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8.1 STORAGE ROUTING
Reservoirs In many applications of engineering hydrology, it is necessary to calculate the variation of flows in time and space. These applications include reservoir design, design of flood control structures, flood forecasting, and water resources planning and analysis. A reservoir is a natural or artificial feature which stores incoming water and releases it at regulated rates. Surfacewater reservoirs should be distinguished from groundwater reservoirs; the latter store groundwater. Surfacewater reservoirs store water for diverse uses, including hydropower generation, municipal and industrial water supply, flood control, irrigation, navigation, fish and wildlife management, water quality, and recreation. This chapter deals with reservoir routing in surfacewater reservoirs. Reservoir routing uses mathematical relations to calculate outflow from a reservoir once the inflow, initial conditions, reservoir characteristics, and operational rules are known. The classical approach to reservoir routing is similar to that of the storage concept described in Chapter 4. Reservoir routing techniques based on the storage concept are referred to as hydrologic routing methods, or storage routing methods, to distinguish them from the more complex hydraulic routing method. The latter uses principles of mass and momentum conservation to obtain detailed solutions for discharges and stages throughout the reservoir [2]. In practice, however, nearly all applications of reservoir routing have used the storage concept. Reservoirs can be of widely differing sizes. They can range from small detention ponds designed to diffuse flood flows from developed urban sites, to very large reservoirs encompassing substantial segments of large rivers (Fig. 81). For a single reservoir, inflow is dependent on the upstream flows, whether the latter have been modified by human action or not. Outflow, however, may be of the following types: (1) uncontrolled, (2) controlled, or (3) a combination of both. Uncontrolled outflow is not subject to operator intervention; for example, the case of an ungated overflow spillway. On the other hand, controlled outflow is subject to operator intervention, as in the case of a gated outlet pipe or spillway. In certain instances, reservoirs are outfitted with a combination of controlled and uncontrolled outflow devices or structures.
Detention ponds and small floodretention reservoirs are typical examples of uncontrolled outflow. In these cases, an ungated overflow spillway (or, alternatively, a gated spillway that is kept fully open during the flood season) serves as the outflow structure. From a hydraulic standpoint (Chapter 4), outflow from this type of reservoir is solely a function of reservoir stage (pool level, or water surface elevation). There are two types of reservoir routing with uncontrolled outflow: (1) simulated, and (2) actual. Simulated reservoir routing uses mathematical relations to mimic natural diffusion processes in the computational model (or hydrologic software). A typical example of simulated reservoir routing is the linear reservoir method, which is extensively used in catchment routing (Chapter 10). Actual reservoir routing refers to the routing through a planned or existing reservoir, either for design or operational purposes. In this case, the outflow characteristics are determined by the geometric properties of the reservoir and the hydraulic properties of the outflow structure(s). The most widely used method of actual reservoir routing with uncontrolled outflow is the storageindication method. In a reservoir with controlled outflow, gates are used for the purpose of regulating flow through the outlet structure(s) (Fig. 82). The gates are operated following established operational rules. These rules determine the relation between inflow, outflow, and reservoir storage volume, taking into account the daily, monthly, or seasonal downstream water demands. The latter may include a minimum instream flow requirement for water quality or fisheries management. Many large reservoirs operate with controlled outflow conditions.
In certain cases, outflow may be a combination of controlled and uncontrolled typesfor instance, when the reservoir features a combined regulated outflow and emergency spillway designed to operate only above a certain pool level. Flow through an emergency overflow spillway is usually of the uncontrolled type, the outflow being determined solely by the hydraulic properties of the spillway, without the need for operator intervention. Storage Routing The storage concept is well established in flowrouting theory and practice. Storage routing is used not only in reservoir routing but also in stream channel and catchment routing (Chapters 9 and 10). Techniques for storage routing are invariably based on the differential equation of water storage. This equation is founded on the principle of mass conservation, which states that the change in flow per unit length in a control volume is balanced by the change in flow area per unit time. In partial differential form:
in which Q = flow rate, A = flow area, x = space (length), and t = time. The differential equation of storage is obtained by lumping spatial variations. For this purpose, Eq. 81 is expressed in finite increments:
With ΔQ = O  I, in which O = outflow and I = inflow; and ΔS = ΔA Δx , in which ΔS = change in storage volume, Eq. 82 reduces to:
in which inflow, outflow, and rate of change of storage are expressed in L^{3}T^{1} units. Furthermore, Eq. 83 can be expressed in differential form, leading to the differential equation of storage:
Equation 84 implies that any difference between inflow and outflow is balanced by a change of storage in time (Fig. 83). In a typical reservoir routing application, the inflow hydrograph (upstream boundary condition), initial outflow and storage (initial conditions), and reservoir physical and operational characteristics are known. Thus, the objective is to calculate the outflow hydrograph for the given initial condition, upstream boundary condition, reservoir characteristics, and operational rules.
StorageOutflow Relations Unlike in an ideal channel for which storage is a function of both inflow and outflow, in an ideal reservoir storage is a function only of outflow (Section 4.2). The relationship between storage and outflow can be expressed in the following general form:
A common relationship between outflow and storage is the following power function:
in which K = storage coefficient and n = exponent. For n = 1, Eq. 86 reduces to the linear form:
in which K is a proportionality constant or linear storage coefficient, which has the units of time (T ). Real reservoirs usually have a nonlinear storageoutflow relationship; therefore, Eq. 86 is applicable to planned or existing reservoirs. Exceptions are the cases where the storageoutflow relation is indeed linear, as in the case of the proportional weir. The latter is used in connection with irrigation diversions or measurement of small sanitary flows. Simulated reservoirs are usually of the linear type (Eq. 87), although nonlinear reservoirs have also been used in simulation. The use of several linear reservoirs in series leads to a cascade of linear reservoirs, a mathematical procedure that is useful in routing, particularly in catchment routing (Chapter 10). For linear reservoirs, the constant K is the linear storage coefficient. Increasing the value of K increases the amount of storage simulated by the system. In other words, greater values of K result in increased outflow hydrograph diffusion. For routing in actual reservoirs, the nonlinear properties of the storageoutflow relation must be determined in advance. Outflow from an actual reservoir will depend on whether the flow is discharged through either closed conduit(s), overflow spillway(s), or a combination of the two. A general hydraulic outflow formula is the following:
in which
Theoretical values of discharge coefficient C_{d} and rating exponent y are determined using hydraulic principles. For the freeoutlet closed conduit, the conservation of energy between reservoir pool and outlet elevations (neglecting entrance and friction losses) leads to:
in which V = mean velocity, and g = gravitational acceleration. Therefore, the outflow is:
Comparing Eq. 810 with Eq. 88, it follows that y = 1/2, with C_{d} = 4.43 in SI units and C_{d} = 8.02 in U.S. customary units. In practice, these theoretical values of discharge coefficient are reduced by about 30 percent to account for flow contraction and entrance and friction losses. For an ungated overflow spillway, the critical flow condition in the vicinity of the crest leads to:
which reduces to:
Comparing Eq. 812 with Eq. 88, it follows that y = 3/2. Furthermore, the discharge coefficient in SI units, with g = 9.81 m/s^{2}, is: C_{d} = 1.70. In U.S. customary units, with g = 32.17 ft/s^{2}: C_{d} = 3.09. In practice, the discharge coefficient of an overflow spillway varies with hydraulic head, depending on the shape of the spillway crest; see, for example, Fig. 84.
In the proportional or Sutro^{®} weir, the crosssectional flow area, above the rectangular section, grows in proportion to the halfpower of the hydraulic head Therefore, outflow is linearly related to hydraulic head and a spillway rating based on Eq. 87 is applicable.
8.2 LINEAR RESERVOIRS
Equation 84 can be solved by analytical or numerical means. The numerical approach is usually preferred because it can account for an arbitrary inflow hydrograph. The solution is accomplished by discretizing Eq. 84 on the xt plane, a graph showing the values of a certain variable in discrete points in time and space (Fig. 86). Figure 86 shows two consecutive time levels, 1 and 2, separated between them an interval Δt, and two spatial locations depicting inflow and outflow, with the reservoir located between them. The discretization of Eq. 84 on the xt plane leads to:
in which I_{1} = inflow at time level 1; I_{2} = iriflow at time level 2; O_{1} = outflow at time level 1; I_{2} = outflow at time level 2; S_{1} = storage at time level 1; S_{2} = storage at time level 2; and Δt = time interval. Equation 813 states that between two time levels 1 and 2 separated by a time interval Δt, average inflow minus average outflow is equal to change in storage.
For linear reservoirs, Eq. 87 is the relation between storage and outflow. Therefore:
and
in which K is the storage constant. Substituting Eqs. 814 into 813, and solving for O_{2}:
in which C_{0}, C_{1} and C_{2} are routing coefficients defined as follows:
Since C_{0} + C_{1} + C_{2} = 1, the routing coefficients are interpreted as weighting coefficients. These routing coefficients are a function of Δt /K, the ratio of time interval to storage constant. Values of the routing coefficients as a function of Δt /K are given in Table 81. The linear reservoir routing procedure is illustrated by Example 81.
The reservoir exerts a diffusive action on the flow, with the net result that peak flow is attenuated and time base is increased. In the linear reservoir case, the amount of attenuation is a function of Δt /K. The smaller this ratio, the greater the amount of attenuation exerted by the reservoir. Conversely, large values of Δt /K cause less attenuation. Values of Δt /K greater than 2 can lead to negative attenuation (see Table 81). This amounts to amplification; therefore, values of Δt /K greater than 2 are not used in reservoir routing.
In effect, the celerity of short surface waves waves is [4]:
in which u = mean velocity, and h = flow depth. Dividing Eq. 819 by u, and considering only the positive dimensionless celerity c' :
in which the Froude number F = u /(g h)^{1/2}.
In the case of a reservoir, the water surface slope S_{w} ≅ 0,
the mean velocity 8.3 STORAGE INDICATION
The storage indication method is also known as the modified Puls method [1]. It is used to route streamflows through actual reservoirs, for which the relationship between outflow and storage is usually of a nonlinear nature. The method is based on the differential equation of storage, Eq. 84. The discretization of this equation on the xt plane (Fig. 86) leads to Eq. 813. In the storage indication method, Eq. 813 is transformed to its equivalent form:
in which the unknown values (S_{2} and O_{2}) are on the left side of the equation and the known values (inflows, initial outflow and storage) are on the right side. The left side of Eq. 821 is known as the storage indication quantity. In the storage indication method, it is first necessary to assemble geometric and hydraulic reservoir data in suitable form. For this purpose, the following curves are prepared (Fig. 88):
For computer applications, these curves are replaced by digitized tables.
The elevationstorage relation is determined based on topographic information. The minimum elevation is that for which storage is zero, and the maximum elevation is the minimum elevation of the dam crest. The elevationoutflow relation is determined based on the hydraulic properties of the outlet works, either closed conduit, overflow spillway, or a combination of the two. In the typical application, the reservoir pool elevation provides a head over the outlet or spillway crest, and the outflow can be calculated using an equation such as Eq. 88. When routing floods through emergency spillways, storage is alternatively expressed in terms of surcharge storage, i.e., the storage above a certain level, usually the emergency spillway crest elevation (Fig. 89).
Elevationstorage and elevationoutflow relations lead to the storageoutflow relation. In turn, the storageoutflow relation is used to develop the storage indicationoutflow relation (Fig. 88). The storage indication variable is the lefthand side of Eq. 821. In general, the storage indication quantity is [(2S/Δt) + O], with S = storage, O = outflow, and Δt = time interval. To develop the storage indicationoutflow relation, it is first necessary to select a time interval such that the resulting linearization of the inflow hydrograph remains a close approximation of the actual nonlinear shape of the hydrograph. For smoothly rising hydrographs, a minimum value of t_{p}/Δt = 5 is recommended, in which t_{p} is the timetopeak of the inflow hydrograph. In practice, a computeraided calculation would normally use a much greater ratio. Once the data has been prepared, Eq. 821 is used to perform the reservoir routing.
The computational procedure is illustrated in Example 82 using the same data as in Example 81. The results of Example 82 confirm that the storage indication method is applicable to linear reservoir data. Example 83 illustrates the application of the storage indication method to an actual reservoir featuring a nonlinear storageoutflow relation.
8.4 CONTROLLED OUTFLOW
Most large reservoirs have some type of outflow control, wherein the amount of outflow is regulated by gated spillways. In this case, the prescribed outflow is determined by both hydraulic conditions and operational rules. Operational rules take into account the various uses of water. For instance, a multipurpose reservoir may be designed for hydropower generation, flood control, irrigation, and navigation. For hydropower generation, the reservoir pool level is kept within a narrow range, usually close to the optimum operating level of the installation. On the other hand, floodcontrol operation may require that a certain storage volume be kept empty during the flood season in order to receive and attenuate the incomIng floods. Floodcontrol operations also require that the reservoir releases be kept below a certain maximum, usually taken as the flow corresponding to bankfull stage. Irrigation requirements may vary from month to month depending on the consumptive needs and crop patterns. For navigation purposes, outflow should be a nearly constant value that will ensure a minimum draft downstream of the reservoir. Reservoir operational rules are designed to take into account the various water demands. These are often conflicting and, therefore, compromises must be reached. Multipurpose reservoirs allocate reservoir volumes to the different uses. In this way, operational rules may be developed to take into account the requirements of each use (Fig. 811). In general, outflow from a reservoir with gated outlets is determined by prescribed operational policies. In tum, the latter are based on the current level of storage, incoming flow, and downstream flow requirements.
The differential equation of storage can be used to route flows through reservoirs with controlled outflow (Fig. 812). In general, the outflow can be either: (1) uncontrolled (ungated), (2) controlled (gated), or (3) a combination of controlled and uncontrolled. The discretized equation, including controlled outflow, is:
in which Ō_{r} is the mean regulated outflow during the time interval Δt. Equation 823 can be expressed in storage indication form:
With Ō_{r} known, the solution proceeds in the same way as with the uncontrolled outflow case. In the case where all the outflow is controlled, Eq. 823 reduces to:
Furthermore, Eq. 827 is expressed as follows:
by which the storage volume can be updated based on average inflows and mean regulated outflow. Other requirements, such as estimates of reservoir evaporation where warranted (i.e., in semiarid and arid regions) may be implemented to properly account for the storage volumes.
Rating of Gated Spillways A typical rating of a gated spillway is shown in Fig. 813 [5]. Outflow discharge (abscissas) is a function of reservoir water surface elevation (ordinates) and gate opening. Each curve represents a different gate opening. Also shown is the spillway rating when all gates are fully open.
8.5 DETENTION BASINS
As rural areas become urbanized, storm runoff increases in both peak and volume. The paving of formerly rural lands effectively decreases hydrologic abstractions, resulting in marked increases in storm runoff volume. To compound the problem, paving decreases friction and accelerates runoff concentration, shortening the time of concentration and increasing peak flows (Section 2.4). An accumulation of many of these changes in shortterm hydrologic response at the local level may affect the magnitude and frequency of floods at downstream sites. Local governments are enacting regulations to control and manage changes in shortterm hydrologic response which may be attributed to land development. These changes are often referred to as hydromodification. A typical control strategy requires that postdevelopment peak flows do not exceed predevelopment peak flows, for one or more storm frequencies at specified sites. This is accomplished by storing the storm water to decrease the calculated postdevelopment peak flow (prior to attenuation) to a level dictated by local regulation, usually the predevelopment peak flow. A detention basin (Fig. 814) is a small reservoir, built typically in an urban setting, designed to hold and diffuse storm runoff to mitigate and reduce regional downstream floods and channel erosion. As societies learn to recognize the role of anthropogenic activities on floods, attention is increasingly being paid to flood detention and retention as an effective flood control strategy.
The detention basin is a widely used method for controlling peak discharge in urban areas. It is generally the least expensive and most reliable of the measures that are usually considered for controlling storm runoff [6]. It can be designed to fit a wide variety of sites and can accomodate multipleoutlet spillways to meet specific requirements for multifrequency control of outflow. The design of a detention basin, like that of any reservoir, calls for routing of the inflow hydrograph through the structure to determine the required storage volume and the dimensions of the outlet structures. Proprietary (commercial) and nonproprietary (government) software packages are available for routing floods through detention basins, either as standalone structures, or as part of a network of detention basins, channels, and other structural flood control measures.
TR55 Storage Volume for Detention Basins The USDA Natural Resources Conservation Service (NRCS) has developed a method to estimate storage volume for detention basins. The method, referred to as TR55 detention basin, is recommended for preliminary design, in lieu of more elaborate routing techniques. The method provides an expedient way to estimate the effects of temporary detention on peak discharges. It may be adequate for final design of small detention basins. The following are defined:
The storm runoff volume V_{r} is obtained by multiplying the storm runoff depth times the catchment area. The peak inflow discharge Q_{i} is taken as the postdevelopment peak flow, prior to attenuation with the detention basin. The peak inflow discharge is calculated with the TR55 graphical or tabular methods (Chapter 5) [6]. The peak outflow discharge Q_{o} is normally taken as the predevelopment peak flow. Figure 815 is used to estimate V_{s} when V_{r}, Q_{i} and Q_{o} are known. Alternatively, this figure can be used to estimate Q_{o} when V_{s}, V_{r}, and Q_{i} are known.
The TR55 detention basin method is based on average storage and routing effects for many structures. The curves shown in Fig. 815 depend on the relationship between available storage, outflow device, inflow volume, and shape of the inflow hydrograph. When the required storage volume (V_{s}) is small, the shape of the outflow hydrograph is sensitive to the rateofrise of the inflow hydrograph. In this case, parameters such as rainfall volume, curve number, and time of concentration become especially significant. Conversely, when the required storage volume is large, the shape of the outflow hydrograph is little affected by the rateofrise of the outflow hydrograph. In this case, the outflow hydrograph is controlled by the hydraulics of the outflow device, and the procedure yields more consistent results [6]. The procedure is recommended for final design if an error in storage of 25 percent may be tolerated. The method may significantly overestimate the required storage capacity, because it is biased to prevent undersizing of outflow devices. Detailed hydrograph analysis and reservoir routing (Section 8.3) will generally result in reduced project costs.
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