ESTIMATED LOW-FLOW FREQUENCIES FOR SELECTED STREAMS IN SOUTHEASTERN MONTANA
A report to Yellowstone Tongue Areawide Planning Organization, Broadus, Montana
From Earth Science Services, Inc., Bozeman, Montana
January, 1977
INTRODUCTION
This report covers the second phase of an investigation of low-flow frequencies of selected streams in southeastern Montana. The first phase was concerned with examining the accuracy of a practical method of analysis for ungaged streams and streams with short periods of record; that phase also included the production of a flow-duration curve and a 7-day low-flow frequency curve for Little Powder River near Broadus, Montana. The purpose of the second phase of the investigation was to produce estimated flow-duration curves for four additional streams -- Great Porcupine Creek near Forsyth, Little Porcupine Creek near Forsyth, South Sunday Creek near Miles City, and North Sunday Creek near Miles City. These streams are all ungaged.
The methods used in this study were limited to those that required a relatively low level of effort. This limitation excluded (1) regional approaches that require much data gathering and processing, (2) statistical techniques that require a large amount of data processing and computation, (3) methods that require additional streamflow measurements, and (4) computer simulation.
In the following report, the method of investigation willl be covered first; then the results wtll be described and discussed.
METHOD
Flow-duration curves for the gaged streams were prepared by using the method described by Searcy (1959) with the modification proposed by Willeke (1967). Willeke's modification reduces the data-processing labor by using a random sample of the streamflow data for each stream instead of using the entire record. The random sample is large enough to produce a good approximation of the flow-duration curve that would be produced by using all of the data. The accuracy of this approximation was checked by comparing the approximate flow-duration curve for the Little Powder River with the actual curve based on the complete set of data. The computation of the points for the actual curve was provided by the Water Resources Division of the U.S. Geological Survey, Helena, Montana. The approximate curve was nearly identical with the actual curve.
Flow-duration curves for streams with short periods of record (Sand Creek, for example) were adjusted to long periods of record by the method described by Searcy (1959). In this method, a stream with a long period of record is chosen as an index station on the basis of similarity of basin characteristics. The flow-duration curve for the stream with a short period of record is then adjusted to the long period of record by correlation of flow rates on the respective flow-duration curves.
Flow-duration curves for ungaged streams were prepared by using a method described by Wisler and Brater (1959). In this method, flow-duration curves for nearby gaged streams are used to estimate the flow-duration curve for an ungaged stream. The estimation is based on similarity of drainage basin characteristics; and the following considerations are useful as aids in making the estimate:
When the flow-duration curve is made dimensionless by using the discharge ratio (discharge/mean discharge) as the ordinate, the area under the flow-duration curve must be one hundred.
Drainage basin size and annual precipitation are likely to have a considerably greater influence on the shape of the flow-duration curve than other drainage basin characteristics. This conclusion is supported by the studies of Thomas and Benson (1970). It is also consistent with plots of flow-duration curves for gaged streams that have been studied in the present investigation (phase I and phase II).
The large basins have lower peak discharge ratios and higher low-flow discharge ratios than the smaller basins when mean annual precipitation is about equal. One example of this influence of basin size is shown in Figure 3, where Big Dry Creek (2554 square miles) and Sand Creek (317 square miles) may be compared. These basins are in an area where the mean annual precipitation is less than thirteen inches. Another example may be seen in the report on phase I of this investigation (Earth Science Services, Inc., 1976, Figure 1), where Box Elder Creek (1,092 square miles) is plotted with Little Beaver Creek (587 square miles). Box Elder and Little Beaver Creeks are in an area of about fourteen inches mean annual precipitation. The influence of basin size on the shape of the flow-duration curve is not surprising, because theoretical studies (Butler, 1967) suggest that the slope of the baseflow recession curve (where it is not greatly affected by evapotranspiration and underflow) should be less steep for large basins than for small basins if the basins are otherwise homogeneous. Furthermore, it is this investigator's opinion that the average slope of this part of the baseflow recession curve is represented by the middle part of the flow-duration curve, and that the part of the baseflow recession that is most affected by evapotranspiration and underflow is represented by the lower, steeper part of the flow-duration curve. However, this interpretation of the parts of the flow-duration curve has not been established and should be regarded only as a hypothesis.
The flow-duration curves of large basins and small basins will bracket the curves of intermediate sized basins if other drainage basin characteristics (including mean annual precipitation) are homogeneous. This conclusion is a consequence of items 2 and 3. The flow-duration curve of an ungaged stream is plotted closest to the most similar gaged stream.
All of the basin characteristics except basin size and precipitation are used only for selecting the most similar gaged streams for use in estimating flow-duration curves of ungaged streams and streams with short periods of record. Using these characteristics to refine the estimates by multiple regression analysis is beyond the scope of this investigation.
The middle part of the flow-duration curve should plot as a nearly straight line on log-normal probability paper.
When the dimensionless flow-duration curves are converted to dimensioned curves, a value for mean annual discharge is needed. Since no such values exist for the ungaged streams, they were estimated. These estimates were made graphically from Figure 1. In this graph the mean annual discharge for selected gaged streams in the area are correlated with drainage basin size and mean annual precipitation. The curves were fitted by eye, and they are based on only a few points. Consequently, a high degree of confidence cannot be associated with the lines on this graph. However, the points fit the lines well; and the relationship shown is reasonable.