 Namaste and welcome back to the video course on watershed management. In module number 4 on watershed modeling today in lecture number 17, we will discuss numerical watershed modeling. So, some of the important topics covered in today's lecture include physically based watershed modeling, numerical modeling, final defense method, final term and methods, then computer modeling. These are the topics covered and some of the important keywords in today's lecture include physically based watershed modeling, numerical modeling, final defense methods, final term and methods. So, as we discussed in the last few lectures, so we have to go for modeling of the various processes taking place within a watershed and these processes are very complex. So, most of the time we cannot go for any simplified models like analytical solutions or even field based experimentation is very very difficult. So, most of the time, so we have to go for computer based modeling. So, that way we have to say as we discussed in the last lecture, we have to formulate the problem, we have to conceptualize the model, then a mathematical model should be developed and the corresponding governing equations like Sainte-Vinand's equations for overland flow or channel flow or other kinds of equations we have to solve numerically since analytical solutions are only available for very simplified cases. So, we have to solve these equations numerically using a computer, using a numerical technique and then apply the boundary conditions to get a solution. So, that we can identify the various processes and then we can get the results. So, why this kinds of modeling is very important as far as watershed is concerned. So, as we discussed various processes like transformation of rainfall into runoff over a watershed or generation of flow hydrograph at the outlet of the watershed. So, and then say for example, use of the hydrograph at the upstream and into route to the downstream and then hydraulic simulations say in all these cases, we need to simulate the various processes using the corresponding mathematical equations or governing equations to calculate the say like runoff volume, peak runoff, time to peak, etcetera. So, the advantage of this kinds of numerical modeling or computer models include, say this models allow parameter variations in space and time with use of numerical methods. Since, we cannot solve this partial differential equations which we have discussed earlier like Sainte-Vinand's equation in one dimensional, two dimensions, so that way we have to go for computer models is using say numerical methods. So, numerical methods are used to approximate this partial differential equations and then we apply the boundary condition to get a solution. So, say the main purpose is in simulation of complex rainfall pattern and then say the by considering the various heterogeneity of the watershed. So, as we discussed the various processes or various parameters within the watershed are very heterogeneous and it is varying from one point to another point. So, we have to consider all these aspects and then we have to develop a computer model say for example, rainfall to runoff say for this figure shows say for the given rainfall at the outlet of the watershed, we have to identify say for the given rainfall how much should be the runoff, so that various watershed management measures can be undertaken. So, this is the main purpose of this kind of watershed modeling. So, as we discussed in the last few lectures, there are number of types of models are available like black box model, lambda models, then distributed models etcetera. So, as say for example, now if you are looking to distributed models, so it is so complex process since we have to solve the given equations by using number of parameters. So, that way say depending upon the objectives of the study and then depending upon the data availability and expertise, so we can go for such a models. So, as we discussed earlier say here I have shown the various types of hydrologic models these models we have already discussed earlier also. So, broadly as we have seen earlier say the model can be either lambda parameter models like in the unit hydrograph, then distributed like kinematic wave, then or even based models like HEC1 or SWMM or continuous simulations like a stand for watershed model, then HSPF, then physically based model like HEC1 or SWMM, then stochastic model like synthetic stream flows, then in all these many of the times we have to go for numerical techniques. So, it can be numerical or analytical, so numerical like dynamic wave or diffusion wave or kinematic wave or say sometimes say depending upon the equations and problems some simplified analytical solutions are available like instantaneous unit hydrograph or the analytical solution to kinematic wave as we discussed earlier. So, that way we will be looking for numerical watershed modeling or computer based models. So, we have already seen earlier why such models are required, so necessity of distributed models include as we discussed the flow of water say for example, water as a resource, flow of water in a watershed is a distributed process. So, rainfall is totally spread over the watershed depending upon the conditions and then from the rainfall to runoff number of transformations functions will be there and then the runoff is also distributed, so that way it is always better to go for distributed modeling. So, even though the black box models or the lambda models need a say very less data the data requirement will be less and it is much easy to use, but those kinds of models cannot capture what is really happening within the watershed. Since it is these models are not simulating the various hydrological processes what is happening just like in a physically based model. So, distributed models are required to identify how the runoff is distributed or how the various processes are taking place within the watershed, so that way we need physically based distributed models. So, generally say physically based distributed models are based upon the Sainte-Venant equations or corresponding Navier-Stokes equations and then say as I mentioned earlier, so these equations we cannot solve directly, so we need a computer models, computer based models, so we have to approximate these governing equations of Sainte-Venant by numerical techniques and then we have to solve using computer. Then so the computer models allow computation of flow rates and water level as functions of space and time. Say for example, if this is the watershed then overland flow you can see here and then this overland flow component is joining to the channel, so channel flow is taking place. So, if you want to identify with respect to space and time how the variation is taking place, we have to solve this Sainte-Venant equation either in one dimension or two dimensions and then we can identify how the flow rates, what is the water level and then time to peak, then say peak discharge etc. So, these models, so the advantages distributed models, these models are closely approximates the actual and steady non-uniform nature of flow propagation in the overland and channel. So, that is the advantage of these models, but of course, number of disadvantages like the model modeling is so cumbersome and then we need a large amount of data to develop such models and then expertise also required. So, that are some of the disadvantages, but then say to capture the entire hydrology processes taking place within a watershed, we have to go for the distributed models or physically based models. So now, say this physically based models, we can classify them into either hydrologic or hydraulic models. So, the hydrological or hydraulic models, say these are the conceptual or physically based procedures and numerically solving the hydrologic processes and then diagnose or forecast the processes. So, the various hydrologic processes we can say consider in this starting from rainfall to the evapotranspiration, interception to infiltration and then runoff and then we can diagnose the or the forecast the various processes. So, the models are called physically based since the description of natural system using basic mathematical representation of flows, flows of mass, momentum and various forms of energy. So, here we call it as physically based models since we are using the loss of physics like conservation of mass, conservation of momentum and conservation of energy. Most of the time we will be using the continuity equation based upon the conservation of mass and then the q-sstrom motion based upon the conservation of momentum and then say we may use the simple Bernoulli's theorem based upon the conservation of energy. So, we consider these types of models as distributed since the special variation of variables and parameters are undertaken in the model. So, that is why we call these kinds of models as distributed models. And then as we discussed in earlier lectures also large number of applications are there for such a hydrologic or hydraulic modeling if I mean the physically based or distributed modeling like rainfall to runoff, then surface water, ground water assessments, then flood door drought predictions, then evaluation of watershed, then catchment management strategies, then river base in our agricultural water management etcetera. So, number of applications are there so to get a understanding of these various processes what is taking place in a watershed we have to go for the physically based distributed models either hydrologic or hydraulic modeling where say as far as the watershed is concerned by considering the particular say depending upon the objectives we have to choose particular the governing equations and then develop a mathematical model and then develop a computer model and solve for the various inputs to get the appropriate outputs and then we have to analyze for that. So, that way now we have already discussed say when we discussed about the physically based models. So, we have to systematically develop the model and then we have to do the modeling say step by step in a by a step by step procedure. So, here in the slides say for example, when we consider watershed based modeling say rainfall to runoff so here various steps I have identified in this slide say as I mentioned say the problem definition that means if the objective is depending upon the objectives we can define the problem and then we can go for the solution. So, first step is problem identification with respect to the objectives set. Then say we have to conceptualize a model so this conceptualization is very important as I mentioned earlier also so the real challenges to the engineer or scientist is to conceptualize the model. So, the conceptualization include say identify the watershed area then say whether we are going for one-dimensional or two-dimensional or three-dimensional modeling and then we have to identify what are the boundary conditions and other various parameters which are say governing the system and then also what kind of assumptions we have to utilize depending upon the problem. So, now once we conceptualize the model then next step is say mathematical modeling. So, mathematical modeling means we can identify the governing equations for that particular system say whether it can be say three-dimensional fully three-dimensional form like Navier-Stokes equations or the St. Wiener's equations or two-dimensional or one-dimensional form of these equations and then say for the given say domain which we have already defined as the watershed then we consider the appropriate initial and boundary conditions. So, once the all these are defined so then our mathematical model is ready. So now then the next step is say so as I mentioned mathematical model say most of the time we have to solve this complex partial differential equations then we do not have analytical solutions most of the time. So, we have to go for numerical modeling so numerical modeling means we have to choose particular numerical techniques which can be appropriately used to solve this partial differential equations like St. Wiener's equations it can be like a finite difference scheme or finite element method or boundary element methods like that and then we choose that particular technique or method and then we transform the governing equations into that particular scheme a finite difference or finite element like that and then we apply the boundary conditions and then solve the system of equations so that way we will get the solution. And that procedure once the numerical technique is chosen that the development of the code and then putting to computer model that is so called a computer modeling. So for next step is computer model so most of the time this computer models we have to say before applying to the particular problem of the watershed we have to verify and see that whether this model is working fine say especially if you are going to develop your own model mathematical model and computer model it is always better to verify with respect to some analytical or field observations and see that the model is giving appropriate results. But say some standard models like mod flow or the watershed models or mic 11 etcetera so those models may not need such kinds of verifications but if you are developing your own models definitely we have to go for model verification with respect to either available analytical solutions or the field problems. Then once we do all these steps then say we can identify whether the model developed is adequate for the particular problem to be solved so as I mentioned for while developing watershed management plans we are doing this watershed modeling so the model developed whether it is adequate to deal with the particular objectives set so that we can verify. So now once the model is we identify that model is adequate then model can be constructed by considering the particular say the data set of the watershed which you are considering so there we need the field data. So field data like rainfall pattern rainfall intensity or the various watershed parameters like size of the watershed land use the manning's roughness coefficients then the saturated hydraulic conductivity etcetera. So the field data is required to construct the model so this model construction means the model which we have to construct for the particular watershed to deal with the particular objectives which we have set. So that is the model construction here and then once the model is constructed then we can run the model and see the performance of the model and then say maybe for the particular watershed if some field data is available our field observations are available say for the given rainfall condition if some runoff measurements are available we can run the model and then see that whether we are getting the same results same so that way we can be very confident whether the model is giving appropriate research. So that is that step is called the performance criteria of the model. Then say with respect to the field conditions most of the parameters like hydraulic conductivity porosity then say the manning's roughness coefficient etcetera. So these parameters are varying drastically from one location to another location for the watershed. So we have to go for a calibration and then validation process. So this calibration and validation process we do with respect to the field data and then compare with respect to some of the observed data. So that process is so called a calibration and validation. So this is very important for the particular watershed. So once this is done then we can easily run the model for various scenarios. So then the next step is model predictions. So this model predictions means say for example if you are going to construct a dam or a check dam for a particular watershed, a particular location of the watershed. So we have to identify say the annual average rainfall conditions, maximum rainfall conditions, minimum rainfall conditions and then we have to say generate various scenarios whether how much water will be available, how much will the storage possibilities then what kind of the storage is available for the given conditions. So all these are so called we can run the model and then we can do the model predictions. So then the next step is we can do various analysis like sensitivity analysis. So there the variation say of the various parameters we can identify how the model will behave with respect to those parameter extremes and then modeling parameters like time step and then grid size also we can vary and then check. And then finally say with respect to the various scenarios we have generated or with respect to the results which we got. So we have to also do a post audit analysis I mean say if those kinds of events happens or we say with respect to the say after say few months or after few season we have to see whether the model is given appropriate results which we are looking for. So that is so called a post audit analysis. So that way now we can develop the model say a distributed model numerical watershed model depending upon the objectives what kind of objectives you are setting and then for that say like as I mentioned rainfall to run off or development of a soil erosion model or say we are developing contaminant transport model. So like that say depending upon the objectives particular mathematically model can be developed and then corresponding computer models can be developed using a numerical tool numerical technique and then various processes as we discussed we can follow and then we can go for a predictions mode for various scenarios. So that is the way we develop physically distributed models as far as a watershed is concerned. So now coming back to physically based distributed models say for example if you are going for rainfall to run off. So depending upon the objectives as I mentioned we may consider say for example only some of the important hydrologic process say for when rainfall to run off same then in a agriculture watershed for even based modeling say for example then we do not have to worry the small quantity process like interception or say even for even based modeling we can neglect the evapotranspiration. So that way the important process like the rainfall to run off the overland flow channel flow components and then infiltration parameters we can consider. So this is the conceptual model which we have developed IIT Bombay. So by considering the overland flow channel flow and the infiltration modeling. So here this is an even based model so that way we have neglected the even evapotranspiration aspects since the rainfall is for few hours and generally we will be simulating this kinds of events after few rainfall seasons or few events of rainfall. So that way other parameters other hydraulic process we can neglect. So that way we can develop a typical physically based distributed model. So then now based upon the discussion so far now we will discuss some more details about the numerical modeling as far as watershed modeling is concerned. So as I mentioned when we deal with the physically distributed model so we have to either solve the governing equations such as Navier-Stokes equations or the simplified forms like St. Venant's equations in one dimensions, two dimensional, three dimensions and then St. Venant's equations also considered we have already seen it can be full form of the St. Venant's equations like a dynamic waveform or the approximated form like a diffusion waveform or the further approximated form like kinematic waveform. So now we will see the governing equation which we have already discussed in the previous lecture also. So the governing equations for overland flow modeling and then the governing equations for channel flow modeling. So this is as far as rainfall to runoff prediction is concerned. So here in the slides so as I mentioned the St. Venant's equations are given for one dimensional condition. So the governing equations are the continuity equations as given by this equation. So del A by del t plus del V A by del x minus q is equal to 0. So where V is the velocity, q is the inputs like rainfall, excess rainfall coming to the watershed then t is the time and A is the with respect to area of flow. And the momentum equation can be written here is the momentum equation. So this is del q by del t plus del V q by del x plus g A into del y by del x minus A 0 plus S f is equal to 0 where q is the discharge say g is the accession to gravity, y is the depth of flow, S 0 is the bed slope, S f is the energy slope. So as we discussed say this is for overland flow modeling. So here you can see that say here say when we are discussing the only the this say with the continuity equation when we are equating the bed slope to energy slope then that kind of modeling is the model say is called kinematic wave. And when we are considering only the continuity equation and then this much parts I mean the this portion of the governing equation then it is called diffusion wave modeling. And then as we discussed earlier if it is only up to this part of the momentum equation then we call it as quasi steady dynamic wave equation otherwise when we consider the entire equation we call such kinds of modeling as dynamic wave model. So then with respect to this we have to this governing equations we have to consider the initial and the boundary conditions. So initial conditions at a period since we are going for time dependent modeling or transient modeling. So the initial conditions can be for the beginning at the beginning of the time step t is equal to 0 whether depth of flow or the discharge can be either can be taken as 0 or if values are known that those values can be taken. And then boundary conditions are considered same say here say for example if this is our watershed. So at the on the ridges of the watershed throughout the simulation we can consider all the time h is equal to 0, head is equal to 0 and the discharge per unit width also 0 on the ridges. And then at the outlet if the some values are known for either head or the discharge or depth of flow discharge we can consider those values. So now this constitute this governing equations initial conditions and boundary conditions and say when the system is defined like the boundary of the watershed is defined. So now our mathematical model is complete as far as overland flow is concerned for this particular watershed. So that way for the given watershed once it is conceptualized now the mathematical model is ready. So the next question is how to solve this governing equations. So we will be discussing about the numerical techniques. Then say the other component is the channel flow. So here in this slide you can see the governing equation this is also we have discussed in the last lecture. So channel flow is concerned. So as far as the watershed is concerned we separate into overland flow and the channel flow. So this is the overland and this is the channel. So here for channel flow the various overland flow components will be joined to the channel and then we have to route the flow through the channel. So here again say if we consider say for example the same variance equations in one dimensions. So most of the time the channel flow one dimension modeling is sufficient. So the equation of continuity as given by this equation del Q by del x plus del A by del t minus small q is equal to 0. So here we can see that this q is the discharge at any location of the watershed. So t is the time A is the cross section flow and small q is the overland flow components joining at various locations of the channel. Then the moment the equation is again del Q by del t by del x of q square by A is equal to g A into s 0 minus s of minus g A into del h by del x. So as I mentioned this equation various forms are available in the literature. So one particular form is like this. So here s 0 is the bed slope of the channel, s f is the energy slope which we can obtain through the Manning's equation as given here. And then corresponding the approximations like diffusion wave form as we discussed in the previous slide also by given by these two equations called equation of continuity and equation of momentum and the kinematic wave form the continuity equation and the energy slope is equal to bed slope. So these three forms either one or the typical form we can solve for the given watershed and then say for the given channel condition of the channel of the watershed. So then we supplement with the initial conditions for the given channel. So initial condition can be the depth of flow discharge for the watershed is known from the inlet or with respect to outlet. So for the initial time so that is the initial condition and boundary condition is concerned if the if there is some flow is coming from the out the inlet of the watershed I mean if a stream is continuing that can be our boundary condition or we can consider the depth of flow and the discharge at the beginning of the stream as 0 depending upon the condition. Here say for example this location and then say the boundary condition also can be applied maybe at the outlet. So now as far as channel flow modeling is concerned the given equations are defined. Then the various forms of the given equations like the St. Vincent's equation dynamic wave form diffusion wave form then the kinematic wave form one of the form we can utilize and then the boundary conditions and the initial conditions are defined. So our mathematical model is ready. So now when we are going for total watershed modeling of course we cannot separate the overland flow component and channel flow component. So we have to couple the overland flow component and the channel flow component together. So that we will have a coupled model for overland and channel flow. So once both the models are coupled so once both models are approximated given equations are approximated using particular numerical technique and then we can couple together. So that we have a complete model and then of course various processes like infiltration we can model through Philips models or Greenham's model or various other models and then evapotranspiration if we are considering we can model using Penman method or any other method. So like that various hydraulic process also can be combined within the distributed model which we are concentrating. So that way now the definition of the mathematical model is ready as far as the considered watershed. So now once the mathematical model is ready for the considered watershed now next question is how to solve these given equations. So we have already seen these given equations are partial differential equations and then it is most of time non-linear type equations so that way we have to go for numerical techniques. So here I have mentioned about the solution methodologies. So solution methodologies generally available solution methodologies include analytical methods, physical method and computational method. So analytical method for the given mathematical formulation and analytical expression involving the parameters and independent variables are obtained using various mathematical procedures. So it can be either integration or integration by parts and then various schemes we can adopt. But it is obviously a lot of since depending upon the given equations it is a complex process to develop such analytical solutions and then say these kinds of developed analytical solutions are only applicable for very simplified problems. So main limitations only for a small class of mathematical formulations with the simplified governing equations, boundary conditions and geometry. So analytical solutions can be obtained. So that way for most of the field problems we cannot apply the analytical solutions. So once if you are developing a computer model or a numerical model then we can verify your numerical model using such analytical solutions. So that is the advantages of these analytical solutions. So then the second method is physical method. So physical method means this is not a computer based model or other kinds of model. Physical means physically we are developing a scaled model or we are developing a say we are doing it in the field. So physical method as the mathematical model represents a real physical system or the certain idealized assumptions and variables and parameters of the model can be considered as having physical dimensions and can be analyzed sometimes in the laboratory or in the field itself. So this is possible but when we consider the complexities of a watershed. So the watershed itself is so complex by considering various processes or various parameters. So that when these physical models are used in a very limited way to develop such a by considering all these complexities in a lab it is very very tedious job and even to go to the field and say while various hydrology process like rainfall happening then to measure and then identify the various parameters are very difficult. So that way the physical models are used less frequently since it is expensive, cumbersome and difficult in practice. Then the next say our next methodology is computational modeling or computer modeling. So that is what we generally use as far as the solutions of these kinds of equations and watershed modeling is concerned. So in computational method the solution is obtained with the help of some approximate methods such as numerical techniques using a computer. So as I mentioned commonly numerical methods are used to obtain the solution in the computational method. So depending upon the governing equations depending upon the boundary conditions we can use particular type of numerical technique. So wider class of mathematical formulations and advent of fast computers, computational models have become the most widely used valuable tool for solving the engineering problems. So we can see that the last 50 years 5 decades a large number of numerical techniques have been developed and these are all these were all possible due to the advancement in the computer technology. So actually the first numerical method was finite difference method, then finite element method came, then boundary element method came and now nowadays we are same developing mesh free methods. So that way the developments have taken place for the last 50 years and these were only possible due to the advancement in the computers and the development of past computers. So variety of numerical methods are available as I mentioned. So depending upon the governing equations, depending upon the problem we can choose particular numerical technique for the solutions of the governing equations. So this can be either for one dimensional problem, two dimensional problems or three dimensional problems. So some of the important numerical methods I have listed here like method of characteristics, finite difference methods, finite volume methods, finite element methods, boundary element methods and now latest addition is mesh free methods. So these are some of the important numerical methods available for the solutions of these types of partial differential equations such as same variance equations either one dimension, two dimensions or three dimensions. So now we will briefly discuss these numerical tools and then most of the time we use either finite difference method or finite element method. So we discuss some what details about these two techniques of finite difference method and finite element methods. So let us have a brief look into these numerical tools which we can utilize as far as the solutions of same variance equations or Navier-Stokes equations for the watershed modeling. So first one is the finite difference method. So here say what we do the finite difference method is concerned the continuous variation of function concerned by a set of values at points on a grid of intersecting lines. So if this is our domain so we discretize this domain into rectangular or square grids like this and then say we use the Gaviani equations so that we discretize the Gaviani equations. So you can see that if say if you consider a domain like this then what we do we can discretize the domain using the grids rectangular or square grids like this and then we can say and we will consider the Gaviani equations. So here if this is x so for two dimensional modeling so x y so this is delta x this is delta y. So that way we will do a discretization and then the gradient of the function are then represented by differences in the values at neighboring points and a finite difference version of the equation is formed. So we can see that in our Gaviani equations like we are having the term like del h by del x so this we can represent as delta h by delta x. So this again we can represent as h i minus h i minus 1 divided by delta x. So this is say for example if this is i this is i minus 1 so its difference is taken. So that is as far as the derivative say first order derivative say in spatial wise. So similarly time wise also say for example del h by delta t we can write as delta h by delta t. So there again we can write say we can have the variation with respect to the time domain. So here again you can have this is x this is t so we can consider the variations with respect to the x t plane. So that is x x x plane give the variation in space and this t represent the time. So this will be now delta t and this will be delta x so that way we can consider. So we can then represent delta h by delta t with respect to say if we consider this as j and this as j plus 1 so we can write with respect to h j plus 1 minus h j divided by delta t. So this details we will discuss further. So finally at the points in the interior of the grid this equation is used to form a set of simultaneous equations given the value of the function at a point in terms of values at nearby points. So you can see that nearby points once we consider this particular point the nearby points with respect to nearby points we can use various schemes like backward forward or center different schemes. So that finally we can form a system of equations. At the edges of the grid the values of the function is fixed or a special form of final difference equation is used to give the required gradient of the function. Say for example if this is a free surface which we consider so in final difference this is one of the disadvantage we have to consider like this but then we use certain special type of approximations to deal this kinds of regular domain. So that is about the final difference method. So then another important say method which we use in water resource is so called method of characteristics. So method of characteristics is also one variant of final difference method. So here the method of characteristics here in this slide you can see the details are given. MOC reduces a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hyper surface. Say for example this is an xt plane then we can consider we will do some transformation so and then we will be having a C plus or C minus characteristic lines or characteristic curve and with respect to that the approximation will be done. So for example for a first order partial differential equation the method of characteristic discurses, discovers curves so called characteristics curve or characteristics along which partial differential equations become an ordinary differential equations. So we do certain transformations and then accordingly we do. So then it is solved along the characteristics curves and transformed into a solution of the solution for original partial differential equations. So the first step is to transform the partial differential equation to ordinary differential equation by defining the characteristics and then we solve through the characteristics this the governing the transformed governing equations. So as you can see this is also a variant of final difference scheme suitable for solving especially for hyperbolic equations. So I am due to lack of time not going to the ender details of this method of characteristics. So MOC is used to simulate like it is very useful to simulate adduction dominated transports and then also it can be used to track idealized particles through the flow field and this is this method is efficient and minimize numerical instabilities. So that is about the method of characteristics. Then the next numerical true method is so called finite element method. So this is one of the most and widely used finite element methods. So here compared to finite difference methods. So here the region of interest is divided in much more flexible way say for example same earthen dam which we have seen the previous slide for final difference scheme. So here instead of rectangular grid or square grid we can use triangle elements like this so that we can represent the variation the regular boundary very easily. So the nodes at which the values of the function is found to found how to lie on a grid system or a flexible measure. So this junction is called node and this is called elements. So the boundary conditions are handled in a more convenient manner. So that is the advantage of the finite element methods. So here various schemes like a direct approach variation principle or weighted result approach are available as far as the finite element method is concerned. So now say for example the say if you consider a rectangular domain like this. So or irregular domain also it is much easier to deal with finite element method. So here what we can do say we can use say triangle elements or rectangular elements or combinations of this elements and then this is so called triangle elements in 2D. So this is so called rectangle or square element in 2D. So here you can see that for triangle element there will be 3 nodes. So these are called nodes and for rectangle elements 4 nodes when we consider linear variations. So like that various schemes either 2D or 1D one dimension is concerned we can consider linear line element like this. So the variation will be we can consider linearly varying like this. So that is so linear variation for 1 dimension. So this is for 2 dimensions and then also 3 dimension like prisms, triangular prisms or rectangular prisms like that we can consider as far as 3D is concerned. So various elements, various schemes are available in a much more flexible way as far as finite element method is concerned. So the method is much more flexible and very easy to deal compared to finite different schemes. But of course mathematically it is not so easy we have to have a number of transformation process to be considered. But due to the flexibility now finite element method is the most popularly or most widely used numerical tool in the solution of various problems not only in civil engineering but mechanical aerospace or even bioengineering. So now the next method which I would like to briefly discuss here is so called boundary element method. So this is further development with respect to finite element method. So here we first discretize only the boundary like this. So this is the domain so we consider the elements like this and then we initially discretize the boundary and then we consider the internal nodes. So actually the partial differential equations describing the domain is transformed into an integral equation relating only to boundary values. So the method is based upon Green's integral theorem and the boundary is discretized instead of the total domain. So you can see that this is the boundary. So first we discretize the boundary and then to identify various values on the internal domain so we consider various nodes at various locations. So that way a three dimensional problem reduces to a two dimensional problem and a two dimensional problem will be reduced to a one dimensional problem only computational problem is of course the same dimension but computationally a three dimensional problem become two dimensional and two dimensional problem become one dimensional. So this boundary element method is ideally suited to the solution of many two and three dimensional problems in elasticity and potential theory but as far as water shut modeling is concerned due to the various complexities we rarely use the boundary element method. So water shut modeling either we can go for finite difference method or the finite element method. So that way we will discuss further more aspects of these two methods in this lecture. So now coming back to the water shut modeling so say for example if we consider the kinematic wave form of the same Green's equations. So as I mentioned so we can either go for the analytical solution. So here the analytical solution for one dimensional case I have given as given by Jaybar and Mokhtar as published in Advanced Water Resources in 2003. So an analytical solution for one dimensional kinematic wave equation can be derived but this is only suitable for very simplified problem rectangle domains with very simple boundary conditions. So here this Q can be used to identify the flow variation and then T c is the time of concentration and T r is the rainfall duration and T f is the simulation time and L w is the length of water shut in the direction of main slope. So other than this very simplified form of the analytical solutions which we have seen now no other analytical solutions are available for field applications as far as water shut modeling is concerned. So that way we have to go for numerical modeling like a finite difference scheme or finite element scheme. So now let us see some more aspects about these two techniques finite difference methods and the finite element method. So as I mentioned already in finite difference methods the calculations are performed on a grid placed over the domain. So if it is one dimensions it can be x t plane or in two dimensions it can be x y t plane or in three dimensions it can be x y z and t plane. So this is say for example one dimensional x plane and t. So flow and water surface elevation are obtained for incremental time and distances along the channel. So if we consider channel modeling we can do overland and channel also. So if we consider channel so with respect to x and with respect to time this space and time we can do the modeling. So this finite difference scheme is concerned generally we can have two types of schemes one is called explicit method and second one is called implicit method. So in the explicit methods we calculate the values of velocity and the depth over the over a grid system based on a previously known data for the river or channel which we consider. So that way from one step to another step we are directly calculating the values either flow depth or discharge or velocity. So that is why it is called explicit method and the implicit method we set up a series of simultaneous numerical equations over a grid system for the entire stream or river and equations are solved at a at each a time step. So that way the methodology will be stable and you get a better solution. So that is so called a implicit method. So in finite difference method we can typically follow a step by step procedure. So for the given governing partial differential equations the say and boundary conditions we can divide the domain into grids as shown here and then put the at nodal junctions at grid say this junctions we can identify various values like i minus 1 j or i j this we can write 1 2 in force of the rows wise and 1 2 n the column wise also. Then next is transformation by finite difference method as I mentioned either first order or second order partial differential terms can be transformed and then finally we can form a system of difference equations for the total grid points which we consider and then we can apply the boundary conditions and then we can solve either by direct or iterative schemes by applying the boundary conditions so we get the solutions. So as far as the finite difference scheme is concerned generally 3 schemes are available which are commonly used first one is called a backward difference. So backward difference means say for example if we consider del h by del x so here we consider the node in the backward direction of the node at which gradient is sourced. So here h i minus h i minus 1 divided by delta x so similarly we can have forward difference scheme so we consider the forward direction of the node at which gradient is sourced so h i plus 1 minus h i by h i by delta x and then we can have central difference scheme so we consider the central scheme like h i plus half minus h i minus half divided by delta x. So one of the scheme we can utilize as far as the finite difference modeling is concerned. So now for example if we are using the finite difference scheme while doing a river or channel modeling so here one dimension model in x direction and that t is the time so we can see that we can discretize like this with respect to spatial x direction and time and then say correspondingly we can identify various terms like depth of flow or discharge say that particular grid points like this. So this shows a typical finite difference scheme which is applicable for the channel flow modeling using the finite difference approximation. So as we have seen the finite difference is concerned we can have explicit scheme and implicit scheme so in the explicit scheme the temporal derivative is mentioned like this and special derivative is mentioned by this equation. So from one time step to another time step we use the previous values and march forward. But in the case of implicit scheme we write with respect to all the grid points the governing equations is the terms of various terms are written like this and then we use a weighting factor where theta is a weighting factor varying from 0 to 1 and then we can put say for example when theta is 0.5 it is called the Krang-Nickerson scheme semi-implicit scheme when theta is equal to say 0 then we will be having same fully explicit or fully implicit scheme when theta is equal to 1 we can have either 1 or 0 we can have explicit or implicit scheme. So that is the difference between explicit and implicit finite difference scheme. So then the next method I would like to discuss briefly is called a finite element method. So here we again use the kinematic and diffusion waveforms of our line flow to demonstrate the finite element scheme. So we have already seen the governing equations as far as the kinematic wave scheme is concerned. So the continuity equation and the momentary equation kinematic waveform is s0 is equal to sf. So the continuity equation we can use scheme like Galerkin finite element method in one dimension. So we can use linear line elements. So first we use a shape function here n is a shape function or interpolation function and then multiply by that the governing equation and then integrate over the domain and then equate to 0 and we use the Galerkin scheme here and then we can integrate and then transform the equation as in equation number 2. This shows the expansion that form of this equation. So then here the shape function with different types of shape function we can use. So some of the shape function like linear line elements which is based upon polynomial. So this is 1 n1 is 1 minus x by L or n2 or nj is x by L and then we can discretize the governing equation for one element and then we can discretize the form for the given element is shown here for the continuity equation in the kinematic waveform and corresponding various terms are given here and then finally we can assemble say one by considering all the elements line elements and then time is concerned here this model was developed in IIT Bombay by one of my Ph.D. student. So we used implicit finite difference scheme as we mentioned here. So the final system equation is given here and then say after rearranging the term we will get a final system like this. So here if you are using the say in Krang-Nikolson scheme here omega will be 0.5 and in the scheme we apply the boundary conditions and initial conditions and then we can solve the system of equations. So we can find the unknowns like depth of flow or velocity or discharge at given location by solving the system of equations in the final term method. So here the overland flow final term formulation I have demonstrated. So very similar way we can go for the channel flow also. So using this channel flow formulation and for overland flow formulation both we can combine together or couple together so that we can identify how the flow variations and depth of flow or various parameters can be identified and the system of equation can be either solved directly or using creative techniques. So here say for example if this is a typical watershed then this is a channel. So here in one dimension model we use consider the strips like this. So this is the discretization using the final term method and this is the channel discretization. So that way we can solve this system of equations. So these details we can see some of our publications ready and others published in 2007 in hydrologic process and then water resource management etc these formulations are available in the journal publications which you can refer. So this ready my student has say developed a model by considering the flow chart is given here say for example for overland flow. So with input data and then various hydrologic process making reception infiltration we consider then we generate the element matrix using final term and Galerkin final term method then we generate the global matrix by assembling the element matrices and apply the boundary conditions and we continue the modeling until the time step the entire time period is considered. So this shows the typical flow chart say for example for overland flow using final term method. So before closing today's lecture we will briefly discuss one case study which was done by my student Wenger Reddy. So here the watershed which we consider is so called Harsul watershed. So this also for rainfall to runoff modeling using the final term and approach the using the diffusion wave and kinematic wave approach. So the watershed is located in the Asigdi-Stichmar after state area is about 10 points 9 to 9 square kilometer. So this is the watershed area and this is the boundary and this is the outlet of the watershed and the major soil class is Gavri loam and here we use the remote sensing data given by IRS 1D and then also the thematic maps like detail elevation model, slope map, land use, land cover were developed using ArcGIS software. So these details we will be discussing in a later lecture when we discuss the geographic information system and remote sensing applications as far as the watershed modeling is concerned. So here for this typical watershed we consider overland flow elements of 144 as shown here and overland flow nodes of 188 channel flow elements. So this is the channel. So channel flow elements of 22 are used with element length of 0.25 kilometer and average bed width is considered as 18 meter and then this overland flow slope and channel slope are also considered and then with respect to the land use, land pattern we identified the Manning's reference coefficient and then put into the particular these trip or the element or model wise which we consider and then as I mentioned we developed the scheme and then we solved the system equation to identify for a given rainfall condition how the runoff will be taking place at any location of the watershed. So here some results I am presenting say this is based upon the diffusion wave model done by Wenger Reddy and the infiltration model is green amped model. So he used 3 rainfall events for calibration and 2 rainfall events for validations. So here the event date are given here and as far as the infiltration parameters like saturated hydraulic conductivity, suction head, saturated water condense, initial water condense were identified by using standard values available values and then calibrating with respect to the observed results. So this shows at the outlet of the watershed the for the calibrated events, 3 calibrated events and 2 validated events with respect to the rainfall pattern which is shown with respect to rain fine density in millimeter per hour how the discharge is varying at the outlet of the watershed and then these results were compared with respect to the observed data also. So you can see that in the distributed model we may not get the correct fit, the accuracy of the model depends upon the accuracy of the data and then we need a huge data for such modeling. So of course modeling is quite complex, we have to get all this if you get complete data in an accurate way of course the model results will be also better. So these are some of the references used for today's lecture as I mentioned this paper gives the model which we final term model which we discussed today. Then before closing the lecture some tutorial assignments and self evaluation questions. So for the tutorial question is illustrate the necessity physically based watershed modeling and develop a conceptual model for a typical watershed for physically based modeling describe the merits and demerits of physical modeling. So based upon today's lecture you can answer this question. Then some self evaluation questions like why distributed modeling required for watershed modeling illustrates various solution methodologies for problem solution differentiate between explicit and implicit final difference schemes describe final term and solution methodology with the serian features. So these questions you can easily answer based upon today's lecture. Then few assignment questions like with the help of flow chart illustrate hydrologic and hydrologic modeling then describe final defense method solution methodology with serian features then differentiate between final defense method and method of characteristics and describe boundary and method solution methodology with the serian features. So these details you can answer these problems you can answer based upon today's lecture. So finally an unsolved problem study the serian features and problems of your watershed area identify how various physically based models can be used for various problem solutions such as rainfart runoff, flooding, drought management, rainwater harvesting schemes, soil erosion etc. So you can study your watershed in detail and then you can come up with the physically based model which you can develop appropriately for your watershed. So with this today's lecture on physical model and numerical watershed modeling is over. So in the next lecture we will be discussing the groundwater and subsurface flow as far as watershed modeling is concerned. The details will be discussed in the next lecture. Thank you.