Discharge As Related to Stream System Morphology

DISCHARGE AS RELATED TO STREAM SYSTEM MORPHOLOGY

JOHN B. STALL and YU-SI FOK Engineer and Hydrologist Illinois State Water Survey Urbana, Illinois

SUMMARY The basin of the Embarras River in Illinois is 2,400 square miles in size and about 90 percent of the land is cultivated. The streams have relatively shallow gradients and the basin is typical of much of the mid-western USA. The Embarras has a drop of 317 feet throughout its course of 202 miles, for an average slope of 1.6 feet per mile. The morphological structure of the stream system has been analyzed using the Horton-Strahler stream ordering system. The stream network conforms to an excellent degree to the following laws of Horton: the law of stream numbers, the law of stream lengths, and the law of stream slopes. The sinuosity of streams in the basin is shown to increase geometrically with the stream order. Discharges at eight stream gaging stations within the basin are shown to be related to stream order and to the frequency of occurrence of the discharge. Discharge measurements of the eight stream gaging stations provide rating curves through which other hydraulic factors of the stream can be related to stream order.

RÉSUMÉ Le bassin de ia rivière Embarras en Illinois est grand de 2.400 milles carrés dont 90% sont cultivés. Les rivières ont des pentes relativement faibles et le bassin est typique pour le Middle-West. L'Embarras a une chute de 317 pieds sur son cours de 202 milles avec une pente moyenne de 1,6 pied par mille. La structure morphologique du système de la rivière a été analysée en utilisant le système de classification Horton-Strahlef. Le réseau fluvial de la rivière se conforme excellement aux lois suivantes de Horton : la loi du nombre de rivières, la loi de la longueur et la loi des pentes. La sinuosité des rivières dans le bassin augmente géométriquement avec l'ordre de la rivière. Les débits dans huit stations de jaugeage dans le bassin sont en relation avec l'ordre de la rivière et avec la fréquence de production du débit. Ces mesures de débits donnent des courbes de tarage grâce auxquelles d'autres facteurs hydrauliques peuvent être mis en relation avec l'ordre de la rivière.

INTRODUCTION

Some of the important hydraulic characteristics of a stream channel are the depth, width, velocity, cross-sectional area, and channel slope. These factors at a particular stream cross-section are inherently related to the amount of stream flow or discharge. In a pioneering study in 1953, Leopold and Maddock (>•) showed that there appears to be a consistent pattern in which many of these characteristics change along the course of a stream. These channel characteristics of natural streams are shown to constitute an interdependent system, Leopold and Maddock (*) illustrated how these factors could be evaluated quantitatively. They described this pattern of variability of stream factors as the hydraulic geometry of the stream system. In Illinois a comprehensive study has been made to evaluate and document the morphology of Illinois river systems; and secondly, to associate channel system para meters with hydrologie factors such as discharge. In this paper the Embarras River Basin is used to illustrate these associations. Qualitative Channel Morphology The physical laws of mechanics and hydraulics were used by Horton (-) in a pio neering and comprehensive study of importance to hydrology. Here, Horton revealed

224 a consistent pattern under which stream systems develop and to which they continually adjust. He showed that the number of streams, the length of streams, and the slope of streams were all related consistenily to stream order throughout any existing stream system. Later, revisions to the Horton stream ordering system were made by Strahler (s). The Strahler system of stream ordering has been used in this report. This stream ordering system provides a means of evaluating numerically the structure of a stream system.

Importunes to Hydrology

Several writers have illustrated that the Horton's system of stream orders and the associated laws ofstream development have hydrologie implications. In 1963 Wong (4| developed an empirical equation using multivariate analysis in which he showed that the mean annual floods in 90 basins in New England were associated mathematically with two parameters: 1) an indicator of basin size (from Horton's first and second laws) and 2) the other an indicator of basin slope (Horton's third law). Wong used a multivariate mathematical model and explained the variance of the flood flows to a coefficient of determination, B2 of 0.80. Studies of reservoir sedimentation in Illinois by Stall and Bartelli (5) confirmed the applicability of the Horton relationships on some Illinois basins. They also show the importance of such channel factors in explaining sediment movement. For 20 watersheds in the Springfield Plain physiographic division of west-central Illinois, the mean slope of the third order streams was found to be an important quantitative measure in explaining the sediment delivered to a reservoir. Writings by Roehl (s) and Miller (7) also discussed the promise which morphological factors offer in understanding sedi mentation.

THE BASIN

The Embarras River is located in south-central Illinois in midwestern USA. It is considered representative of much of the agricultural land of the midwest. The Embar ras has a drainage basin of 2,400 square miles. About 90 percent of the land is cultivated. Stream gradients are relatively shallow; the Embarras lias a drop of 317 feet throughout its course of 202 miles for an average slope of 1.6 feet per mile, in figure 1 is shown a generalized map of the stream system of the Embarras River. Shown in figure 1 are the locations of eight stream gaging stations within the basin. Continuous records of discharge at these stations are available for time periods ranging from three years to 50 years.

Suspended Sediment The Embarras River system carries a considerable load of sediment. For 35 years a comprehensive research program has been carried out to measure and evaluate the amount of sediment carried by streams in Illinois and deposited in reservoirs. Results from this program have shown that the sediment carried by streams in Illinois is fine in texture and size, and is carried by the stream as wash load. Because of this, the amount of sediment carried in suspension by a stream is more dependent on the soil, farming conditions, and the occurrence of erosion-producing storms on the watershed than it is on the composition of the bed of the inflowing stream. The complex of factors affecting reservoir sedimentation in a Illinois reservoir has been described in detail for Crab Oichard Lake (8>. Size distribution analyses made on 303 sediment samples collected from 32 reser voirs in Illinois confirmed the very fine nature of the sediment. This showed that two- thirds of the sediment deposited tn Illinois reservoirs is predominently silt and clay

225 wr '&

.3-. .4/- r-C.3, V3-. '3; 4/ >

/ V

; 3* 3,'

NEWTON

STE. MARIE..

MILES IO o

IO O IO uu L-mi I I —1 KILOMETERS

Fig. 1 — Embarras River Stream System, Stream Orders and Gaging Stations. sized particles. The movement of this sediment through Illinois streams and reservoirs has been described in an earlier paper by Stall (9). In the present study of the channel morphology of the Embarras River system a separate evaluation has not been made of the bed sediments of the river and their effects on the channel morphology. Because of the extremely fine nature of the sediment and the fact that it is carried as wash load'it is felt to have a relatively minor effect upon the channel shapes. Because of the extremely low gradients of the Embarras River system, it is felt reasonable to neglect, essentially, the effects of the sediment carried through the river system as wash had. In most areas of the USA it would not be possible to neglect the sediment factor as done in this study; because in most other places sediment and the nature of the bed material would have a highly important bearing on the channel morphology. In Illinois and the Embarras River basin, however, the effects of sediment have been considered small and have not been subject to a separate evaluation in this study.

3 4 STREAM ORDER

Fig. 2 — Channel Morphologic Factors as Related to Stream Order.

227 CHANNEL MORPHOLOGY

The streams of the Embarras River basin were assigned stream order numbers based upon the method described by Strahler (3). The maps utilized for this stream ordering were the standard fifteen-minute topographic maps published by the U.S. Geological Survey. These maps had a scale of about I inch to I mile. First order streams were considered to be those shown on the topographic maps in blue as unbranched tributaries. A second order stream was formed by the joining of two of these unbranched tributaries. The second order stream extends on downstream to the point where it is joined by another second order stream to form a third order stream, etc. After the stream system was ordered, the same USGS topographic maps were uti lized to determine the number of streams in each order. The length of each stream was measured, and the slope of every stream segment was measured for third order and higher streams. The results of this analysis provided the channel morphology factors needed to evaluate the Horton stream system parameters for the Embarras River basin. These factors arc shown in table I arid are shown graphically in figure 2. The graph in figure 2 shows on the vertical scale the number of streams, stream length, and stream slope and on the horizontal scale the Horton order number. As can be seen from the upper three straight lines in figure 2, the stream numbers, the stream lengths, and the stream slopes for the Embarras River basin follow straight lines very well. This shows the relatively high degree to which the channel system of the Embarras River basin conforms to Horton's law of stream numbers, the law of stream lengths, and the law of stream slopes. These three laws can be written in the form of empirical equations as follows: Law of Stream Numbers Log Nu = 8.85-1.47 U Law of Average Stream Lengths Log£ = -2.8S-H.25 V Law of Average Stream Slopes LogS = 2.82-0.81 V Law of Average Sinuosity Log Sinuo = -0.04 + 0.05 U where

Nu number of sreams ; U Horton-Strafiler stream order; L stream length, miles; S stream slope, feet per mile; Sinuo stream sinuosity, ratio.

In order to provide an understanding of the variability of some of these channel morphology factors, results are shown in the graph in figure 3 of the variability of the stream slopes for the third order streams. While the average channel slopes follow Horton's law of stream slopes very well, it has been noted by others (10) that the variability of the stream parameters themselves may be badly skewed to the right. The graph in figure 3, shows a distribution curve of the stream slopes for the third order streams in the Embarras River basin. This is a sampling of the total of 85 streams as shown in table I. The horizontal scale is logarithmic which provides for a reasonably well-shaped bell curve representing the distribution of these stream slopes. Without the logarithmic transformation on the horizontal scale this distribution is badly skewed to the right as described earlier. As has been pointed out, however, this skew is pro bably present in many of the stream parameters measured and, in spite of the presence

228 of this skew, the stream parameters do follow Horton's taw very well as shown in the graph in figure 2.

^ t S

ENT OF OCCURRENCE SI S '0 7/ 0, \\\ el

••" \ - HISl [AtfTWflf

n 1 Il ' ' 1 0 OS 1.0 15 2.0 2.S 3.0 3.5 4.0 LOGARITHM OF 3(d-0RDER STREAM SLOPES

Fig, 3 — Variability of Slopes of Third-Order Streams.

TABLE I Channel Morphology Factors, Embarras River Basin

Number Average Average Average Stream of Length, Slope, Sinuosity, Order Streams miles feet per mile ratio

1 2 367 3 85 4.1 10.5 1.11 4 18 8.1 5.7 1.11 5 4 23 3.2 1.30 6 1 126 1.3 1.35

Simiosily

Other writers (n) have investigated the sinuosity of a stream as being a reflection of the hydraulics of a channel. In the belief that the sinuosity of the various stream orders within the Embarras River basin might be a reflection of important hydrologie para meters, an evaluation was made of this factor throughout the stream system. In figure 4

229 Fig. 4 — Examples of the Determination of Sinuosity. are shown examples of the determination of sinuosity within the Embarras River System. The four maps shown in figure 4 represent a section of the Embarras River channel system for a third order stream at the top, a fourth order stream, a fifth order stream, and a sixth order stream. In each case the channel proper is shown, as well as the flood plain of the stream. In the lower two graphs, for the fifth and sixth order streams, the figure also shows a generalized centerline of the flood plain. This centerline was used to measure the average down valley distance. For any particular stream segment the sinuosity was computed as being the actual stream length divided by the down valley distance. The sinuosity for the various stream orders within the Embarras River basin are shown in table I. This was shown to be 1.11

230 for both the third order and fourth order streams. The sinuosity increased to 1.30 for the fifth order stream and 1.35 for the sixth order stream. Shown as the lower-most straight line in figure 2, is the plotting of average sinuosity versus stream order. It does seem that, for the Embarras River basin, the sinuosity tends to be associated with stream order. Further studies of sinuosity are underway and it is felt to be a promising parameter in the better understanding of the stream system development.

DJSCHARGE

In order to associate various discharge phenomena with channel morphology in the Embarras stream system use was made of the discharge records at the eight stream gaging stations shown in figure 1. One of the most basic methods of evaluation of the variability of discharges is by developing flow-du ration curves. This was carried out for the eight gaging stations where records were available. In figure 5, is shown a flow duration curve for the Embarras River at St. Marie, Illinois, based on records from the stream gaging station as shown in figure 1. At St. Marie, the drainage area is about 1,540 square miles. The flow-duration curve shown in figure 5, is for the fifteen year- period, water years 1950-1964. As can be seen from the curve in figure 5 the variability of flow in the Embarras River is moderate. The position and slope of this line is fairly representative of streams in Illinois.

100

ems ARRAS RIV «w ^ AT ST. MARIE 's

X N \J

\ \

0.1 1 10 SO 10 11 11 5 TIME IN PERCENT OF TOTAL PERIOD

Fig. 5 — Flow-Duration of Daily Discharges, Embarras River at St. Marie.

From the eight flow duration curves available in the Embarras River basin it was possible to read the discharge expected at the eight gaging stations for various frequen cies of recurrence. In order to evaluate the association of discharge with the channel morphology parameters derived earlier, a comparison was made between the gages at a constant frequency of discharge. For example, the median discharge at each of the eight stations was determined. The association of these eight discharges were then associated with the channel morphology factors. An example of this is shown in figure 6. In figure 6 on the horizontal scale, is shown the stream order. On the vertical scale is shown the discharge incu.ft per sec. for the Embarras River stream system. The middle

231 curve in figure 6, represents tile association of the median discharge at the 8 stations, with stream order. As can be seen, there is a general association between these two para meters. The upper curve in figure 6, shows the association of discharges, each expected at a 10 percent frequency, at the 8 stations. As can be seen in figure 6, the discharges representing 10 percent frequency cluster about the line better than those representing the 50 percent frequency. The lowermost line in figure 6, shows the discharges expected to occur 90 percent of the time. These points determine a line that is less well designated than at the higher frequencies. The three lines shown in figure 6 are only three of a total of ten which were associated in this manner. The lines were fitted by eye in such a manner as to fit best the entire rangeof frequencies. The fact that the lOpercent frequency discharges seem to be associated with stream order to a better extent than the 90 percent discharges, suggests that perhaps the relatively infrequent flood events have a greater importance

STREAM ORDER

Fig. 6 — Discharge as Related to Stream Order and Frequency.

232 in forming the stream system than do the common events expected 90 percent of the time. The relations shown in figure 6 are far from being well defined, but it is believed significant that a definite (rend is noticeable. Exploratory studies, only, have been made in efforts to associate other hydraulic channel factors with channel morphology. An example of the direction of this explo ratory effort is shown in figure 7. In figure 7 are shown a series of rating curves, which have been developed for the Embarras River at Newton, at the location as shown in figure 1. The data utilized for the development of these rating curves were made avail able from field measurements of discharge which were collected by the U.S. Geological Survey over the period of years at a permanent, complete-record stream gaging station, operated at Newton. The plotted points in figure 7 were derived from 38 discharge measurements thus available. At the time the discharges were measured, the width of the stream, the average velocity, and the cross-sect ion a I area of the stream were also measured directly in the field. The upper curve in figure 7 shows the cross-sectional area as a function of discharge. The next curve shows the stream width; the lowermost curve shows the average velocity of the stream.

10 100 1,000 OISCHABGE IN Cf5

Fig. 7 — Hydraulic Factors for Embarras River at Newton.

The fourth curve in figure 7, is derived by dividing the stream cross-sectional area by the width, to obtain a value of average depth. Because of the variability of local conditions over the period of 3 or 4 years during which time these 38 discharge measurements were made, there is some variability in the hydraulic parameters plotted in figure 7. A general trend is evident, however. In spite of the variability of the measured factors, it is felt there is a reasonable association between these factors and discharge. Further evidence as to the realism of the actual measurements used to derive figure 7 is gained by plotting the actual shapes of the cross- sections measured at the various discharge measurements. By this means it was possible

233 to determine the bankfull capacity of the stream; to notice the slightly higher discharges during which the water impinged on the flood plain, and the very high discharges at which the flow covers the entire flood plain to a relative deep extent. The curves shown in figure 7 are generalized curves drawn by eye to represent the points. Great care has been given, however, to provide curves which are consistent with each other. For example, if a vertical section is taken through figure 7 at any par ticular discharge, it will be found that the values read from the curves will satisfy the physical laws that width times depth must equal area; and area times velocity must equal discharge. It is by means of generalized rating curves such as shown in figure 7 that it is hoped to show by further study that some of these hydraulic parameters may be associated with channel morphology.

CONCLUSIONS

Analysis of the stream system of the Embarras River reveals that the structure of the stream system follows to a reasonable degree the following laws: the law of stream numbers, the law of stream lengths, and the law of stream slopes. Measurements of the sinuosity of streams provide information which suggests that stream sinuosity increases with stream order. Sinuosity is suggested as a parameter which is highly revealing of the nature of the stream system and the pattern to which the stream system is continuously adjusting. Flow-duration curves provide information on the frequencies of discharges expected throughout the Embarras River basin. Discharge amounts are shown to be generally associated with stream order. The relatively rare events seem to be best associated with channel morphology. Discharge events having a frequency of 10 percent of the time are best associated with stream order. Median discharges are fairly well associated with stream order; and discharges expected 90 percent of the time are rather poorly associ ated with stream order. It is suggested that the discharges occurring 10 percent ofthe time, the relatively rare events, seem to be most important in the development of the stream system. This conclusion is due to the fact that these discharges seem to be best associated with channel morphology, as reflected by the stream order. At a particular location on a stream, generalized relations have been developed for hydraulic parameters such as cross-sectional area, width, average depth, and average velocity. By means of these generalized associations, it may be possible to reveal asso ciations between these hydraulic parameters and channel morphology.

AC KNOWLËDGMENT

This investigation was supported in part by grant No. 14-01-0001-1021, Office of Water Resources Research, U.S. Dept. of Interior.

REFERENCES I1) LEOPOLD, Luna B. and MADDOCK, Thomas, The Hydraulic Geometry of Stream Channels and Some Physiographic Implications, U.S. Geological Survey Prof. Paper 252, Washington, D.C. 1953. (s) HORTON, Robert E., Erosional Development of Streams and Their Drainage Basins—Hydrophysical Approach to Quantitative Morphology, Geological Society America Bulletin, Vol. 56, No. 3, pp. 275-370, 1945. (3) STRAHLER, A,N,, Quantitative Analysis of Watershed Geomorphology, Trans-, actions American Geophys. Union, Vol. 38, No. 6, pp. 913-920, December 1957.

234 (4; WONG, Shue Tuck., A Multivariate Statistical Model for Predicting Mean Annual Flood in New England, Annals Association American Geographers, Vol. 53, No. 3, pp. 298-311, September 1963. (°) STALL, John B. and BARTELLI, L.J., Correlation of Reservoir Sedimentation and Watershed Factors, Springfield Plain, Illinois: Illinois Stale Water Survey Kept. of Inms., 37, 24 pp., 1959. (6) ROEHL, John W., Sediment Source Areas, Delivery Ratios, and Influencing Morphological Factors, Pubi. No. 59, International Association Scientific Hydrology, pp. 202-213, 1963. ('I MILLER, Carl R,, Advances In Sedimentation Relevant to Watershed Problems, Transactions American Society Agricultural Engineer, Vol. 8, No. 1, pp. 146-152, 1965. (8) STALL, John B., et a!., Water and Land Resources of the Crab Orchard Lake Basin, Illinois Slate Water Surrey Bulletin, 42, 1954. (s) STALL, John B., Sediment Movement and Deposition Patterns in Illinois Impounding Reservoirs, Journal American Water Works Assoc, Vol. 56, No. 6, June 1964. (I0; STRAHLER, Arthur N., Quantitative Geo morphology of Drainage Basins and Channel Networks, Section 4-11, Handbook of Applied Hydrology, McGraw-Hill, New York 1964. (ll) SCHUMM, S.A., A Tentative Classification of Alluvial River Channels, U.S. Geological Surrey Circular 477, Washington, D.C. 1963. DISCUSSION

Intervention of Mr. P. O. WOLF Question: Expressed his appreciation of the research reported and asked for explanations on the diagrams showing that intermediate stream orders were assumed and that the results of channel flow should not be plotted on the same diagrams as those of overland flood flow. As to the choice of independent variables against which to plot the dependent ones, it was suggested that the discussion at the recent Fort Collins Symposium, on Marisa was's paper, would be of interest to the authors. Answer: Our study was limited to a range of flow frequency where 0.1 < F < 0.9. This means that the overbank flood flows were excluded. We will study Morisawa's paper as suggested whenever the publication becomes available. Intervention of Dr. Joseph P.B. M. OUMA (E. Africa, Kenya) Question: Mr Stall, I am interested in your sinuosity —number curve. 1 believe you have tested the significance of the regression lines you have presented. If you have, how significant was your sinuosity-number curve ? Answer: We have not tested the significance of the linear regression line of the sinuosity —stream order relationship. However, for the fourteen river basins we studied, the linear relationship between sinuosity and stream order in a semi-log paper seems, reasonable.

Intervention of Dr. 1. DOUGLAS Question: May 1 point out that work on the morphometric analysis of streams using stream orders of fractional sizes, e.g. 6.8, was carried out by geographers under Professor J. Tricart at Strasbourg and published in the Revue de Géomorphologie Dynamique and Annales de Géographie in about 1962. Answer: We are pleased to have information from Dr. Douglas regarding the report of Prof. Tricart in the use of fractional stream order; this matter was also questioned by Mr. Wolf. Our use of the fractional stream order is still under study and refinement. We believe that this concept is not critical in the current paper and that the presented results will not be invalidated in this respect.

235