REAL-TIME FLOOD FORECASTING BY TRACING DEPTH CONTOURS

Louis M. Laushey

Emeritus Professor & Consultant; Dept. of Civil & Environmental Engineering

University of Cincinnati; 9066 Eldora Dr., Cincinnati, Ohio, 45236, USA

Baoguo Huang

Senior Engineer of The Flood Control Headquarters Office, Liaoning Province,

China; No.5 ShiSi Wei Road, Shenyang, 110003, China

Abstract: Real-time reliable announcements of downstream flood stages can be made during rising depths being measured upstream. All that is needed are simple gages recording instantaneous depths that are being transmitted to an engineer making rapid calculations; then broadcasting the future depths at downstream times and locations. This provides an ultimate warning system, with credibility, to follow early warnings of only probable rainfall and stream flows. The flood wave velocity is the only variable needed, measurable with a string of depth gages spaced at known intervals. Rainfall intensity, rating curves, channel dimensions and roughness are not necessary to develop a theory. The real-time measurements of distance between gages divided by time of arrival of a constant incremental depth to be traced is indeed the correct wave velocity. This timely measured wave velocity will be found within the spectrum bounded by the theoretical kinematic and dynamic waves. The method involves a movable frame of reference (imagine walking along the bank) at the speed necessary to “see” a constant depth, like a surveyor tracing a ground contour of locations with equal elevations. Examples describe the method and results. The first is a computer simulation of a thunderstorm over a flashy tributary; the second uses observed depths measured in a large river. The method of real-time contour tracing calculated eventual depths that agreed with those measures later. The method can be used similarly for planning, estimation, and design of flood control systems.

Keywords: real-time flood forecasting, depths and crests, wave velocities, depth contour tracing, moving reference frames

1  INTRODUCTION

Early warning of intense precipitation has become a reality that needs to be extended to predictions of the crests and the real-times of their occurrences in streams experiencing rises. Weather forecasters have developed extensive high-tech systems to inform  the public by radio and impressive TV simulations of storms moving over a wide area. At the earliest warnings, the probability of the flood damage in selected streams cannot be given with real-time certainty. Hydraulic engineers must extend the early warning system of precipitation to later warnings for accurate and reliable announcements of the impending magnitudes of stream elevations.

Needed is a system of stream gages that automatically and instantly transmit rising water surface elevations to a hydraulic engineer. He will use a simple method of tracing constant depth contours with times of their arrival downstream in real-time and magnitudes as he observes the event taking place upstream.  The method simply uses the wave speeds to trace contours of equal water depth, like a surveyor rodman moving along and tracing a path of equal ground elevations. The wave velocities can be approximated by a simple theory, but more easily and accurately from field measurements on simple gages showing the early rates of rise of the depths. No estimates of stream characteristics are needed. Rating curves are needed only if flow rates must be estimated. With yet unknown crests, monitoring and informing is very beneficial.

This paper proposes any early warning system for rainfall be extended to a later warning system for stream stages, to begin when a rise is actually observed, to give a high predictability of magnitudes and times of crests downstream.

2  WAVE VELOCITY: THEORY

An understanding of "wave speeds" is often murky and ambiguous. Flood waves are "long waves" having no meaningful wave length or frequency of oscillatory waves. A wave speed must be specific, defined in terms of the depth, discharge, velocity, energy, etc. The wave speed for the depth w_y is the speed that a moving frame of reference must travel to keep abreast of a selected constant depth, from which a rider would "see" and imagine the depth phase to be constant. Other moving frames, at different speeds, might be needed to see other phases appear to be constant.

In summary, the purpose of a moving frame of reference is to transform an unsteady motion into an apparent steady motion. Simply, if a person walks along the bank at the speed necessary to always see the same depth, that walking speed is the depth phase velocity, w_y (2). More clearly, w_y is DL/Dt, where Dt is the time to trace that contour of constant depth a DL distance downstream. Depth gages at known distances can easily and accurately measure and transmit the time intervals between their occurrences of the same depth. Similarly, w_q would be the frame speed to see the discharge appear to be constant.

3  COMPUTATIONS

The computer program that simulated the flood routing used the Method of Characteristics. The program by Streeter and Wylie (1) was used, but supplemented by the author (2), and by Grover (3) who wrote a subroutine to calculate the wave velocities by finite differencing the computer solution for the unsteady depths and discharges. Twenty subdivisions of the channel length were used. All computer results tested accurate and stable.

4  EXAMPLE: FLASHY STREAM

4.1  Real-time forecasting

The method is illustrated using a typical small flashy tributary stream in the eastern USA. A thunderstorm produces a rise and fall in a small stream for 60 minutes. The initial steady uniform depth was 0.61 meters. The increase in the discharge ratio (q_max/q_n) was 5, in 20 minutes (T₀).  Fig. 1 shows the computer solution for the depths and times at four downstream locations after the initial rise at x=0 and T=0.

4.2  Wave speeds

Fig. 2 plots the useful wave speeds at the upstream location (x=0) from T=0 to the cresting depth of y_max at T=22 min. The two component wave speeds, w_y and w_q, are practically equal to each other during the full rise in depth. At the beginning of the rise, they are equal to the theoretical dynamic wave velocity w_d, when L_w and x are both equal to zero. As L_w increases with time, they decrease rapidly and tend to approach the theoretical kinematic velocity that is always equal to or greater than the particle velocity v_o, as shown on Fig. 2. Of special importance is the wave velocity at T=22 minutes when the depth crests at x=0. This special wave velocity (2.5mps) will be shown to be the wave velocity to trace the later crests as they travel downstream. The wave speeds were within the spectrum bounded by these limits:

4.3  Dynamic wave speed

This upper boundary of the wave spectrum is only at the leading edge where the advancing unsteady flow is encroaching into an undisturbed steady uniform flow at a velocity of

w_d = v_n + (gy_n)^0.5                                                      (1)

4.4  Kinematic wave speed

The lower bound at uniform flow, far upstream is

w_k = v_n (1+ mB/P)                                                          (2)

5  METHOD: TRACING STREAM DEPTH CONTOURS

Tables 1 and 2 show the field procedure during the actual flooding of the example flashy stream having several depth gages. Table 1 covers the time from the beginning of a rise upstream to its crested depth. The sequence of operations follows in these columns with increasing numbers, that are numbered (1) through (10) below:

(1) At the upstream station, record several depths to be traced downstream. In this example; 0.61, 0.76, 1.07, 1.37 and 1.60 meters; incremental rises from the initial steady depth of 0.61 meters.

(2) At the upstream gage, record the time of observance of the depth contours to be traced. The initial rise upstream should be noted as starting at both T=0 and the real clock time.

(3) Select a trial wave speed w_y for the first increment of rise at the first downstream gage, from previous observations, theory, or judgment. The exact speed is relatively unimportant since it will soon be measured and corrected by iteration. The speed must be less than the dynamic and more than the kinematic wave speeds, both calculable and discussed earlier in the paper.

(4) Immediately predict and announce the expected arrival time downstream of the first contoured depth by dividing the reach length by the trial wave velocity, Col. 3. This correct wave velocity, and later measured wave velocities can be used with accuracy for later times and reaches.

(5) The anticipated arrival time of later contoured depths will simply be a summation of time increments, Col. 2 plus Col. 4.

(6) The measured arrival times are recorded in Col. 6. They become the correct and adjusted times for subsequent announcements.

(7) The corrected wave speeds x/T will now be the anticipated wave speeds for following reaches and depth contouring.

(8) For the second reach, the incremental time for a contour is the reach length divided by the new wave velocity, in Col.7.

(9) The predicted time of arrival of each contour at the downstream end of the second reach is Col. 6 plus Col. 8.

(10) These arrival times are from a computer solution that is presumed "measured", The agreement between Cols. 9 and 10 shows the continual replacement of measured wave velocities for assumed velocities is reasonable, easily done, and accurate.

Table 2 shows probably the most important information: the times and maximum depths (crests) to be expected downstream based on the actual rising hydrograph as it is being observed upstream. Col. 1 of Table 2 is a listing of selected distances downstream. Col. 2 is the computer solution for the crests, and Col. 3 is the time of the respective occurrences. Col. 4 shows an important variable, the increasing wave length L_w=w_dT. The ratios x/L_w and (w_d-w_c)/w_c are the variables in Eq. 3,defining the ratio of the crests at distances downstream. Cols. 5 and 6 show the good comparison of Eq. 3 with the computer solution for the downstream crest ratios, (y_x/y₀), The time of the downstream crests is found by dividing the channel distance by the wave velocity w_max at the upstream gage.

6  DEPTHS AT CRESTS

The equation for the successive crests at downstream locations is

(y_max at x) = (y_max at x=0)[1- (x/L_w)]^R                                               (3)

     R=(w_d-w_c)/w_c  and  L_w=w_dT                                                   (4)

Fig. 4 shows the agreement of the equation with the calculated crest depths.

7  SUMMARY

Figs.3 and 4 supplement Tables 1 and 2, presenting the most important results from simulated field measurements.

(1) The diagonal lines on Fig.3 are distances to the downstream stations listed in Col. 1 of Table 2. On the leftmost diagonal (x=0), the symbols 0 represent the selected depth contours observed at times T to be traced.

(2) The horizontal lines are the traces of constant depth. The symbols x locate the times when the traced depths appear at the downstream locations defined by the diagonal lines.

(3) The wave speeds of each depth contour between the downstream locations are inserted over the depth traces.

(4) Fig. 4 shows probably the most valuable information: the announcement of the imminent height and time of crests, for this warning is most important to minimize personal safety and economic damage.

8  EXAMPLE: AN APPLICATION TO A LARGE NATURAL RIVER

The previous example of a flashy stream has shown the method of real-time forecasting in a channel of constant cross sections. The following example will be a natural channel of irregular cross sections. The example flood happened in 1995, on the Hun River in Liaoning, China. The river is a large seasonal river that flows through the central area of Liaoning province. The river is a meandering stream with six big cities and large areas of farmland. The data all involved in it came from field measurement done by Liaoning Hydrology Bureau. Length of the reach is 48.63 kilometers, and the average slope of the bottom in the reach is 0.000208, and n averages 0.025. The average channel widths upstream and downstream are 267 and 140 meters. The average water surface widths after overflowing banks at peak are 1260 and 878 meters for upstream and downstream, see Fig. 5 and 6. At the beginning of the rise, the profile of the water surface was steady. The full rise and fall occurred over one month, with several peaks. Fig. 7 shows that portion that contained the maximum peak between  T=95 and 150 hrs, the figure shows the upstream and downstream depths were different at T=95 hrs. (the start of the rise to the maximum ) because of an initial steady backwater curve. That difference of 2.8 meters was added to the upstream gage to make them equal and to cancel out the backwater difference and deal with only the same incremental depths in tracing incremental depth contours, as in the first example that started with an uniform depth along the channel. Fig. 8 shows a replot using adjusted common equal gages settings. Now Fig. 8 is similar to Fig. 3 where the wave velocities that were measured are shown over depth contours in Fig. 8. Fig. 9 is a similar plot of the discharge during the rise and fall with the peak discharge downstream being slightly smaller and later than the peak discharge upstream. The initial steady depth is 2.10 meters at the upstream station. For the later contoured depths of 5.49, 6.45, 7.15, 7.78 meters, the wave speeds at each depths are 1.35, 1.35,1.93,3.38 mps respectively. Figs.8 and 9 show the time of about four hours to trace peak from upstream to downstream.

The method could be applied to predicate future design of other frequencies and the design of channel improvements. For all applications to real-time or future determinations:

(1) Trace constant depth differentials between the cross sections upstream and downstream.

(2)   Measure the speeds of the depths waves and adjust by iterations during the process of tracing depth contours over time and space along the channel.

(3) Add additional gages to divide the stream into significant reaches to increase the usefulness and accuracy of the proposed method.

9  SUMMARY

This example shows the versatility and wide application to all types of streams and rivers that are instrumented with simple gages that transmit depths to a command headquarters. The channel sections can be irregular, the roughness variable, the initial profile can have a backwater curve, and the inflow hydrographs can have multiple peaks and valleys.

10  CONCLUSIONS

Trustworthy announcements of times and depths of imminent flooding at downstream locations can be made in real-time while measurements of rising stages are occurring upstream. The method involves tracing contours of equal incremental depths to downstream locations. Only simple depth gages are required. Channel and watershed data and rating curves are not required. The real-time projections by an engineer can be simple, rapid, and accurate. The addition of the proposed late-warning method to existing early-warning systems can easily protect public safety and minimize economic damage.

Acknowledgements

The authors thank the University of Cincinnati for the temporary appointment of the junior author as a Visiting Scholar, and also the Liaoning Hydrology Bureau, Liaoning, P.R China for providing the data for the practical example.

References

[1]  Streeter, V.L. and Wylie, E.B. (1967). Hydraulic Transients. McGraw-Hill, N.Y.

[2]  Laushey, L.M. (1994). "Multiple Axis Theory for Unsteady Flow and Waves." Wave Modeling Symposium. University of British Columbia, Vancouver, Canada.

[3]  Grover, T.A. (1982) "Loop Rating Curves Using a Modified Form of the Saint Venant Equation," M.S. Thesis, University of Cincinnati, Cincinnati, Ohio, U.S.A.

Appendix I. Notation

The following symbols are used in this paper

B = Breadth of rectangular channel

C = Chezy roughness coefficient

g = Gravitational acceleration

k = Multiple of initial flow rate at gage, in stated time

L_w = Length of long flood wave, w_dT

m = coefficient, 1/2 for Chezy or 2/3 for Manning

P = Wetted perimeter

q = Discharge per unit width of rectangular channel

R = Ratio of crest depths, with x/L_w

S_b = Slope of channel bed

S_f = Slope of friction loss

T = Time from beginning of a rising depth

T₀ = Rise time to crested depth at x=0

v = Velocity of the particles, q/y

w = Velocity of imposed frames of reference, with subscripts:

    y for depth phase, q for discharge phase, and c for crests

w_k = Kinematic wave velocity

w_d = Dynamic wave velocity

w_c = Crest wave velocity

x = Distance downstream from the upstream gage

y = Depth of flow

y_n = Initial normal uniform steady depth at T=0

Subscripts:( maxx is max at x); (max0 is max at x=0)

Table 1  Field measurements and computations for depth contours and wave velocities during the rising part of the inflow hydrograph.

Table 2  Real-time computations for the cresting depths and times of arrival at downstream gaging sites.

Figure List:

Fig.1  Depth profiles of the illustrative example:

B=30.5, S_b=0.00080, n=0.017, y_n=0.61m, v_n=1.2m/s

Fig. 2  Significant waves and particle velocities at upstream site of inflow hydrograph, x=0.

Fig. 3  Real-time field measurements of wave velocities, while tracing contours of depths to downstream gaging sites.

Fig. 4  Real-time field announcements of expected downstream crests and times of arrival.

Fig. 5  Upstream section for practical application, Hun river, Liaoning province.

Fig. 6  Downstream section for practical application, Hun river, Liaoning province.

Fig. 7  Profile of upstream and downstream depths at peak time.

Fig. 8  Upstream and downstream crests and wave velocities during tracing of contours.

Fig. 9  Upstream and downstream discharges during tracing of contours.