Nonlinear response of a small drainage basin model

Journal of Hydrology 14 (1971) 29-42; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NONLINEAR

RESPONSE

OF A SMALL

DRAINAGE BASIN MODEL* RAM A. RASTOGI U.P. Agricultural University, Pantnagar, U.P., India and BENJAMIN A. JONES, Jg. University of Illinois at Urbana-Champaign, Illinois, U.S.A. Abstract: A surface flow model utilizing the kinematic wave theory was applied to a thirdorder stream system which represented conditions in Williamson and Johnson Counties, Illinois. The model drainage basin was considered as a distributed hydrologic system with the stream network, channel characteristics and overland flow lengths as distributed hydrologic variables. Different order channel lengths, channel cross-sections and channel slopes were treated as spatially distributed. The results of the hydrologic response of the model small agricultural drainage basin showed that various time parameters are affected by change in rainfall-excessintensity. The peak flow rates for a given rainfall-excess duration showed a nonlinear response with the rainfall-excess intensities. The base or time duration of a direct run-off hydrograph increased with an increase in rainfall-excess intensity and duration. Introduction

S h e r m a n 1) i n t r o d u c e d the c o n c e p t o f the unit h y d r o g r a p h (or u n i t - g r a p h ) which is defined as a h y d r o g r a p h resulting f r o m one inch o f rainfall-excess o f a specified d u r a t i o n occurring u n i f o r m l y in time a n d space over the entire d r a i n a g e basin. Since its i n t r o d u c t i o n the unit h y d r o g r a p h has b e c o m e a f r e q u e n t l y used m e t h o d to p r e d i c t p e a k flood flows for the design a n d o p e r a t i o n o f h y d r a u l i c structures. V a r i o u s a t t e m p t s have been m a d e to utilize this m e t h o d to derive the m a t h e m a t i c a l t h e o r y for the i n s t a n t a n e o u s unit h y d r o g r a p h and to extend the t h e o r y to u n g a g e d d r a i n a g e basins by m e a n s o f " s y n t h e t i c unit h y d r o g r a p h " a n d " d i m e n s i o n l e s s h y d r o g r a p h " techniques. These h y d r o g r a p h s m a y be d e v e l o p e d if the basins have h o m o geneous h y d r o l o g i c c o n d i t i o n s a n d it is a s s u m e d t h a t the conversion o f rainfall-excess into surface r u n o f f is a linear process. The u n i t - g r a p h c o n c e p t a n d the basic a s s u m p t i o n s c o n c e r n i n g it, as * Contribution of project 10-319 of the Agricultural Experiment Station, University of Illinois, Urbana as a part of the work of NC-66 regional committee on hydrologic characterization of small watersheds. 29

30

RAM A. RASTOGI AND BENJAMIN A. JONES, JR.

discussed by Chow") and most standard textbooks, provide the basis for the unit hydrograph theory. Since the hydrologic phenomenon of direct runoff is so varied in time and space, it is practically impossibleto satisfy all of the assumptions. Hence the validity of the unit hydrograph theory has been questioned in recent years. Results of research with laboratory models by Amorocho4), Pabst s), Morgali 6) and Marcus 7) indicate that nonlinearity does exist. These results are applicable to laboratory channels and the degree of nonlinearity has not been established especially for complete drainage basins. This paper presents the results of a study of a mathematical model of flow from a complex hydrologic system which is space distributed, time invariant and nonlinear. The mathematical model is based on the equations of continuity and momentum and routes spatially varied unsteady flow through the overland and channel system of a third-order drainage basin which represented conditions for an area of 160 acres in Williamson and Johnson Counties in Illinois. The model has not been verified with field data because they are not available for this watershed.

The drainage basin model A drainage basin model was developed for a five square mile area and a drainage density of 14 mi/sq mi with an entire drainage network of different order streams and defined channel characteristics, using morphological relationships. These relations were developed from geomorphic characteristics of six small ungaged drainage basins in Illinois which had different drainage densities. A detailed description of the development of the basin model was given by Rastogi and Jones s). Although the basin model included one fifth-order stream, restraints which had to be placed on the study led to the decision to use a third-order basin for routing purposes. Figure 1 shows the details of the third-order drainage basin which was the basic unit of the model which represented an area of 160 acres. In nature, many variations exist in the length of the different order stream tributaries. Therefore, to closely correspond to the condition that existed in nature, two types of first- and second-order stream lengths, i.e. large and small, were used in the model to provide large and small first-order basins. In general, two first-order basins formed a second-order stream system. To form a third-order basin system, twelve large and four small first-order basins were used. Table 1 gives the details of the components of the third-order drainage basin model. Soil Conservation Service, U.S.D.A. data 9) showed that n values of 0.15 and 0.07 would be representative for overland and channel flow respectively for the drainage basins studied. The U.S. Geological Survey topographic map

NONLINEAR RESPONSE OF A SMALL DRAINAGE BASIN MODEL

31

B

D F

C

E

H

1

K

Fig. 1.

T h e m o d e l drainage basin representing third-order s t r e a m system.

for Williamson and Johnson Counties showed that the average overland slope was 15.95 percent. Therefore, these values were used for the drainage basin model. A general equation for Wagon Creek in Williamson and Johnson Counties, which represented a drainage density of about 14, was developed to determine the average slope of various order streams. The equation was of the form logSo = 1.374 - 0.32 o

(1)

in which So is the slope, in percent, of the streams of order o. The coefficient of correlation was - 0 . 9 3 and the standard error of estimate from the regression was 44.34 percent. This relationship was used to define slopes of various order streams in the model. In similar fashion an equation was developed for the channel cross-section at any location in the model drainage basin.

32

RAM

A. RASTOGI

AND

BENJAMIN

A. JONES,

JR.

"o

--

e4

0 0

0

I'-1

"0

0

e~

~.o.

0 0

O

O

e~ 0

8

d

--;o

od

O e~

.~= ,¢ eq

¢q ¢q

=_

NONLINEAR RESPONSE OF A SMALL DRAINAGE BASIN MODEL

33

Routing flow in the drainage basin model Precise solutions of the equations of motion and continuity describing the physical phenomenon of surface runoff from a drainage basin with exact boundary and initial conditions are unavailable at present. Nevertheless the work of Lightbilll°), Woodingn), Henderson12), Brakensiek13), Larson and Machmeier14), Woolhiser and LiggetOS), and HilP ~) resulted in the decision that the kinematic formulation for one-dimensional flow can be used to describe qualitatively the basic nature of the rainfall-runoffprocess. Woolhiser and Liggett 15) showed that the kinematic wave solution gives very accurate results for K > 10. (K is a dimensionless parameter suggested by them.) In this study K was found to vary between 25.85 and 1.41 x 104 indicating that the method was applicable to the assumed conditions. The kinematic method solves the continuity equation

aQ

~A

0x + a t

q

(2)

Q = Q (A, x)

(3)

and a rating function

in which Q is the flow rate, A is the flow area, x is the length coordinate, t is the time and q is the lateral inflow per unit length. The detailed development of a mathematical model utilizing the kinematic formulation for flow from the model drainage basin was presented by Rastogi17). Rainfall-excess was routed through the model in two parts simultaneously, namely overland flow, and channel flow. The overland flow came from the unit drainage basins that made up the model. The model consisted of large and small first-order basins of 0.01782 and 0.009043 square miles respectively. The overland flow lengths for these basins were 422.40 and 675.32 feet respectively. Rainfall-excess, ranging from 0.25 to 2.0 inches per hour intensities and 2 min to 120 min durations, was given as input to the overland flow strips of unit width. For all the intensities and durations, runoff hydrographs at the downstream end of the overland flow strips were computed during and after the lateral inflow periods by performing iteration solutions of the continuity equation in finite difference form with the flow rating functions given as the downstream boundary for the overland phase. As an example, the runoff hydrographs from these overland flow strips for a rainfall-excess intensity of 0.25 inch per hour and rainfall-excess durations of 60 and 120 min are shown in Figs. 2 and 3 respectively. In the numerical channel routing process each channel was treated separately. Various order streams were divided into channel sections in such

34

RAM A. RASTOGI AND BENJAMIN A. JONES, JR.

,4

'7,~ (ZZ)c~-

LJ(~ Z~. I

LLI

Oq--"

.oo

50.00

1oo.oo

iso.oo

200.00

2so.oo

~o.oo

~so.oo

,4oo.oo

TIME IN MINUTE5

Fig. 2. Overland flow hydrographs for a rainfall-excess intensity of 0.25 in/hr and a rainfall-excess duration o f 60 min. (I) Flow length = 0.08 mi; (2) flow length = 0.1279 mi.

2 z $ w

I

8

.~

i

~.~

i

,L.~

n

,~.®

~.®

~.~

~.~

,

t

I

350.130

I

qO0.1~

TIMF IN MINIITFS

Fig. 3. Overland flow hydrographs for a rainfall-excess intensity o f 0.25 in/hr and a rainfall-excess duration o f 120 min. (1) Flow length = 0.08 mi; (2) flow length = 0.1279 mi.

NONLINEARRESPONSEOF A SMALLDRAINAGEBASINMODEL

35

a way that resulting reach lengths were approximately equal to 105 feet in all order streams. The runoff from the overland phase entered into the first-order stream along the entire length. No flow entered at the upstream end and a flow rating function for a first-order stream served as the downstream boundary for routing flow up to junction points C and L Thus, at the upper end of each channel the input hydrograph was known. The flow rating function served as the downstream boundary and by performing iteration procedures solutions for runoff rates and times were made at the lower end of the channel. Outflow hydrograpbs from the various tributaries were combined by direct summation of the channel junctions which served as the inflow hydrographs at those junctions. Since the channel slopes were steep, ranging from 2.6 to 11.3 percent, it was assumed that any backwater effect was negligible and that no change in storage took place at the junction points. In this manner the outflow was routed to the basin outlet (K), for various rainfall-excess intensities and durations. For both the overland and channel routing a tolerance value of 0.000001 sq ft was used for iteration cutoff. A constant time increment, At, of one minute was used in routing. Discussion of results The nonlinear response of the basin model in relation to the assumption of linearity of the unit hydrograph theory is apparent in Table 2. This table presents the numerical values of various parameters at different locations in the drainage basin model for a selected rainfall-excess duration of 20 min.

TABLE 2

Numerical values of various time parameters for a rainfall-excess duration of 20 min at different locations in the drainage basin model Channel location

D r a i n a g e Intensity area (sq mi) (in/hr)

D

0.03564

E

0.07128

F

0.1069

J

0.2138

K

0.25

0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0

Lag time (min)

T99 (min)

T97 (rain)

32.99 25.34 35.24 27.08 34.21 26.29 38.94 29.95 46.59 35.79

58.80 43.64 59.59 44.67 59.61 44.40 61.89 46.37 66.87 50.58

47.39 35.77 48.56 36.80 48.20 36.58 50.50 38.52 54.92 42.19

Tp Qp (min) (in/hr) 24 22 25 23 24 22 27 24 31 27

0.329 0.785 0.316 0.764 0.307 0.753 0.301 0.738 0.280 0.694

36

RAM A. RASTOGI AND BENJAMIN A. JONES, JR.

At all the locations as the rainfall-excess intensities increase, the time to the peak decreases and the peak rate increases proportionally more than the increase in rainfall-excess intensity. This does not agree with the unit hydrograph theory. The results of this study showed a nonlinear response of the peak flow rates with rainfall-excess intensities as also shown in Fig. 4. This nonlinearity may be associated with the change in the flow velocity of the .j-

1

n-

!.i •

5 4

o

t

:0o

10.00

20.00

30.00

qO.O0

50.00

60.1)0

70.00

!

I

80.00

TIME IN MINUTES

Fig. 4. Influence of rainfall-excess intensity on the time distribution of surface runoff. Runoff hydrographs at drainage basin outlet. Rainfall-excess duration ~ 20 min. ; rainfallexcess rates (1) 0.25 in/hr, (2) 0.50 in/hr, (3) 1.00 in/hr, (4) 1.50 in/hr, (5) 2.00 in/hr

runoff water which may result from an increase in stage height caused by the change in the rainfall-excess intensity. This increase in the flow velocity reduces the time to the peak. Again the curves in Fig. 5 show that the rate of increase in the peak discharge is higher for higher rainfall-excess intensities and for shorter durations thus exhibiting non-linear behaviour. The peak flow rates for the drainage basin model for all of the durations and intensities tested and given in Table 3 were converted to the unit hydrograph peak flow rates as listed in Table 4. This was done by reducing the volume of runoff to one inch. These peak rates were plotted in Fig. 6 against rainfall-excess intensities for all the durations. By unit hydrograph theory the peak flow rates for a unit duration and various intensities should be equal and thus the lines should be horizontal. The changes in unit hydrograph peak

NONLINEARRESPONSEOF A SMALLDRAINAGEBASINMODEL

37

flows for the same duration and various intensities show a nonlinear relation. Izzard is) used the time to equilibrium as the time required for the outflow to reach 97 percent of the inflow rate, designated as 7"97. Larson and Machmeier 14) called this term the time to virtual equilibrium. Time to equilibrium may also be defined as the time for the outflow to reach 99 percent of the inflow rate, termed T99. According to the unit hydrograph theory both the 400

360 INTENSITY

IN/HR

2.00

320

280

03

o

240

1.5

Z t,..~ 200 tr
1.00

"xLLI

Q_ 120

80

o 0,50

40

0.25

%

,'5

3b

RAINFALL-EXCESS

,15

6b DURATION

:,'5

9'o

IN M I N U T E S

Fig. 5. Effect of rainfall-excess duration on the peak discharge rate of runoff hydrograph at the drainage basin outlet.

RAM A. RASTOGI AND BENJAMIN A. JONES, JR.

38

TABLE3 Peak flow rates for the model drainage basin Duration

(min) 2 5 7 10 14 20 30 60 90

0.25

1.305 3.671 5.301 7.769 11.053 17.435 26.930 38.147 40.013

Rainfall-excess intensity, in/hr 0.50 1.00

2.873 6.834 11.381 18.909 28.804 45.168 63.905 79.331 80.567

(c~) 6.164 19.270 31.170 50.856 77.958 111.992 143.992 160.852 161.322

1.50

2.00

9.588 34.588 55.637 90.067 134.768 185.676 226.083 241.773 241.999

13.781 22.288 83.733 133.896 196.722 262.821 308.552 322.542 322.669

TABLE4 Unit hydrograph peak flow rates for the model drainage basin Duration

(min) 2 5 7 10 14 20 30 60 90

Rainfall-excess intensity, in/hr 0.25

0.50

1.00

1.50

2.00

156.61 176.21 181.75 186.46 189.48 209.22 215.44 152.59 106.70

172.38 188.02 195.10 226.91 246.89 271.01 255.62 158.66 107.42

(cfs) 184.92 231.24 267.17 305.14 334.10 335.97 287.98 160.85 107.55

191.76 276.70 317.92 360.27 385.05 371.35 301.44 161.18 107.56

206.72 313.73 358.85 409.13 421.55 394.23 308.53 161.27 107.56

lag time a n d the time to e q u i l i b r i u m (T97 or T99) should be c o n s t a n t for all rainfall-excess intensities of given duration. However, in this study the lag time, defined as the time from the center of mass of the rainfall to the center of mass of the r u n o f f from the drainage basin, decreased with a n increase in b o t h the rainfall-excess intensity a n d duration. A g a i n a n increase in rainfallexcess intensity increases the stage a n d thus the flow velocity a n d decreases the lag time. Similarly, at all of the locations in the drainage basin (shown in T a b l e 2 also) b o t h T97 a n d T99 decreased with an increase in rainfall-excess

39

NONLINEAR RESPONSE OF A SMALL DRAINAGE BASIN MODEL

.....

500

• '

"i . . . .

'~

,

=

I

450 DURATION

MINUTES 14

IO 400

20

O3 L~

7 Z

350

Ld I--Q~

30O 0

~:~ UJ

250

I 2

~00

(_9 0 a I

_.~..

-~

e

-~

o

60

~,,

90

150

Z o..--

,..o

•

I00

50

0

' 0

I 0.5

,

I 1.00

,

I

RAINFALL-EXCESS INTENSITY IN Fig. 6.

,

,

1.5

I 2.00"

IN/HR

Effect of rainfall-excess intensity on unit hydrograph peak flow rates for the model drainage basin.

intensity. This occurred because as the rainfall-excess intensity increased the travel time decreased. In this study T97 was related to the rainfall-excess intensity at the basin outlet to the - 0 . 3 7 power. Larson and Machmeier 14) proposed an exponent o f - 0 . 2 3 , The larger size o f drainage basin used by them may be one reason for this difference. In general, the exponent decreased with basin size.

40

RAM A. RASTOGIAND BENJAMINA. JONES,JR.

To determine the effect of rainfall-excess intensity and duration on the time duration of runoff hydrographs, the time required for the flow to attain a value of 0.1 cfs on the recession segment of the hydrographs at the basin outlet was found for all of the test runs. The values, as shown in Fig. 7, were plotted against rainfall-excess intensity on logarithmic paper for 2 to 20 min duration only because the hydrographs for longer durations did not have sharp peaks. The assumption was made that the time from 0.1 cfs until the runoff ceases is independent of rainfall-excess intensity and duration. Table 5 shows that the time difference between 0.1 and 0.01 cfs flow rate is almost constant for all rainfall-excess durations and intensities and that the assumption was valid. 350

I

'

I

'

'

'

'1

I

DURATION

MINUTES

20 14 I0 7 5

O3 kd I-Z

2

Z O3 h t~

o O I" hi

200

I--

¢_

150 O.I

J 0.2

,

,

RAINFALL-EXCESS

,

I 0.5

,

,

,

,I

] ;).00

I.OO INTENSITY

IN

IN/HR

Fig. 7. Effect of rainfall-excess intensity on the time base of runoff hydrographs, Tb at the drainage basin outlet.

41

NONLINEAR RESPONSE OF A SMALL DRAINAGE BASIN MODEL TABLE 5

Time required for a decrease in flow rate from 0.1 to 0.01 cfs on the recession segment of the runoff hydrographs for the model drainage basin

Duration

(rain) 2 5 7 10 14 20 30 60 90

Rainfall-excess intensity, in/hr 0.25

0.50

1.00

1.50

2.00

186 185 184 183 185 183 183 183 182

186 183 183 182 183 182 183 182 183

(min) 183 182 183 183 183 183 182 183 183

183 184 182 182 182 182 182 182 182

182 182 182 183 182 183 182 182 182

Another assumption of the unit hydrograph theory is that the base or the time duration of the hydrograph of direct runoff is constant for all rainfallexcess storms of unit duration. The results of this study showed that the base or time duration of a direct runoff hydrograph increased with an increase in rainfall-excess intensity and duration and the equation relating these parameters was of the form logTb = 2.328 + 0.1 logD + 0.184 l o g / -

0.108 logD (logI)

(4)

where D is the rainfall-excess duration, and I is the rainfall-excess intensity. The r value for this equation was + 0.993. This result is in complete contrast with Minshalla), who analyzed data from small experimental drainage basins and showed that the base of the unit hydrograph increased with a decrease in rainfall intensity.

Summary The hydrologic response of a third-order drainage basin system representing a 0.25 square miles upland area with steep ground and channel slopes was investigated through the use of a mathematical model. The model used overland and channel flow routing based on the kinematic wave method. In describing the effects of rainfall characteristics on runoff hydrographs emphasis was given to the general behaviour of rainfall-runoff phenomenon rather than to absolute values. Verification of the mathematical model was not possible because rainfallrunoff records of small gaged drainage basins having identical tributaries

42

RAM A. RASTOGI AND BENJAMINA. JONES, JR.

were n o t available. H e n c e the m o d e l should only be used for theoretical studies.

Acknowledgements T h e a u t h o r s wish to t h a n k Dr. Ven Te C h o w o f Civil E n g i n e e r i n g D e p a r t ment, U n i v e r s i t y o f Illinois at U r b a n a - C h a m p a i g n an d Dr. K. P. Singh o f the Illinois State W a t e r Survey, C h a m p a i g n , for m a n y useful discussions.

References 1) L. K. Sherman, Steam flow from rainfall by the unit graph method. Engineering News Record, 108 (1932) 501 2) V. T. Chow, Runoff. Section 14 in Handbook of Applied Hydrology edited by V. T. Chow (McGraw-Hill, New York, 1964) 3) N.E. Minshall, Predicting storm runoff on small experimental watersheds. J. hydraulics Division, Proc. Amer. Soc. Civ. Eng., 86 No. HY-8 (1960) 17-38 4) J. Amorocho, Measures of the linearity of hydrologic systems. J. Geophys. Res. 68 (1963) 2237-2249 5) A. F. Pabst, Hydrograph linearity in an elementary channel. M.S. Thesis, University of Minnesota (1966) 6) J. R. Morgali, Hydraulic behaviour of small drainage basins. Tech. Report No. 30, Stanford University (1963) 7) N. Marcus, A laboratory and analytical study of surface runoff under moving rainstorms. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1968) 8) R. A. Rastogi and B. A. Jones, Jr., Simulation and hydrologic response of a drainage net of a small agricultural drainage basin. Trans. Amer. Soc. Agr. Engnrs. 12 (1969) 899-908 9) R. W. Martin, Soil Conservation Service, Champaign, Illinois (1968) private communication 10) M. J. Lightbilt and G. W. Whitham, On kinematic waves. I. Flood movement in long rivers. Proc. Royal Soc. Series A, 229 (1965) 281-316 1l) R. A. Wooding, A hydraulic model for catchment-stream problem. 1. Kinematic wave theory. J. Hydrol. 3 (1965) 254-267 12) F. M. Henderson, Flood waves in prismatic channels. J. Hydraulics Division, Proc. Amer. Soc. Civ. Eng. 89 No. HY-4 (1963) 39-67 13) D. L. Brakensiek, Kinematic flood routing. Trans. Amer. Soc. Agr. Engnrs. 10 (1967) 340-343 14) C. L. Larson and R. E. Machmeier, Peak flow and critical duration for small watersheds. Trans. Amer. Soc. Agr. Engnrs. 11 (1968) 208-213 15) D. A. Woolhiser and J. A. Liggett, Unsteady, one dimensional flow over a plane - the rising hydrograph. Water Resources Res. 3 (1967) 753-771 16) I. K. Hill, Runoff hydrograph as a function of rainfall excess. Water Resources Res. 5 (1969) 95-102 17) R. A. Rastogi, Morphological analysis and hydrologic response of small agricultural drainage basins. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1968) 18) C. F. Izzard, The surface profile of overland flow. Trans. Amer. Geophys. Union VI (1944) 959-968