 Welcome to this lecture number 33 and in this particular lecture, we will talk about salt water intrusion in aquifers and topics to be covered is analytical solution of salt water intrusion in coastal aquifer and we will continue from the previous lecture class and we will also talk about density dependent flow density dependent salt water intrusion models. So, this is our lecture number 33 and in this one, we will talk about salt water intrusion in aquifers and topics to be covered are first one is analytical solution of saline water intrusion in coastal aquifer will continue from previous lecture class and next one is density dependent salt water intrusion model. We have already talked about salt water intrusion in confined and unconfined aquifers. Now, let us derive the relationship that exists in oceanic island. So, interface in an oceanic island. So, in this one, let us say we have an oceanic island which is surrounded by ocean from all the sides and the interface is like this and we have ground water table which is also symmetric in nature. This is central line and we have infiltration or precipitation from the top of the oceanic island and this state is n and let us consider this is having equal radius with r for a arbitrary location. We have this distance below the ocean level denoted as psi r because this is radially varying for oceanic islands and this is the distance r from center and our fresh water head h f which is again a function of r because this is again varying radially. So, this is our ground water table and this is location of interface and this is location of fresh water and we have salt water in this position. So, in this case if we write our mass balance equation. So, we can write our basic equation and this basic equation shows that the q f radially that should be equated with the recharge or infiltration or precipitation from the top of the island as this equation. This is varying with the radial direction where we have this q f or our flux is 2 pi r into k f, k f is the fresh water hydraulic conductivity h f plus psi d h f and d r. So, in this case also we will utilize our Geibn-Artsberg approximation. So, with Geibn-Artsberg approximation we have h f equals to psi by delta and delta is our rho f by rho s minus rho f. So, with this approximation we can write it as our d q f by d r is equals to we have 2 pi r n and from this one we can write this q f as we have pi n r square plus c 1 where we have integration constant. Now, for oceanic island we can specify certain conditions. So, number one condition for oceanic island we can specify that at r equals to 0, we have q f equals to 0. So, it implies that we have c 1 equals to 0. So, thus we can write our equation as q f equals to pi n r square and we can again replace our q f from our previous relationship as 2 pi r k f h f plus psi d h f d r equals to pi n r square or if we replace this h f with our Geibn-Artsberg approximation then we can write it as 2 pi r k f and this as psi by delta plus psi and this one again psi by delta r square or we can write it as minus pi by cancels from both the sides. We have r and we can write it as d psi by d r 2 psi equals to n delta square pi k f into 1 plus delta into r. So, with this thing delta square and n r we have. So, with this thing we can write it as minus d psi square d r equals to. So, this is equals to we have this n delta square k f 1 plus delta r or if we simplify this just exchange in the sign both the sides. Psi square with integration we can write it as n delta square k f 1 plus delta and r square by 2 plus c 2. Again we have another integration constant. So, another assumption we have for this particular thing is that if r equals to capital R then we have psi value is equals to 0. So, at the end of the oceanic island we have this psi value 0. So, with this let us say write this as 33.1 and next one we can just write it as 0 minus n delta square k f 1 plus delta and this is capital R square plus c 2 and we can write it as 33.2. So, if we subtract this equation 33.2 from 33.1 then we can directly write it as I square is to minus n delta square k f 1 delta r square plus this is n delta square 2 k f 1 plus delta r square or finally, we can write it as n delta square 2 k f 1 plus delta r square minus r square. So this shows the relationship between the location of interface, the recharge rate and recharge rate and different radius. This is radius of island. This is hydraulic conductivity and interestingly in this case if we have n which is our recharge rate, if n equals to 0 then the psi value is always 0. That means if there is no recharge in oceanic islands, so it means there will be intrusion from the bottom. So, next thing we will discuss another important solution in case of our salt water intrusion problem that is again sharp interface approximation of the problem. So, in this case this is popularly known as Stratx solution. In Stratx solution we consider two different zones for modeling our salt water intrusion problem. In our three consecutive derivations that is for confined, unconfined aquifers and our oceanic islands we have talked about single zone modeling, but in case of Stratx solution we consider a two zone model for problem solving in salt water intrusion modeling. So, we have location of sea surface or ocean surface, we have interface located as toe or tee and we have ground water table. So, in this one the problem is divided based on the location of toe. So, from the left hand side we denote this particular zone as zone one and right hand side we denote it as zone two. So, in this zone one and zone two we have two different situations zone two consist of only fresh water aquifer and in zone one we have both salt water and fresh water thing that exists with a sharp interface. So, Stratx what it did it defined certain potential for defining different piezometric heads. So, we can simplify or we can write it directly according to our previous modeling conventions this is our H F this is xi and q prime that is entering from zone one to zone two and we have thickness of this free attic aquifer below sea level or ocean level is B and this is our sea water position, this is our fresh water location and this is impervious bottom with in Stratx solution the coordinate system is taken from left to right and it starts at the beginning of the interface. So, in this case we can say that the Stratx that is he derived this particular formulation in 1976 developed a model that describes the fresh water flow in both zones using a harmonic potential as the single variable of state for both zones or both zones. So, what is this Stratx solution in common boundary common boundary it requires that. So, on the common boundary in each zone we require the continuity of both the piezometric head and the flux. So, at the interface level we require that continuity of both piezometric head and flux. So, using this thing two things can be defined for both zone one and zone two. So, for zone one zone one we have free attic flow above interface. So, in this case Stratx potential is defined as equals to h f plus i into h f if we utilize our things. So, we can see that this is basically our half this is again we have this for Geiben-Arzberg principle for free attic aquifer we have xi equals to delta into h f. So, in this case for in case of h f we can replace it with delta h f plus h f into h f. So, we have half 1 plus delta h f square and for flux continuity we have q prime. So, which is in vector form is k h f plus xi this is h f and k del xi del phi and this thing is basically fresh water discharge for unit aquifer width. In case of zone two in case of zone two which is fresh water only we can define our Stratx potential as phi h f plus b square 1 minus delta by delta b square and again we can define our flux thing as q prime equals to minus k this is h f plus b prime h f minus k phi. So, at location t at location t or at tau location we have b which is equals to h f into delta as per our continuity of zone one and zone two and by using Geiben-Arzberg approximation. So, in this case we have phi which is equals to phi tau and we can directly write it as b square by 2 delta square. So, if you utilize this equation or this particular equation we will get same result because this is at the transition of zone one and zone two. So, using this thing or continuity of the flow is considered and for steady state flow steady state flow condition we have dot q prime in each zone. So, from this one we can directly write the Laplace equation for this particular case. So, this means that phi is harmonic function in the respective zones. So, it satisfies our Laplace equation. So, strike utilized this particular approximation and theory for derivation of coastal aquifer modeling with single pumping well and he showed that phi can be constructed as phi can be constructed as phi equals to q naught prime x plus w 4 pi k ln x minus x w which is location of x w is the location of pumping well and this is can be utilized for modeling of pumping wells in coastal aquifers with simple approximation. Again this phi can be replaced with phi tau and which is equals to 1 delta p square delta square equals to x 2 delta square where q w we have is equals to pumping rate from the aquifer with pumping well q naught x prime and this is the flow that is occurring horizontal flow that is occurring from coastal aquifer phrase zone towards C. So, if we write this thing we have one top view this is basically top view in top view we can see that this is y axis this is x axis and this is the location of pumping well. So, with this location of the pumping well this is x w for different pumping values we can have different location for this is different location for tau this is ocean and this side we have this flow horizontal flow that is coming from fresh water aquifer towards C ocean q naught x this is prime and this is this thing is basically k by x. So, as it is in our previous thing. So, initially if we have this q w equals to 0 the location may be like this however if we have if we have higher pumping values. So, the location of interface that will change and that will be symmetric with this x axis and finally it has been observed that for specific value it shows a critical pumping location which will signify the end of no flow zone this is the influence zone for pumping and at this point it shows 0 flux situation and it is called as q c or critical pumping. So, next thing we will discuss is this density dependent flow models what is this density dependent density dependent salt water intrusion model. So, in this model a transition zone is considered instead of a sharp interface that is used for our sharp interface modeling using Gaibman Herzberg principle. Let us say we have some semi pervious location and we have another this is again impervious bed or impervious bottom this is the location of transition zone. So, in transition zone it is considered that the density is in between fresh water and salt water and during this kind of modeling we need to consider the composition of sea water because the composition of sea water is important for exact modeling of using its exact modeling of salt water intrusion using density dependent flow models. So, in this case this density is considered to be a function of pressure, temperature and salinity. So, this density is a function of pressure, temperature and salinity. So, if we see the variation of density this is density of water kg per meter cube this is temperature is 0, 5, 10, 15, 5, 10, 15 and degree Celsius. So, this is with difference of 1 kg per meter cube this is 1014, 1015 like that we have. So, with this thing it has been observed that this density varies like this red line where we have a chloride concentration of is considered to be 1.2 percent of chloride. Next if we increase this we decrease this chloride percentage 1.1 percent of chloride and again this is 15 say in between also there will be different curves. So, we can see that there is some kind of non-linear relationship between density this is temperature and this chloride which signifies the salinity for salt water. So, depending on the chloride concentration depending on this chloride concentration type of ground water we can define. So, classification of ground water on the basis of chloride concentration type of ground water and milligram chloride per liter. Next category is oligo halion we have the range 0 to 5, oligo halion fresh we have 5 to 30, oligo halion fresh we have 3 to 150, 30 to 150, fresh brackish that is 150 to 300 then we have brackish which is 300 to 1000 then brackish saline we have 1000 to 10000 then saline we have this range 10000 to 20000 and the final range is hyper saline or brine this is greater than 20000. So, this is the classification of ground water on the basis of chloride concentration. So, different geochemical composition is required for modeling of detailed modeling of salt water geochemical components seawater seawater has a uniform chemistry due to the long residence time of the major constituents. So, for this one to know something about salinity salinity chloride is important thing. So, time series steadily increasing chloride concentration can indicate the early evaluation of salinity breakthrough from seawater can indicate the early evolution of salinity breakthrough from seawater. So, there are other components or other indicators exist that can be used for identification and evaluation of salinity level. So, we will discuss those topics in the next lecture class. Thank you.