Hypothetical flood compution for a stream system 1

Transcription

1 Hypothetcal flood computon for a stream system 1 Leo R. Beard 2 ABSTRACT: One of the computer packages beng developed n The Hydrologe Engneerng Center of the Corps of Engneers s the Flood Hydrograph Package, whch performs all common operatons n dervng and usng flood hydrographs entrely n core storage of computers as large as the IBM One of the problems managed by ths program sflood computaton throughout an entre stream system n one operaton. Ordnarly, a desgn or hypothetcal flood mght be derved from a storm centered n the trbutary area. After floods are computed n ths manner for several sub-areas wthn a stream system, t s often necessary to compute floods for the combned areas at varous locatons. The orgnally computed floods cannot be routed and combned, snce the storm cannot center over all sub-areas at once. It s ordnarly necessary to recompute each sub-area flood for every combnng pont downstream, and to label each recomputaton as the contrbuton to that partcular locaton. In a complex system, such as a storm dran or major rver basn, ths procedure would lead to great amounts of computaton and consderable confuson. Essental features of the computer package and the smplfed stream system analyss are descrbed. RÉSUMÉ : L'un des programmes des calculateurs en tran d'être développé au Centre d'hydrologe du Géne, est le Programme Hydrogramme des crues qu accomplt toutes les opératons communes pour trouver et utlser les hydrogrammes des crues qu se trouvent dans un ensemble de calculateurs auss vaste que l'ibm L'un des problèmes résolu par ce programme est le calcul des crues d'un bout a l'autre d'un cours d'eau en une seule opératon. Normalement, une crue hypothétque peut être dérvée d'une averse concentrée dans la regon trbutare. Lorsque les crues ont été évaluées de cette façon pour pluseurs endrots d'un cours d'eau, l est alors nécessare d'évaluer les crues pour l'ensemble de ces dfférents ponts. Les crues calculées à l'orgne ne peuvent pas être drgées et combnées, car l'averse ne peut pas être concentrée partout à la fos. Il est normalement nécessare de recalculer chaque crue pour chaque endrot combné en aval et d'étquetter chaque nouveau calcul pour une contrbuton à cet endrot partculer. Dans un système complexe comme un fossé d'écoulement d'averse ou un majeur bassn fluval, ce procédé aboutrat a d'énormes sommes de calcul et auss à une confuson consdérable. Les caractérstques essentelles de ce programme du calculateur et l'analyse smplfée du système de cours d'eau y sont décres. INTRODUCTION A desgn flood, defned as the flood for whch a hydraulc structure s desgned, can be selected n many ways, one of whch conssts of a detaled economc analyss of costs and benefts assocated wth protecton aganst varous magntudes of floods. When desgn flows are requred for many locatons n a stream system, t s often desrable to compute varous hypothetcal floods from ranfall and snowmelt n order to assure a degree of hydrologe consstency n the selecton of desgn magntudes. In the computaton of such hypothetcal floods, certan general relatonshps exst between szes of trbutary area and sze of flood, partly because average storm ntenstes decrease as area covered ncreases, and for varous other reasons. 1. Prepared for presentaton at the Internatonal Assocaton of Scentfc Hydrology Symposum, Tucson, Arzona, 8-15 December Drector, The Hydrologe Engneerng Center, Corps of Engneers, Sacramento, Calforna. 258

2 Hypothetcalfloodcomputon for a stream system A comprehensve computer program has been developed n The Hydrologe Engneerng Center of the Corps of Engneers to manage the varous problems assocated wth computaton of runoff from ranfall and snowmelt quanttes. The hydrologe features of ths computer program, partcularly regardng a new technque for decreasng flood magntudes per unt area wth sze of dranage basn, are descrbed heren. SNOWPACK ACCUMULATION AND DEPLETION Those basns where snow accumulaton and snowmelt are sgnfcant factors n flood producton are dvded nto elevaton zones of 1000-foot range n order to account for marked changes of temperature wth elevaton. Temperatures specfed for each nterval correspond to the bottom elevaton of the lowest zone and are decreased wth elevaton n accordance wth a specfed lapse rate. If desred, they are also modfed for changes n lattude. In each elevaton zone, precptaton s consdered to occur as snowfall f the temperature s below a specfed value. Snowmelt wthn each elevaton zone s computed n accordance wth generalzed relatons used by the U. S. Army Corps of Engneers, wth an opton between a straght degree-day computaton and a comprehensve energy-budget computaton, usng methods descrbed n the Corps of Engneers manual. 3 The former method usually gves more satsfactory results, because the detaled data requred for energy-budget computatons are usually not avalable durng each nterval of the storm. Snowfall s added to the snowpack n each elevaton zone and snowmelt s subtracted durng each storm nterval. Ranfall s added to snowmelt to obtan the total water nput for each nterval. NON-LINEAR LOSS RELATION A great many studes relatng basn-average ranfall amounts for short ntervals n a storm to observed runoff ndcate a dstnctly non-lnear relatonshp between ranfall ntensty and loss rate (nfltraton). Consderng the non-homogenety of sols, vegetaton and precptaton throughout a basn, ths non-lnearty between basn-average (lumped) amounts s logcal. The followng equaton has been used n these studes, wth a constrant that loss does not exceed water nput (ranfall plus snowmelt) for each nterval: n whch L loss n nches per hour durng nterval ; K coeffcent decreasng wth ncreased ground wetness ; P ranfall plus snowmelt n nches per hour durng nterval; E constant exponent between zero and 1.0. L = KP E (1) It wll be noted that a value of zero for E would correspond to loss rates ndependent of ranfall ntensty (the tradtonal assumpton) and that an exponent of 1.0 would correspond to loss rates drectly proportonal to ranfall ntensty. Hydrograph reconsttuton studes have ndcated the exponent to range ordnarly between 0.5 and 0.9, and frequently an average of 0.7 has been adopted for purposes of unformty. Loss rates durng snowmeltfloodsare small n comparson wth observed losses durng ranfoods. Accordngly, separate loss functons are used n the computer program for 3. Runoff from Snowmelt, EM , Corps of Engneers, 5 January

3 L.R. Beard snow-covered areas and for snow-free areas. In each case, the loss coeffcent K n equaton 1 decreases wth ground wetness durng a storm n accordance wth the followng equaton: where K 0 C IL K = K 0 C- (ILI10) (2) loss coeffcent at start of storm (dfferent for snow-free and snow-covered areas; coeffcent controllng rate of decrease of K (dfferent for snow-free and snowcovered areas) ; accumulated loss durng storm (dfferent for each elevaton zone). Fgure 1 llustrates the use of equatons 1 and 2. FIGURE 1. Example of loss functon Durng long-duraton snowmelt floods, ranfall and snowmelt losses from snowcovered areas decrease n ths manner, but ranfall losses n snow-free areas are consdered to ncrease gradually throughout the flood perod, nasmuch as temperatures are gradually ncreasng and the ground surface becomes drer. Ths rate of ncrease n K s arbtrarly set at one percent of the ntal value per day. 260

4 Hypothetcalfloodcomputon for a stream system LINEAR RUNOFF TRANSFORM The excess quanttes of ranfall and snowmelt durng each nterval are added for all elevaton zones and expressed as basn-mean excess n nches. These quanttes are then transformed to runoff at the concentraton pont by use of the unt hydrogaph technque developed by C O. Clark. 4 Ths method s selected because t provdes a means of drect computaton from only 2 coeffcents and a tme-area curve. The advantage of usng the tme-area curve s that changes n basn confguraton, such as dversons, reservor constructon, etc., can be mmedately reflected n the modfed tme-area curve. For use n studes where a tme-area curve s not obtaned from detaled maps, for any of varous reasons, a typcal tme-area curve s provded n the computer program. Ths s usually satsfactory, snce the shape of the tme-area curve has lttle nfluence compared wth tme and storage coeffcents. Also, a unt hydrograph wth specfed Snyder coeffcents (T p and C p ) 5 can be obtaned by automatc successve approxmatons of the Clark coeffcents (TC and R). = = = 120 FIGURE 2. Example of multplefloodnterpolaton 4. CO. Clark, storage and the Unt Hydrograph, Trans. ASCE, Vol. 110, Flood Hydrograph Analyses and Computatons, EM , Corps of Engneers, 31 August

5 L. R. Beard The computaton procedure used does not nclude separaton of surface flows from sub-surface or ground-water return flows. All of these phases of flows are ncluded n a sngle unt hydrograph. However, runoff from antecedent storms that would have occurred n the absence of the current storm (or snowmelt event) s added as a recesson flow to runoff computed from the current storm. Also, after the total runoff recedes below a specfed value, runoff computed by use of the unt hydrograph s constraned from recedng below a specfed maxmum exponental rate of recesson. Ths devce precludes the necessty of treatng sub-surface and return flows separately, whch s otherwse necessary because of strong non-lneartes n the relaton of recesson flow to ranfall and snowmelt amounts. The lnear unt-hydrograph transform or convoluton process s ordnarly adequate, f the tendency for change n unt hydrograph characterstcs wth flood magntude at a gven locaton s recognzed. Non-lneartes of ths convoluton process wthn a flood, whch are known to exst, are extremely dffcult to determne accurately and are beleved to have mnor effect relatve to uncertantes of precptaton and loss functons. DERIVATION OF RUNOFF FUNCTIONS A routne s ncluded n ths computer program to derve the unt-hydrograph and loss-rate coeffcents that best reconsttute an observed flood hydrograph by a leastsquares procedure. The gradent optmzaton procedure used s descrbed n a prevous publcaton 6 by the wrter. Provsons are ncluded to fx any of the varables desred n order to smplfy a regonal correlaton analyss of the resultng coeffcents. For example, the exponent of precptaton n the loss rates functon would ordnarly be fxed so that coeffcents derved for dfferent locatons can be compared drectly. There s also provson for permttng the engneer to nfluence the exact nature of the hydrograph reconsttuton to some extent by temporarly dstortng the observed hydrograph and thus forcng a reconsttuton of the undstorted hydrograph that would be more acceptable than the one automatcally derved n the least-squares operaton. DERIVATION OF ROUTING COEFFICIENTS A provson s also ncluded n the program to derve routng coeffcents that best reconsttute downstream hydrographs from observed upstream hydrographs and estmates of ntermedate runoff. In all cases, ths s a process of successve approxmatons. In the case of contnuous functons, such as Muskngum coeffcents, the optmzaton routne used s that employed for reconsttutng hydrographs from ranfall and snowmelt, dscussed above. In the case of dscrete functons, such as straddle or stagger, successve values of the parameters are examned untl the standard error of reconsttuton begns to ncrease. An optmum combnaton of a dscrete parameter and any other parameter s determned by sub-optmzng the other parameter for each successve value of the dscrete parameter untl the sub-optmum standard error of reconsttuton decreases. HYDROGRAPH ROUTING AND COMBINING Hydrographs of runoff computed for varous sub-basns can be routed and combned to form hydrographs at downstream locatons. In order to account for changes n shape and 6. BEARD, Leo R. Optmzaton Technques for Hydrologe Engneerng, Water Resources Research, Vol. 3, No. 3,

6 Hypothetcalfloodcomputon for a stream system tme of the hydrographs as they travel downstream, a varety of commonly-used routng procedures are ncluded n the computer program. These nclude the Muskngum (coeffcent) method, the storage-lag method, the multple-storage method, and the Straddle-stagger method. 7 Also, a reservor routng (modfed Pus) routne s ncluded. In order to compute hydrographs for a large number of ponts wthn a stream system, storage space wthn the computer core s released as soon as a hydrograph has been routed or combned wth another hydrograph. It s, of course, prnted out before removal from storage. In order to requre the smallest number of hydrographs n storage at any one tme, t s necessary to start the computaton at the most remote upstream locaton. Proper combnng of hydrographs requres that, once a hydrograph s computed upstream of a locaton, all operatons upstream of that locaton must be performed before performng the operaton for a locaton that s not above that combnng pont. Ths s because the latest computed, routed or combned hydrographs are those used n each new combnng operaton. Channel losses must be expressed as a lnear functon of flow n each routng reach (a constant loss plus a rato of the remanng flow). MULTIPLE-FLOOD COMPUTATION In order to evaluate the effects of changes at any locaton wthn a rver basn on flows at a downstream pont, t s necessary to dstrbute precptaton throughout the trbutary area n a balanced manner, f computaton and management feasblty s to be obtaned. Otherwse, a great many storm centerngs must be used for each successve downstream evaluaton, n order to reflect a reasonable range of potental events at that locaton. In computng a balanced flood, t s recognzed that averagng technques mght obscure rare combnatons that should possbly be consdered n specal analyses. Snce the average depth of precptaton over a trbutary area generally decreases wth ncreasng sze of area, t would ordnarly be necessary to recompute a decreasng balanced-flood quantty contrbuted by each sub-area to successve downstream ponts. In order to avod ths prolferaton of hydrographs, t s proposed that dfferent floods correspondng to varous depths of precptaton be computed for the entre rver basn complex. The depths of precptaton selected should correspond to desgn quanttes for specfed dranage basn szes coverng the range of nterest. Thus, by provdng a table of dranage area vs. desgn precptaton depth, and a tme dstrbuton pattern, varous floods can be computed. To mnmze confuson, these can be called flood 1, flood 2, etc. Each of thesefloods would correspond to a specfed dranage basn sze, and would be the desgn flood magntude for any pont n the basn whose trbutary area corresponds to that sze. Above such ponts, there would be a balanced contrbuton from all parts of the sub-basn. In order to obtan desgn floods for locatons whose trbutary area does not exactly correspond to the selected precptaton amounts for a numbered flood, the desgn flood hydrograph s obtaned by nterpolatng between the two numbered floods whose precptaton corresponds to area szes nearest the trbutary area sze. An llustratve example s shown n fgure 2. The nterpolaton routne used s lnear wth respect to the logarthm of area sze. The average rates of precptaton for very large areas are small, and adopted loss functons mght ndcate very low runoff volumes for such precptaton ntenstes. However, t s recognzed that some locatons wthn the basn are experencng much hgher rates of precptaton, whle other locatons m a y be obtanng lttle or no precptaton. In order to account for ths, stream system computatons are madefrstfor the large precptaton amounts (correspondng to small areas) and contnued successvely for 7. Routng of Floods Through Rver Channels, EM , Corps of Engneers, 1 March I

7 L.R. Beard floods 2, 3, etc., each representng successvely larger desgn area szes (lower precptaton depths). The excess amounts obtaned for each flood are retaned n the computer, and used as a proportonal contrbuton for the next flood system computaton. The proporton used s the rato of dranage area szes that precptaton amounts represent on the desgn depth-area curve. In effect, the ncremental desgn-crteron precptaton volume for the second flood computaton s consdered to occur over the ncremental desgncrteron area sze that such precptaton represents. Ths combnaton s appled to all sub-basns, thus mantanng a balance of runoff wthn the entre basn. As an example, f desgn precptaton s 10 nches for 100 squares mles (flood 1) and 9 nches for 500 square mles (flood 2), precptaton of flood 2 for a 10-square mle sub-area would consst of 10 nches over 2 squares mles and (90 20)/8 or 8.75 nches over 8 square mles. Where orographc effects apprecably nfluence precptaton amounts, the "depth-area" curve of desgn precptaton can represent ratos to some base precptaton pattern, such as normal annual precptaton, and these would be multpled by normal precptaton amounts specfed for each elevaton zone of each sub-area. It s possble that a storm centered over one trbutary can produce a larger flood than would result from the storm spread over both trbutary basns above a confluence, despte the addtonal area of precptaton. Ths occurs n the system computaton occasonally. Although each base flood below a confluence s computed by drect addton of trbutary hydrographs, the nterpolaton procedure used can produce a smaller desgn flood below the confluence than on the larger contrbutor of two trbutares. When ths occurs, the program provdes for an examnaton of the peaks and volumes of trbutary floods, and assures that the flood below the confluence s nterpolated such that quanttes below the confluence are at least as large as for each trbutary flood. ASSESSMENT OF REGULATORY EFFECTS One of the most dffcult problems n the functonal evaluaton of a system of flood control reservors s determnng the over-all effect that a reservor located on a remote trbutary has on floods at a down-stream pont. A storm mght center manly on ths remote trbutary, n whch case the effect s large; or t mght produce lttle precptaton n that trbutary area, n chch case the effect could be trval. Consderng that all magntudes of storms can occur and that a varety of centerngs can also occur more-or-less ndependently, the expected (average) effect of a reservor s ordnarly approxmated closely by evaluatng ts effect on a balanced type of flood for each general range of magntude at the downstream pont. Thus, the expected (average usable for economc evaluatons) relaton between regulated flows and unregulated flows of the same exceedence probablty, for any confguraton of reservors, can be derved from the effects demonstrated n the set of balanced floods computed generally as dscussed heren. Ths would requre only two complete multple-flood system computatons: one for unregulated condtons and one for regulated condtons for a gven plan of development. EFFECTS OF URBANIZATION AND CHANNEL IMPROVEMENT Some dramatc changes have occurred n certan stream systems where urbanzaton and channel mprovement have greatly modfed the surface and stream channel characterstcs. These changes generally result n reduced losses (ncreased volumes of runoff) and more rapd concentraton of streamflows. Both natural percolaton and pondng can be greatly reduced by urbanzaton and channel mprovement. To account for reducton n nfltraton loss, a coeffcent of mpervousness s used, whch smply assumes 100% 264

8 Hypothetcalfloodcomputon for a stream system runoff for the mpervous proporton of the dranage basn and usual losses for the remander. In order to account for reduced storage effects and for more rapd concentraton of flows, the unt hydrograph coeffcents are reduced n proporton to the estmated ncrease n average velocty of travel through the stream system. An example s shown nfgure 3.,/ ' J 1 / \, FIGURE 3. Example of urbanzaton effect / f\ TC=.70 R\.7C -" 20% mpervous X- V \ n Tme n hours TC- loo 00 y-~ó'f. In per* ous!s S COMPUTER OPERATION The flood hydrograph computer package s wrtten n FORTRAN IV and requres about 32,000 words of memory. On an altra-hgh-speed computer such as the CDC 6600, tme requrement s nomnal, snce all operatons are performed wthn core. The greatest tme requrement s assocated wth reconsttuton of snowmelt floods where 100 to 200 complete computatons of snowmelt, losses, etc., n 10 elevaton zones and 120 daly ordnates requres about one mnute of central processor tme. For ranflood reconsttuton, only a few seconds s requred. A complete multple-flood system computaton of hydrographs nvolvng snowmelt n a few dozen sub-areas requres about one mnute. A great varety of operatons s performed by the program. Each of the general operatons ndcated above, such as dervaton of unt hydrograph or loss rate or routng coeffcents, computaton of runoff, routng, etc., can be manpulated n many ways. For example, precptaton can be specfed as basn-mean values or ratos or staton values or can be computed accordng to some standard crtera. Unt hydrographs can be furnshed or computed. Precptaton can be accompaned by snowmelt or not, and n the latter case t s automatcally treated as ranfall. Ths great varety of operatons can be talored to each specfc need by use of control numbers n the nput data. Whle many of the operaton sequences are ntrcate and requre consderable thought, the controls are desgned to smplfy use of the program as much as possble. An automatc plot routne for showng hydrographs and related data n graphc form on the prnted output s ncluded. Plottng for each operaton s completely pre-programmed, ncludng selecton of tems to be plotted, ttlng and selectng scales to be used, and arrangement on the paper. An example s shown nfgure

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10 Hypothetcal flood computon for a stream system A summary of pertnent system flows s provded for ease of revew and apprasal. An example s shown n table 1. TABLE 1. Example of summary RUNOÏT SUMMARY, AVERAGE CFS 2 COMBINED 2 COMBINED 2 COMBINED 3 COMBINED 2 COMBINED 1+ COMBINED 2 COMBINED PEAK 591^ IOI V ^ ^ ^100 6-HOUR M * U O HOUR ^ ^ O II HOUR II * ! CONCLUSION The comprehensve flood hydrograph computer package descrbed performs vrtually all types of flood hydrograph studes needed for the functonal evaluaton of flood control mprovements. A large-memory, hgh-speed computer s requred. Computatons are rapd and easly controlled by smplfed nput data. The program provdes a unque procedure for computng hypothetcal floods for a complex stream system, whether t be a major rver basn such as the Oho Rver basn or an urban storm dran system. Provsons are ncluded for evaluatng the effects of structural mprovements and urbanzaton on flood runoff ACKNOWLEDGMENT Development of the work descrbed heren was accomplshed n The Hydrologe Engneerng Center of the Corps of Engneers, Unted States Army. However, vews expressed and procedures descrbed n ths paper do not necessarly reflect polces of the Corps of Engneers. 267