 Namaste and welcome back to the video course on watershed management. In module number 8 on storm water and flood management in lecture number 33, today we will discuss about the flood routing. So, somehow the important topics covered in today's lecture include flood routing through channels, reservoir routing, hydrologic routing, hydraulic routing, lambered flow routing, musking gum methods, same venance equations. So, some of the keywords for today's lecture flood routing, channel, reservoir, hydrologic and hydraulic routing. So, we were discussing about the flood problems say on a watershed basis we have seen what will be happening say with respect to the rainfall and no processes and then with respect to channel flooding and then with respect to the the overland flow conditions, urban flow flood management, urban drainage system all these details we are discussing in this module in the last two lectures. So, in today's lecture we will be mainly focusing on the flood routing. So, let us first look into what is flood routing. So, as we discussed when we look into on a watershed basis say watershed receives the rainfall as inputs and then say through various hydrologic processes say this input of rainfall is transformed into runoff. So, that is the output. So, that way as we discussed in some of earlier lectures. So, we we can try to get the discharge or the the flow depth at particular location with respect to time or the discharge versus time that is so called hydrograph we can get. So, as we can see that when this rainfall runoff process taking place and then the the runoff as our flow hydrograph will be coming from various locations through various channels to a main channel or through various drains to a main channel and then this will be say this hydrographs or the the flow will be merged and then the the flood depth will be increasing or the discharge will be increasing the main channel which we consider. So, as we can see as far as a watershed based same say the flow routing is concerned we can see that the the hydrograph differs in shape duration and magnitude. So, as the flow is keep on coming from various locations and then merging into the main channel. So, there will be the the shape and duration and a magnitude will be keep on changing. So, this attribute to the storage properties of watershed system. So, what is the storage properties within the watershed or what is the storage properties within a channel accordingly the things will be changing I mean the the hydrograph say the pattern will be changing its peak will be changing or time to peak will be changing or its shape will be changing. So, within this context let us look what is a flood routing or flow routing. So, flood routing or flow routing means it is a procedure to compute output hydrograph when input hydrograph and physical dimensions of the storage are known. Say for example, if you consider a a a river like this. So, the flow will be keep coming from various locations through as overland flow or say as inflow to a particular location and then say we can see that this hydrograph say if you consider any location with respect to time the discharge versus time we can see that this the shape and other properties will be changing. So, the flood routing or flow routing means this is a procedure to compute say the output hydrograph at any location with respect to time for the given input hydrograph and then what will be the physical dimensions and storage properties of this river channel or the overland flow or the watershed so like that. So, flood routing means it is a the process of say what we are say we we know what is the inflow coming and then say at various location at particular location we are identifying what will be the outflow I mean the discharge versus time the hydrograph we are identifying. So, this depends upon what will be the inflow coming to the system when we consider say between two sections and then what will be the the the channel properties or the storage properties of the channel or the river or the watershed which we are considering. So, this procedure is so called a flood routing or flow routing. So, this flood routing or flood routing is very important in many of the hydraulic calculations since when we are going for flood predictions flood warning system we have to calculate how how this outflow hydrograph at particular location with respect to given inflow hydrographs. And then also when we are going for design of various hydraulic structures we should know this outflow hydrograph. So, flood routing or flow routing is very important procedure in hydraulic engineering and of course, within the conductor watershed management also this is very important since we have to identify how this outflow hydrograph will be changing with respect to the storage system between two structures or two sections of a channel or a river with respect to the inflow hydrograph. So, this flood routing is used for flood forecasting as I mentioned it is very important we have to identify the flood routing to see how the flood will be taking place or flood forecasting. Then design of spillways if you consider a Zerovoyer or a say hydraulic structures say like while designing we should know how this given for given inflow how the outflow or flood routing is required. And then also for various flood protection works we should know say how this outflow hydrograph is changing or how what is the pattern. So, that way flood routing is very important in watershed management say to identify say whether there will be any flooding problems or to while designing various structures in a river or in a channel we need to say identify how the flood is proceeding or we have to go for flood routing. So, now let us look what are the important motivations for this flood routing. So, flood routing as we identify for the given inflow conditions say at particular location we want to know how the outflow condition is taking place or outflow hydrograph or output hydrograph so we should identify. So, why we should go for this as we have already seen say for various flood conditions like a flood forecasting or hydraulic structure design we need. So, some of the important motivations I have listed here say when we consider the floods we should know the say we have to predict the flood propagation. So, how the flood is as you can see in these photographs how the flood is moving from one location to another location. So, that is so called flood propagation and then if you want to take for some protection measures with respect to the flooding. So, we should know and then say to go for flood warning. So, we should know how this the flood movement or flow movement takes place. So, that process is the flood routing. So, flood routing is very important as far as the floods are concerned with respect to prediction of floods or flood warning system or to go for protection from the flooding like that. And then it is very important to do flood routing to for the design of various structures like a water conveyance systems say like channels, scanners or what kind of conveyance system. We should know for the given inflows say how the outflow will be at particular locations. So, water conveyance system design then protective measures like if you are going to construct a flood protection say embankments or the protection walls then we should know how this flood outflow hydrographs. So, that way flood routing is very important. And then hydro system operations say while operating a reservoir system we should know say we should go for reservoir routing and then we should know how the water level will be is going to rise within the reservoir system. So, for the hydro system operation we should do this flood routing. And Theodomand is the water dynamics. So, say some of the many of the rivers will be engaged. So, we should know how the flow can propagate or flood can propagate. So, for engaged rivers then peak flow estimation. So, many of the times we should know what will be the peak of flow or the hydrograph with respect to the given rainfall conditions or with respect to given releases from a reservoir. So, peak flow estimation is very important. And also sometimes when we deal with river aquifer interaction say whenever the water level rises in the river say the flow will be taking place from the river to the aquifer system. And when the water level resides or going down within a river system then the aquifer will be recharging to the river. So, for this kind of calculation also we should go for flood routing. So, that way flood routing is very important as far as where you say when we consider various hydrology or hydraulic phenomena like here floods, design of various hydraulic structures, water dynamics etcetera. So, now let us see say when we discuss about flood routing say we can classify the flood routing into mainly into two one is the reservoir routing, second one is the channel routing. So, say as I mentioned say when as far as a reservoir when there is a dam and then there will be reservoir. So, to the reservoir what will be flow will be keep on coming in the upstream sections say from various channels and then we should know how the depth of flow is changing. So, that is so called a reservoir routing and as far as channel or rivers or canal circumsents for the given inflow condition say on the downstream sides we should identify how the outflow or how the discharge or the flow depth variations. So, that way we can classify the flood routing into reservoir routing and channel routing. So, as I already mentioned reservoir routing considers modulation effects on a flood wave when it passes through a water reservoir results in outflow hydrographs with attenuated peaks and enlarge of time basis. So, that way say as far as a reservoir is concerned with respect to given inflow conditions how the flood depth variation taking place and with respect to the space and time that will be identifying through reservoir routing. So, variations in reservoir elevation and outflow can be predicted with the time when the relationships between elevation and volume are known. So, as far as a reservoir is concerned if we know the at various flood depth conditions what will be the volume within the reservoir then we can easily identify say how much is the outflow or how much storage is there how much is the possible outflow from the reservoir system. So, that way reservoir routing is very important and the second one is channel routing. So, channel routing considers changes in the shape of input hydrograph while flood waves pass through a channel downstream. So, whatever inflow is taking place from upstream side and then there will be storages and then various changes taking place within the river or channel section. So, channel routing means we are considering say on the downstream side say what will be the hydrograph or what will be the flood depth or what will be the discharge. So, flood hydrographs at various sections predicted when input hydrographs and channel characteristics are known. So, here the reservoir characteristics or channel characteristics are important since accordingly there will be some storage will be taking place within the reservoir or within the channel. So, that way when we go for reservoir routing or channel routing we should know the channel characteristics and the reservoir characteristics for these kinds of routing. So, that way we can classify the flood routing into reservoir routing and channel routing. So, now when we consider say flood routing so, we have to systematically say consider the inflow and then the various channel properties or reservoir properties and that way a flood routing we have to go systematically. So, flood routing methods we can classify into hydraulic routing in which generally both continuity and dynamic equations are considered. So, hydraulic routing means with respect to channel with respect to time and space how the flood depth or the how the discharge will be varying. So, generally we will be solving the continuity equation which is based upon the conservation of mass and the dynamic equations. So, called Sainte-Vinand's equation generally by considering the conservation of momentum and then that is with respect to the hydraulic routing with respect to channel or hydraulic routing which generally uses the continuity equation alone. So, that way the flood routing can be hydraulic routing methods generally which we consider the conservation of mass or equation of continuity and then hydraulic routing methods generally which we consider the conservation of mass or continuity equation and the conservation of momentum or these are called the momentum equations which are say we call as Sainte-Vinand's equations. So, that way we can we have to systematically consider the system within a control volume approach generally either with respect to the continuity equation or the continuity equation and the momentum equation. So, with respect to the hydraulic routing or the hydrologic routing. So, as we mentioned so these kinds of routing is very important as far as flood forecasting, flood protection, reservoir design or design of spillway and various outlet structures. So, that way say either hydrologic routing or hydraulic routing we have to do as far as design and planning of say various say structures or say when we go for watershed management this routing either hydraulic routing or hydraulic routing are essential. So, now say as flood routing when we consider so we have seen that we are trying to identify how the flow depth is varying or discharge is varying with respect to given inflow conditions and the storage conditions of the channel or the reservoir. So, that way the flood routing say as we discussed it is a technique of determining the flood hydrograph at a section of a river or a channel by utilizing the data of flood flow at one or more upstream sections. So, you can see that here this is the hydrograph discharge versus time. So, if this is the inflow hydrograph coming and then say you can see that due to attenuation and then various changes. So, this can be the outflow hydrograph. So, similarly say there can be storage taking place within the river or the reservoir then you can see that say as far as reservoir is concerned release from storage then accumulation storage. So, that way we can identify with respect to the hydrograph. So, that way when we deal with flood routing we can have two types of flow routing or flood routing one is lumbered flow routing and second one is the distributed flow routing. So, lumbered flow routing so here flow is a function of time only at particular location. So, the spatial aspects we are not considering, but with respect to time. So, that is the lumbered flow routing and then distributed flow routing is concerned flow is a function of space and time throughout the system. So, that way we have to see the channel section and then slope and various conditions whether it is channel routing or river routing or as far as reservoir is concerned we should know the aerial extension and then with respect to the depth conditions how the volume changes. So, that way we should know that also. So, that way flood routing can be either lumbered flow routing or flood routing can be lumbered flow routing or distributed flow routing. So, as I mentioned when we deal with the lumbered flow routing so generally we will be going for the solution of continuity equation. So, that we can express as the rate of change of storage is equal to inflow minus outflow or input minus output. So, here input or inflow is say as a function of time and then outflow or output is a function of again time. So, that inflow minus outflow that gives the change in storage. So, here S is the storage. So, if you know the inflow then the unknowns. So, are the outflow and storage or Q and S. So, that way other than this continuity equation like the change in storage is equal to inflow minus outflow. So, we should have some second relationship as a storage function and this is needed to relate this storage inflow and outflow. So, that way we will be dealing with the lumbered flow routing. So, specific form of storage functions say like say when we deal with the reservoir then reservoir capacity with respect to the depth variation within the reservoir or with respect to channel how the as a can either linear or non-linear variations with respect to the storage function. So, that we can get. So, specific form of storage function depending upon the nature of the system either a channel or river or a reservoir we have to get. So, that way the lumbered flow routing is mainly based upon this continuity equation. So, lumbered flow routing some of the methods as I mentioned one equation is the continuity equation and then the other relationship which we need to identify how the storage is varying or how the discharge or the outflow is varying. So, some of the important relationships which we can get is like level pulled reservoir routing where storage is a non-linear function of the outflow only. So, like a storage say if we consider say for example, a reservoir storage will be a function of the outflow. So, S is equal to F cube and then say for example, in channel routing say using Muskingum method which is one of the commonly used method. So, here flow routing in channels so the storage we can relate linearly. So, linear relates inflow to the outflow. So, there can be inflow to outflow we can say relate linearly. So, that is say which we generally do in Muskingum method which will be discussing in detail later. Then also say like other type of relationships like a linear reservoir models where storage is a linear function of the outflow and its time derivatives. Then the effect of storage so that way we can relate to or redistribute the hydrograph by shifting the centroid of the inflow hydrograph to the position of that of the outflow hydrograph in time. So, like that so depending upon the conditions or the non-relationships so in the lumped floor routing the main equation is the consideration of mass or continuity equation which gives us change in storage is equal to inflow minus outflow. Then as a second relationship say there are two unknowns one is storage and second one is the outflow. So, we should have two relationship one is the continuity equation. Second is say like the relationship which I mentioned like the level pool reservoir routing where storage is a function of the outflow. Then Muskingum method where storage is linear related with respect to inflow and outflow and then linear reservoir models where the storage is a linear function of outflow and its time derivatives. So, like that we can connect and get another relationship and then we can solve for the outflow or the storage depending upon the condition. So, that way this lumped floor routing we can consider say in two methods. So, first method is case one we consider invariable relationship between the storage and outflow s and q. So, here there is it is invariable relationship. So, like storage versus outflow q versus s we can put like this and based upon that we can get a relationship second relationship. Storage and outflow are functions of water surface elevation when reservoir has a horizontal water surface elevation. So, s can be written as a function of q or combination of these two functions. So, then as we can see the peak flow occurs at intersection of inflow hydrograph and outflow hydrograph. So, say if this is the outflow hydrograph this is the inflow hydrograph. So, here with respect to the peak flow we can identify peak outflow this is peak outflow and we can identify and then this case one we consider the invariable relationship between the storage and the outflow. And the second case is where we can have variable relationship between the storage and outflow. So, this applies to long narrow reservoirs and open channels. So, depending upon the the conditions whether the channel is narrow and the reservoir is also narrow or long. So, there we can apply this variable relationship between the storage and outflow. So, here the outflow is in the y axis storage is this x axis. So, then we can see that this is not considered as a single line as we consider the last slide, but here this will be varying like this. So, there is a variable relationship. So, the water surface profile is curved due to the backwater effects. So, there is supposed to be backwater effects. So, that way the back the water surface profile may be curved one. Peak outflow occurs later than point of intersection. So, you can see that if this is the inflow then the peak flow you can see that that is a later stage than the point of intersection between inflow hydrograph and outflow hydrograph. So, this will be the location of peak outflow. So, replacement of loop with dashed line. So, as you mentioned the outflow and storage it is a loop like this. So, we can consider we can replace this loop with respect to a dashed line when backwater effect is not significant. So, that way also we can consider. So, in the case 2 we consider the variable relationship between storage and the outflow. So, that is about the Lumberder say floor routing. So, now we will be discussing in detail about the reservoir routing and the channel routing with respect to some of the important techniques which we generally use as far as the flood routing or flow routing. So, now let us look into first into the reservoir routing. So, reservoir routing as we discussed it is a procedure for calculating the outflow hydrograph from a reservoir with a horizontal water surface. So, if this is a reservoir. So, this is a procedure for calculating the outflow hydrograph. So, flow of flood waves from rivers streams keeps on changing the head of water in the reservoir and say. So, the head in the reservoir is a function of time. So, since flow is keep on coming in the upstream, so that way this is a function of time. So, we are required to find variations of the storage storage within the reservoir outflow and this depth of flow within the reservoir or depth the flow depth within the reservoir for given inflow with the time. So, the problem in the reservoir routing is that we know the inflow what is coming to the reservoir, then we know the the reservoir characteristics. So, we have to identify what will be the storage within the reservoir and then what is the outflow possible from the reservoir and then say the the depth of flow or the water depth available within the reservoir. So, if you consider a small time interval. So, here again if you can use the continuity equation we can write the inflow into delta t minus outflow q into delta t is equal to change in storage delta s equation number 1. So, if you consider small time interval say say for example, few minutes then we can write inflow into delta t minus outflow into delta t is equal to change in storage delta s. So, if you consider the as far as inflow is concerned if you consider say an average with respect to time. So, that we can write i 1 plus i 2 say if i 1 is the say the inflow at time t 1 and i 2 is the inflow at time t 2. So, if the delta t is t 2 minus t 1 then we can write i 1 plus i 2 divided by 2 into delta t minus similarly the outflow is q 1 at time t 1 and outflow is q 2 at time t 2. So, q 1 plus q 2 divided by 2 into change in time delta t. So, that will be the change in storage s 2 minus s 1. So, this average inflow we consider here in time t and average outflow in time t and then the change in storage in t. So, that way we can rewrite this equation when we go for reservoir routing. So, now when we deal with the reservoir routing say we should have some important data with respect to the various characteristics of the reservoir and then we should also know the inflow pattern coming to the reservoir. So, accordingly only we should be able to say predict the outflow conditions or the storage variations within the reservoir. So, let us look some of the important data cared as far as reservoir routing. So, data like elevation versus storage. So, when the water level in the reservoir raises how much storage is available. So, elevation versus storage then elevation elevation versus outflow discharge and hence storage versus outflow discharge. So, that data also needed then we should know how much inflow is coming coming to the reservoir. So, that inflow hydrograph should be known and then we should also know initial values of inflow, initial values of outflow and then what is the initial storage s at time t is equal to 0. So, if the time starting time is t is equal to 0 and at that time since this is a time dependent at that time what is the inflow, what is the outflow and what is the storage. So, based upon that only we will be going for the prediction as far as the next time step is considered. So, here as I mentioned when we go for reservoir routing the time step delta t must be shorter than the time of transit of the flood wave through the reach which we consider saying you know within the reservoir reach. So, variety of methods say you can see as far as reservoir routing mainly based upon this continuity equation variety of methods we can see in the literature. So, like Wendy Chow et al have given in the test book. So, some of the two important methods which are generally used we will discuss briefly for reservoir routing one is the Pulse methods and second one is the Good Riches method. So, these are two important methods which are generally used as far as reservoir routing is concerned. So, as I mentioned what we are trying to do. So, we know the inflow we want to identify how is the outflow or then storage with respect to time. So, that is what we are trying to solve in reservoir routing. So, now let us look into these two methods Pulse method and the Good Riches methods. So, now in reservoir routing using Pulse methods. So, here now we have earlier seen the continuity equation as given here. So, this equation number 2 we will be rewriting as like this I 1 plus I 2 by 2 you know delta t plus S 1 minus Q 1 you know delta t divided by 2 is equal to S 2 plus Q 2 you know delta t and divided by 2. So, here we can see that if you consider this equation which is obtained from equation 2 which is the continuity equation. Here all terms on the left hand side are known at the starting of the routing. So, all these terms are known. So, what are the unknowns are this S 2 and Q 2. So, right hand side is a function of elevation h for a chosen time interval delta t. So, now we can prepare graphs say which can give same the h that means, the flow depth or the depth of water in the reservoir versus the outflow h versus Q and h versus storage and then we can also get h versus S plus Q into delta t by 2. So, this way we can prepare graphs with respect to various given conditions and then from that we can try to get this the for the given time with respect to the given time we can identify what is the possible storage and then with respect to the given time we can also identify how much is the outflow which can take place. So, this procedure is repeated for full inflow hydrograph. So, we can accordingly we can prepare curves and then based upon that curves we can easily identify for the given inflow what is the possible outflow and then what will be the possible storage. So, through these graphs like h versus Q h versus S and h versus S plus Q into delta t by 2. So, this is so called a pulse method. Then the second method is so called a Goodrich method. So, here again the continuity equation equation number 2 this equation we will be rewriting in another form here in Goodrich method. So, here i 1 plus i 2 plus 2 S 1 by delta t minus Q 1 is equal to 2 S 2 by delta t plus Q 2. So, where i 1 i 2 are the inflows with respect to time t 1 and t 2 and S 1 S 2 are the storage with respect to time t 1 and t 2 and Q 1 Q 2 are the outflow with respect to time t 1 and t 2. So, now we prepare graphs for h versus Q and h versus S and h versus 2 S delta t by plus Q. So, as in the previous step here we prepare graphs h versus Q h versus S and h versus 2 S delta t plus Q. So, flow routing through time interval delta t all times on the left hand side and hence left hand side is known from that we can get for what is there for the right hand side. So, value of outflow Q 4 say with respect to 2 S delta t by Q can be read from the graph. So, we can prepare the graph from that say for this 2 S by delta t plus Q for the value of outflow say we can the outflow can be obtained. So, value of 2 S plus delta t minus Q is calculated by. So, I mean for the next time step this time we calculate by means of say 2 S by delta t plus Q will deduct 2 times Q for next time interval. So, that this again we for the next time interval this is repeated the procedure repeated. So, we will get the left hand side and based upon that we can obtain the right hand side from these graphs. So, repetition of the computations for subsequent routing periods we can continue and then we can identify what will be the storage or the what will be the outflow for the given inflow conditions. So, this is so called a good riches good rich method. So, that way when we deal with the Muskinga when we deal with the the reservoir routing two important methods one is pulse method and second one is a good rich method. So, this is mainly we prepare graphs for various conditions and from that we can identify what will be the outflow conditions or the storage possibilities. So, that is about the reservoir routing. So, there are number of other techniques available in literature, but we say due to lack of time we will not be going through all these techniques. So, here we considered only two techniques with respect to pulse method and the good rich method. So, now, we will discuss the channel routing. So, channel routing is concerned and we will be discussing one Lambert approach and then another one is the distributed approach. So, called a solution of Saint Viennese equations. So, first let us look into the Muskinga method which is again a Lambert approach. So, here as you can see the hydrologic routing method. So, this is a hydrologic routing method for handling variable discharge or discharge and storage relationship. So, that is the essence of the Muskinga method. So, this is a hydrologic routing based upon the continuity equation. So, storage here is a function of both outflow and inflow discharges. So, as we have seen for reservoir routing. So, this is mainly for channel routing. So, here again the outflow is a function of say or storage is a function of outflow and inflow characteristic or inflow discharges. So, water surface in a channel rich is not only parallel to the channel bottom, but also varies with the time. So, based upon these assumptions say model storage in a channel is a combination of wedge and prism. So, as you can see in this figure the bottom one we consider as a prism and then above that there is a wedge. So, this is say with respect to the uniform flow condition is a prism storage and then above this is a wedge storage. So, which will be the floor depth will be keep on changing. So, this is the wedge storage. So, this model storage we consider as wedge and prism storage. So, prism storage is the volume that would exist if uniform flow occurred at the downstream depth as I mentioned here. Another wedge storage wedge is like a volume formed between actual water surface profile and top surface of the prism storage. So, this wedge storage is due to the change in depth between two reaches. So, prism storage is with respect to the uniform flow conditions. So, this is the inflow to the channel. So, here is outflow from the channel and here we consider two section. So, this is the prism storage and this one is the wedge storage. So, now in Muskinga method so, we consider say this continuity equation say with respect to the change in storage. So, during the advance of flood wave inflow exceeds outflow. So, whenever the flood is advancing so, the inflow will be more. So, inflow will be exceeding the outflow. So, we say that the there is a positive wedge. During the recession when flow is say the depth is reducing say with respect to inflow and then during recession outflow exceeds inflow. So, there is a negative wedge. So, and also some of the assumptions like a cross section area of the floods flow section is directly proportional to the discharge at this section. So, based upon this assumption only this Muskinga method is working. So, that way when we consider this assumption we can put volume of prism storage is equal to k into o where here o is the outflow. So, where volume of the wedge storage is equal to k into x into i minus o where i is the inflow o is the outflow and k is the is a proportionality coefficients depending upon the channel characteristics and then x is a weighting factor generally that varies from 0 to 0.5. So, here this is the channel inflow and this is the outflow and say prism storage we can consider as this the proportionality coefficient k multiplied by the outflow k into o and then wedge storage is k into x into i minus o where x is a weighting factor which varies from 0 to 0.5. So, these details and the details are given in this test book by Chau and others of 1988. So, then say we can write the total storage with respect to the wedge storage and prism storage total storage is equal to s is equal to k into xi plus 1 minus x into o. So, this equation is called a Muskingam storage equation. So, now here we consider a linear model for routing flowing the streams. So, that way the value of x depends upon the shape of model wedge storage. So, x is equal to 0 for level pool storage. So, that we have to consider only s is equal to k into o and x is equal to 0.5 for a full wedge. So, that way we can consider say with respect to say the last figure like this. So, now say as I mentioned here k is the time of travel of actually k is the time of travel of flood wave through channel reaches and values of storage at time say if we consider j and j plus 1 we can rewrite this Muskingam equation as follows s j s subscript j is equal to k into x into i j i subscript j plus 1 minus x into o subscript j and s at j plus 1 time we can write k into x i j plus 1 plus 1 minus x into o j plus 1. So, at two time two time intervals j and j plus 1 we can rewrite the Muskingam equation in this portion. So, now the storage change in storage between these two time steps will be the difference between this j plus 1 and j time step. So, s j plus 1 minus s j is equal to k into x into i j plus 1 minus i j plus 1 minus x into o j plus 1 minus o j. So, this is the change in storage when we consider the Muskingam methods. So, now we have considered earlier the continuity equation. So, continuity equation is inflow minus outflow is equal to change in storage. So, that we can write with respect to jth and j plus 1 time step we can write i j plus i j plus 1 divided by 2 into delta t minus o j plus o into o j plus 1 divided by 2 into delta t is equal to s j plus 1 minus s j. So, we can equate this equation and then we can equate with respect to this continuity equation. So, when we equate these two equations we say we finally, get the equation like this. So, now this is our final equation as far as Muskingam charm routing method is concerned. So, this we can simplify as o j plus 1 is equal to c 1 into i j plus 1 plus c 2 i j plus c 3 o j. So, this is the Muskingam routing equation. So, where this c 1 is equal to 0.5 delta t minus k x divided by k into 1 minus x plus 0.5 delta t and then c 2 is equal to 0.5 delta t plus k into x divided by k into 1 minus x plus 0.5 delta t and c 3 is equal to k into 1 minus x minus 0.5 delta t divided by k into 1 minus x plus 0.5 delta t and this c 1 c 2 c 3 coefficients. So, summation of this should be equal to 1 c 1 plus c 2 plus c 3 equal to 1 as explained in the test book of Chow and others. So, that way we can derive this Muskingam routing equation. So, here say when we consider the time step in Muskingam method it should be such a way that delta t should be chosen such a way that k should be greater than t should be greater than 2 times k x for best results. So, through various k studies these are shown and if delta t is less than 2 k x then coefficients c 1 will become negative. So, that way some restrictions of the methodology are there. So, now this Muskingam method is Muskingam routing method is one of the commonly used routing method as far as channels routing or river routing is concerned. So, some of the important data required as far as the Muskingam routing method is concerned like inflow hydrograph through a channel reach that should be known then values of k and x for the reach. So, this k the coefficient k and x we can determine through either some empirical relationships or through trial and error approach especially x. So, that way we can determine. So, this values should be known for the given channel reach. Then value of the outflow oj from the reach at the at the starting time. So, based upon that only we proceed for outflow at j plus 1 time step. So, for a given channel reach k and x are taken as constants and then k is determined empirically. So, as far as as I mentioned this k we can determine empirically like Paris method where it is defined as k is equal to c into L divided by s to the power 0.5 where c is a constant L is the length of stream and a c is mean slip slope of the channel or this also k can be also determined for given inflow outflow condition we can do the plot a graph and from that graph also and we can determine this k. And generally x is determined by trial and error procedure for the given channel reach. So, that way once we know k and x and then systematically with respect to inflow and say at the initial time step the outflow is known for the other time steps. We can find the outflow and the storage using the Muskinga method. So, this method is one of the commonly used method for channel routing and the systematic procedure is there. So, the routing procedure here I have put in this slide. So, knowing k and x this coefficients we can select an appropriate value of delta t. So, the time step we can choose then we can calculate c 1 c 2 and c 3. So, like here this c 1 c 2 c 3 using k x delta t we can identify what will be c 1 c 2 and c 3. Then starting from the initial conditions known inflow outflow we can calculate the outflow for the next time step. So, we can repeat the calculation for the end air inflow hydrograph. So, that way with respect to time we can identify how the outflow hydrograph will be varying. So, that way we can do the flat routing using the Muskinga method. So, you can see that this is also a lumbered approach as far as channel routing is concerned. So, now before closing today's lecture we will discuss the flat routing say flat routing a same entity can be distributed approach or lambed approach. So, we have already seen the lambed approach for server based upon the continuation and then the lambed approach for the channel routing or river routing we have seen the Muskinga methods. So, these are all say we are not concerned the spatial variations. So, these are also called a lambed approach. So, now say in the distributed approach. So, we have to generally solve the governing equations say it is based upon the physics of the problem. So, we have to say obtain the governing equations. So, either one dimensional, two dimensional, three dimensions depending upon the conditions. So, generally say river routing or channel routing is concerned most of the time we rely upon one dimensional modeling. So, we will be having the continuity equation one continuity equation and one momentary equation. So, based upon that say for the given channel conditions non characteristics of the channel or river and the inflow condition we can identify how the flow depth or how the discharge is varying with respect to space and time. So, this is the essence of a flat routing in the distributed approach. So, generally we use so called Saint-Vinand equations regarding Saint-Vinand equations we have discussed earlier in some of the lectures earlier. So, this is as I mentioned this is a physically based theory of flood propagation. So, from the Saint-Vinand equations say for gradually varying. So, this is generally used for gradually varying flow in open channels. So, this is so called hydraulic routing method. So, here you can see that if this is the watershed which we consider. So, what we do? So, we know the inflow coming through this channels and then from various the sub water sheds or from various say drainage systems or the small streams like this what will be the inflow coming to the main channel. This here in this figure this is the main channel. So, this inflow we know at various locations. So, however the main question is what will be the outflow or the flood hydrograph or the discharge versus time at particular location like here at this location or at the outlet of the watershed. So, generally for this kinds of problem we consider flow as one dimension flow and then the assumption is gradually varied flow condition. So, generally say we consider the hydrostatic pressure distribution condition and also we consider the flow is as incompressible like that some of the important assumptions. So, here we consider the consideration of mass, the continuity equation and the consideration of momentum the dynamic wave equation. So, that is so called a Saint-Venier equations. So, these equations we have already discussed earlier. So, the governing equation for flow routing or flood routing is the equation of continuity given as del q by del x plus del A by del t minus q is equal to 0 where q is the discharge at any location and A is the cross sectional of the flow and then t is time x is the distance and small q is the inflow coming from say that from overland say or the lateral inflow coming to the channel. Then the second equation is so called a momentum equation. So, momentum equation is del q by del t plus del A by del x of q square by A is equal to g into A into s 0 minus s f minus g into A into del h by del x. So, h is the flow depth and A is the cross sectional area, g is the accession due to gravity, s 0 is the bed slope of the channel, s f is the energy slope. So, this energy slope which we generally find using the Manning's equation. So, that way we have to say in one dimension we have to solve these two equations. So, that we can identify what is the discharge or what is the flow variation with respect to space and time. So, that way this is a distributed model. So, this dynamic so whatever the equation which we consider here these are so called dynamic wave equations or the full form of the Saint Venant equations. So, there in literature we can see two kinds of approximations one is so called a diffusion wave approximation and second one is so called a kinematic wave approximation. In both of these approximations the continuity equation is same. So, the same continuity equation is used and in diffusion wave approach. So, here the dynamic momentum equation we consider only like this where del h by del x is equal to s 0 minus s f whereas s 0 is the bed slope s f is the energy slope which we can identify by using the Manning's equation. So, to solve this system so the diffusion equation appropriate initial and boundary conditions to be applied and then kinematic wave form is further simplification of this model. The continuity equation is same equation, but here say if the slope is not drastically varying we can write s 0 is equal to s f or energy slope is equal to bed slope. So, we in kinematic wave form we will be solving the continuity equation and this we will be putting the energy slope is equal to the bed slope. So, this is one simplified form. So, that way we can either solve the dynamic wave form or the diffusion wave form or the kinematic wave form depending upon the requirement or depending upon the problem with appropriate initial conditions like what will be the initial flow depth or the discharge and then boundary conditions what will be or the upstream boundary what will be the condition or downstream boundary what will be the condition. So, accordingly with respect to this governing equations initial conditions and the boundary conditions we can solve this system of equations. So, that we can do the flood routing or flood routing. So, that the output will be say we know the discharge or flood depth for the given conditions at any location of the river or the channel. So, as we discussed earlier as far as solution is concerned for this governing equations say in the distributed model or the St. Menon's equation they say generally either we can go for analytical methods or computational method. So, as I mentioned earlier these say analytical methods very difficult to get only for simplified governing equations boundary conditions and geometry we can have some very simple analytical solutions, but that is not applicable for most of the field problem. So, that way these equations either in dynamic wave form or the diffusion wave form or kinematic wave form we have to solve using the numerical techniques. So, here in computational methods solution is obtained with the help of some approximate methods using a computer. So, commonly used numerical techniques like a finite defense method, finite element method or finite volume method are used to obtain the solution in the computation method. So, this we have already discussed earlier. So, we are not going to details. So, say for example, in finite element method if you want to solve this diffusion wave form of the equation we can consider the channel to be say constitutive of number of linear line elements and then we can consider the element properties and then say for example, using Galerkin approach and we can approximate this continuity equation and the final form of the equation by considering the an implicit form for the time variation the final form of the equation can be written like this. So, here in the finite element formulation. So, for the given condition say we can solve this boundary condition we can solve this system of equations to get the unknowns of flow depth or the discharge at the given condition. So, that way we can go for. So, we have developed some flood routing model for as far as channels are concerned with respect to the overland flow and then say for example, tidal boundary conditions on urban watershed basis. So, this kind of flood routing or flood routing the essences the inflow conditions are known from that flow or the beginning of the channel and then we want to identify how the flood or the outflow is taking place. So, output will be times of discharge or the depth of flow. So, before closing today's lecture we will just briefly go through one case study. The case study is a catchment called Calamboli watershed urban watershed in Nami Mumbai near Mumbai. So, this is the watershed. So, there is a main channel through which the drainage taking place. So, the question here is how the flood routing to be done through this channel. So, catchment area is about 8.47 square kilometer elevation varies from 0.5 meter to 227 meter above the MSL. So, here this through this channel the flow is reaching to the creek or the sea. So, here we consider say Lambert model approach as far as the overland flow by considering the continuity equation. So, we consider about 31 sub catchments to get the overland flow coming to these channels at various locations and length of this channel is about 5.271 kilometer and here we used a finite element method as we discussed earlier. So, about 80 channels remains as shown here is taken and here the tidal boundary condition is there at this location which varies from 3.25 meter to minus 1 meter depending upon the spring tidal or nip tidal or the tidal variation, daily tidal variation at this outlet of the channel. So, here we used an indirect approach of remote sensing GIS. So, we prepared the detailed elevation map, then slope map and then land use land cover map as shown in this figure figures. Then say this is a rainfall which we simulated this to identify the flood routing. So, for 26 July 2005 the rainfall was about 670 mm for a duration of 12 hours 675 mm. So, say using the model which we developed. So, we routed this flow. So, at various locations of the channel like node number 18 here, node number 33 here, node number 48, node number 63, node number 80. So, like that the time versus stage of flow is or the hydrograph is shown here. So, from this we can identify how the flooding is taking place. You can see that say for example, for this problem some location here there is some flooding flood depth is shown with respect to the bank elevation. So, then again this for this area the model was run again for another rainfall of 15 July 2009. So, the rainfall pattern is shown here and then the channel the stage variation with respect to the chain age from beginning to the creek ends is shown here. So, you can see that and this is the channel diagonal level. So, there is no flooding and here we did some measurement with respect to the flow depth with respect to time. So, that is we have compared our model results with observed and simulated. So, that way what I want to say here is this flood the flow, flow routing or flood routing is very important in the flood assessment or flood control warning systems for urban watershed or agricultural watershed. So, that way flood routing is very important. So, some of the important references as to this Chau and others applied hydrology which is some of these aspects are used in today's lecture. So, before closing today's lectures some of the questions like a tutorial question study the various flood routing methodologies details and suggest applications of each and then what are the software available for flood routing. So, for example, HEC, HMS HEC RAS all these details are available in this website evaluate the application for various problems such as reservoir routing or the channel routing. So, then a few self evaluation questions what is flood routing and where it is used as plain reservoir routing differentiate between pulse method and good rich method and describe the muskinga method of flood routing and describe the prism storage and wet storage in a channel and what are the input data required for muskinga routing. And then if we assign many questions what are the motivations for flood routing describe the different types and advantages of flood routing illustrate the channel routing procedure and describe the lumped flow routing discuss physically based flood routing in channels by using the same millions equation. All these questions you can answer by going through today's lecture. So, today what we are discussing was about the flood routing or flood routing in channels and reservoirs and say on a watershed basis also. So, flood routing is or flood routing is very important same in the watershed management. Thank you.