 In the previous lecture, we had seen some alternative ways of expressing the stream flow data. For example, we have looked at the hydrograph, where we plot the stream discharge versus time. We have looked at a flow duration curve, which tells us a certain discharge will be exceeded or equal to how much percentage of time. We have also seen the mass curve, which represents the accumulated volume passing through that point in the river versus time. So, we briefly introduced the flow duration curve and the mass curve, and we said that we can use them to find out the dependable discharge and also to find out the storage requirement to maintain certain demand. So, let us look at flow duration curve and the mass curve in details. We will start with the flow duration curve, and then we will move on to the mass curve. Also, we would look at some situations where enough data may not be available, and we have to estimate the yield of the stream. For example, if you do not have enough data available, we can derive a synthetic unit hydrograph, and based on that we can find out runoff for any given precipitation. But, there are other empirical or semi empirical relationships, which are available to estimate the runoff for any given rainfall. So, we would look at some of these techniques also in this lecture. So, let us start with the flow duration curve and see how we can obtain the flow duration curve for any stream at a given station. As we saw in the previous lecture, we plot the discharge versus the percentage of time it is equaled or exceeded. Now, suppose we have the data for the runoff in the stream over a period of let us say 10 years, 20 years, 30 years. So, over a long period we have the data for discharge in the stream. We can decide on an interval, this may be daily, weekly, monthly or sometimes we take an interval of 10 days and average this charge over that period. So, if we take let us say daily discharge and we want to find out average daily discharge suppose we have a 10 year data. So, we can pick up a certain date for example, 1st January and find out what is the average discharge on that day over this period of 10 years. So, that will give us an average daily flow or if we want to go for let us say weekly flow we say in the first week of the year what is the average over all this 10 years. So, depending on what is our aim for using this data we can go for a small duration which may be daily or we may go for a longer duration which may be monthly. Typically, when we take smaller durations we get more variation in the data, but when you average it over a period of let us say 7 days or 10 days then the variations are averaged out and the data does not show that much variability. Let us take an example of this daily data. So, if we have obtained this average daily flow for all 365 days in the year, we have these 365 values of the daily discharges. Now, we can arrange them in descending order. So, the first value would be highest now this may occur let us say on 15th of July then we go to second highest and so on and this way we can arrange all the 365 data in the descending order and assign a rank to them. So, the first data will have a rank of 1, the second data will have a rank of 2 and this way till 365. Now, suppose we want to find out the percentage of time a particular discharge is equal or exceeded we can just take the rank which we denote by let us say m and the total number of points let us say are n. In this case n would be 365 and the rank for the first 15th July discharge would be 1 then the next rank would be 2 and so on. So, we say that the percent p for any given discharge the percentage of time it is equal or exceeded is given by this plotting position m over n plus 1 into 100 percent. So, in this way we can find out a certain discharge versus its percent and get the flow duration curve which would look like. So, from this data once we prepare this curve the flow duration curve from this data we can answer questions like what is the 50 percent dependable discharge or what is the 85 percent dependable discharge and so on. So, this is the way we can predict dependability of certain given discharge. Now, let us look at the mass curve and see how we can decide the storage capacity for a given demand. So, in order to maintain certain demand we need to store water and then release it during the dry periods and we can also look at for a given storage what will be the demand which can be maintained. So, let us look at these two questions first as we have seen a mass curve is a plot of time versus cumulative volume. So, accumulated volume let us say in meter cubes the time may be in days or months. Now, the mass curve as we have seen earlier looks like this when the volume is increasing at a very fast rate this means that this charge in the stream is high and then we have these dry periods where the mass is increasing at a very slow rate indicating that this charge is very small. So, over this period we can see that there are lots of highs and lows or ridges and valleys in the curve. In order to find out the maintainable demand. So, there are really two kinds of problems which we can solve one is given demand find a storage and the second problem would be for a particular storage suppose our storage capacity is limited then we say that what will be the demand which can be maintained. So, given a storage find maintainable demand so if our storage capacity is limited or given to us fixed from some other considerations what demand can we maintain at that point. Now, when we say demand it also includes whatever water we have to release downstream of that point. So, this is the channel and we are looking at this point a the hydrograph at a will give us the amount of water which is coming upstream from the upstream side at the point a and the demand would mean whatever demands are existing in the nearby area as well as water amount of water we have to release downstream. So, we will include everything in the demand and we have already seen that the maximum demand which we can maintain would be obtained by the average annual yield of the stream. So, let us say that the demand let us say that this is the volume v the demand we can plot on this inset and let us say the demand which we want is this let us call it q. So, again this is on the same scale time versus volume. So, we are maintaining let us first assume that the demand is constant. So, for maintaining a particular demand q it means that we are withdrawing from the storage at a constant rate and therefore, the curve will be a straight line. Now, let us assume that this is the start of the dry period and let us assume that the reservoir is full at this point. So, we have some reservoir here created let us say we create reservoir here and it is full at the start of the dry period. Now, this is dry period because after this the slope is very flat that means this charge in the river is very small. Now, at this dry period if we draw a tangent which is parallel to our demand line this will include all the demands as we have already seen and this is the tangent at the high point here or the ridge here. Now, in order to find out the storage let us look at what is happening here our demand is constant at this rate the supply is small. Therefore, there is a deficit between the supply and the demand which has to come from a storage. So, at this point this is the deficit which must come from a storage at this point this is the deficit which must come from the storage and therefore, whatever the maximum difference between these two curves will give us the required amount of storage. In this case to find out the maximum difference we take a tangent which is again parallel to the demand line and the difference between these two lines is the required storage s. So, if we have this storage we start with the full reservoir at this point and by the time we reach here the reservoir becomes empty because we have utilized all the storage s, but after this point as you can see from this figure the supply is more than the demand and therefore, the reservoir again starts filling. Now, we can do the same thing on the second curve also we can draw a line tangent at the ridge and find out what is the storage requirement for this part of the curve let us call it s prime. Now, we can do it at all the or throughout the mass curve and find out all these ordinates and whatever is the maximum ordinate we choose that as the storage capacity. So, if we have let us say this two year data and in this case s is more than s prime. So, this s will be our required storage which will be able to maintain this demand q. If you look at the reservoir filling and emptying pattern reservoir is full at this point empty here and it will again become full at this point. After this there is some water coming in the reservoir which may spill also, but at this point again the reservoir is full. So, our assumption that at the start of the dry season the reservoir is full is valid throughout this curve. Now, let us look at what happens if the demand is variable accumulated volume v versus time t. So, we again have this mass curve and then we can plot the demand curve on this which is now variable. So, it may look like this and again the philosophy is the same that we take whatever is the maximum difference between these two curves, but in this case we cannot do or we cannot use the same method which we used earlier because now we cannot draw a parallel line at the point. So, we have to estimate the maximum ordinate for example, in this case this s may be the maximum ordinate. So, for a variable demand we have to compute the maximum ordinate and that will be the required storage s. Now, let us look at the other problem in which for a given storage we find out what is the queue which we can supply from the river at that point. So, let us have a similar mass curve and the storage is given which is let us say I am showing it here s. So, this s value is given to us and we want to find out what is the demand which we can maintain at the point at the outlet point a draw lines let us say at the ridge at some point q 1 such that our demand here is met by storage of s. This s is known to us. So, it will give us an idea about what is the queue which we can maintain for this s. Similarly, from here if we have certain s what queue can be maintained let us call it q 2 and so on. So, in all these values we have to find this q 1 q 2 q 3 and so on and choose whatever is the smallest queue that will be our maintainable demand. The smallest of these q 1 q 2 q 3 etcetera will be the maintainable demand. Now, the procedure of finding out this q 1 involves a little bit of trial because this point at which we are drawing the tangent is not fixed and we have to adjust the q 1 line. So, that this storage available is exactly equal to the given storage s, but the way we do it is do it for all the values and find the minimum of the queues that will be the maintainable demand. Now, in some cases we may not have enough data or sometimes we may want to extrapolate the values for some other conditions for example, extreme conditions for which the data may not be available. In that case we have to come up with some either empirical or semi-empirical relationships. So, we would look at these methodologies to estimate the yield of the stream at a point. So, estimation of yield typically represents the annual flow in the stream at a particular point. So, if we have this catchment area of the stream at the point a what yield generally refers to although it can be over any period, but generally it refers to annual flow at the point a from the stream. Now, if we have data as we have seen earlier we can find out yield from the mass curve, but sometimes we may not have data. In that case either we have to use some synthetic hydrograph to obtain the yield or there are some methodologies which can be used which may be written as rainfall runoff correlation or we can use some empirical equations or if data is available and of a very good quality sometimes people have to use water budget modeling to estimate the runoff given the catchment properties. So, if we know the rainfall over the catchment we can estimate the evapotranspiration infiltration. So, all the abstraction we can estimate and do a water budget modeling to find out the runoff. So, we would look at these three different methods in details. Let us start with this rainfall runoff correlation. As we know rainfall or precipitation is the driving force behind runoff. So, naturally the runoff will depend on the rainfall, but the relationship is not very clear because sometimes there are other factors which affect the runoff. So, for the same rainfall we may have a different runoff depending on the existing if you plot the record of the rainfall and runoff. So, if we have the precipitation let us say in centimeter or millimeters and the runoff can also be converted into centimeter units over the whole catchment area. So, let us say that the runoff also has units of centimeter. Now, if you have data for let us say yearly amount of precipitation in a catchment and the yearly runoff we can plot it on the curve and for a number of years if you plot the data it is seen that there is a definite trend it may not be a very very defined relationship, but there is a trend between the precipitation and the runoff. So, based on this observed trend we use rainfall runoff correlations or typically we write P R correlations P for precipitation and R for runoff. So, precipitation versus runoff correlations can be obtained for any given catchment if we have this data available for a large number of years. So, over a duration if we have this data available we can plot them and typically a straight line is fitted suppose we have a lot of data available for data. So, we can use a lot of techniques are available for fitting a straight line through data. So, we will not go over those techniques, but once you fit the straight line you have a relationship like this which relates the runoff with the precipitation on the area. Now, even if there is no precipitation there may be some base flow. So, that is why for 0 precipitation also you may have a fixed or finite amount of runoff a 0 and a 1 they would be obtained from the data. So, once you develop this relationship for let us say we want to estimate the runoff for a precipitation which has not been measured in the field, but which is expected to be the maximum precipitation which can occur over the area and for that we can find out what is the probable runoff even though we do not have data for that point because that rainfall has not occurred. We can based on the existing trend we can extrapolate the value of the runoff to estimate what is the maximum flood likely in the river if extreme rainfall occurs. So, that is the advantage of having a relationship like this typically the precipitation data is available for a very long duration compared to runoff and also runoff data for very high discharge discharges or very high stage in the river is not very reliable. Therefore, r versus p relationships are very useful since p is known for a longer duration and p is also more reliable compared to r for high values of runoff. This is not the only possible correlation there are sometimes people use a power law in which you relate r with some power of p typically for larger basins a power law is found to be better than the linear relationship. Some other options are that instead of relating it with the rainfall of that time for example, that year or that month you relate the runoff with precipitation for that time and also a few previous times. For example, if in a year there is some rainfall the runoff will also depend on what is the rainfall in the previous year because if the rainfall in the previous year is more the ground is likely to be wetter and therefore, there will be more runoff also the ground water level is likely to be higher and therefore, there will be more base flow. So, not only the precipitation in that time period for example, that year, but precipitation in the previous year or year before that may also influence the runoff and therefore, sometimes people have tried to develop a correlation like this. Also, sometimes we do not have a very defined straight line or power law relationship and we really do not want to find out the relationship we just want to find out the runoff for any given precipitation event. In that case artificial intelligence models like ANN artificial neural network they also have been used in which we put the input values here as P t P t minus 1 may be the temperature also and all these inputs result in one output which is the runoff at time t and in between we have these various layers of neurons which convert this input into this output R t, but ANN models are little more advanced we will not discuss them here in this course. The second method of estimating the yield is based on empirical equations which are nothing but, equations of similar kind for example, R equal to k p, but this constant k now is an empirical constant not really based on analysis of observed values, but based on analysis of for similar catchment areas we can say that this k its value will depend on the type of catchment area. So, the k typically is a function of type of catchment and some other parameters and different investigators have related k with different parameters for example, there is a Barlow equation where he says that k would be a function of type of catchment as well as the season. So, k will be a function of the season then there is a strange empirical equation which relates k with amount of monsoon rain. So, type of catchment and monsoon rain. So, there could be a number of empirical relationships like this, but they will be useful only in catchments which are similar to the catchments for which the relationships have been derived. There are some regional relationships also which have been derived for example, English for Deccan plateau has derived a relationship in which the runoff is given by in centimeters and p also in centimeters. So, these are again everything is based on observed values and for a given region you can use this kind of equation. For example, for the Garts a similar relationship have been obtained as r equal to 0.85 p minus 30.5. So, these kind of relationships are available, but for a very limited area. So, in that area we can use these, but not everywhere. Typically these are for annual runoff and annual precipitation, but there is an equation by Khozla which computes the monthly values of rainfall and runoff. So, the Khozla empirical equation is given by r equal to p minus l. So, again r is now monthly. So, we find out the precipitation in a month which we can obtain from the rain gauge l is the loss and this loss typically depends on the mean monthly temperature at that area and Khozla has given a relationship which says that r l will be equal to 0.48 T, where T is the mean monthly temperature. So, mean monthly temperature of the area will be known to us from meteorological records. We can find the losses from here precipitation will also be known to us. So, we can find out the monthly value of runoff and then we add this for the entire year to get the annual yield. The water budget model is quite a straight forward mathematically, but typically we will need computers to implement this model because there are lot of competitions involved and a lot of data has to be collected which can be used to obtain the abstractions from the precipitation. The equation which is used in water budget models is this which is a simple mass balance equation saying that the runoff is nothing but precipitation minus the abstractions. The abstractions are evapotranspiration and delta s is the change in soil moisture. So, what the model says is that out of total precipitation, some part goes into evapotranspiration and some part goes to increase the soil moisture. So, it infiltrates into the ground and therefore, the soil moisture is increased. Now, if delta s is negative then of course, it means that there is decrease in soil moisture and therefore, precipitation minus delta s would the runoff would really be higher than what it would be for a delta s which is positive, but for this model we need to estimate evapotranspiration. We have seen in some other lectures that evapotranspiration can be obtained from the atmospheric conditions like wind velocity, temperature and so on. So, all those factors have to be input here to obtain the evapotranspiration and we can do this water budget modeling over a daily or weekly or monthly time interval. Delta s has to be estimated based on the crop pattern, the crop water requirement or if soil moisture data is directly available we can utilize that. So, next thing which we would like to look at is once we have an idea about the stream flow either a hydrograph or a flow duration curve or a mass curve how to obtain the extreme conditions because our interest as a water sources engineer is in the extreme conditions of either very small flow or very large flows. So, the main thing which we are interested in is these conditions of drought or flood. So, if the flow in the river is at an average rate and our demand is also average rate we do not have to worry about these things, but in extreme cases when there is a drought means lack of water. Now, this drought may be defined in various ways we will look at what are these ways, but drought in general indicates a lack of water and similarly the flood indicates more than or a lot more water than desired and both of these are not good for a water source engineer point of view and we need to control them. So, if there is a drought condition we must be able to find out what is the deficit and we should be able to account for it we should be able to carry water from some other place to that place where there is a drought. Similarly, for flood we need to be able to device flood control measures we can store water out of that flood or we can have some other measures upstream. So, that the entire water is not coming at the same time there is some delay and therefore, the flood peak is reduced. So, drought and floods these two characteristics we should look at in details now. So, when we say drought the general understanding of this term is that there is a lack of water. So, there is not enough water to satisfy our demands, but the demands are also different from different point of view. So, a drought may be for a person who uses that water for drinking purposes or for other household purposes drought means that he is not getting enough water, but if you look at it from let us say agricultural point of view then a drought means that the plants are not able to get enough water. So, a drought can be defined based on our point of view and the definitions vary from one point of view to the other and therefore, we typically talk about a meteorological drought or a hydrological drought or an agricultural drought. So, as the name implies meteorological drought means a lack of rain. So, if we have some precipitation p and there is some normal precipitation in the area let us call it p normal. If p is smaller than p normal then naturally we have less than average conditions, but just p little bit less than p normal would not really be defined as drought. So, there is a limit that we say when p is less than 75 percent of the normal rainfall then meteorologically speaking there is a drought. So, any rainfall annual rainfall over the catchment which is less than 75 percent of normal would be classified as a drought and again the less than 0.75 there is a wide range of precipitation below this range. So, again it can be classified as a severe drought or a moderate drought depending on what are the actual values of p compared to p normal. So, there are this ratio of p over p normal it can be let us say 0.5 to 0.75 or we can say that if it may even be less than 0.5. So, 0.5 would be a severe drought indicating that the rainfall in that particular year is less than half of what is normally expected and on that area and this up to a 25 percent deficiency or up to the 75 percent of rainfall. So, 50 to 75 percent we can call it a moderate drought. So, meteorologically speaking the drought only depends on the rainfall and the ratio of the actual rainfall to the normal on that area will define whether the drought is severe or moderate. Now, just because the rainfall is less than normal does not mean that let us say stream flow will also be less than normal because there may be a stream flow which is contributed by the base flow the groundwater discharge and that may happen even if there is a smaller than normal rainfall. So, only meteorological drought does not indicate a drought in terms of hydrological variables. Therefore, hydrological drought is defined based on stream flow being less than normal or we can use some other parameters. For example, soil wetness, soil moisture or we can have groundwater. So, hydrological drought will indicate the lack of stream flow or soil moisture or groundwater compared to their usual values. So, this encompasses a lot more variables than just the precipitation although typically precipitation is what is driving these values. Therefore, meteorological drought typically implies hydrological drought also, but not always. The agricultural drought is over a very short period typically about a week or so because when we say agricultural drought it means that the plants are not able to get enough water to sustain and they can die even within a short period of a week. So, even if annually let us say we have an annual drought in which annual precipitation is less than 75 percent of normal, but suppose this precipitation occurs when the plants need water then they will not die. So, depending on what is the temporal distribution of this precipitation the plants may be able to survive or sometimes even when there is normal precipitation the plants may die because they do not get water when they need it. So, for agricultural drought we have to take a very short interval typically a week to say that during that one week time whether the requirements of the plants are being met or not. If they are not being met then we say that this is an agricultural drought. Typically what we do for agricultural drought is define a term which is known as aridity index and aridity index is typically defined on the basis of the potential evapotranspiration. Because potential evapotranspiration it represents the requirements of the plants also and therefore, we use the P E T this is the potential evapotranspiration. If water is available in abundance then this would be the rate of evapotranspiration and we divide it by the rainfall and based on this aridity index we can classify the areas as arid, semi arid, hyper arid and so on. So, this aridity index takes the potential takes care of the amount of rainfall versus the requirement of the plants for that week. So, typically over a week so we can say that during that time the area can be classified as arid if the requirements of P E T are not met by P. The other term which is used sometimes is the actual evapotranspiration. So, sometimes the aridity index is also defined on the basis of the ratio of actual and the potential evapotranspiration. So, if P E T is the potential evapotranspiration, A E T is the actual evapotranspiration then the deficit as a percent of P E T this is an alternative definition of the aridity index. In fact, there are 5 or 6 different kinds of definitions for the aridity index and we will just go over these two and we will not discuss the other definitions here. So, this is about the drought which is of three kinds and can be specified based on these indexes. The floods for a water source engineer actually the floods are more relevant than the drought periods although droughts are also significant for the floods. We have to find out really or extrapolate the values which we have not even seen. So, sometimes we may have to estimate what is let us say 100 year flood, what is a 1000 year flood because floods will damage the structure. So, if we have a river and we have a dam here there is a certain amount of flood which the dam can safely pass, but if the flood coming in becomes more than that there is a danger that this water level goes very high and the structural failure of the dam may occur. So, to avoid this we need to know what is the flood expected in the channel there are lots of methods of estimating the maximum flood in the river. We will look at a few of them for example, rational method or we can have some empirical methods we have already seen and we have discussed in details the hydrograph theory. So, we can use the hydrograph rather the unit hydrograph method to estimate the maximum flood or sometimes we use what is known as a flood frequency analysis. So, these methods are typically used to estimate what is the maximum flood likely to occur in the river at a particular point. In the rational method we estimate the maximum flood as a coefficient C and intensity I and the catchment area A. Now, this runoff coefficient C varies widely from area to area and there are values given for this I is the intensity of rainfall occurring over the catchment. So, if you look at this equation I into A gives you the volume of runoff if there is no loss. So, I is the intensity A is the catchment area, A I is the volume of the precipitation occurring over the area and therefore, this C value is really a loss coefficient or 1 minus a loss coefficient which gives you the runoff for a total amount of precipitation. So, it is similar to the rainfall runoff relationship which we looked at earlier R equal to k into p, but this runoff coefficient it depends on the type of area. For example, if the area is very impermeable, paved area is lot C will be very high, if the area is very pervious then lot of water will go as infiltration and C will be smaller. Tables of C are given for different kinds of areas and we can look at that table little later on. In the empirical equations the flood is given as some function of the catchment area. So, the intensity is not there, but the area is some function of the catchment area. Some of the relationship which have been used are like area to the power 3 by 4 or area to the power 2 by 3 and so on, but these are again derived for a particular location and they would be typically applicable only for similar catchments. Hydrograph method we have already looked at in detail. So, we will not discuss it again and flood frequency analysis is similar to what we did in the flow duration curve that we take the values of annual floods, arrange them in order and based on that we can decide how frequent is a particular flood. So, we can find out a 100 year flood or 50 year flood depending on the importance of the structure. For example, if there is a dam it is failure may be catastrophic, it may include a lot of loss of life or property downstream of the dam. Therefore, when we design a dam we should take a flood which has a very high what is known as return period. So, when we say a 1000 year flood it typically means the flood which is expected only once in 1000 years. And therefore, for structures which are very critical which involve a loss of human life, it is generally accepted practice to take a very high return flow for example, 100 to 1000 years, but for some structures for example, there is a culvert in the road. And if we want to design that we do not need to take very large flood or a very high return period, we may be able to do a once in 20 years or once in 50 year flood, because if the flood is exceeded it will only lead to submergence of the culvert may be traffic will be affected for a few days or a few hours, but it is not so critical that we take a very high flood, because when we take a very high flood for design it will increase the cost of the structure, because we have to pass that amount of flood through the culvert. And the utility probably will not be that high we may be able to live with that inconvenience for a few hours or a few days, but there will be lot of saving in the cost. So, these aspects come into picture and therefore, flood frequency analysis becomes very important, it also gives us an idea about the risk involved in the design. So, we would look at little later on we would look at the risks reliability. So, what is the probability of failure of a structure within the next 10 years or next 50 years, what design flood to take for its design, how to find out what is 100 year flood, because we may not have data available to us for a 100 year period. So, we have to take some kind of we have to have some extrapolation to find out what is the flood, which occurs let us say once in 1000 years. It every flood will have some risk attached with it. So, if we take 100 year flood there is a risk that it may come next year or in let us say 5 or 10 years. So, we can find out doing this frequency analysis and risk analysis, we can find out what is the chances of failure of a structure within the next 5 years or 10 years or whatever is the useful life of the structure, how reliable the structure would be. So, all these concepts we can define and we can analyze the flood frequency to define these concepts. So, this drought and flood we will be dealing with little later. So, to summarize today's lecture, we have looked at alternative ways of presenting the stream flow information, the hydrograph information we have discussed in details earlier. We looked at the flow duration curve, which represents the dependability of flow, how much percentage of the time that particular flow will be available or what we can say will be equal or exceeded. The flow duration curves they give us an idea about what this charge will be available for what percentage of time. So, we can find out an 85 percent dependable flow or 50 percent dependable flow and so on. The mass curve is another way of representing the stream flow, which plots the accumulated volume of water passing through that point versus time. The mass curve is helpful in finding out the storage requirement. So, if we have certain demand which we have to satisfy, we can find out the required storage from the mass curve or for a given storage we can also find out what is the demand which can be maintained. After looking at the stream flow record presentation in different formats, we looked at the extreme conditions of very small flow which is drought or very large which is flood. So, droughts we classified based on from which point of view we are looking at the data as meteorological, hydrological or agricultural drought. For the floods, we looked at various methods of finding out the floods. We introduced them, but we will look at more details little later. We also looked at water various drawbacks or advantages of different approaches. We will discuss that in details little later.