Routing
Content
CEL251 Hydrology SURFACE FLOW : Flood and Flood Routing
Introduction A flood is an unusually high stage in a river – normally the level at which the river overflows its banks and inundates the adjoining area. The damages caused by floods in terms of loss of life, property and economic loss due to disruption of economic activity are very high. Flood peak values are required in the design bridges, culvert waterways, spillways for dams, and estimation of scour at a hydraulic structure. At a given location in a stream, flood peaks vary from year to year and their magnitude constitutes a hydrologic series. To estimate the magnitude of a flood peak the following methods are adopted: 1. Rational method, 2. Empirical equations, 3. Flood frequency studies, and 4. Unit hydrograph technique. The use of particular method depends upon (i) the desired objective, (ii) the available data, and (iii) the importance of the project. Rational Method The rational method is found to be suitable for peak flow prediction in small size (< 50 km2) catchments. It finds considerable application in urban drainage designs and in the design of small culverts and bridges. At the start of a rainfall event, the portions nearest the outlet contribute runoff first. As rain continues, farther and farther portions contribute runoff, until flow eventually arrives from all points on the watershed, “concentrating” at the outlet. An isochrone is a line on the catchment joining points having equal time of travel of surface runoff from the point to the A catchment outlet. catchment can have infinite number of isochrones but time of concentration isochrone is the last isochrone on the catchment. For a rainfall of uniform intensity and very long duration over a catchment the runoff increases as more and more flow from remote areas of the catchment reach the outlet. If the rainfall continues beyond the time of concentration (t > tc), the runoff will be constant and at the peak value (Qp) equal to
Q p = CiA
1
where i = rainfall intensity; A = catchment area; C = runoff coefficient = runoff/rainfall. The runoff coefficient represents the integrated effect of the catchment losses and hence depends upon the nature of the surface, surface slope and rainfall intensity. The rational formula assumes a homogeneous catchment surface. If the catchment is non-homogeneous but can be divided into distinct sub areas each having a different C, then the runoff from each sub area is calculated separately and merged in proper time sequence. If the rainfall is uniformly distributed over such a non-homogeneous catchment for t > tc then a weighted equivalent runoff coefficient Ce can be determined and used.
Ce = ∑ C j A j A
Time of concentration is assumed to be independent of rainfall intensity. Watershed parameters that may affect tc are: (1) Length of channel and overland flow plane, (2) average slope of channel or watershed, and (3) retardance or roughness characteristics of the watershed. Empirical Formulae The empirical formulae used for the estimation of the flood peak are essentially regional formulae based on statistical correlation of the observed peak and important catchment and storm properties. To simplify the form of the equation, only a few of the many parameters affecting the flood peak are used. Most of the formulae use the catchment area as a single parameter affecting the flood peak and other factors are clubbed in a region specific constant parameter. For example Dickens formula:
Q p = C D A3 / 4
in which Qp is in m3/s and A is in km2. CD = Dicken’s constant with value between 6 to 30 depending upon the region (catchment type and average rainfall). It is used in the central and northern parts of the country. Ryves formula:
Q p = C R A2 / 3
in which CR = Ryves’s constant with value between 6.8 to 12 depending upon the region (catchment type and average rainfall). It is used in Tamil Nadu, parts of Karnataka and Andhra Pradesh. Inglis formula: Q p = 124 A A + 10.4
It is used in Western Ghats in Maharastra. These rational formulae are applicable only in the region from which they were developed and when applied to other areas they can at best give rough estimates. Envelope curves: In this method the available flood peak data from a large number of catchments which do not significantly differ from each other in terms of meteorological and topographical characteristics are collected. The data are then plotted on a log-log paper as flood peak vs catchment area. This would result in a scattered data plot. If an enveloping curve that would encompass all the plotted data points is drawn, it can be used to obtain maximum peak discharges for any given area. Envelope curves are useful in getting quick rough estimations of peak values. If equations are fitted to envelop curves, they provide empirical flood formulae of the type Qp = f(A).
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Flood Frequency Studies Hydrological processes such as floods are exceedingly complex natural events. They are resultants of a number of component parameters (e.g. floods depend upon the characteristics of the catchment, rainfall and antecedent conditions, each one of these factors in turn depend upon a host constituent parameters) are therefore very difficult to model analytically. An alternate approach to the prediction of flood flows (and other hydrologic processes) is the statistical method of frequency analysis. The values of the annual maximum flood from a given catchment area for large number of successive years constitute a hydrologic data series called the annual series. The data in the series are then arranged in descending order of magnitude and the probability P of each event being equaled to or exceeded (plotting position) is calculated by the plotting position formula
P = m ( N + 1)
where m = order number of the event and N = total number of events in the data series. The recurrence interval or return period or frequency T is given by
T = 1/ P
For small return periods or where limited extrapolation is required, a simple best fitting curve through plotted points can be used as the probability distribution. A logarithmic scale for T is often advantageous. However, when larger extrapolations of T are involved, theoretical probability distributions have to be used. The general equation of hydrologic frequency analysis is due to Chow:
xT = x + Kσ
where xT = value of the variate X of a random hydrologic series with a return period T, x = mean of the variate, σ = standard deviation of the variate, K = frequency factor which depends upon the return period and the assumed frequency distribution. Some of the commonly used frequency distribution functions for the prediction of extreme flood values are: (i) Gumbel’s extreme value distribution, (ii) log-Pearson type III distribution, and (iii) log normal distribution. In frequency analysis of floods, the usual problem is to predict extreme flood events. For this, specific frequency distribution functions are assumed and the required statistical parameters are calculated from the available data. Using these parameters, the flood magnitude for a specific return period is estimated. The results of the frequency analysis depend upon the length of data. The minimum number of years of record required to obtain satisfactory estimates depends upon the variability of data and hence on the physical and climatological characteristics of the basin. Generally a minimum of 30 years of data is considered as essential. Frequency analysis should not be adopted if the length of records is less than 10 years. Flood frequency studies are most reliable in climates that are uniform from year to year. In such cases a relatively short record gives a reliable picture of the frequency distribution. Unit Hydrograph Method and Design Flood The unit hydrograph technique can be used to predict the peak flood hydrograph if the rainfall producing the flood, infiltration characteristics of the catchment and the appropriate unit hydrograph are available. The hydrograph of extreme floods and stages corresponding to flood peaks provide valuable data for purpose of hydrologic design. For design purposes, extreme rainfall situations are used to obtain the design storm, viz the hyetograph of the rainfall excess 3
causing extreme floods. The known unit hydrograph of the catchment is then operated upon the design storm to generate the desired flood hydrograph. Design flood is the flood adopted for the design of a structure. In the design of a hydraulic structure it is not practical from economic point of view to provide for the safety of the structure at the maximum possible flood in the catchment. The type, importance of the structure and economic development of the surrounding area and associated damages in case of failure dictate the design criteria for choosing the flood magnitude of a certain return period. Standard project flood (SPF) is the flood that would result from a severe combination of meteorological and hydrological factors that are reasonably applicable to the region. Extreme rare combinations of factors are excluded. While probable maximum flood (PMF) is the extreme flood that is physically possible in a region as a result of severe most combinations, including rare combinations of meteorological and hydrological factors. The PMF is used in situations where the failure of the structure would result in loss of life and catastrophic damage and as such complete security from potential floods is sought. SPF is often used where the failure of a structure would cause less severe damages. Typically the SPF is about 40 to 60% of the PMF for the same catchment. To estimate the design flood for a project by the use of a unit hydrograph, one needs the design storm. This can be the storm producing PMF or SPF as per the design case.
FLOOD ROUTING The flood hydrograph is in fact a wave. The stage and discharge hydrographs represent the passage of waves of stream depth and discharge respectively. As this wave moves down, the shape of the wave gets modified due to channel storage, resistance, lateral addition or withdrawal of flows etc. When a flood wave passes through a reservoir its peak is attenuated and the time base is enlarged due to effect of storage. The reduction in the peak of the outflow hydrograph due to storage effects is called attenuation. Further the peak of outflow occurs after the peak of the inflow; the time difference between the peaks of inflow and outflow hydrographs is known as lag. Modification in the hydrograph is studied through flood routing. Flood routing is the technique of determining the flood hydrograph at a section of a river by utilizing the data of flood flow at one or more upstream sections. Flood routing is used in (i) flood forcasting, (ii) flood protection, (iii) reservoir design, and (iv) design of spillway and outlet structures. Food routing types: (a) reservoir routing, and (b) channel routing. A variety of routing methods are available and they can be grouped into (1) hydrologic routing, and (2) hydraulic routing. Hydrologic routing methods employ essentially the equation of continuity, on the other hand hydraulic methods use continuity equation along with the equation of motion of unsteady flow (St. Venant equations) hence better than hydrologic methods.
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Basic Equation for hydrologic routing The passage of a flood hydrograph through a reservoir or a channel reach is a gradually varied unsteady flow. If we consider some hydrologic system with input I(t), output O(t), and storage S(t), then the equation of continuity in hydrologic routing methods is the following:
I −O = dS dt
If the inflow hydrograph, I(t) is known, this equation cannot be solved directly to obtain the outflow hydrograph, O(t), because both O and S are unknown. A second relation, the storage function is needed to relate S, I, and Q. The particular form of the storage equation depends on the system; a reservoir or a river reach. Level Pool Reservoir Routing The effect of reservoir storage is to redistribute the hydrograph by shifting the centroid of the inflow hydrograph to the position of that of the outflow hydrograph in time. When a reservoir has a horizontal water surface elevation, the storage function is a function of its water surface elevation or depth in the pool. The outflow is also a function of the water surface elevation, or head on the outlet works. By combining these two functions, S = f(O) we get a single valued storage function (for rivers it becomes a loop: not single valued). For such reservoirs, the peak outflow occurs when the outflow hydrograph intersects the inflow hydrograph. Because maximum storage occurs when
I −O = dS =0 dt
As the horizontal water surface is assumed in the reservoir, the reservoir storage routing is known as Level Pool Routing. The outflow from a reservoir (over a spillway) is a function of the reservoir elevation only. The storage in the reservoir is also a function of the reservoir elevation. Further due to passage of the flood wave through the reservoir the water level in the reservoir changes with time h = h(t) and hence the storage and discharge change with time. It is required to find the variations of S, h, and O with time for given inflow with time. In a small time interval ∆t the difference between the total inflow and outflow in a reach is equal to the change in storage (∆S) in that reach I ∆t − O ∆t = ∆S
5
where I = average inflow in time ∆t, O = average outflow in time ∆t. If suffixes 1 and 2 denote the beginning and end of the time interval ∆t then the above equation becomes
I1 + I 2 O + O2 ∆t − 1 ∆t = S 2 − S1 2 2
The time interval ∆t should be sufficiently short so that the inflow and outflow hydrographs can be assumed to be straight lines in that time interval. Further ∆t must be shorter than the time of transit of the flood wave through the reach. For reservoir routing the following data are known (i) elevation vs Storage (ii) elevation vs outflow discharge and hence storage vs outflow discharge (iii) inflow hydrograph, and (iv) initial values of inflow, outflow O, and storage S at time t = 0. There are a variety of methods available for routing of floods through a reservoir. All of them use the above equation but in various rearranged manners. In Pul’s Method the equation is rearranged as O ∆t O ∆t I1 + I 2 ∆t + S1 − 1 = S 2 + 2 2 2 2 At the starting of flood routing, all the terms on the left hand side are known and hence right hand side at the end of the time step ∆t. Since S = f(h) and O = f(h), the RHS is a function of elevation h for a chosen time interval ∆t. Graphs can be prepared for h vs O, h vs S and h vs (S + O∆t / 2) , which enable one to determine the reservoir elevation and hence the outflow discharge at the end of the time step. The procedure is repeated to cover the full inflow hydrograph. In Goodrich’s method the rearranged equation is
(I1 + I 2 ) + 2S1 − O1 = 2S 2
∆t
+ O2 ∆t
For known S = f(h) and O = f(h), graphs are prepared for h vs O, and h vs S. Since S = f(h) and O = f(h), the RHS (2S / ∆t + O ) is a function of elevation h hence a function of outflow O as well
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for a chosen time interval ∆t. Another graph may be prepared for O vs (2S / ∆t + O ) . In routing the flow through time interval ∆t, all terms on the LHS and hence RHS are known, and so the value of outflow O for (2S / ∆t + O ) can be read from the graph. To set up the data required for the next time interval, the value of (2S / ∆t − O ) is calculated by (2S / ∆t + O ) − 2O . The computations are then repeated for subsequent routing periods. The attenuation and lag of a flood hydrograph at a reservoir are two very important aspects of a reservoir operating under a flood control criteria. By judicious management of the initial reservoir level at the time of arrival of a critical flood, considerable attenuation in the floods can be achieved. The storage capacity of the reservoir and the characteristics of spillways and other outlets control the lag and attenuation of an inflow hydrograph.
Channel Routing In very long channels the entire flood wave also travels a considerable distance resulting in a time redistribution and time of translation as well. Thus, in a river, the redistribution due to storage effects modifies the shape, while the translation changes its position in time. In the reservoir the storage was a unique function of the outflow discharge S = f(O). However in channel the storage is a function of both outflow and inflow discharges and hence a different routing method is needed. The water surface in a channel reach is not only parallel to the channel bottom but also varies with time. The total volume in storage for a channel reach having a flood wave can be considered as prism storage + wedge storage. Prism storage is the volume that would exist if uniform flow occurred at the downstream depth i.e. the volume formed by an imaginary plane parallel to the channel bottom drawn at the outflow section water surface. Wedge storage is the wedge like volume formed between the actual water surface profile and the top surface of the prism storage. At a fixed depth at a downstream section of a river reach the
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prism storage is constant while the wedge storage changes from a positive value at an advancing flood to a negative value during a receding flood. Assuming that the cross sectional area of the flood flow section is directly proportional to the discharge at the section, the volume of prism storage is equal to KO where K is a proportionality coefficient, and the volume of the wedge storage is equal to KX(I - O), where X is a weighing factor having the range 0 < X < 0.5. The total storage is therefore the sum of two components
S = K ( XI + (1 − X )O )
which is known as Muskingum storage equation representing a linear model for routing flow in streams. The value of X depends on the shape of the modeled wedge storage. It is zero for reservoir type storage (zero wedge storage or level pool case S = KO) and 0.5 for a full wedge. In natural streams mean value of X is near 0.2. The parameter K is the time of travel of the flood wave through the channel reaches also known as storage time constant and has the dimensions of time. From the Muskingum storage equation, the values of storage at time j and j+1 can be written as
S j = K (XI j + (1 − X )O j ) and S j +1 = K (XI j +1 + (1 − X )O j +1 )
I-Q
Q
So change in storage over time interval ∆t is
S j +1 − S j = K (X (I j +1 − I j ) + (1 − X )(O j +1 − O j ))
From the continuity equation the change in storage for the same time interval ∆t is
O + O j +1 I j + I j +1 ∆t = S j +1 − S j ∆t − j 2 2 Equating these two equations, O + O j +1 I j + I j +1 ∆t ∆t − j K (X (I j +1 − I j ) + (1 − X )(O j +1 − O j )) = 2 2 Collecting similar terms and simplifying
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O j +1 = C1 I j +1 + C 2 I j + C 3O j
which is the Muskingum’s routing equation for channels where C1 = C2 = C3 = 0.5∆t − KX K (1 − X ) + 0.5∆t 0.5∆t + KX K (1 − X ) + 0.5∆t K (1 − X ) − 0.5∆t K (1 − X ) + 0.5∆t
C1 + C 2 + C 3 = 1 For best results the routing interval ∆t should be so chosen that K > ∆t > 2KX. If ∆t < 2KX, the coefficient C1 will be negative. Generally negative values of coefficients are avoided by choosing appropriate values of ∆t. To use the Muskingum equation to route a given inflow hydrograph through a channel reach, the values of K and X for the reach and the value of the outflow Oj from the reach at the start are needed. The procedure is (i) knowing K and X, select an appropriate value of ∆t (ii) calculate C1, C2, and C3 (iii) starting from the initial conditions known inflow, outflow calculate the outflow for the next time step. (iv) Repeat the calculations for the entire inflow hydrograph.
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Introduction A flood is an unusually high stage in a river – normally the level at which the river overflows its banks and inundates the adjoining area. The damages caused by floods in terms of loss of life, property and economic loss due to disruption of economic activity are very high. Flood peak values are required in the design bridges, culvert waterways, spillways for dams, and estimation of scour at a hydraulic structure. At a given location in a stream, flood peaks vary from year to year and their magnitude constitutes a hydrologic series. To estimate the magnitude of a flood peak the following methods are adopted: 1. Rational method, 2. Empirical equations, 3. Flood frequency studies, and 4. Unit hydrograph technique. The use of particular method depends upon (i) the desired objective, (ii) the available data, and (iii) the importance of the project. Rational Method The rational method is found to be suitable for peak flow prediction in small size (< 50 km2) catchments. It finds considerable application in urban drainage designs and in the design of small culverts and bridges. At the start of a rainfall event, the portions nearest the outlet contribute runoff first. As rain continues, farther and farther portions contribute runoff, until flow eventually arrives from all points on the watershed, “concentrating” at the outlet. An isochrone is a line on the catchment joining points having equal time of travel of surface runoff from the point to the A catchment outlet. catchment can have infinite number of isochrones but time of concentration isochrone is the last isochrone on the catchment. For a rainfall of uniform intensity and very long duration over a catchment the runoff increases as more and more flow from remote areas of the catchment reach the outlet. If the rainfall continues beyond the time of concentration (t > tc), the runoff will be constant and at the peak value (Qp) equal to
Q p = CiA
1
where i = rainfall intensity; A = catchment area; C = runoff coefficient = runoff/rainfall. The runoff coefficient represents the integrated effect of the catchment losses and hence depends upon the nature of the surface, surface slope and rainfall intensity. The rational formula assumes a homogeneous catchment surface. If the catchment is non-homogeneous but can be divided into distinct sub areas each having a different C, then the runoff from each sub area is calculated separately and merged in proper time sequence. If the rainfall is uniformly distributed over such a non-homogeneous catchment for t > tc then a weighted equivalent runoff coefficient Ce can be determined and used.
Ce = ∑ C j A j A
Time of concentration is assumed to be independent of rainfall intensity. Watershed parameters that may affect tc are: (1) Length of channel and overland flow plane, (2) average slope of channel or watershed, and (3) retardance or roughness characteristics of the watershed. Empirical Formulae The empirical formulae used for the estimation of the flood peak are essentially regional formulae based on statistical correlation of the observed peak and important catchment and storm properties. To simplify the form of the equation, only a few of the many parameters affecting the flood peak are used. Most of the formulae use the catchment area as a single parameter affecting the flood peak and other factors are clubbed in a region specific constant parameter. For example Dickens formula:
Q p = C D A3 / 4
in which Qp is in m3/s and A is in km2. CD = Dicken’s constant with value between 6 to 30 depending upon the region (catchment type and average rainfall). It is used in the central and northern parts of the country. Ryves formula:
Q p = C R A2 / 3
in which CR = Ryves’s constant with value between 6.8 to 12 depending upon the region (catchment type and average rainfall). It is used in Tamil Nadu, parts of Karnataka and Andhra Pradesh. Inglis formula: Q p = 124 A A + 10.4
It is used in Western Ghats in Maharastra. These rational formulae are applicable only in the region from which they were developed and when applied to other areas they can at best give rough estimates. Envelope curves: In this method the available flood peak data from a large number of catchments which do not significantly differ from each other in terms of meteorological and topographical characteristics are collected. The data are then plotted on a log-log paper as flood peak vs catchment area. This would result in a scattered data plot. If an enveloping curve that would encompass all the plotted data points is drawn, it can be used to obtain maximum peak discharges for any given area. Envelope curves are useful in getting quick rough estimations of peak values. If equations are fitted to envelop curves, they provide empirical flood formulae of the type Qp = f(A).
2
Flood Frequency Studies Hydrological processes such as floods are exceedingly complex natural events. They are resultants of a number of component parameters (e.g. floods depend upon the characteristics of the catchment, rainfall and antecedent conditions, each one of these factors in turn depend upon a host constituent parameters) are therefore very difficult to model analytically. An alternate approach to the prediction of flood flows (and other hydrologic processes) is the statistical method of frequency analysis. The values of the annual maximum flood from a given catchment area for large number of successive years constitute a hydrologic data series called the annual series. The data in the series are then arranged in descending order of magnitude and the probability P of each event being equaled to or exceeded (plotting position) is calculated by the plotting position formula
P = m ( N + 1)
where m = order number of the event and N = total number of events in the data series. The recurrence interval or return period or frequency T is given by
T = 1/ P
For small return periods or where limited extrapolation is required, a simple best fitting curve through plotted points can be used as the probability distribution. A logarithmic scale for T is often advantageous. However, when larger extrapolations of T are involved, theoretical probability distributions have to be used. The general equation of hydrologic frequency analysis is due to Chow:
xT = x + Kσ
where xT = value of the variate X of a random hydrologic series with a return period T, x = mean of the variate, σ = standard deviation of the variate, K = frequency factor which depends upon the return period and the assumed frequency distribution. Some of the commonly used frequency distribution functions for the prediction of extreme flood values are: (i) Gumbel’s extreme value distribution, (ii) log-Pearson type III distribution, and (iii) log normal distribution. In frequency analysis of floods, the usual problem is to predict extreme flood events. For this, specific frequency distribution functions are assumed and the required statistical parameters are calculated from the available data. Using these parameters, the flood magnitude for a specific return period is estimated. The results of the frequency analysis depend upon the length of data. The minimum number of years of record required to obtain satisfactory estimates depends upon the variability of data and hence on the physical and climatological characteristics of the basin. Generally a minimum of 30 years of data is considered as essential. Frequency analysis should not be adopted if the length of records is less than 10 years. Flood frequency studies are most reliable in climates that are uniform from year to year. In such cases a relatively short record gives a reliable picture of the frequency distribution. Unit Hydrograph Method and Design Flood The unit hydrograph technique can be used to predict the peak flood hydrograph if the rainfall producing the flood, infiltration characteristics of the catchment and the appropriate unit hydrograph are available. The hydrograph of extreme floods and stages corresponding to flood peaks provide valuable data for purpose of hydrologic design. For design purposes, extreme rainfall situations are used to obtain the design storm, viz the hyetograph of the rainfall excess 3
causing extreme floods. The known unit hydrograph of the catchment is then operated upon the design storm to generate the desired flood hydrograph. Design flood is the flood adopted for the design of a structure. In the design of a hydraulic structure it is not practical from economic point of view to provide for the safety of the structure at the maximum possible flood in the catchment. The type, importance of the structure and economic development of the surrounding area and associated damages in case of failure dictate the design criteria for choosing the flood magnitude of a certain return period. Standard project flood (SPF) is the flood that would result from a severe combination of meteorological and hydrological factors that are reasonably applicable to the region. Extreme rare combinations of factors are excluded. While probable maximum flood (PMF) is the extreme flood that is physically possible in a region as a result of severe most combinations, including rare combinations of meteorological and hydrological factors. The PMF is used in situations where the failure of the structure would result in loss of life and catastrophic damage and as such complete security from potential floods is sought. SPF is often used where the failure of a structure would cause less severe damages. Typically the SPF is about 40 to 60% of the PMF for the same catchment. To estimate the design flood for a project by the use of a unit hydrograph, one needs the design storm. This can be the storm producing PMF or SPF as per the design case.
FLOOD ROUTING The flood hydrograph is in fact a wave. The stage and discharge hydrographs represent the passage of waves of stream depth and discharge respectively. As this wave moves down, the shape of the wave gets modified due to channel storage, resistance, lateral addition or withdrawal of flows etc. When a flood wave passes through a reservoir its peak is attenuated and the time base is enlarged due to effect of storage. The reduction in the peak of the outflow hydrograph due to storage effects is called attenuation. Further the peak of outflow occurs after the peak of the inflow; the time difference between the peaks of inflow and outflow hydrographs is known as lag. Modification in the hydrograph is studied through flood routing. Flood routing is the technique of determining the flood hydrograph at a section of a river by utilizing the data of flood flow at one or more upstream sections. Flood routing is used in (i) flood forcasting, (ii) flood protection, (iii) reservoir design, and (iv) design of spillway and outlet structures. Food routing types: (a) reservoir routing, and (b) channel routing. A variety of routing methods are available and they can be grouped into (1) hydrologic routing, and (2) hydraulic routing. Hydrologic routing methods employ essentially the equation of continuity, on the other hand hydraulic methods use continuity equation along with the equation of motion of unsteady flow (St. Venant equations) hence better than hydrologic methods.
4
Basic Equation for hydrologic routing The passage of a flood hydrograph through a reservoir or a channel reach is a gradually varied unsteady flow. If we consider some hydrologic system with input I(t), output O(t), and storage S(t), then the equation of continuity in hydrologic routing methods is the following:
I −O = dS dt
If the inflow hydrograph, I(t) is known, this equation cannot be solved directly to obtain the outflow hydrograph, O(t), because both O and S are unknown. A second relation, the storage function is needed to relate S, I, and Q. The particular form of the storage equation depends on the system; a reservoir or a river reach. Level Pool Reservoir Routing The effect of reservoir storage is to redistribute the hydrograph by shifting the centroid of the inflow hydrograph to the position of that of the outflow hydrograph in time. When a reservoir has a horizontal water surface elevation, the storage function is a function of its water surface elevation or depth in the pool. The outflow is also a function of the water surface elevation, or head on the outlet works. By combining these two functions, S = f(O) we get a single valued storage function (for rivers it becomes a loop: not single valued). For such reservoirs, the peak outflow occurs when the outflow hydrograph intersects the inflow hydrograph. Because maximum storage occurs when
I −O = dS =0 dt
As the horizontal water surface is assumed in the reservoir, the reservoir storage routing is known as Level Pool Routing. The outflow from a reservoir (over a spillway) is a function of the reservoir elevation only. The storage in the reservoir is also a function of the reservoir elevation. Further due to passage of the flood wave through the reservoir the water level in the reservoir changes with time h = h(t) and hence the storage and discharge change with time. It is required to find the variations of S, h, and O with time for given inflow with time. In a small time interval ∆t the difference between the total inflow and outflow in a reach is equal to the change in storage (∆S) in that reach I ∆t − O ∆t = ∆S
5
where I = average inflow in time ∆t, O = average outflow in time ∆t. If suffixes 1 and 2 denote the beginning and end of the time interval ∆t then the above equation becomes
I1 + I 2 O + O2 ∆t − 1 ∆t = S 2 − S1 2 2
The time interval ∆t should be sufficiently short so that the inflow and outflow hydrographs can be assumed to be straight lines in that time interval. Further ∆t must be shorter than the time of transit of the flood wave through the reach. For reservoir routing the following data are known (i) elevation vs Storage (ii) elevation vs outflow discharge and hence storage vs outflow discharge (iii) inflow hydrograph, and (iv) initial values of inflow, outflow O, and storage S at time t = 0. There are a variety of methods available for routing of floods through a reservoir. All of them use the above equation but in various rearranged manners. In Pul’s Method the equation is rearranged as O ∆t O ∆t I1 + I 2 ∆t + S1 − 1 = S 2 + 2 2 2 2 At the starting of flood routing, all the terms on the left hand side are known and hence right hand side at the end of the time step ∆t. Since S = f(h) and O = f(h), the RHS is a function of elevation h for a chosen time interval ∆t. Graphs can be prepared for h vs O, h vs S and h vs (S + O∆t / 2) , which enable one to determine the reservoir elevation and hence the outflow discharge at the end of the time step. The procedure is repeated to cover the full inflow hydrograph. In Goodrich’s method the rearranged equation is
(I1 + I 2 ) + 2S1 − O1 = 2S 2
∆t
+ O2 ∆t
For known S = f(h) and O = f(h), graphs are prepared for h vs O, and h vs S. Since S = f(h) and O = f(h), the RHS (2S / ∆t + O ) is a function of elevation h hence a function of outflow O as well
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for a chosen time interval ∆t. Another graph may be prepared for O vs (2S / ∆t + O ) . In routing the flow through time interval ∆t, all terms on the LHS and hence RHS are known, and so the value of outflow O for (2S / ∆t + O ) can be read from the graph. To set up the data required for the next time interval, the value of (2S / ∆t − O ) is calculated by (2S / ∆t + O ) − 2O . The computations are then repeated for subsequent routing periods. The attenuation and lag of a flood hydrograph at a reservoir are two very important aspects of a reservoir operating under a flood control criteria. By judicious management of the initial reservoir level at the time of arrival of a critical flood, considerable attenuation in the floods can be achieved. The storage capacity of the reservoir and the characteristics of spillways and other outlets control the lag and attenuation of an inflow hydrograph.
Channel Routing In very long channels the entire flood wave also travels a considerable distance resulting in a time redistribution and time of translation as well. Thus, in a river, the redistribution due to storage effects modifies the shape, while the translation changes its position in time. In the reservoir the storage was a unique function of the outflow discharge S = f(O). However in channel the storage is a function of both outflow and inflow discharges and hence a different routing method is needed. The water surface in a channel reach is not only parallel to the channel bottom but also varies with time. The total volume in storage for a channel reach having a flood wave can be considered as prism storage + wedge storage. Prism storage is the volume that would exist if uniform flow occurred at the downstream depth i.e. the volume formed by an imaginary plane parallel to the channel bottom drawn at the outflow section water surface. Wedge storage is the wedge like volume formed between the actual water surface profile and the top surface of the prism storage. At a fixed depth at a downstream section of a river reach the
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prism storage is constant while the wedge storage changes from a positive value at an advancing flood to a negative value during a receding flood. Assuming that the cross sectional area of the flood flow section is directly proportional to the discharge at the section, the volume of prism storage is equal to KO where K is a proportionality coefficient, and the volume of the wedge storage is equal to KX(I - O), where X is a weighing factor having the range 0 < X < 0.5. The total storage is therefore the sum of two components
S = K ( XI + (1 − X )O )
which is known as Muskingum storage equation representing a linear model for routing flow in streams. The value of X depends on the shape of the modeled wedge storage. It is zero for reservoir type storage (zero wedge storage or level pool case S = KO) and 0.5 for a full wedge. In natural streams mean value of X is near 0.2. The parameter K is the time of travel of the flood wave through the channel reaches also known as storage time constant and has the dimensions of time. From the Muskingum storage equation, the values of storage at time j and j+1 can be written as
S j = K (XI j + (1 − X )O j ) and S j +1 = K (XI j +1 + (1 − X )O j +1 )
I-Q
Q
So change in storage over time interval ∆t is
S j +1 − S j = K (X (I j +1 − I j ) + (1 − X )(O j +1 − O j ))
From the continuity equation the change in storage for the same time interval ∆t is
O + O j +1 I j + I j +1 ∆t = S j +1 − S j ∆t − j 2 2 Equating these two equations, O + O j +1 I j + I j +1 ∆t ∆t − j K (X (I j +1 − I j ) + (1 − X )(O j +1 − O j )) = 2 2 Collecting similar terms and simplifying
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O j +1 = C1 I j +1 + C 2 I j + C 3O j
which is the Muskingum’s routing equation for channels where C1 = C2 = C3 = 0.5∆t − KX K (1 − X ) + 0.5∆t 0.5∆t + KX K (1 − X ) + 0.5∆t K (1 − X ) − 0.5∆t K (1 − X ) + 0.5∆t
C1 + C 2 + C 3 = 1 For best results the routing interval ∆t should be so chosen that K > ∆t > 2KX. If ∆t < 2KX, the coefficient C1 will be negative. Generally negative values of coefficients are avoided by choosing appropriate values of ∆t. To use the Muskingum equation to route a given inflow hydrograph through a channel reach, the values of K and X for the reach and the value of the outflow Oj from the reach at the start are needed. The procedure is (i) knowing K and X, select an appropriate value of ∆t (ii) calculate C1, C2, and C3 (iii) starting from the initial conditions known inflow, outflow calculate the outflow for the next time step. (iv) Repeat the calculations for the entire inflow hydrograph.
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