 So, welcome to lecture number 35. In this particular lecture, we will start this modeling and management of groundwater. And under this, we will cover groundwater simulation models ground water management models, particularly for confined aquifer. So, under this modeling and management of ground water, it has got two aspects. One is simulation aspect, another one is management aspect. In simulation aspect, we try to see the physical processes by using our governing equations that we have derived in our previous lectures. And in management model, we try to see what if kind of scenarios, if we do certain kind of management or we put certain kind of restriction on the ground water use or restriction on variables, then what will be the situation in near future or in long term aspect. So, in flow equations under modeling that has got two components. One is the temporal component and other two parts are spatial based components. So, in this one, the first one this epsilon, this is basically porosity, then we have water saturation and fluid density. So, basically if this combined term, the multiplication of porosity, water saturation, fluid density, this is temporarily varying, then we can say that either we can model that density dependent or density independent phenomenon. For density independent phenomenon, the equation will be slightly modified because it will not be varying with the density aspect. So, other parameters in this equation, solid matrix permeability tensor, we have fluid viscosity, this is basically dynamic viscosity of the fluid, then relative permeability and fluid pressure, we have fluid mass source. So, in the flow equation, this rho value or this G value, this is having only component in the vertical direction. So, this is basically a modified form of our Darcy's equation if we consider it with the fluid pressure term. Next word is saturated, unsaturated transport model. So, this is basically porosity, water saturation density and this is dissolved mass fraction. Dissolved mass fraction, then volumetric adsorbate source, other things that is average fluid velocity, then we have apparent molecular diffusivity and identity tensor and this is dispersion tensor and this is our mass, solute mass in source fluid due to production reaction. And this is Q p into C star, this is C star is basically solute concentration of fluid sources. So, our solution methodology is basically solving the basic governing equations. So, what are the basic governing equation we have covered in this particular course? First one is Darcy's law, this is our most fundamental equation that is relates the equation of motion or if we say that is one form of our Navier-Stokes equation if we compare it with fluid mechanics. Then we have equation of continuity that is mass balance equation and finally, we have solute transport equation or advection dispersion equation. In advection dispersion equation also we can have other components like adsorption loading isotherms or our radioactive decay and these terms will be there to model the whole thing. Now, these three equations are our fundamental equations these are basically if you consider it in one dimensional then it will be mostly if you are considering a steady state case then it will be normal differential equations or ordinary differential equations, but most of the cases it depends on both space and time. So, these equations are basically partial differential equations then we need boundary conditions. So, boundary conditions and initial conditions. So, for a particular problem we can have either a initial value problem IVP partial differential equation IVP or initial value problem or BVP or boundary value problem or IBVP or initial boundary value problem. So, our equations are mostly in these two forms in these two forms. So, solution can be obtained for these two equations using either analytical or numerical methods. So, to solve these two or solve this equation in form of analytical solution we need to have some kind of simplified form of the equation otherwise it is difficult to find out the complex analytical solution for a complex hydrogeological system. Numerical solutions are easier to find out because we generally used a discretized form of the equation and we try to solve it with algebraic equations. So, in case of numerical methods either we can use this finite difference, finite element, finite volume, SPH that is smooth particle hydrodynamics, spectral methods or any mesh free methods for this purpose. So, what is the conceptual form of calibration? So, first we need to have some kind of definition or we should have proper definition of our purpose. What is our objective? Whether we want to model the thing or we want to model it for some kind of management strategy. So, we will have some kind of field data and from field data we can create the conceptual model that is conceptualization of the mathematical model. So, we need to identify the governing equations that will be required for a particular problem. So, identification of governing equation then boundary condition initial condition. So, next thing is numerical formulation. Numerical formulation we can discretize the equation we can write the computer program code verification. This part is important because we need to have some kind of verification of the code with the existing solutions. So, solutions can be either existing analytical solutions. So, or existing solution for a two dimensional problem, but two dimensional problem or three dimensional problem the problem is that if we start with the reality thing then there will be certain kind of errors. So, it is better to use that analytical solution for code verification. So, if the code is not verified then we can go back to the computer program and we can try to rectify or correct our errors. So, code selection is important. So, either one can write is or hard own code or one can select a proper code which is either a open source or a commercial one. So, if code verification is complete or code identification or code selection is complete then we can go to this model design part. So, with the field data we can design the model then calibration, calibration comparison with the field data. So, one way model verification is done and calibration is done and this calibration should be supported by the sensitivity analysis. With the sensitivity parameters, with the sensitivity of the parameters there should not be much change in the model results because if we are identifying the parameters in such a way that it is not a proper identification for which there will be not much variation then it will be a problem. So, we should have proper identification and sensitivity analysis with our problem and then after model verification we can go for simulation. With simulation also we can correct our sensitivity analysis and final is presentation of the results. If there is some kind of monitoring network then finally, we can monitor the thing and we can correct our assumptions or conceptualization with those monitoring data and again we can start from the initial part and we can conceptualize the thing and we can redo the whole process so that we can correctly adjust the model to get the proper results. So, a code selection in this case is an important part of the modeling exercise either one can have their own code or they can readily select some codes which are available in public domain. So, standard simulation models that are available in the public domain or with some restriction or in commercial domain this is one comprehensive list of those models. So, first model is the mod flow this is basically three dimensional finite difference based simulation flow simulation model this is FEM water this is 3D finite element model for saturated unsaturated flow through porous medium see what this is saturated ground water modeling or density dependent ground water modeling in coastal aquifers. Sutra this is saturated unsaturated flow and transport model with this we can model it the ground water flow and transport in any kind of aquifers. Empty 3D MS this is for contaminant transport with multiple contaminants RT 3D this is reaction with reaction mod plot is flow path identification thing in ground water then sharp this is two dimensional quasi 3D simulation model for salt water intrusion modeling and this is based on sharp interface then H S 3 3D this is this includes also that heat transport part other models that can be aimed here are this hydro term fast this also include the chemical reaction part tough to flow through fractures media flow trun ita the inverse tough thing bio plume this is one hydras this is also important software to model the flow in unsaturated porous medium. So, the identification or selection wise we can select any of these models for our modeling purpose or we can discretize our governing equations and we can use our standard discretization methods to get the solution. So, this is the starting of our modeling exercise. So, in modeling and management of ground water we have two aspects one is the modeling aspect in modeling aspect will see what is the simulation results and the policy aspect or management aspect we try to formulate some kind of management strategy. So, for hydraulic management I am talking about the hydraulic management because if we are talking about the flow part only then only we say it as hydraulic management otherwise we need to have certain kind of management models with transport mechanism. So, in case of hydraulic management hydraulic management we can have equations in the form of embedding technique embedding approach in embedding approach the governing equations are directly discretized and these are used within the optimization model or decision models to get the solution. Next is our linked simulation optimization simulation optimization and linked simulation optimization the simulation model is directly linked with the optimization model or decision model to get the solution. So, first approach in embedding approach we generally write the equation in discretized form and we directly use it in the decision models to get the results, but the problem with the embedding approach is that as we are discretizing it for a number of points or for a number of grids so number of decision variables, number of variables that is crucial for this kind of approach. But in case of linked simulation optimization the simulation model is doing the simulation part separately and only with a limited number of variables we can solve the decision model. So, first approach concern is variable number of variables, the second approach the concern is simulation time. If we are setting up a complex simulation model then for each simulation it will take significant amount of time and that will be a limitation for this linked simulation optimization. Then the third approach which can overcome these two limitations that is called meta-model approach, meta-model approach. In this meta-model approach basically the third approach basically meta-models are trained with the simulation results from original simulation model and we try to minimize the meta-model errors between the meta-model output and the output from original simulation model. So, at each iteration there will be correction in terms of correction of errors and that can be assessed with certain kind of indicator reference functions. So, in meta-model approach the most primitive approach is response matrix, most primitive approach is this response matrix. So, response matrix approach what is the advantage advantage is these are basically linear, linear type. With the linear type of response matrix thing we can easily form some kind of linear model based decision model and we can quickly get the solution, but the problem is that response matrix will always give you some kind of linearized results and one way we are compromising with the accuracy of the simulation model. So, for a complex non-linear general equations or general problem we cannot represent it properly with our response matrix approach. So, we need to have certain kind of approach in which we can consider that linear model thing, non-linear model thing. So, nowadays people are using this ANN SVM GP based models. So, what are these ANN is artificial neural network SVM is support vector machine and GP is genetic programming. These models can consider or can represent the non-linear behavior of the complex hydrogeological system. So, now we can have the approach where we should be defining our equations, then some optimization methodology and then we should have the solution. So, first part is symbolic. Simulation model, simulation model that is original simulation model or a meta model, then identification of objectives, identification of objectives, constraints. So, from simulation model to identification of objectives, constraints, then based on constraints and objectives whether they are linear or non-linear we can find out the optimization approach slash algorithm. So, this is the process where first identification of simulation model, then identification of objectives, constraints, then depending on linear or non-linear behavior of the objectives or constraints, we can identify the optimization algorithm. So, let us start with the simulation model. So, for as per Darcy's law we know that Darcy and flux V for any confined aquifer that is for any confined aquifer can be defined as V del H by del L. So, this is basically head loss per unit length per unit length per unit length. So, now with this we can have one finite difference grid. So, these are basically nodes. So, black dots are nodes. This is level say J plus 1, this is J, this is J minus 1, this is Ith cell. So, del X I and let us say that Q 3 is entering, then Q 2 is leaving, Q 4 is leaving and Q 1 is entering and this is your del Y. So, basically this is cell to cell water transfer. So, here we can say that this Q 1 is basically minus T X I minus J del Y J by del X, this is 1, this is 1, this is 1, this is 1, this is Q 2 is basically T X I J del Y J del X second one, this is Q 3 is minus Y I J plus 1 del X I del H by del Y third one and the fourth one and the final one. So, this is T Y I J that is del X I del H by del Y 4. So, in this case T X J is the transmissivity in the X direction of the element I J. So, and T T I J is the transmissivity of the I T I J and this is two element I plus 1 J. So, from this point to this point, so in this direction this T X I J that is acting. So, rate at which water stored that is rate at which water is stored that is considered as Q 5. So, S I J where S I J is your storage coefficient, this is del Y J and this is del H by del T. And in addition to that flow rate flow rate Q 6 for constant net with draw all withdrawal or recharge. So, Q 6 is basically Q I J and T. So, Q 6 is basically Q I J and T. So, Q this is varying with T. So, by continuity equation may be with the continuity of the flow we can write it as Q that is entering Q 2 that is leaving plus Q through Q 3 that is entering minus Q 4 that is leaving the thing and Q 5 is again that is the rate at which it is stored. So, it is equivalent to the storage plus Q 6. So, with this if we replace all this Q 2 previous expressions for Q 1, Q 2, Q 3 and Q 4 we will get this is T X this is del H by del X 1 this is del H by del X 2 this is del X I and this is T Y with this del H by del Y del H by del Y 4 and this is del Y J. So, S I J del H by del T plus Q I J T divided by del X I and del Y J. So, for X I and del X I and del X I and del X I and del Y J. So, in finite similarly in finite similar values of del X and del Y we can write the equation as T X del 2 H by del X 2 plus T Y del 2 H by del Y 2 S del H by del T W. So, if we talk about finite difference thing then we can write this equation as del H by del 1 and this is basically J T and H I J T del X I del H by del Y 2 S del H by del T W. So, del X for second phase that is I J T minus I plus 1 J T divided by del X I. So, del H by del Y third one or third phase that is I J plus 1 T minus I J minus I J minus I J T divided by del Y J and the last one that is the fourth one is I J T minus H I J minus 1 T del J. So, with this and del H by del T is H I J T minus H I divided by del X I J T minus 1 divided by del T. So, if we substitute these 5 expressions in our previous equation, then we will get a compact form of the equation in terms of A I J H I J H I J by del X I T plus B I J H minus 1 J T plus C I J and H I plus 1 J T plus D I J I J plus 1 T plus E I J H I J minus 1 T plus F I J T plus D I J plus 1 T plus E I J H I J minus 1 T plus I plus 1 J T plus D I J I J plus 1 T plus E I J H I J minus 1 T plus F I J T equals to 0, where these terms are basically coefficients involving our known values of the equation or provided information. So, this can be solved using any standard numerical technique like A D I or alternating direction implicit method, alternating direction implicit method. So, this is the form of the discretized equation, these can be directly used for embedding purpose. Now, we are concerned about the management model and ground water management model, there we are concerned about the confined aquifer. So, in case of confined aquifer, let us say that our configuration is like this, this is the base or lower portion, then we have in between the impermeable part q 1, this is q 3 and this is q 4. So, basically, this is the aquifer material there and this is the water material. Water height which is H 5 at this level and all are having equal del X difference and on the left hand side, we have H naught value of the water level. So, it is a bounded problem, both cases we have specified boundary condition, this is also called as Dirichlet kind of boundary condition. So, for this kind of equation, we have del h by del t, del h by del x equals to w by t x, where for a steady state condition we have t equals to del h by del t equals to 0. So, for this kind of equation, let us discretize it using our normal finite difference method. For second order finite difference, we will get del x square w i t x. So, this is the discretized form of governing equation and we have 1, 2, 3, 4, 5, 6, 7, 8, 3, 4, 4 pumping from this particular aquifer and this H naught and H 5 bar to water levels on left hand, right hand side. So, our objective is to maximize. So, first we have completed the simulation part. Now, we are concerned about identification of our object. Now, objective function. So, objective function is maximization of our total head, total head for i, for i that belongs to the set of i, that is the set of wells, set of wells, set of wells. And this is subject to or subject to or s t, we can write it as this i, this w i is greater than w min and h i is greater than equals to 0, del h i is greater than w i is greater than equals to 0, again i belongs to that set i, that w min is the minimum total production rate. So, rate for the well. So, the total rate should be greater than or equal to the minimum rate that is specified. So, unknown for this problem are unknowns H and w. So, once the model is solved, this can be determined w min, w min, w min, w min, w min can be determined as q i divided by x i del x s square. So, head objective that thing is for managing the aquifer. So, the above formulation one limitation is there that is it considers, considers negligible well diameter and negligible well losses. So, for any example problem, we can define the same thing, we can define the same thing using our this particular approach. So, for our original problem, we can write the equation as the maximize z equals to h 1 plus h 2, h 3 and h 4 and this is subject to our finite difference equations and that is minus 2 h 1 plus h 2 minus del x s square by t w 1 minus h naught. This is h 1 minus 2 h 2 plus h 3 minus w x s square by t w 2 equals to 0. So, this is h 1 plus h 2 plus 2 h 3 h 4 minus del x s square q 3 0. This is h 3 minus 2 h 4 del x s square t w 4, this is minus h 5. So, we have defined our we have defined our constraints, we have defined our constraints and the final constraint is this one that is there is a minimum value of total pumping in production wells and h i greater than 0 that is i equals to 1 to 4 and w i that is also positive. So, in this model additional constraints, we can put that h 4 should be greater than h 5, then h 3 h 4 greater than 0 that means, h 2 h 3 greater than 0 and h 2 h 3 h 2 h 3 h 1 minus h 2 that is greater than 0 and finally, h 1 is greater than or less than h naught. That means, your head on the upstream direction or you can say that where we have a higher head value that cannot exceed or that cannot be a lower one compared to your down gradient value. So, these are the additional constraints that will be required for solving the thing and the full equation as because this is linear in nature, we can solve it using a p p. So, this is all about the management of confined aquifers. Thank you.