CIVE 430 Mass Curves

If historical flows could be considered to be representative of all possible future variations that some project will experience during its lifetime, there would be little need for synthetic hydrology. The historical record is seldom adequate for predicting future events with certainty. The exact historical pattern is unlikely to recur, sequences of dry years (or wet years) may not have been as severe as they may become, and the single historical record gives the planner limited knowledge of the magnitude of risks involved. Hydrologic synthesis techniques are classified as (1) historical repetition methods, such as mass curve analyses, which assume that historical records will repeat themselves in as many end-to-end repetitions as required to bracket the planning period; (2) random generation techniques, such as Monte Carlo techniques, which assume that the historical records are a number of random, independent events, any of which could occur within a defined probability distribution; and (3) persistence methods, such as Markov generation techniques, which assume that flows in sequence are dependent and that the next flow in sequence is influenced by some subset of the previous flows. Historical repetition or random generation techniques are normally applied only to annual or seasonal flows. Successive flows for shorter time intervals are usually correlated, necessitating analysis by the Markov generation method. As with most subfields of hydrology, a number of computer programs for timeseries analysis and hydrologic data synthesis have been developed. One of the first, and one of the most widely applied, was the U.S. Army Corps of Engineers' model HEC-4 (see Section 12.1), published in 1971 [50]. Its use is limited, though, to synthesizing sequences of serially dependent monthly streamflow,s in a river reach. Other codes [56],[57] are available to the hydrologist. Additional models and descriptions of theory and applications of time-series analysis of precipitation and streamflow are detailed in a number of available texts and publications [3],[25],[58]-[61].

TABLE 12.19 Streamflows for Example 12.2 Flows (thousands of acre-ft) 1 14 14

Year Inflow Cumulative inflow

2 10 24

3 6 30

5 12 50

4 8 38

7 10 74

6 14 64

8 6 80

9 8 88

10 12 100

Solution. A 10-year sequence of synthetic flows, using Rippl's assumptions, is shown in Table 12.19. Inflows are set equal to the historical record repeated twice. When cumulative inflow and cumulative draft are plotted, the maximum deficiency shown in Fig. 12.19 is 4,000 acre-ft. Thus a reservoir with a 4,000-acre-ft capacity should be placed in the stream. Starting with a full reservoir at the beginning of year 1, the reader should verify the adequacy of the reservoir by "simulating" a draft of 9,000 acre-ft per year for 10 years. 100,000

Beginning of dry period

90,000 80,000 70,000 Storage required for 9,000-acre-ft/yr draft is maximu deficiency

60,000 50,000

Beginning of dry period

Mass Curve Analysis

40,000

One of the earliest and simplest synthesis techniques was devised by Rippl [62] to investigate reservoir storage capacity requirements. His analysis assumes that the future inflows to a reservoir will be a duplicate of the historical record repeated in its entirety as many times end to end as is necessary to span the useful life of the reservoir. Sufficient storage is then selected to hold surplus waters for release during critical periods when inflows fall short of demands. Reservoir size selection is easily accomplished from an analysis of peaks and troughs in the mass curve of accumulated synthetic inflow versus time [63]-[65]. Future flows can be similar, but are unlikely to be identical to past flows. Random generation and Markov modeling techniques produce sequences that are different from, although still representative of, historical flows.

30,000

Cumulative draft, slope of 9,000 acre-ft/yr

4,000-acre-ft storage required

20,000 10,000

0

0

1

3

4

5 6 Year

8

10

FIGURE 12.19 Mass curve for Example 12.2: — cumulative inflow; — cumulative draft.

Example 12.2

Streamflows past a proposed reservoir site during a 5-year period of record were, respectively, in each year 14,000, 10,000, 6,000, 8,000, and 12,000 acre-ft. Use Rippl's mass curve method to determine the size of reservoir needed to provide a yield of 9,000 acre-ft in each of the next 10 years.

A

WATER-RESOURCES ENGINEERING

Area-1000 acres 48

44

40

36

32

28

24

20

16

12

8

4

0

20

22

24

26

Fourth Edition Ray K. Linsley David L. Freyberg

Irwin McGraw-Hill

Joseph B. Franzini George Tchobanoglous

—

January 1 flood control level

—

Minimum operating level

—

. —

RESERVOIRS

A water-supply, irrigation, or hydroelectric project drawing water directly from a stream may be unable to satisfy the demands of its consumers during low flows. This stream, which may carry little or no water during portions of the year, often becomes a raging torrent after heavy rains and a hazard to all activities along its banks. A storage, or conservation, reservoir can retain such excess water from periods of high flow for use during periods of drought. In addition to conserving water for later, use, the storage offloodwater may also reduce flood damage below the reservoir. Because of the varying rate of demand for water during the day, many cities find it necessary to provide distribution reservoirs within their watersupply system, Such reservoirs permit water-treatment or pumping plants to operate at a reasonably uniform rate and provide water from storage when the demand exceeds this rate. On farms or ranches, stock tanks or farm ponds may conserve the intermittent flow from small creeks for useful purposes. Whatever the size of a reservoir or the ultimate use of the water, the main function of a reservoir is to stabilize the flow of water, either by regulating a varying supply in a natural stream or by satisfying a varying demand by the ultimate consumers. The general aspects of reservoir design are discussed in this chanter 7.1 Physical Characteristics of Reservoirs Since the primary function of reservoirs is to provide storage, their most important physical characteristic is storage capacity. The capacity of a reservoir of regular shape can be computed with the formulas for the volumes of solids. Capacity of reservoirs on natural sites must usually be determined from topographic surveys. An area-elevation curve (Fig. 7.1) is constructed by planimetering the area enclosed within each contour within the reservoir site. The integral of the area-elevation curve is the elevation-storage, or capacity, curve for the reservoir. The increment of storage between two elevations is usually computed by multiplying the average of the areas at the two elevations by the elevation difference.1 The summation of these increments below any elevation is the storage volume below that level. In the absence of adequate topographic maps, cross sections of the reservoir are sometimes surveyed and the capacity computed from these vertical cross sections by use of the prismoidal formula.

10

12

14

16

18

Volume-100,000 acre-ft FIGURE 7.1 Elevation-storage and elevation-area curves for Cherokee Reservoir on the Holston River, Tennessee.

Normal pool level is the maximum elevation to which the reservoir surface will rise during ordinary operating conditions. For most reservoirs normal pool is determined by the elevation of the spillway crest or the top of the spillway gates. Minimum pool level is the lowest elevation to which the pool is to be drawn under normal conditions. This level may be fixed by the elevation of the lowest outlet in the dam or, in the case of hydroelectric reservoirs, by conditions of operating efficiency for the turbines. The storage volume between the minimum and normal pool levels is called the useful storage. Water held below minimum pool level is dead storage. In multipurpose reservoirs the useful storage may be subdivided into conservation storage and flood-mitigation storage in accordance with the adopted plan of operation. During floods, discharge over the spillway may cause the water level to rise above normal pool level. This surcharge storage is normally uncontrolled, i.e., it exists only while a flood is occurring and cannot be retained for later use. Reservoir banks are usually permeable, and water enters the soil when the reservoir fills and drains out as the water level is lowered. This bank storage increases the capacity of the reservoir above that indicated by the elevation-storage curve. The amount of bank storage depends on geologic conditions and may amount to several percent of the reservoir volume. The water in a natural stream channel occupies a variable volume of valley storage (Sec. 3.18). The net increase in storage capacity resulting from the construction of a reservoir is the total capacity less the natural valley storage. This distinction is of no importance for conservation reservoirs, but from the viewpoint of flood mitigation the effective storage in the reservoir is the useful storage plus the surcharge storage less the natural valley storage corresponding to the rate of inflow to the reservoir (Fig. 7.2).

Surcharge storage

Pool level during design flood

... Storage in reservoirs subject to marked backwater effects cannot be related to water-surface elevation alone as in Fig. 7.1. A second parameter such as inflow rate or water-surface elevation on a gage near the upper end of the reservoir must also be used. Storage volume under each profile can be computed from cross sections by the methods used for earthwork computations.

Stream bed Natural stream surface before dam' FIGURE 7.2 Zones of storage in a reservoir

The preceding discussion has assumed that the reservoir water surface is level. This is a reasonable assumption for most short, deep reservoirs. Actually, however, if flpw is passing the dam, there must be some slope to the water surface to cause this flow. If the cross-sectional area of the reservoir is large compared with the rate of flow, the velocity will be small and the slope of the hydraulic grade line will be very flat. In relatively shallow and narrow reservoirs, the water surface at high flows may depart considerably from the horizontal (Fig. 7.3). The wedge-shaped element of storage above a horizontal is surcharge storage. The

5681-

FIGURE 7.3 Profiles of the water surface in the Wheeler Reservoir on the Tennessee River. (Data from TV A)

shape of the water-surface profile can be computed by using methods for nonuniform flow (Sec. 10.4). A different profile will exist for each combination of inflow rate and water-surface elevation at the dam. The computation of the water-surface profile is an important part of reservoir design since it provides information on the water level at various points along the length of the reservoir from which the land requirements for the reservoir can be determined. Acquisition of land or flowage rights over the land is necessary before the reservoir can be built. Docks, houses, storm-drain outlets, roads, and bridges along the bank of the reservoir must be located above the maximum water level expected in the reservoir.

7.2

Reservoir Yield

Probably the most important aspect of storage-reservoir design is an analysis of the relation between yield and capacity. Yield is the amount of water that can be supplied from the reservoir during a specified interval of time. The time interval may vary from a day for a small distribution reservoir to a year or more for a large storage reservoir. Yield is dependent on inflow and will vary from year to year. The safe, or firm, yield is the maximum quantity of water that can be guaranteed during a critical dry period. In practice, the critical period is often taken as the period of lowest natural flow on record for the stream. Hence, there is a finite probability that a drier period may occur, with a yield even less than the safe yield. Since firm yield can never be determined with certainty, it is better to treat yield in probabilistic terms. The maximum possible yield during a given time interval equals the mean inflow less evaporation and seepage losses during that interval. If the flow were absolutely constant, no reservoir would be required; but, as variability of the flow increases, the required reservoir capacity increases. Given a target yield, the selection of reservoir capacity is dependent on the acceptable risk that the yield will not always be realized. A reservoir to supply municipal water should have a relatively low design yield so that the risk of a period with yield below the design value is small. By contrast, an irrigation system may tolerate 20 percent of the years with yield below the nominal design value. Water available in excess of safe yield during periods of high flow is called secondary yield. Hydroelectric energy developed from secondary water may -be sold to large industries on a "when available" basis. Energy commitments to domestic users must be on a firm basis and should not exceed the energy that can be produced with the firm yield unless thermal energy (steam or diesel) is available to support the hydroelectric energy. The decision is an economic one based on costs f and benefits for various levels of design.

7.3 Selection of Distribution-Reservoir Capacity for a Given Yield Often a project design requires the determination of the reservoir capacity required to meet a specific demand. Examples are found in municipal water supply or in irrigation when it is desired to irrigate a specified area. Since the yield (outflow) is equal to the inflow plus or minus an increment of storage, the determination of the capacity to supply a given yield is based on the storage equation [Eq. (3.12)]. In the Jong run, outflow-must equal inflow less waste and unavoidable losses:-This ' ^ ' i f c ^ $"t is another way of saying that a reservoir does not make water but merely permits its redistribution with respect to time.

A simple problem involving the selection of distribution reservoir capacity is given in Example 7.1. Here the required yield is based on an estimate of the maximum daily demand by the consumers. The inflow rate is fixed by a decision to pump at a uniform rate. The reservoir capacity must be sufficient to supply the demand at times when the demand exceeds the pumping rate. A similar solution would be used if a variable pumping rate were assumed.

1000

Example 7.1. The water supply for a city is pumped from wells to a distribution reservoir. The estimated hourly water requirements for the maximum day are as follows. If the pumps are to operate at a uniform rate, what distribution reservoir capacity is required? Pumping Required from Demand, Hour reservoir, m3 rate, m/h ending m/h 0100 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

Total

273 206 256 237 257 312 438 627 817 875 820 773 759 764 729 671 670 657 612 525 423 365 328 309

12,703

529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 529.3 12,703

0 0 0 0 0 0 0 98 288 346 291 244 230 235 200 142 141 128 83 0 0 0 0 0

2426

Solution. The average pumping rate is determined by dividing the total pumped by 24. The required reservoir capacity is the sum of the hourly requirements from storage, or 2426 m3. This is also shown graphically in Fig. 7.4; the required storage is given by {* (0 - I) dt, where 0 is the outflow (demand) and / is the inflow pumping rate.

7.4

Selection of Capacity for a River Reservoir

The determination of required capacity for a river reservoir is usually called an operation study and is essentially a simulation of the reservoir operation for a period of time in accord with an adopted set of rules. An operation study may analyze only a selected "critical period" of very low flow, but modern practice favors the use of a long synthetic record (Sec. 5.16). In the first case the study can do no more than define the capacity required during the selected drought. With the synthetic data it is possible to estimate the reliability of reservoirs of various capacities.

12 Time—hours

FIGURE 7.4 Graphical illustration of the computation of required reservoir capacity.

An operation study may be performed with annual, monthly, or daily time intervals. Monthly data are most commonly used, but for large reservoirs that carry over storage for many years, annual intervals are satisfactory. For very small reservoirs, the sequence of flow within a month may be important and a weekly or daily interval should be used. When lengthy synthetic data are to be analyzed, computer analysis is indicated and the sequent-peak algorithm1 is commonly used. Values of the cumulative sum of inflow minus withdrawals (including average evaporation and seepage) are calculated (Fig. 7.5). The first peak (local maximum of cumulative net inflow) and the sequent peak (next following peak that is greater than the first peak) are identified. The required storage for the interval is the difference between the initial peak and the lowest trough in the interval. The process is repeated for all cases in the period under study and the largest value of required storage can thus be found. A mass curve (or Rippl diagram) is a cumulative plotting of net reservoir inflow. Figure 7.6 is a mass curve for a 4-yr period. The slope of the mass curve at any time is a measure of the inflow at that time. Demand curves representing a uniform rate of demand are straight lines. Demand lines drawn tangent to the high points of the mass curve (A, B) represent rates of withdrawal from the reservoir. Assuming the reservoir to be full wherever a demand line intersects the mass curve, the maximum departure between the demand line and the mass curve represents the reservoir capacity required to satisfy the demand^ A\e vertical the spillway. If the demand is not uniform, the demand line becomes a curve

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Before finalizing the decision regarding reservoir capacity, it is usually desirable to perform a detailed operation study on one or more periods of data. These detailed analyses should consider seepage as a function of reservoir level, evaporation as a function of reservoir area and variable evaporation potential, and operating rules that may be dependent on natural inflow, reservoir storage, or other factors. It is generally convenient to use a computer1 for the operation study since this permits a number of trials using various assumptions as to operating rules, etc. Construction of a reservoir increases the exposed water-surface area above that of the natural stream and increases the evaporation loss. At the same time, there is an increase in runoff from the area occupied by the reservoir water surface because all precipitation falling on the water becomes available (i.e., k = 1.0), whereas only a fraction of the precipitation became runoff previously (i.e., k < 1.0). Usually there is a net loss of water flowing past a dam. Thus, in terms of depth of water, disregarding seepage losses, Net loss of water = Ew — (P — q)

(7.1)

where Ew is the free water evaporation, P is the precipitation, and q is the runoff from the area inundated by the reservoir. As an example, employing information from Figs. 2.5, 2.12, and 2.16, comparing a reservoir near the southwest corner of Utah with one in central Georgia gives annual averages (in inches) as follows:

7.5

We may estimate the reliability by generating stochastically (Sec. 5.16) 500 to 1000 traces, each trace equal in length to the adopted project life. Each trace may then be said to represent one possible example of what might occur during the project lifetime, and all traces are equally likely representatives of this future period. If the storage required to deliver a specified demand is calculated for each trace, the resulting values of storage can be ranked in order of magnitude and plotted as a frequency curve, or the theoretical curve can be calculated from the data. The Gumbel extreme-value distribution appears to be the appropriate one for this purpose. The result is a reliability curve (Fig. 7.8) that indicates the probability that the demands during the project life can be met as a function of reservoir capacity. For the stream of Fig. 7.8, a reservoir capacity of 615,000 acre-ft (758 x 106 m3) is required if a reliability of 99.5 percent is desired while 550,000 acre-ft (678 x 106 m3) are adequate if a reliability of 95 percent is acceptable. Zero risk or 100 percent reliability is impossible and the traditional concept of safe yield or firm yield has no meaning. Use of reliability analysis permits one to compare the costs of achieving various levels of reliability and to determine whether an increase in reliability is warranted. 7.6

Southwest Utah Central Georgia

60 43

46

Reservoir Reliability

The reliability of a reservoir is defined as the probability that it will deliver the expected demand throughout its lifetime without incurring a deficiency. In this sense lifetime is taken as the economic life, which is usually between 50 and 100 yr.

Sediment Transport by Streams

Every stream carries some suspended sediment and moves larger solids along the stream bed as bed load. Since the specific gravity of soil materials is about 2.65, the particles of suspended sediment tend to settle to the channel bottom, but

0.3 15

From Eq. ^7.1) the respective water losses are as follows: Southwest Utah Central Georgia

60 - (8 - 0.3) - 52.3 in. 43 - (46 - 15) - 12 in.

The volume of water loss per year is calculated by multiplying the depth loss by the average area of the reservoir water surface during the year. More accurate results are possible if the calculations are conducted on a monthly or weekly basis. It should be noted that a net gain in water at a reservoir is possible where the precipitation is considerably greater than the evaporation. In arid regions, however, the loss may be so great as to defeat the purpose of the reservoir. Lake Powell behind Glen Canyon Dam on the Colorado River has reduced the runoff in the Colorado River by over 500,000 acre-ft/yr, equivalent to about 4 percent of the runoff from the entire Colorado River basin. The reservoir, though reducing the volume of water available, permits control of the flow in the river, provides a water-surface elevation drop for the generation of hydroelectric energy, and has resulted in recreational benefits. To justify this project, these benefits had to be balanced against the depletion of volume of

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90 95 9798 Probability

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99.8 99.9 T, '

in the stream channel, to remain until a subsequent storm washes it downstream. Portions of the drainage area may be more susceptible to erosion than others, and higher sediment loads may be expected when a storm centers over such areas. Thus, the rate of suspended-sediment transport and the rate of streamflow^ are rarely closely correlated. Despite these inaccuracies, the sediment rating provides a useful tool for estimates of suspended-sediment transport. The total sediment transport may be estimated by adding a suitable amount to the suspendedsediment transport to allow for the bed-load contribution. In the absence of suspended-sediment data, the total sediment transport of a stream may be estimated by comparison with similar watersheds whose sediment transports have been previously determined from suspended-sediment-load data or from studies of reservoir-sediment accumulation. The total amount of sediment that passes any section of stream is referred to as the sediment yield or sediment production. Rates of sediment production for typical watersheds in the United States ar£ presented in Table 7.1. Mean annual sediment-production rates generally range from 200 to 4000 tons/mi2 (70 to 1400 t/km 2 ).

,Turbid inflow Water surface

Delta

FIGURE 7.9 Schematic drawing of the sediment accumulation in a typical reservoir.

upward currents in the turbulent flow counteract the gravitational settling. When sediment-laden water reaches a reservoir, the velocity and turbulence are greatly reduced. The larger suspended particles and most of the bed load are deposited as a delta at the head of the reservoir (Fig. 1.9). Smaller particles remain in suspension longer and are deposited farther down the reservoir, although the very smallest particles may remain in suspension for a long time and some may pass the dam with water discharged through sluiceways, turbines, or the spillway. The suspended-sediment load of streams is measured by sampling the water, filtering to remove the sediment, drying, and weighing the filtered material. Sediment load is expressed in parts per million (ppm), computed by dividing the weight of the sediment by the weight of sediment and water in the sample and multiplying the quotient by 106. The sample is usually collected in a bottle held in a sampler (Fig. 7.10) that is designed to avoid distortion of the streamlines of flow so as to collect a representative sample of the sediment-laden water. Most of the available sediment-load data have been gathered since about 1938. Because of poorly designed samplers, many of the early data are of questionable accuracy. No practical device for field measurement of bed load is now in use. Bed load may vary from zero to several times the suspended load. More commonly, though, it lies in the 5 to 25 percent range. Einstein1 has presented an equation for the calculation of bed-load movement on the basis of the\size distribution of the bed material and the streamflow rates. The relation between suspended-sediment transport Qs and streamflow Q is often represented by a logarithmic plot (Fig. 7.11), which may be expressed mathematically by an equation of the form

Q, =

(1.2)

where n commonly varies between 2 and 3, though values of n as low as unity have been observed on some streams. A sediment-rating curve such as Fig. 7.11 may be used to estimate suspended-sediment transport from the continuous record of streamflow in the same manner that the flow is estimated from the continuousstage record by use of a stage-discharge relation. The sediment rating is much less accurate than the corresponding streamflow-rating curve. Rates of erosion vary from storm to storm with variations in rainfall intensity, soil condition, and vegetal development. Sediment eroded from a basin during one storm may be deposited

Suspended sediment discharge in metric tons per day 10 10,000 EE^

100

1,000

10,000

100,000'

' 1

1,000

10 100 1,000 10,000 100,000 Suspended sediment discharge in tons per day FIGURE7.il Sediment-rating curve for Powder River at Arvada, Wyoming. (L. B. Leopold and T. Maddock, Jr., The Hydraulic Geometry of Stream Channels and Some Physiographic Implications, U.S. Geol. Surv. Prof. Paper 252, 1953)