 Welcome to this lecture number 10, so we are continuing with the previous lecture, so wherein we derived an expression for the two dimensional general ground water flow equation for an isotropic aquifer, so which is of the form d square h by dx square plus d square h by dy square is equal to s by t into dh by dt where here h represents the head, x and y represents the flow directions, s is the storativity, t is the transmissivity and since there is an isotropic aquifer, the transmissivity in both the directions tx and ty are same equal to t and this dh represents the rate of change of head with time, the partial derivative of the rate of change of head with time. Suppose it is a three dimensional flow and the flow the aquifer is isotropic, so in this case there will be the third term also and so therefore this for an isotropic aquifer the general 3D ground water flow equation will be d square h by dx square plus d square h by dy square plus d square h by dz square is equal to s by t into dh by dt. So, this is the expression and obviously in this case the left hand side is denoted by del square h, so which is the combined notation for these 3 second order partial derivative terms in with respect to each of the directions and so this is when the flow is unsteady flow. So, this is of the form this is known as the diffusion equation and when the flow is steady then this dh by dt will become 0, so that the right hand side will be 0 and then this equation attains the form of the Laplace equation. So, when the flow the ground water flow is steady that case this dh by dt is equal to 0, so therefore d square h by dx square plus d square h by dy square plus d square z by I am sorry d square h by dz square is equal to 0. And obviously, so this is del square of h, so del square h which is equal to d square h by dx square plus d square h by dy square plus d square h by dz square which is equal to 0. So, this is the famous Laplace equation and so this is the general ground water flow equation, general 3D 3 dimensional ground water flow equation, here we can add one more reason 3D ground water steady flow equation for an isotropic aquifer. So, initially when the aquifer was anisotropic in that case it will be different this the terms will be different. So, for anisotropic aquifer the 3D ground water steady flow equation will be kx into d square h by dx square plus ky into d square h by dy square plus kz into d square h by dz square this is equal to 0. So, this is when the flow is steady and the aquifer is anisotropic and the flow is steady and 3 dimensional and so when the flow is anisotropic 3 dimensional through an anisotropic aquifer and the flow is unsteady. In that case the there will be the right hand term the right hand side will not be 0. So, here we can write on say for unsteady flow the 3D general ground water flow equation through an anisotropic aquifer will be this kx into b into d square h by dx square plus ky into b into d square h by dy square plus kz into b d square h by dz square. So, this is equal to s into dh by dt. So, this is the expression here the flow is unsteady and 3 dimensional and the aquifer is anisotropic. So, in that case this is the general form and when the flow becomes steady the right hand side term becomes 0 and the aquifer is isotropic in that case kx is equal to ky is equal to kz in that case it will be the Laplace equation. So, therefore, so this is how the general ground water flow equation it varies. Now, the same expression the for say in radial coordinates that is axis symmetric ground water flow. So, the axis symmetric ground water flow through an isotropic aquifer the equation will be the ground water flow equation will be d square h by dr square plus 1 by r dh by dr this is equal to s by t into dh by dt. So, this is the in the cylindrical coordinates one of the radial coordinates where r is the radial distance and h is the head and the radial coordinate and obviously, this s is the same storativity or storage coefficient t is a transmissivity. So, when the flow is unsteady. So, here ground water unsteady flow. So, and the axis symmetric ground water steady flow in the equation for axis symmetric ground water steady flow through an isotropic aquifer is d square h by dr square plus 1 by r dh by dr this is equal to 0 in this case the flow is steady. So, therefore, this dh by dt this term becomes 0. So, the right hand side becomes 0. So, this is the so, this is the obviously, the Laplace equation it is equivalent in the cylindrical coordinate system or the radial coordinate system. So, now let us consider say this is the confined ground water flow between two water bodies. So, here let us say this is the upstream water body. So, let this be x and this is x is equal to 0 and here we have the so, this is the water level at the upstream water body where the head is h 0 and of course, here we have the top and bottom confining layers that is the impervious layers of this confined aquifer having a thickness b and so, here this is the downstream water body where the head is h 1. So, this is the so, here this is the upstream water body. So, this is the downstream water body and the flow is taking place between and here let us say so, this is the general the ground between these two and obviously, so, this is how the ground water flow takes place and so, this is the hydraulic grade line and so, this is the downstream water body this is the upstream water body and so, in this case obviously, consider steady flow an isotropic aquifer and in this case this is the flow is one dimensional. So, therefore, the governing equation will be d square h by dx square is equal to 0 because the flow is only in the x direction. So, therefore, the this d square h by dy square as well as d square h by dz square. So, they will be 0. So, only we get this is d square h by dx square is equal to 0. So, this is the governing equation. So, on integration we get so, this is h which is equal to say c 1 x plus c 2 and now, we need to substitute the boundary conditions to evaluate this c 1 the constants of integration c 1 and c 2. So, the boundary conditions are at x is equal to 0 h is equal to h 0. So, therefore, h 0 is equal to c 2 the second boundary condition is at x is equal to l I am sorry here I just forgot to show this one. So, this is x is equal to l. So, this is the distance between the upstream water body and the downstream water body. So, here at x is equal to l then h is equal to h 1. So, this when c 2 is equal to h 0. So, therefore, we get the expression that is x is equal to I am sorry this h is equal to c 1 x plus h 0. Now, let us substitute the second boundary condition. So, this therefore, h 1 will be equal to c 1 into l plus h 0. Therefore, c 1 is equal to h 1 minus h 0 divided by l which we can also write this as minus of h 0 minus h 1 divided by l. Since, h 0 is bigger the head in the piezometric surface in the upstream water body. So, it is having this one. So, we can write that if we write that first. So, therefore, we get the expression after evaluating both the constants we get this h is equal to h 0 minus h 0 minus h 1 by l into x. So, this is the expression for head at any general section which is at a distance of x from the upstream water body. So, this is the here you can say this is the equation for. So, this is the equation for the hydraulic grade line h g l which is assumed to be linear. And let us also write down an expression for the discharge per unit width of the through the confined aquifer between two water bodies. So, the by Darcy's law discharge per unit width for ground water flow between for here you can say the confined ground water flow between two water bodies is given by this q is equal to minus the hydraulic conductivity k into d h by d x k into minus d h by d x where this minus d h by d x is the hydraulic gradient into the area since this is unit width. So, this will be 1 into b or simply b. So, in this case. So, this is minus k into b into the hydraulic gradient is minus of h 0 minus h 1 divided by l that is q is equal to k into b into h 0 minus h 1 divided by l. So, this is the expression for discharge per unit width for confined ground water flow between two water bodies. So, the discharge per unit width for confined ground water flow between two water bodies and obviously. So, here this we all start with this steady flow confined ground water steady flow between two water bodies. So, in this case the discharge per unit width is given by this k into b which we can also replace this by transmissivity t the transmissivity or transmissibility and h 0 and h 1 have the heads in the upstream and downstream water body and l is the distance along the flow direction between the two water bodies. So, if you know the either the transmissivity or the hydraulic conductivity and the thickness of the confined aquifer through which the flow takes place between the two water bodies the upstream water body having a head of h 0 in the downstream water body having a head of h 1 and with the distance of l separating them. So, then this q is given by k into b into h 0 minus h 1 divided by l. So, now let us come to the unconfined flow that means flow through an unconfined aquifer where in the Dupitz assumptions for unconfined ground water flow. So, in 1863 so, this Dupitz assumption Dupitz is in hydraulic engineer and in the year 1863 he proposed two assumptions for the unconfined ground water flow. The first assumption is the curvature of the free surface is very small. So, that the stream lines assume to be horizontal at all sections at all sections. So, this is the first assumption and as per this assumption even though there is slight inclination for the stream lines. So, that inclination is neglected and the stream lines are assumed to be horizontal at all sections and the second assumption is the slope of the hydraulic grade line that is h e l is equal to the free surface slope it does not vary with the depth. So, this is a second Dupitz assumption. The first assumption in which the stream lines are assumed to be horizontal in all directions. So, that whatever little inclination the stream lines is there that is neglected and the second assumption is the slope of the hydraulic grade line is equal to the free surface slope and it does not vary with the depth. So, these are the two Dupitz assumptions and which are applicable for ground water flow through unconfined aquifers. And now using these Dupitz assumptions let us determine the ground water flow through unconfined aquifer. So, here let us consider the ground water flow through unconfined aquifer. So, let us consider this is the z direction, this is the x direction and this is the y direction and here let us consider. So, this is the water table. So, this dimension along the y direction. So, this is delta y and this is the origin and this dimension. So, this is delta x and this head is h and here let us consider the ground water inflow. Let us consider the flow to be in the positive direction of x and y. So, let this be mx1 and the ground water outflow. The mass outflow let us take this to be mx2 in the x and y direction. Similarly, the ground water inflow in the y direction let us take this to be my1 and the ground water outflow through the other phase in the y direction. Let us take this to be my2 and this is the water table. Now, let us write down the expressions for the mass flux entering the element. So, this is given by mx1. So, this is rho into say vx which is the velocity ground water flow velocity in the x direction into h into delta y. h into delta y is the cross sectional area. So, this vx into the cross sectional area of flow that will be discharged into rho that will be the mass flux or the mass rate of flow mass rate of inflow. Similarly, mass flux leaving the element. So, this is mx2. So, this is given by rho vx h delta y plus let us add the mass rate of change term in the x direction that is d by dx of rho vx h delta y into this delta x. So, therefore, now let us write down this is a net mass flux in x direction. So, this is mx1 minus mx2. So, this will be given by minus d by dx of rho vx h delta y into delta x. Similarly, net mass flux in y direction. So, this is given by my1 minus my2. So, this will be given by minus d by of rho vy into h dx h delta x into delta y. Therefore, since there is neither inflow nor outflow in the z direction. So, the continuity equation gives obviously. So, here this mx1 minus mx2 is denoted by this is delta mx. Similarly, my1 minus my2 is denoted by delta my. So, therefore, this delta mx plus delta my plus delta my is equal to 0. Therefore, we get. So, after simplification we get d by dx of vx into h plus d by dy of vy into h is equal to 0. And obviously, so here this delta x delta y as well as rho. So, that can be cancelled. So, this is the expression we get. And we know that say by Darcy's law we get this vx is equal to minus k into dh by dx and this vy is equal to minus k into dh by dy. Therefore, if we call this equation 1. So, this equation 1 becomes d by dx of minus k dh by dx into h plus d by dy of minus k into dh by dy into h this is equal to 0. And again here we can take out this k outside the 1 can cancel out. So, therefore, we are left with this is d square by dx square of h square plus d square by dy square of h square is equal to 0. And this we can denote this as del square of h square. So, this is the governing equation for steady ground water flow unconfined aquifer. So, this unconfined aquifer for the steady flow this it satisfies Laplace equation in h square whereas, the confined aquifer will satisfy the Laplace equation in h. In case of unconfined aquifer it satisfies the Laplace equation in h square. So, now let us consider the unconfined aquifer that is the unconfined ground water steady flow with recharge. Suppose, there is some recharge in the z direction say when there is due to rainfall or precipitation. So, this the ground water is getting recharged. So, in that case what will be the expression let us say. And so here let us draw the basic figure with x, y and z directions. And so this is the water table and it is getting a recharge at the rate of r here. So, this is the origin and this is the dimension in the x direction and this is the dimension of this element in the y direction that is delta y and the variable head h and this case. So, let this the mass inflow through x direction let it be m x 1 and the mass inflow let this be m x 2. And similarly the mass inflow in the y direction let it be m y 1 and the mass inflow in the y direction mass outflow along the y direction be m y 2. And obviously, so this is the water table and this r is the recharge in the z direction. Now, in this case it is all similar to the previous case except that in this case. So, there will be an additional term that is mass efflux in the z direction that is the mass efflux in z direction that is delta m z is given by rho into r which is the rate of recharge into the area of flow perpendicular to the z direction that will be delta x into delta y. So, therefore, for steady incompressible flow the continuity equation becomes delta m x plus delta m y plus delta m z is equal to 0. So, that is we get minus d by dx of rho v x h delta x delta y minus d by dy of rho v y h delta x delta y. And the this recharge that is the net efflux in the z direction that is given by rho r delta x delta y this becomes 0. So, again let us substituting substitute v x is equal to minus k into d h by dx and v y is equal to minus k into d h by dy and simplify. So, here we get this is of course, here this is rho delta x delta y that can be taken out. And in this case we are left with that is d square by dx square of h square plus d square by dy square of h square this is equal to minus 2 r by k. So, here this term r is there. So, therefore, it will be h square by 2 and then so it will be minus 2 r by k. So, this is the general ground water steady flow equation through unconfined aquifer with recharge. So, the general ground water here this is the governing equation. So, in case of steady flow through an unconfined aquifer the it satisfies Laplace equation in h square whereas, when there is a recharge through the this one the unconfined aquifer at the rate of r along the z direction. The general governing equation for steady flow through unconfined aquifer will be given by this is d square by dx square of h square plus d square by dy square of h square. So, this is equal to minus 2 r by k. So, this is how the ground water governing flow equation it changes its form in case of an unconfined aquifer when there is no recharge as well as when there is recharge. Now, let us consider the one dimensional Dupitz flow with recharge between say 2 water bodies. So, let us say this is a 1 d ground water flow between 2 water bodies 1 d ground water unconfined flow with recharge. So, let us consider unconfined ground water flow and let this be the upstream water body and let this be the ground surface and let this be. So, this is the upstream water body where the head is h 0 and ground stream water body where the head is h 1 and there is a constant recharge. Here this is the constant recharge of rate r and in this case the water table will assume a shape like this there will be a peak somewhere in between and wherein the h is h m or the h max and this is the water table and. So, the aquifer permeability is hydraulic conductivity as k and in this case there is a obviously flow into the upstream water body as well as flow into the downstream water body this is q 0 this is q 1 and in the next lecture and obviously at this so at any general section h. So, here this is the general section h. So, this is x is equal to 0 x is equal to l and so obviously this is l and at general section h. So, this case this is q x. So, in the next lecture we will discuss this one. Thank you.