 Namaste and welcome back to the video course on Watershed Management. In module number 4, lecture number 18, today we will discuss subsurface and groundwater flows. So, some of the important topics which will be discussed today include subsurface flow, infiltration, aquifers, groundwater flow, groundwater flow modeling, numerical modeling and groundwater quality. So, some of the important keywords are subsurface flow, infiltration, aquifer, groundwater flow, groundwater flow modeling, numerical modeling, groundwater quality. So, as we discussed earlier in some of my earlier lectures, so as we discuss about the water source in a watershed. So, we are related to watershed management. So, we have to see the total availability of the water. So, as far as water is concerned, water can be either surface water or groundwater. So, in any of the watersheds, since groundwater is a major resource as water, so that we have to say utilize it properly, we have to plan it properly, we have to understand its behavior within the watershed and then also we have to, while we use the groundwater, we have to replenish the groundwater through the various schemes like rainwater harvesting or artificial recharge and other ways. So, within this contest today, we will discuss the important aspects of groundwater flow and its modeling within the perspective of watershed management. So, as we have already seen groundwater is say is a part of the subsurface flow. So, subsurface water is the all water beneath the air surface. So, we can see that whenever rainfall takes place, there is the infiltration takes place and then water will be percolated down and that movement, so all water beneath the air surface, we can call it as subsurface water. So, the water will be is recharged by infiltration either directly on the land surface or in the beds of streams, lakes and oceans. So, you can see in this figure, so when the rainfall takes place, the recharge is taking place with respect to the infiltration and then also say from the surface water sources, lake lakes then rivers and oceans also the water is infiltrated down to the subsurface and that becomes the subsurface water. And then as far as the subsurface water discharge is concerned, it is discharged through evaporation, transpiration and then from springs, seeps on land surface or beds of surface water bodies and then we can get this water through pumping wells, gravity drains etcetera. So, as far as subsurface water is concerned, there is say the water recharge taking place with respect to rainfall or with respect to the movement of water from the surface water sources like lakes, rivers, ponds or ocean that is the recharge taking place to the subsurface as subsurface water and a discharge is concerned with the water which we are taking through pumping wells or water coming out as springs or say the gravity drains which we give, so that way the discharge is taking place. So, as far as subsurface environment is concerned, you can see that some of the, so this depends the subsurface flow, the subsurface water movement depends upon the porous material. So, water moves within the pores of the soil, so you can see that if this is the soil media, so within the soil media lot of pores will be there. So, water will be moving through this say the pores and that water is coming as the subsurface water. So, most of the terrestrial hydrologic activities takes place within a few feet from the surface of the earth or within the root zone. You can see that whatever is happening is that say water is recharging and then say it is the immediately it is going through the soil and then it will be reaching the aquifer system or the water table, it can be say aquifer can be confined or unconfined nature. So, generally to be initially within the unconfined layers and then say most of these activities will be taking place within the root zone and then further this ground water may percolate down deep, so that deep percolation also can take place. So, then say as far as subsurface water is concerned say what we discussed now is say within the context of soil water. So, this soil water we can divide into 3 parts. So, say how the water is available within the subsurface or within the soil, so accordingly we can classify into 3 types. So, first one is the drainable water. So, that is the water that readily drains from soil under the influence of gravity and water occupying pores larger than the capital size. So, that way that will be drained out. So, that is say with respect to the recharge and then what is available say within the subsoil or the subsurface that kind of water is called a drainable water. And then a second class is so called a planned available water. So, planned available water is the volume of water released from soil between a soil water pressure head of about minus 1 by 3 bar which is so called generally called field capacity and about minus 15 bars which is so called a wilting point. So, this water is a generally detained in storage by capital forces. So, you can see that when this water is infiltrating down, so we can generally classify the water available in subsurface or in the soil as fully saturated which is generally called as ground water. And then there will be a capillary fringe like this and then there will be unsaturated zone where the water will be available within the soil which is not fully saturated. So, that way then the water which is generally available to the plants I mean between the field capacity and the wilting point that is so called a planned available water. And third one is say the water still existing the soil that is so called unavailable water which we cannot drain or which we cannot which is not will not be available to the plants. So, that is so called a hygroscopic water. So, water held tightly in film around the individual soil particles. So, that is will not be available to the to the plants or even it will not be possible to drain. So, that way the water subsurface water we can classify into drainable water, planned available water and the unavailable water. So, as we discussed earlier say within the hydrologic processes then say with respect to the rainfall to run off various processes are there like infiltration, then evaporation, evapotranspiration, interception like that various processes will be there. So, what is available as water a subsurface water or ground water is coming from the infiltration of the water which is coming from the rainfall. So, this infiltration process is very important as far as ground water or subsurface water is concerned. So, the about the infiltration we have discussed already earlier. So, again say as subsurface water we will have a brief discussion further. So, infiltration is a process by which what on the ground surface and there is the soil as we already seen earlier. Then the infiltration capacity of soil determines how much water is infiltrated down and then how much is the time distribution of rainfall access for runoff from a storm. So, accordingly the infiltration capacity varies and say it is a very important to estimate the infiltration within the contest of surface runoff and then the availability of water as far as the subsurface water is also concerned. So, subsurface flow and storage of water within the watershed depends upon the infiltration. So, as we discussed in one of the earlier lecture the controlling factors say that control the infiltration are the soil type like size of particles, then degree of aggregation between particles, arrangements of particle and then what kind of vegetative cover is there, then surface crusting, then season of the year, then say what is the soil moisture level available within the soil or so called andesite moisture, then rainfall hyctograph, subsurface moisture conditions etcetera. So, we had a detailed discussion on infiltration in one of the earlier lectures. So, we will not be discussed further on infiltration. So, now we will further discuss about the ground water say within the contest of the subsurface water and a ground water flow. So, as we discussed just in the previous slides, so water is the water available within the soil we can classify into unsaturated and saturated. So, the flows within the soil can be unsaturated flow or saturated flow. As you can see here say the once the water is infiltrated down. So, here the we can classify unsaturated soil and saturated soil. So, unsaturated soil means the water moves primarily in small pores and through fillings located around and between solid particles as water content decreases, cross sectional area of the fillings decreases and flow paths become more limited. So, this results in hydraulic conductivity function that decreases rapidly with water content. So, this unsaturated flow or unsaturated soil is a very important as far as especially as far as the water availability to plants are concerned. And then saturated soil or saturated flow is related to the soil pores are considered full with water. So, it may not be completely full due to air entrapment within the soil, but say wherever we consider the hydraulic conductivity will be constant with respect to the head edge as far as saturated flow is concerned or saturated soil is concerned. So, here you can see that as shown in this figure. So, say once the water is infiltrating down this is the unsaturated flow region and here this is the so called water table then this is the groundwater or the saturated flow. So, that way now this so called unsaturated flow that depends upon how much is the infiltration, how the infiltration process take in place and then say how much is the what kind of soil is that accordingly the water availability within the unsaturated zone and water movement through the saturated flow condition takes place. So, as far as the soil water movement is concerned. So, this is the response to a gradient generally. So, depends upon how much is the pressure or how much is the gradient accordingly the soil water movement takes place. So, for example, if the soil is fully wet. So, wet soil to dry soil. So, generally low soil moisture tension to high soil moisture tension and high soil water potential to low soil potential when we consider wet soil to dry soil. And when we say that the soil is fully saturated or saturated condition means water is moving mainly in the macro pores. So, that all of the pores are filled with water. Then the unsaturated flow is concerned unsaturated conditions are concerned the macro pores are full of air micro pores filled with water and air. So, the moisture tension gradient creates the unsaturated flow. And then very similarly the saturated flow is concerned the it is we can also called saturated flow as gravitational flow. So, this occurs under saturated conditions when the force of gravity is greater than forces holding water in the soil. So, from capillary flow onwards the saturated flow starts. So, capillary flow occurs in unsaturated soil. So, here this is the unsaturated soil and just below there is a capillary fringes there and then below is the saturated flow. So, we can measure this the soil content the water content in the soil. So, the measurement of soil moisture various methodologies are available like gravimetric methods, then tensiometer, then electrical resistance method etcetera number of methodologies are available. So, using these techniques we can identify how much moisture content is there within the soil. So, anyway we will not be discussed further on these aspects. Now, we will go to the saturated flow conditions or the ground water flow. So, the ground water flow is say fully saturated flow condition. So, infiltrated water say some replenishes soil moisture deficiency if soil is not saturated and when the soil is fully saturated then we are having the shallow ground water system. So, as you can see within the hydrologic processes. So, you can see that with respect to rainfall the water is infiltrating down and then where the shallow ground water system is after the saturation we are having the shallow ground water system. So, then water then percolates down until it reaches the saturated zone called a aquifer or deep ground water system. So, here once it is say fully saturated and then water is further percolating down to the aquifer system. So, the upper water surface of saturated zone is called generally a water table. So, when we consider an unconfined aquifer we are having the say the unsaturated zone then the capillary fin zone and then the saturated zone. So, that saturated zone that say the surface is so called a water table. So, then say as we can see that say here say for example, this is the water table and this is the unsaturated zone and this further to the saturated flow condition. So, as we can see that as I mentioned earlier ground water is one of the important source of water for the humanity for say domestic and all types of uses. So, it is an important part of the hydrologic cycle. So, we cannot separate the ground water flow with respect to the runoff or the surface water. So, say after the available fresh water resources the major source is the ground water. So, an estimate says that the ground water constitute more than 80 times amount of fresh water in rivers and lakes combined. So, you can see so the importance of ground water. So, ground water is one of the most important say resource water resource available for humanity. So, only difficulty is that the deeply percolated water it is very difficult to pump out, but say in most of the places we can the whatever is especially shallow water or the somewhat deep zone water we can say pump it out and then use accordingly. So, ground water is one of the important source of water. So, as we discussed say once the water is percolating down we are have the water through the saturated zone once it is reaching the aquifer system or where the water is stored or water is moving through the soil. So, that formation is so called aquifer. So, accordingly now let us see the different types of classifications as far as the aquifers are consents. So, here in the slides the classification of aquifers are given. So, aquifer as I mentioned it is a formation that contains sufficient saturated permeable material to yield a significant quantity of water to wells or springs. So, this is the definition of aquifer. So, say for example, if a sand formation where significant amount of water water is holding within the soil and that can be drained say by using wells or it will coming through springs. So, that is so called a aquifer. And then say there are classification like a acute clout, acute clout is a saturated, but relatively impermeable material and it does not yield appreciable quantities of water. Say for example, clay formation. So, clay formation once it is saturated contains so much of water, but we cannot drain the water easily. So, it is relatively impermeable material. So, we cannot drain the water. So, that is that kind of formation is soil formation is called acute clout. And then aquifuge is a relatively impermeable formation it neither contains nor transmits water say for example, granite formation. So, that is so called aquifuge. And then acute clout, acute clout is saturated, but poorly permeable stratum. So, like sand clay mix or sandy clay. So, that is that kind of formation is called an acute clout. So, that way when we look the ground water as a source is available mainly from aquifer or acute clout. And then say other formations even the water is available we cannot get a substantial amount of water. So, then aquifers again further can be classified into confined or unconfined. So, you can see that say an unconfined aquifer means say water is say infiltrating down and this is the water table as shown in this figure and then there is an impermeable layer below. So, there is a water table or a free surface as you can see here in this figure. So, that kind of formation is called unconfined aquifer. And then confined aquifer means the formations or the aquifer between which is confined between two layers you can see that here this is one layer of impermeable layer and this is another layer of impermeable layer. So, the confined aquifer is this soil this material where a considerable amount of water is there which is also moves with respect to gradient. So, that formation is so called a confined aquifer. So, generally the aquifers can be classified into either confined aquifer or unconfined aquifer. So, that depends upon the hydrogeological conditions of the particular area. So, now let us further discuss what are the important characteristics as far as the aquifers are concerned. So, we have seen that the rainwater or the precipitate precipitate water is say infiltrating down and then that is further percolating down to the aquifer system. So, the water movement within the subsurface or within the soil depends upon various parameters like porosity, specific historic coefficient, hydraulic conductivity etc. So, let us see some of these important definitions as given in these slides. So, first one is porosity. So, those portions of soil not occupied by solids. So, that is say the meaning of porosity. So, generally the porosity is expressed as ratio volume of pores or industries to total volume. So, that is the porosity. So, you can see that in this soil medium say large number of pores you can see. So, porosity is the ratio volume of pores to pores or industries to total volume. Then further percolation, percolation means rate at which water moves downwards through the soil. So, once it the water is infiltrated down. So, what is the rate at which the water is moving? So, that is so called percolation. And then another important time is so called permeability. Permeability is an expression of movement of water in any direction. So, percolation is generally we mentioned as downwards movement, but permeability means it can be in lateral directions also. So, that is the so permeability and expression of movement of water in any direction. And then as far as the aquifer is concerned say the times like specific yield, specific yield especially we mentioned about form unconfined aquifers. So, it is a ratio of volume of water that after saturation can be drained by gravity. So, that is the definition of specific yield. And then storage coefficient or generally called it as storativity. So, storativity is the volume of water that an aquifer releases from or takes into storage per unit surface area of aquifer or per unit change in head normal to that surface. So, generally we mentioned the storage coefficient for confined aquifers or sometimes we mentioned for an unconfined aquifer also depending upon the way which is mentioned within specified equations. And then another important time is so called hydraulic conductivity. So, hydraulic conductivity which is generally mentioned as say the simply scale. So, this is a constant that serves as a measure of the permeability of the porous medium. So, how much water can move through the soil medium say from one location to another location. So, it is that care represent or hydraulic conductivity represent the measure of the permeability of the soil media or porous media. So, accordingly the movement takes place within the soil media and its unit is the velocity the unit of velocity like meter per day or meter per second like that. Then another time is called transmissivity of the aquifer media. So, transmissivity is the rate at which water is transmitted through a unit width of aquifer under unit hydraulic gradients. So, generally transmissivity is equal to hydraulic conductivity multiplied by B where B is the saturated thickness of aquifer. So, these are some of the important times which we have we will be using as far as the subsurface flow or ground water flow is concerned. We have seen what is subsurface water or ground water and what is aquifer and what are its important characteristics. So, now let us discuss what are the governing laws as far as the ground water flow or ground water movement is concerned. So, one important say governing law was proposed by Darcy in 19th century Darcy is a French engineer Darcy was a French engineer who conducted large number of experiments in 19th century and put forward certain laws called Darcy's law. So, Darcy defined how water moves through a saturated porous media with analogy of a cylinder fitted with inflow and outflow pipes Darcy showed that velocity was a function of difference in head H over a finite distance L. So, if we consider the say for example, if this is a soil media. So, between say one location to another location within the flow direction say what is the head difference and what is the distance between those two points accordingly he proposed the Darcy's law. So, Darcy's law is say velocity of flow v is equal to minus k dh by dl where v is so called Darcy's velocity or specific discharge and k is the hydraulic conductivity and this dh by dl is the hydraulic gradients. So, here in this equation minus sign indicates the flow of water is in the direction of decreasing head and once we find the Darcy's velocity we can find the actual velocity by dividing Darcy's velocity divided by the porosity. So, this Darcy's law is one of the most important law as far as ground water is concerned and based upon that only all the further theories on ground water flow movements and transport process were put forward later time. So, Darcy's law is very very important. So, generally the Darcy's law combined with mass balance or the conservation of mass we derive the governing equations as far as the ground water flow is concerned. So, that will be discussing later and as per this Darcy's law generally the flow should be in very low Reynolds number or laminar flow condition. So, Reynolds number which is the ratio of inertia force to viscous force generally this law is valid when Reynolds number is less than 1, but still it may be valid within the range of say even Reynolds number up to 10. So, now say this hydraulic conductivity which we have seen one of the most important parameter as far as the porous media is concerned. We can find the hydraulic conductivity by various experiments like pumping test within the aquifer or tracer test or say depending upon the soil media we can use certain formulas and then also based upon laboratory methods we can determine the hydraulic conductivity. So, now let us further discuss say how the ground water flow is moving within the porous media and then what way we can quantify or what way we can model this ground water flow movement. So, let us see in the coming slides. So, as I mentioned earlier also so as far as soil or the porous media is concerned it is very heterogeneous and then the its properties like the porosity or the hydraulic conductivity is varying drastically from one location to another location and from one direction to another direction. So, the soil media or the porous media is totally heterogeneous and anisotropic. So, anisotropic means the parameters are varying with respect to direction. Heterogeneous means the parameters are varying from one location to another location. So, due to this say generally it is very difficult to say get to the correct value of this hydraulic conductivity or porosity. So, generally we use some approximations depending upon say some pumping test or some of the field test and we determine some values and then we do through a calibration process. We identify the hydraulic conductivity and other parameters as far as the aquifer consumption. So, as I mentioned earlier also say as far as the soil is concerned say or the aquifer is concerned various types of soils forms the aquifer system like a alluvial deposits, limestone, volcanic rock, sandstone, igneous and metamorphic rocks. So, accordingly since all these varieties taking place and sometimes say one variety or most of time a mixture of these varieties. So, that when the porous media characteristics drastically changes from one location to another location. So, that is why we call the porous media is heterogeneous and then from say either vertical direction or lateral direction also these parameters are changing. So, that is why we call this the porous media as anisotropy. So, accordingly say when we discuss about the groundwater flow movement. So, due to the heterogeneity or due to the anisotropy the movement rate also in the groundwater movement rate also varies. So, that way when we discuss about the groundwater flow analysis. So, the process is very complex due to the complexity of the porous media and then various other parameters. So, most of the time when we are looking for groundwater conductivity or quality investigations say the we have to deal with very complex hydrogeological systems. So, you can see that as in as shown in this figure. So, there are number of layers of material different types of materials as far as porous media is concerned. So, that way it is very difficult process very complex to say study the groundwater flow movement or the groundwater quality analysis. So, that also clear from this figure also where a watershed is concerned the parameters are changing from one location to another location. We have the limitations as far as the field investigations are concerned. So, we cannot say generally field investigation means through bore holes or bore wells we have to get the data. So, that is also not so easy since we cannot have so many bore wells or bore holes say as far as a watershed is concerned. So, generally say the data available through few through some of the bore wells or bore holes we get the data and then we average as far as the various parameters are concerned. So, that way groundwater flow modeling the computer based modeling are the only solutions as far as the groundwater flow investigations or groundwater quantity investigations or quality investigations. So, now another issue the other than the groundwater quantity and the important issue is the quality of the available groundwater. So, when we discuss about the water resource plans within a watershed, watershed management plans say water resource is concerned. So, the available water should be the not only the quantity, but quality also should be good for various uses like the domestic uses or the agriculture uses. So, groundwater quality is a major issue. So, earlier times say with the there was a thinking that the groundwater is a good quality and we can directly utilize, but due to the industrial revolution or say overuse of fertilizers and other things for the agriculture. Now, last few decades it has been identified that groundwater is also very much amenable to pollution and many locations groundwater pollution have been identified. So, groundwater pollution has become a major problem in many countries. So, the reasons are indiscriminate disposal of industrial waste, extensive use of chemicals in agriculture like fertilizers and pesticides and a host of other human interventions are causing the groundwater pollution. So, effluents in water bodies after affecting soils. So, if say the untreated effluents or treated effluents to certain extent are putting into the in the river or lakes. So, this water will be percolating down to the groundwater system through down downwards gravitational movements, lateral dispersion and adductive migration. So, that way you can see that like in this figure. So, this groundwater pollution space and that it will be further moving and further the fractures, fissures, joints etcetera what are there in groundwater or the aquifer system provide additional preferred pathways for fast migration of pollutants. So, with increase in industrialization and increasing use and reliance on groundwater it is very important to assess the water quality and the steady the movement of contaminants in an aquifer system to predict the migration. So, it is not only the issue of groundwater quantity or the quantity of the water available as groundwater, but quality is very important. So, after identifying that many locations groundwater is polluted. So, we now we have to understand how the pollution is taking place and what extent the pollution has been taken place and then what kind of remedial activities can be done. So, that is nowadays a major questions. So, as far as groundwater contamination is concerned as I mentioned earlier some of the bottom sources are mentioned here. So, groundwater pollution can be from natural contamination like seawater intrusion or fluoride or arsenic contamination which is taking place in many locations and then the pollution or contamination can be agricultural contamination by the use of fertilizers pesticides etcetera. Then industrial contamination that means the flue and liquid waste or solid waste which is putting as landfill and that is further say the leachate is seeping down. So, that kind of industrial contamination and then underground storage tanks like in many countries the like petroleum products and other say liquid forms are kept in underground. So, there is always possibility of leakage from these tanks. So, that can be the underground storage tanks then land application and mining. So, what kind of say the fertilizer applications or what kind of solid waste are kept on the land. So, that will be percolating with respect to rainfall this pollutants will be percolating down and then another source is mining. So, you can see that many areas are different types of mines are there. So, mining is another important source of groundwater pollution and then say the septic tanks say most of the houses or many locations the septic tanks will be there and even if it is scientifically constructed there is possibility of leakages. So, that septic tanks can be another source of pollution then waste disposal in injection wells. So, many areas say the liquid waste or solid waste keep it in the soil media or very deep rocks. So, that can be a source of pollution and as I mentioned landfill is another source of pollution where if it is not properly or scientifically constructed landfill then leachate will be percolating down to the aquifer system. So, these are some of the important groundwater contamination sources. So, then as far as once the groundwater is contaminated or the pollutant is introduced to the subsurface or to the aquifer itself. So, there are number of processes taking place. So, that this plume will be or the contamination will be further spreading. So, the changes in chemical concentration or the contamination depends on four distinct processes like advective transports. So, here the dissolved chemicals are moving with the groundwater flow. So, you can see that already groundwater flow is there and then with that flow this contaminant plume is also taken within the flow. So, that is so called advective transports and then a second one is hydrodynamic dispersion. So, since the soil media is porous and then as far as the various way of movements within the soil the plume movement or the the contamination movements. So, hydrodynamic dispersion is another important process. So, this can include mechanical dispersion, hydraulic dispersion, molecular and ionic diffusion. And then another source another concern the groundwater pollution mechanism is fluid sources. So, water on one composition is introduced to and the mixed with a water different composition then that that is the groundwater contamination mechanism like what is happening sea water intrusion. And then reactions say some amount of a particular dissolved chemical species may be added or removed from groundwater as a result of chemical biological and physical reaction in the water or between the water and the solid aquifer materials. So, like that the groundwater contamination various mechanisms are there. So, when we study about the groundwater pollution we have to study this various mechanisms and then accordingly we we have to investigate how the contamination is further spreading. So, now say let us go to the groundwater flow or groundwater quality modeling. So, before that so let us further summarize what are the important work elements as far as groundwater investigation are concerned. So, here I have listed when we go for groundwater investigations we have to go systematically. So, the work elements are well elementary and selection of observation wells then preparation of groundwater level map for the particular watershed or particular area concerns. Then geophysical investigations to decipher the subsurface layers and their characteristics. Then identification of hydro geological features of interest which are likely to control groundwater flow and transports. Then understanding of aquifer geometry. Then detail and periodical water quality analysis. Then periodical monitoring of water levels in the observation wells. So, when we discuss about the the groundwater say say investigations within a watershed we have to go systematically. So, either water quantity investigations or quality investigations we have to study systematically as we are discussed in the work elements. So, various things we have to study systematically and then come up with the investigation report. So, now as I mentioned say we cannot have so much of field investigations to assess how much groundwater is available or how the groundwater quality varies. So, most of the time we have to depend upon mathematical models. So, now let us see some details about the mathematical models as far as groundwater is concerned. So, a model as I mentioned earlier is a representation of a system that means here the groundwater system. So, only effective way to test the effects of groundwater management strategy is through modeling. So, due to especially surface water is concerned we can say directly see what is happening, but groundwater is as you can see that only observation is possible through the wells or the boreholes. So, that way it is it is always better to go for mathematical modeling or mathematical modeling is only way out as far as groundwater investigations are concerned. So, mathematical model simulates groundwater flow and or solute feed and transport indirectly by means of a set of governing equations thought to represent the physical process that occur in the system. So, that is the mathematical model within the context of groundwater. Then the governing equations we can identify governing equations are generally derived from Darcy's law and water balance equation with Hedda's the mainly as the dependent variable. And then of course, we have to define the boundary conditions say if this is the key for system we have to identify what is the Hedda available here or the flux say for example, this is an imber medium water cannot percolate down further. So, that is an imber medium where the flux can be assumed as 0. So, the boundary condition can be in the direct boundary conditions or Dirichlet boundary conditions and then Neumann boundary conditions or in terms of flux. And then when we are going for transient analysis like within an aquifer system when we are pumping water there the Hedda will be continuously varying with respect to time. So, that is called transient analysis. So, there we have to start with some initial conditions what is at time t is equal to 0 how the Hedda is distributed or how the condemnation is spread throughout the aquifer. So, based upon that only we will be doing further the mathematical modeling. So, that way by using the Darcy's law or under the conservation of mass or water balance we can derive the groundwater flow equation. So, as shown in the slide the derivation of groundwater flow equation is we can do by considering a representative elementary volume ReV. So, delta x by delta y by delta z. So, if you consider the flux through the ReV representative elementary volume. So, what is coming out minus what is in is equal to change in storage and then this we can combine with the Darcy's law. So, that way we can derive the groundwater flow equation. So, the derivation of groundwater flow equation is based upon the mass balance and the Darcy's law. So, here this is the say representation as far as Darcy's law and then low mass balance with respect to this. So, finally, using both we get the divergence k gradient h is equal to assess del h by del t this equation we can obtain which is the governing groundwater flow equation. So, in a better mathematical form here I have written the groundwater flow equations generally in 3D form say for example, unconfined aquifer. So, del by del x of k x del h by del x plus del by del y k y del h by del y plus del by del z del h by del z is equal to assess in a del h by del t minus r where this r is the recharge of pumping. So, minus 4 recharge and plus 4 pumping. So, assess is the say specific yield or specific storage coefficients as given here and for confined aquifer we can represent the equation like this and in 2D the same equation can be mentioned like this. So, of course, depending upon like various recharge or pumping times or other sources coming to the aquifer system where this equation can some more times will be there or some changes will be there otherwise general equations are in this form. And then as far as groundwater quality is concerned we have to say study the transport process taking place as far as the aquifer is concerned. So, very similar way we can derive the groundwater transport equation by using the fixed slope. So, here the objective dispersive solute transport equation in groundwater we can write here I am not going through derivation here the equation is written here it is del by del x of d i z del c by del x minus del by del x i c v i plus w by n in the c dash is equal to del c by del t. So, where d is the hydrodynamic dispersion coefficient tensor c is a concentration solute in source or sink fluid n is the porosity dimensions and x i x j r Cartesian coordinates v i is a seepage velocity obtained by solving the groundwater flow equation and then v is obtained by using the Darcy's law and w is the volume of flux per unit volume and c Darcy is the solved concentration. This equation is also concerned different forms are available depending upon various kinds of transport processes, but a general form is written here. So, as I mentioned so, here once the groundwater flow equations are solved we get the head variation and then using the head variation by using the Darcy's law we can obtain the velocity in x y z direction and then from that velocity components we can utilize within the transport equation. And then also we have to specify the initial and boundary conditions as I mentioned initial condition for flow will be flow equation will be say if you know the head variation within the aquifer system that will be initial condition and boundary condition flow for flow equation will be either head prescribed or flux prescribed in terms of Dirichlet boundary condition and the Neumann boundary condition. And as far as the transport equations are concerned initial condition can be the variation of concentration at the starting on the time step and boundary condition can be in terms of the knowledge of concentration at particular boundaries or gradient of the concentration. So, this constitute so, once we define the boundary of the aquifer system and then we identify whether it is confined aquifer or unconfined aquifer and correspondingly we can take the equation for confined aquifer or unconfined aquifer and then we can use the transport equation. So, the modeling can be either in three dimensions or two dimensions or sometimes in one dimension and then say we can also use the Darcy's law with respect to calculate the velocity components. So, that constitute the mathematical model as far as the ground water flow is concerned. And then say the solutions of this mathematical models say we can go for say for simple problems like here some hypothetical or small type of problem where not complexities are there. Analytical solutions are possible say simple steady-stage ground water flow in homogeneous isotropic medium we can go for analytical solution. But as far as field problems are concerned analytical solutions are not at all available not at all possible. So, we have to go for only for numerical solutions. So, numerical solutions as I discussed in the previous lecture. So, various numerical techniques are used to transform this partial differential equations the second order partial differential equations. And the equations are transformed into algebraic form by using techniques like finite difference method, finite element methods, boundary element method or finite volume method etc. So, in the last lecture we discussed the methodologies like same finite element method finite difference methods. Also nowadays a method called analytical element method is also available for the solution of ground water flow and the transport equation. So, now say before discussing little bit further about the ground water the numerical techniques. So, ground water flow modeling is very important as I mentioned. So, this ground water flow modeling is a powerful tool for furthering our understanding of hydrogeological systems and ground water flow and transports. So, that is what we were discussing. So, the importance of ground water flow modeling include construct accurate representations of hydrogeological systems and understand the inter relationships between elements of systems and efficiently develop a sound mathematical representations as we have already seen in the previous slides. Then make reasonable assumptions and simplifications like whether two-dimensional modeling, three-dimensional modeling or one-dimensional modeling, transient modeling or steady state modeling. Then understand the limitations of the mathematical representations. So, we have derived these equations and then we are going for numerical modeling based upon number of assumptions. So, that are our limitations as far as the ground water flow modeling is concerned. Then understand the limitations of the interpretations of the results. So, since the data availability and its accuracy is very much varies with respect to ground water flow investigation. So, that way getting 100% accuracy in ground water flow modeling is impossible task. So, how much is how what kind of assumptions we put and then what kind of limitations are there? Accordingly, the model results will be the accuracy of the model results depends either for flow or transport is consents. So, now say for example, here before discussing the numerical technique like final defense or final tournament. So, here say for example, for simple flow say like we can even simply derive a analytical equation like say if we consider the Darcy's law say for flow movement from one location to another location if it is flow is homogeneous isotropic then we can simply derive equation like this. So, solutions we say for simple problems we can derive the the analytical solutions, but most of the field problems these kinds of analytical solutions will not be valued. So, this for this flow say isotropic homogeneous media we can say the schedule variation is actually linear variation which is obtained by this equation. So, that way we have to go for final defense method or final tournament method or other numerical tools. So, final defense method and final tournament method we discussed in the last lecture. So, further say as I mentioned in the earlier lecture final defense method gives the continuous variation of the function concerned by a set of values at points on a grid of intersecting lines as shown here. So, the gradient of the functions are then represented by the differences in the values of neighboring points and the final difference versions of the equations is formed. At points in the interior of the grid the equation is used to form the set of simultaneous equations giving the value of the function at a point in terms of value such nearby points. At the address of the grid the values of function is fixed or a special form of the final difference equation is used. So, as we discussed in the previous lecture. So, this final difference can be implicit or explicit we can have forward, backward or central final differences. So, according to the formulation which we are doing. So, we can have this various schemes and then we can transform the either the ground water flow equation or ground water transport equation using the final defense method. So, for example, in this slide say the final difference explicit scheme for the ground water flow equation is given. So, this is our ground water flow equation 2 dimension for say for example, for confined flow. So, in explicit scheme using the final difference schemes for a node ij. So, this is ij node for a specific time n using forward discretization time and central difference discretization space we can derive this system of equation. So, these details are available in test books like an induction to ground water modeling using final difference and final term method by Wang and Anderson and say Jacob Beer ground water flow Jacob Beer, Dynamics of porous media by Jacob Beer etcetera. So, this books gives all these details. So, due to lack of time for this this only ground water flow only one lecture is dedicated. So, that way not much time is there. So, that way I am not going to the details of this methodologies, but final difference in explicit scheme we can obtain like a stone in the slide. So, then final element method is another method as we have discussed in the previous slide previous lecture. So, the region of interest is divided in a much more flexible way the nodes at which the value of the function is found how to rely on a grid system or on a flexible mesh as you can see here. Then either like a triangular elements rectangular elements in two dimensions or correspondingly prism elements tetrahedron in three dimensions we can use. So, various schemes are there direct approach, variational principle or weighted threshold etc. So, that way also this given equations like the ground water flow equation or transport equation we can transform and then we can have the system of equation all right system of equations. And then we can apply the boundary initial conditions and boundary conditions and solve those system of equation either iteratively or direct methods to get first the the head variations. And then using the head variations we can get the velocity variation using the Darcy's law. And then if you want to say identify how the transport mechanism then we can solve the transport equation also very similar way using finite element method or finite difference method. So, before closing this lecture let us go through one case study. So, the case study here is the IDI Patanjiru. So, here an industry development area in Patanjiru near Hyderabad the area is in Madag district and the area is about 500 square kilometer. So, this is the the Ikea region and it is spread in three bundles more than 600 industries in this area dealing with the pharmaceutical Spain's and pigments etcetera were established in 1977 since 1977. And then say in 80s say there was no when it was the industries were established there was no say effluent treatment plants. So, these industries directly put this effluence to the nearby streams. So, that way this has become a major problem in this area ground water pollution has been a major problem. So, as a part of this ground water transport mechanism and ground water quality modeling here we have developed a model using visual mode flow. So, I briefly discuss about that. So, here in this area the ground water recharge varies from 100 to 110 mm per year for an annual rainfall of 800 mm. Terminativity values vary from 50 to 80 meter per day and a transmissivity varies from 140 to 1300 meter square per day and offset site data shows that top weather aquifer is having 10 to 15 meter thick underline by fractured layer. So, simulated model domain is above consist 55 rows 65 columns with a 250 meter approximately by 250 meter grid and two layers you can see here two layers and this is the grid which we have generated using the visual mode flow. So, the mode flow is based upon the finite difference scheme and this is the boundary. So, now these cells will be in active cells and only the cells within this boundary will be active cells. So, top layer of 10 to 25 meter thick alluvium of Nakavagu weather zone is granite and is underline by 10 to 20 meter fractured zone. Vertical section simulated in model is having the total thickness of 45 meter and water table in the area has an elevation difference of 75 meter. So, with the southern boundary near this location is about 570 meter above mean sea level and here it is about 495 meter. So, as far as analysis concerned main issue was here the transport modeling. So, that way we conducted a steady state analysis. So, this using the mode flow as I mentioned model flow is a finite difference based package developed by Macdonald and Harbu. So, the aquifer model simulation is carried out. So, model is calibrated between offset data and simulated results. So, water table configuration of November 2003 was adopted for this purpose. So, computer and offset water level for the steady state condition is shown here. So, this is the variation of the. So, once the model is run with model flow is run we obtain the head variation from here to here and that is this contour line shows and then also we can calculate how the velocity variation taking place. So, you can see that this RMR shows velocity variation within the aquifer system. So, using this mode flow good agreement is observed between computer and offset water levels for the aquifer system. And then we run another model called mass transport in three dimension that is also a finite difference based model by assuming dispersivity values 100 meter 1 meter and 0.01 based on field observation. A constant tedious concentration at different nodes of Nakavagu which is this here and varying from 4500 milligram per liter to 1500 milligram per liter has been used. So, the main pollution was on this source. So, this is the initial condemnation at 2003 and then we run the model to find out how the spending will take place in 2007 and further. So, 2003 measure data was available. So, based upon that we run the model to identify how the condemnation spending in 2007. So, this shows the variation. So, like this say by using the mathematical or numerical model like a mode flow or empty 3D we can assess the groundwater say flow condition of the quantity based modeling and the transport modeling how the condemnation spending is taking place. So, that way we can model using the groundwater flow model using finite difference, finite element method or other kinds of numerical techniques. So, for this today's lecture some of the important references are listed here. And then before closing the lecture some tutorial questions so how groundwater condition can be improved in a watershed discuss the importance of groundwater and water management plans discuss groundwater resources improvement by rainwater harvesting and artificial recharge. So, these questions can be answered based upon today's lecture and some of the previous lectures on rainwater harvesting and artificial recharge. Then some self evaluation questions why groundwater is very important in water management describe different types of soil water differentiate between unsaturated flows and saturated flows. What are the important work elements in groundwater investigations discuss groundwater quality issues. So, these all these related answers you can obtain from today's lecture. And then some assignment questions explain how to assess groundwater potential describe different types of aquifer classify aquifer according to characteristics. Discuss fundamental laws governing groundwater in a watershed how to model groundwater flow explain major modeling techniques for groundwater flow. So, these questions also can be answered by going through today's lecture. And finally, as an unsolved problem say study the groundwater potential of your watershed area collect data related to aquifer soil land use land cover etc. obtain hydro geological maps and topo sheets assess the groundwater potential based on available data. Get the data related to number of wells in the watershed and study the head variations within the wells. Discuss how you can improve the groundwater available between the area in terms of rainwater harvesting or artificial recharge. So, today what we discussed is the groundwater flow or subsurface flow conditions. So, we have seen various aspects of groundwater flow and groundwater flow modeling. So, we have say here only one lecture is given for groundwater flow. So, only the introductory aspects only we could discuss today. So, those who are interested in this you can refer to some of the test books and even in the Rappan list. Thank you very much.