation and then get a system of difference equation and so on so here actually we used a package called MT3D so which is which is a package which is based upon the method of characteristics which is also combined with the final difference scheme so the the various parameters which are depending for the transport simulation we should know the dispersion coefficients or dispersivities so these values initially be used and then with respect to some of the offset values we calibrated these parameters and here as I mentioned here the middle where that industry belt is there there are high concentration of TDs total dissolved solid that was the major source of pollution so it is varying from 1500 to 4500 so actually we did a very vigorous field investigations in 2003 and 2004 and then we collected some of the sample water from the available boreholes and then we analyzed and then we found the concentration especially we are trying to model the TDs total dissolved solid variations with respect to time so this values the initial values the initial values are 2003 whatever we measured and that are our initial values and then from that the you can see that the continuous source is there the seat the there is a treatment plant called common effluent treatment plant is there so from that we know how much is the source coming from this so that is the continuous source or as the condition for the transport model so that is the modeling procedure for the transport simulation using this MT3D and then from the given conditions our aim is the flow conditions are known from the head values are known from that we calculate the velocity vectors so that velocities you know that in any kind of transport modeling the major process are the the advection process so that velocity gives the input to the transport equation and using by solving that transport equation we want to find out the concentration variation so the 2003 is our the field data available so we next prediction is for 2007 so these are the initial variations so here you can see that there is a major concentration variation at this location where the industry belt is considered so that is the our initial conditions and then of course the continuous see a common effluent treatment plant is continuously putting this much tedious to the system so that is the the conditions here so this is the 2003 conditions and then we are predicted for 2007 since 2007 as I mentioned now again we have to go for a vigorous appeal investigation to see whether it is how much is the so what we did we run the mod flow model and the empty 3D and then you can see that here the condemnation is seen in this industry but it is keep on increasing and then with respect to the flow conditions it is keep on going to the downstream so this shows the 2007 then 2019 so like that so so many years 5 years 10 years 15 years 20 years etc we have predicted and then the Shajil my student he has done also say the once the the pollution is taken place how say for example if we can cut tile we can stop the condemnation source then how many years it will take it is mind boggling it is say we have shown that if this is the the situation if 2007 if we stop the condemnation that means the condemnation is still going on but if you stop in 2007 it in your time minimum 15 years or a natural attenuation if you don't go for any remedy activities it has to keep it with respect to the flow condition it takes away the transport process and then it needs minimum 15 years to reduce the condemnation to the level of 500 milligram per liter TDS so that is the 500 ppm TDS so that is that much time is required for natural attenuation so then we can plot then the software will give how the time versus concentration at various location we can find out so this is the way we develop a numerical model for a real problem using the finite difference method and of course here we are used the software mod flow and empty 3d so like this say the real field problem when we deal it is so complex even though the methodology symbol the finite difference method but it to develop such a real realistic model it needs lot of input data and then it needs a lot of that using those information we have to do the preprocessing and then we have to run the model and then of course to see how the variation we have to do a post processing so this is about in this session so now next five minutes you can ask any questions we can discuss if you have any questions this is actually as I mentioned the calibration is done so that means we have some observed data so with respect to that observed data we checked how the system is behaving that is already done even the mod flow is a very established package for the given system of course we have to validate it is a process of validation yeah that is what I am saying it is a validation of course is done that is to be done