PRELIMINARY INVESTIGATION OF PRACTICAL METHODS FOR ESTIMATING LOW-FLOW FREQUENCIES OF STREAMS IN SOUTHWESTERN MONTANA
A report to Yellowstone Tongue Areawide Planning Organization, Broadus, Montana
From Earth Science Services, Inc., Bozeman, Montana
PURPOSE OF INVESTIGATION
The purpose of this investigation was twofold. The main purpose was to examine the accuracy of certain practical methods for estimating low-flow frequencies of (1) ungaged streams and (2) streams with short periods of streamflow records. A secondary purpose was to produce low-flow frequency estimates for the Little Powder River near Broadus, Montana.
METHOD FOR UNGAGED STREAMS
A method for estimating the flow-duration curves for ungaged streams was described by Wisler and Brater (1959). In this method, two gaged streams are used to estimate the flow-duration curve for an ungaged stream. The two gaged streams selected are ones that have drainage basin characteristics most like the characteristics of the ungaged stream. Flow-duration curves are plotted for the gaged streams, and the flow-duration curve of the ungaged stream is estimated by interpolating a curve between the two known curves. However, in the present study, it was necessary to extrapolate (instead of interpolate) from the two known curves, because the ungaged stream (simulated by the Little Powder River) was expected to have low-flow characteristics that were not intermediate to the characteristics of the two gaged streams. This unfortunate circumstance existed because no appropriate gaged streams were found that were expected to have flow-duration curves that would bracket the curve of the Little Powder River.
The two gaged streams selected for comparison streams in the present investigation were Little Beaver Creek and Box Elder Creek. These were the only streams in the area which had long periods of streamflow record and which could be expected to have a flow regimen similar to the Little Power River. Other streams with long periods of record have tributaries in high mountain areas, whereas the Little Powder River does not. Melting of the mountain snowpack was expected to produce a significantly different flow-regimen. The drainage basin characteristics of the Little Powder River and the two comparison streams are shown in Table 1.
Table 1. Drainage basin characteristics.
Basin width is the ratio: Drainage area/Main channel length.
Basin shape is the ratio: Main channel length/Basin width.
Average slope of the basin is computed from the average slope along lines across the width of the basin located at one-third and two thirds of the distance from the stream gage to the upstream end of the basin. The slope from the two-thirds point to the upstream end of the basin is also included in the average.
Subsurface volume is average height of the basin land-surface above the gage multiplied by the basin area.
Main valley depth is the average of the depth of the stream below the drainage basin divides at the one-third and two-thirds points.
Main valley slope is an approximation of the gradient of the main stream.
Drainage density index is the number of streams intersected by straight lines drawn in various directions on the 1:250,000 topographic map.
BTF is Bainville-Tullock-Fort Collins; BF is Bainville-Fort Collins; LP is Lismas-Pierre (Southard, 1973)
Tfu is Fort Union Formation; Khc is Hell Creek Formation; Kfh is Fox Hills Sandstone; Kp is Pierre Shale.
Flow duration curves were prepared for the two gaged streams using the method described by Searcy (1959) with the modification proposed by Willeke (1967). In this method, the labor needed to produce the curves is reduced by using a random sampling technique. Little beaver Creek near Marmarth, North Dakota was used as an index station because the period of record at this station is the longest (from 1939). The flow-duration curve of Box Elder Creek was adjusted to a 1934-1974 period of record of the index station by the method described by Searcy (1959).
A flow-duration curve for the Little Powder River was produced by extrapolation from the curves of the two gaged streams. The curve produced this way is shown in Figure 1 along with the curves of the gaged streams. No Little Powder River streamflow data or statistics were used; the Little Powder River was treated as if it were an ungaged stream. The extrapolation was based on the following considerations:
The discharge ratio (discharge rate/mean annual discharge) at high flow rates is likely to be less for the Little Powder River than for either of the comparison streams because (1) its basin area is larger and (2) it appears to have relatively more stream-connected alluvium to provide bank storage.
The middle part of the Little Powder River flow-duration curve is likely to have a lower slope than the comparison streams because of the greater amount of ground water storage in the basin combined with the larger width of the basin. This will tend to cause a slower baseflow recession. Water released from bank storage should also reduce the baseflow recession rate.
The lower part of the Little Powder River curve should tend to have a steep slope because of the effect of the abundant phreatophytes along the main stream. Transpiration by these plants would be expected to rapidly reduce the very low flows to zero when they occur in the growing season.
The area under the flow-duration curve must be equal to the mean annual discharge multiplied by 100, and when the discharge ratio is used, as in Figure 1, the area under each curve on the graph must be equal.
It is noteworthy that some characteristics listed in Table 1 have a relatively small effect on the flow-duration curve. The effect of basin size seems to be much larger. For example, even though Little Beaver Creek has less irrigated acreage per unit area than Box Elder Creek, it also has lower low-flows.
Figure 2 shows the accuracy of the flow-duration curve for Little Powder River that was estimated without using discharge data, as if the stream were ungaged. The greatest error in the low-flow part of the curve appears in the middle part. This type of error could probably be reduced in future estimates for other streams (1) by being more precise in adjusting the curve so that the area beneath it is 100, and (2) by plotting the curve on log-normal probability paper to check whether the middle part plots approximately as a straight line (it should). With these improvements in the method, future flow-duration curves for ungaged streams should tend to be more accurate than the one in Figure 2.
A 7-day low-flow frequency curve was also estimated for the Little Powder River before Little Powder River streamflow data were processed. This estimate was based on extrapolation from curves for Little Beaver Creek and Box Elder Creek. The results are shown in Figure 3 along with the curves used in making the estimate. The estimate was made by (1) extrapolating from Little Beaver Creek and Box Elder Creek curves for the period 1960-1973 on the basis of drainage area and then (2) adjusting the extrapolated curve to the 1939-1974 period by an amount close to the difference between the curves for Little Beaver Creek 1960-1973 and Little Beaver Creek 1939-74.
METHOD FOR STREAMS WITH SHORT PERIODS OF RECORD
A method for estimating the flow-duration curves for streams with short periods of record was 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 than adjusted to the long period of record by correlation of flow rates on the respective flow-duration curves.
The Little Powder River flow-duration curve that was estimated from one year of record (1972) is shown on Figure 2. This estimate adjusts the curve to the period of record of the index station (Little Beaver Creek, 1939-1974). The accuracy of the method is indicated by comparison with the curve that was derived from the entire period of record for the Little Powder River. The low-flows are underestimated. This underestimation is a result of the fact that 1972 was a year of high streamflow and the discharge relationship used in estimating the Little Powder River curve is not the same for periods of high flow as the relationship for more normal periods. It is likely that errors of this sort will occur if flow duration curves for other streams are made using the same method.
The effect of having a longer period of record was studied by using Little Powder River data for 1971 and 1972, a two year period. The results were about the same as for a one year period as may be seen in Figure 2.
LOW FLOW FREQUENCIES FOR THE LITTLE POWDER RIVER
A secondary purpose of this investigation was to produce low-flow frequency estimates for the Little Powder River. These estimates are shown in Figure 2 as the flow-duration curve adjusted from the actual period of record to the period of record for the index station (Little Beaver Creek, 1939-1974). As additional information, the unadjusted flow-duration curve for the period of record of the Little Powder River is also shown in Figure 2.
The 7-day low-flow frequency curve for the period of record of the Little Powder River is shown in Figure 3.
SUMMARY AND CONCLUSIONS
Data from the U.S. Geological Survey stream gage on the Little Powder River near Broadus, Montana were used to test two practical methods for estimating flow-duration curves for streams in southeastern Montana. One method was applied to the ungaged stream case, and the other method was applied to the case of gaged streams with short periods of record. The results are shown in Figure 2 and compared with the actual flow-duration curve for the Little Powder River that was derived from its entire period of record.
A similar experiment was performed to test the possibility of estimating low-flow frequency curves for streams with little or no record. The results are shown in Figure 3.
It is this investigator's opinion that these results are about as good as can be expected for streams in southeastern Montana. Some decrease in the amount of error for ungaged streams would probably be experienced in future estimates because of aforementioned improvements that can be made in the method.
Searcy, J. K. (1959): Flow duration curves; U.S. Geological Survey Water Supply Paper 1542-A.
Southard, A. R. (1972): Soils of Montana; Montana Agricultural Experiment Station Bulletin 621.
Willeke, G. E. (1967): Simplified flow duration curves; Water and Sewage Works, April 1967, p. 129-130.
Wisler, C. O. and E. F. Brater (1959): Hydrology, 2nd ed., John Wiley, 408 p.