 Welcome to lecture number 32. In this particular lecture, we will continue our topic that is saline water intrusion in coastal aquifers and in this particular lecture, we will cover our previous topic that is burden, Gaibin, Herzberg principle. Basically, we will continue from our previous lecture and second topic, we will cover analytical solution of saline water intrusion in coastal aquifer. So, in our last lecture, we have seen that if we have one free attic aquifer, this is basically impermeable bottom, this is the location of ocean with our z axis pointing upward and this is the location of interface that is interface between salt water and fresh water. And we have ground water table located at this position, this is ground water table. So, let us consider any arbitrary section in which this depth below ocean level to the position of interface, we can denote it as xi and head above ocean surface that we can denote it as H f and corresponding head in ocean, we can denote it as H s. We have seen that with our assumption that is Gaibin Herzberg principle, we can express this xi in terms of H f as H f rho f by rho s minus rho f into H f. In this particular derivation, we have assumed that our salt water wedge that is basically stationary or stagnant. With this assumption, we will continue our derivation for saline water intrusion in different coastal aquifer formations that is first one will be confined then unconfined and for oceanic islands. So, let us consider this ratio rho f by rho s minus rho f as delta. So, with this assumption, we can continue our derivation. So, first of all let us see that how to use this Borden-Gaibin Herzberg principle in the field to identify the salt water intrusion in coastal aquifers. So, with this with Gaibin Herzberg principle, it is possible to the salt water fresh water interface. So, what are the steps for identification of this salt water fresh water interface? So, first point is that network of shallow wells are developed observation water table height. So, first of all we need to install some good number of piezometers in the field to identify the location of groundwater table locally. Next thing is that we can use contour plotting for this groundwater table height distribution. So, contour lines showing the fresh water head HF are drawn using some interpolation method. What are the interpolation methods we can use? One interpolation method may be creaking or inverse distance wetting method we can utilize. So, for example, we can use creaking or IDW or any recent techniques for plotting our contour lines for the fresh water head HF. Then the interface depth the interface depth xi which is a function of x and y is represented by the same set of contour lines this is drawn within the plotting. And fourth one the location of aquifer bottom aquifer bottom which is again function B x y is found from geological maps. And final one the intersection of this two surfaces that is the intersection of the two surfaces xi x y and B x y is sought which represents the saltwater tow location. So, with this method let us say we have our coastline like this which is coastline. And this is our ocean and with dotted lines I am representing the fresh water equipotential lines these are basically fresh water zone these lines are fresh water fresh water lines. And the location of tow that is that is the intersection of two surfaces may be located as this. So, this is basically location of tow. So, for this particular location saltwater has entered into coastal aquifer. So, saltwater invaded zone. So, we have saltwater invaded zone then we have fresh water zone and these are our fresh water equipotential lines. So, using Geiben-Hurtsberg principle we can easily identify the tow location in the field. However, for complex geological formation and non-stationary interface condition this is not valid. Geiben-Hurtsberg principle considers stationary saltwater wage assumption for its derivation. However, in complex geological conditions we cannot directly use that assumption for practical modeling or practical use. So, next thing we will try to find out the analytical solutions for our coastal aquifers. So, for analytical solutions for stationary interface. So, basically for these solutions we will utilize our Geiben-Hurtsberg principle and we will try to see the simplified analytical solutions for simple geological conditions. So, first thing with this analytical solution what we can do we can identify the shape of the sharp interface shape of the sharp interface. Next we can find out the relationship between the extent of sea water intrusion and flow of fresh water to the sea and why this relationship is necessary. This relationship is necessary for taking aquifer management decisions and sustainable yield identification for coastal aquifers. So, this relationship is necessary aquifer management decisions or for identification of sustainable yield of a coastal aquifer. So, we will use certain assumptions for our derivation. So, what are these assumptions? First of all we will consider that aquifers bottom is horizontal. So, aquifer bottom is horizontal the flow is assumed to be everywhere perpendicular to coastline with a stationary interface. That means our fresh water is moving sea water is stationary and we will consider that sea water wedge length L. One important assumption is that Dupitz assumption of essentially horizontal flow is valid assumption essentially horizontal flow is valid. So, let us consider a case of confined aquifer and for this condition we will try to find out the nature of interface. So, interface in a confined coastal aquifer. So, interface in a confined coastal aquifer. So, let us draw the conceptual thing we have horizontal bottom which ensures the horizontal flow in the aquifer towards sea. So, thickness of this confined aquifer is B. Again we have the interface this point T or capital T is called as tow location. Then length of this wedge that we have already assumed to be L. Then location of sea surface or ocean surface we have this and location of this ground water table is this green thing. So, for a particular location we have three different distances first one we will denote it with xi. Next one this is constant depth between ocean surface and the aquifer top that is B and this H f is the fresh water head above ocean surface up to ground water table. So, we can consider this zone as fresh water zone and this one is basically salt water zone. So, with this configuration we can consider that there is horizontal flow which is occurring from aquifer towards the sea and our x axis starts from this tow location towards sea. So, with this fundamental assumptions and configuration we can start our derivation. What is this Q f naught denoted in this position? This is the difference between the total inflow to the aquifer its right side boundary the pumping from the aquifer in the aquifer strip to the right of the point T or location of tow. So, next thing in this particular case another assumption is there that is B is much much lesser than our wedge length and B is the thickness of aquifer. So, using our Geiben-Hurtsberg approximation we have we have this H f is the fresh water head above ocean surface up to this ground water table. So, we can express our this xi plus B that is the distance below ocean level up to interface that we can denote as denote it as xi plus B equals to rho f by rho s minus rho f into H f or in simplified form xi plus B equals to delta H f or H f we can directly use it as B plus xi plus B by delta. So, for the small configuration we can say that our fresh water zone we have flow which is K f or we can write it as basic equation of fresh water flow and this is 1 D and only horizontal flow condition we have K f xi by x D h x by D x equals to constant what is this constant this is Q or flow at location x equals to 0 which is equals to Q f naught. So, what is this this is basically e or flow or Darcy and flow is equals to the flow which is coming from right and side boundary and basically we have seen this is the difference between total inflow to the aquifer from the right hand side boundary and pumping from the aquifer. So, using this thing we can write again we will just write K f xi x D h f x D x equals to Q f naught for this particular situation now we can use our Kaibman Hertzberg principle. So, we can replace this H f with xi plus B by delta or K f xi plus D by D x B by delta equals to f or this is K f by 2 2 xi D xi by D x equals to Q f naught or this thing we can directly write it as i square D x and the right hand side will have delta Q f naught by K f with this we can use integration. So, from integration we have i square equals to 2 delta Q f naught by K f to x plus c the c is the integration constant. Now, if we have x equals to l then this psi value is 0 which implies that c is equals to 2 delta Q f by K f into l. So, final equation we can write it as this xi square is equals to 2 delta Q f by K f l minus x. So, important thing is that we have considered our coordinate system from tau location towards left. So, if we have x equals to 0 that is at the location of tau we have xi equals to B which is again B is the depth of the confined aquifer. So, this implies that we can write it as B square 2 delta Q f naught by K f into l because x equals to 0. So, basically this particular relation relates with our depth of the confined aquifer. This is del is the density ratio which is used for Gauguin-Hurtsberg principle K f is the freshwater hydraulic conductivity Q f as we have defined in our assumptions and l is the sea water wedge length. So, this is somewhat parabolic in nature. So, we can say that saltwater interface in confined aquifer is parabolic in nature. So, next thing we will start our interface for interface interface in a free attic aquifer. So, in this free attic aquifer this is also our unconfined aquifer, unconfined coastal aquifer. Again we can draw our basic configuration for the problem we have horizontal impermeable bottom. This is the location of ocean and location of interface can be denoted with this red line and this is T or 2 location with saltwater and freshwater thing and this is width B which is depth below this ocean surface level up to impermeable bottom. Again we have this ground water table which is free attic surface and we can assume some constant recharge rate n. So, with this height of ground water fresh ground water table above ocean level is h f and below this ocean surface level till the interface we have distance xi and saltwater wedge length this is again l and our coordinate system we are assuming from the tow location towards left and we have again this q f naught which is the difference between the flow which is coming from right side boundary and this is the difference between flow and pumping. So, with this situation this is h s and this precipitation rate or recharge rate this precipitation rate we can consider it as n. So, within this interface if we consider one small elemental portion then we will see that for this elemental portion we have this is n which is recharge from the top and q f is a flow coming to this strip and q f plus d q f by d x into del x this is the distance the flow which is coming from left boundary and which is going out from the right boundary and going out from the left boundary. So, if we do mass balance for this particular configuration then we will see that q f minus q f plus d q f by d x into delta x plus we have n into delta x equals to 0 or we have situation where d q f by d x this is n or 0. So, if we use our Darcy and flow conditions then we can easily see that this q f is basically minus k f into h f plus xi into d h by d x. So, this q f at the right hand side boundary we have this value q f is equals to q f naught. So, at this right hand side boundary q f at x equals to 0 equals to q f naught or we can say that minus k f h f plus xi d h by d x equals to q f naught and now if we apply our Guyburn Hertzberg principle we have xi equals to delta plus h f for unconfined aquifer or h f is equals to xi by delta. Now, using this Guyburn Hertzberg principle relationship we can simplify our basic equation as this minus k f this is xi by delta plus xi and this is d by d x by delta or we can say that k f 1 plus delta by delta square into 2 xi d xi by d x this is equals to q f naught or we can simplify this particular term as d i square d x x naught equals to 0 sorry 0 minus 2 q s naught f delta square divided by k s 1 plus delta k f 1 plus delta. Again from our fundamental relationship that is we have derived that is k q f d x n equals to 0 and in this one if we substitute q f then we will get d by d x which is minus k f h f xi d h f d x plus n equals to 0 this is our relation number 1 or 32.1 let us denote it and from this one if we substitute our Guyburn Hertzberg principle relationship. So, k f again this is 1 plus delta divided by delta square d x plus n equals to 0 or we can have a relationship where this d by d x xi square d x equals to minus n 2 n delta square divided by k f 1 plus delta. So, from this one we can derive this square x equals to minus 2 n delta square k f 1 plus delta x plus c 1. So, from our relationship from 32.1 we have seen that d of xi square d x at x 0 or x equals to 0 x equals to 0 this value is equals to 2 q f delta square by k f 1 plus delta. So, if we utilize this thing for our case then we have c 1 is equals to minus 2 q f naught delta square by k f 1 plus delta. So, finally, this particular equation can be written as 2 q naught f delta square k f 1 plus delta. So, again integrating this one we will get phi square equals to minus 2. So, 2 2 will cancel this is x square. So, we have n delta square k f 1 plus delta x square minus 2 q f delta square k f 1 plus delta x plus c 2 this is again one integration constant. So, if we have x equals to 0 we will get xi equals to b. So, with this one we can get that c 2 equals to b square. So, finally, we can write our equation as xi square equals to minus 2 n delta square by k f 1 plus delta into x square by 2. So, 2 2 cancels minus q f delta square by k f 1 plus delta x plus b square or the i square minus b square this is minus delta square by k f 1 plus delta this is 2 q f naught plus n x into x. If we have recharge is equal to 0 then n equals to 0 and we can get simplified relationship with n equals to 0 and if we have xi equals to 0 at x equals to l we can get this is b square equals to delta square f 2 q f naught plus n l into l. So, again we have found out one relationship between depth of this free attic aquifer from ocean surface to impermeable bottom delta is the our Gaiman Herzberg density relationship k f is freshwater hydraulic conductivity q f as we have defined in our assumptions and n is the recharge rate l is the length of the wedge. So, with this relationship we can easily find out the shape of the wedge and this is the final relationship between different parameters in confined aquifer. So, we can utilize this thing for our calculations. So, we can conclude our lecture with this particular confined aquifer, unconfined aquifer condition, unconfined aquifer condition that we have derived here and in the next lecture we will start with case where we will try to find out the ocean island case in which we will try to utilize the condition for symmetric flow symmetric radial flow conditions. Thank you.