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<br />EM 111G-2-H08
<br />1 Much 19GO
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<br />(3) If the baSic data consist of a discharge rating curve at the downstream end of the reach, nu-
<br />merous cross sections in the reach, and i; Manning roughness coefficient, the K value will be
<br />. represented approximately by the slope of a curve representing the volumes under computed
<br />. steady-flow profiles versus the corresponding ou tflow discharges. Obviously the curve a~d
<br />K can be determined also if the storage is computed from sounding surveys for several steady-
<br />flow profiles.
<br />(4) Finally, the K value may be determined from actual hydro graphs by an inverse process of
<br />flood routing. This method is preferable to the preceding methods because it provides a
<br />simultaneous determination of X and K, and because K in .this case represents the change
<br />of total storage per unit of wcighted discharge according to the relationship expressed by
<br />~quations (7), (8), and (9). In this method, equations (5) and (6) are solved for K to obtain,
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<br />K 0:5 6.t [(1,+1,)-(0,+0,)] ____nn____u__uu_n___ (20)
<br />X (I,-I,)+(l-X)(O,-O,)
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<br />The process has been described in a publication of the Engineer School, Fort Belvoir, Va.,"
<br />as follows:
<br />". . . . succes,iye values of the numerator and denominator are accumulated for floods
<br />for which the inflow and outflow are known, with X as a parameter. The accumulated
<br />numerator values are plotted as abscissae and the accumulated denominator values as
<br />ordinates, The result is a series of curves for various values of X. The one approaching
<br />more nearly to a straight line for the entire flood satisfies most closely the equation, and
<br />therefore determines the proper X for the reach, The K value is the reciprocal of the
<br />slope of the curve, Because of the limits of accuracy of the runoff data usually available
<br />it is preferable to compute K and X for several floods and to adopt for routing an average
<br />of tbese values."
<br />The procedure is illustrated by plates Nos.' 3 and 4, which are based on material shown in the
<br />above referenced publication, In plate No.3, column 2 represents the inflow into the reach.
<br />Runoff values for the main stream and one tributary were obtained from U.S. Geological
<br />Survey records and the ungaged tributary runoff was approximated from distribution graphs
<br />and rainfall records. The total inflow was adjusted to equal the outflow volume. The num-
<br />erator and denominator of equation (20) were evaluated for each period using four assumed
<br />values of X, On plate No, 4 the accumulated numerator or storage values in column 9 of
<br />plate No, 3 are plotted against the corresponding accumulated denominator or weighted
<br />discharge values shown in columns 11, 13, 15, and 17 of plate No.3. The best fit is assumed
<br />to be that for which there is the least variation of the valley storage curve from a single line
<br />passing through it. On plate No.4, the fit appears to be satisfied best for X=0.2 and K=l.OO
<br />day, the mean line in this instance being taken as straight throughout the range of discharges.
<br />If, to conform to a criteria that K=6.t=0,5 day, this reach was subdivided into two equal
<br />reaches, the value of K for each reach would be 0.50 day l'ssuming a constant wave celerity
<br />for the two reaches, However, X would not necessarily retain the value of 0.2.
<br />The ratio of reach length to K gives the rate of flood-wave movement. This rate can be
<br />applied to re?,ches of short,er or longer length unless there is a marked difference in "'orage char-
<br />acteristics between the desired reach and the reach for which the K value was derived.
<br />c. Variable K. In the determination of K hy any of the preceding methods, it wilt be found (.hat
<br />K generally varies with stage: A curve of K versus outflow may be prepared. If for a specific routing
<br />a single value of K is to be selected, it may be taken as the value corresponding to the average of the
<br />initial flow and anticipated peak outflow. .
<br />The loop in the storage curves of figure 2 and of plate No.4 for X=O may result in part from the
<br />fact that discharge generally is not a single.valued function of stage but depends on the rate of rise or
<br />fall of the stage hydrograph. If discharges derived from average rating curves are corrected for that
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