A report to Yellowstone Tongue Areawide Planning Organization, Broadus, Montana

From Earth Science Services, Inc., Bozeman, Montana

By Darrel E. Dunn

January, 1977


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.


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:

  1. 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.

  2. 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).

  3. 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.

  4. 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.

  5. 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.

  6. 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.

Mean annual discharge vs. drainage area and precipitation

The low-flow frequency curves were prepared by using the method described by Riggs (1965). This is a distribution-free method.

The Indexes to drainage basin characteristics (Table 1) were defined and measured as follows:

  1. Drainage basin areas for gaged basins were obtained from the U.S. Geological Survey reports on surface water records. For ungaged streams, the areas were estimated from 1:250,000 topographic maps by using a transparent grid overlay. Each grid-square on the overlay had an area of one square mile.

  2. Mean annual precipitation was estimated from a map of average annual precipitation for Montana. The map is one that has beeen prepared by the U.S. Soil Conservation Service, Bozeman, Montana. Its base period is 1953-1967. In addition, precipitation maps of the U.S. Weather Bureau were examined.

  3. Basin width is the drainage basin area divided by the length of the main valley of the basin. This length was measured down the center of the valley starting from the upstream divide.

  4. Basin shape is the main valley length divided by the basin width.

  5. The average slope of the basin was computed by drawing lines across the basin at one-third and two-thirds of the distance from the stream gage (or the stream mouth of ungaged streams) to the upstream end of the basin. These lines were drawn normal to the main valley. Another line was drawn from the upstream end of the basin to the aforementioned two-thirds point. This produced five line-segments. One end of each segment was at a drainage divide and the other end was at stream-level. The slope of each line-segment was computed by subtracting the altitude at the stream-end from the altitude at the divide-end and dividing by the length of the line. The average slope of the five lines was used as the index for the average slope of the basin.

  6. The subsurface volume of the drainage basin is the bulk-volume of earth material above the altitude of the downstream reference point (stream gage or stream mouth) and below the land surface. It was estimated from the 1:250,000 topographic map by (1) estimating the average elevation of the land surface in each township above the basin reference point, (2) weighting these elevations according to the proportion of the township within the drainage basin boundaries, (3) averaging these weighted elevations to obtain an average elevation for the drainage basin, and (4) multiplying this average elevation above the reference point by the drainage basin area to obtain the subsurface volume.

  7. The main valley depth index is the average of the depths of the drainage basin at the one-third and two-thirds points mentioned above. The drainage basin depths at these points were calculated by subtracting the altitude of the stream from the average of the divide at the two ends of the aforementioned line normal to the stream.

  8. The main valley slope is an index produced by subtracting the altitude of the downstream reference point (stream gage or stream mouth) from the altitude of the farthest upstream point where the main stream is clearly larger than its tributaries. The elevation thus obtained is divided by the distance w\between the two points to obtain the slope.

  9. The drainage density index is the number of streams per mile intersected by straight lines drawn in various directions on the 1:250,000 topographic map.

  10. The percent woodland is the percent of green shaded area on the 1:250,000 topographic maps.

  11. Soil types were taken from the soil map of Southard (1973). Detailed soil maps were also studied when they were available.

  12. Bedrock and alluvial area were taken from the state geologic map of Montana, but detailed geologic maps were also studied when they were available.

Table 1 (continued). Drainage basin characteristics.

(1) BA is Badlands; BC is Bainville-Cushman; BFB is Bainville-Flasher-Badlands; LP is Lismas-Pierre.

(2) Tfu is Fort Union Formation; Khc is Hell Creek Formation; Kfh is Fox Hills Sandstone; Kb is Bearpaw Shale; Kjr is Judith River Formation; Kcl is Claggett Formation; Keu is Eagle Sandstone; and Kc is Colorado Shale.


Great Porcupine Creek, Little Porcupine Creek, South Sunday Creek and North Sunday Creek are ungaged. Their flow-duration curves were estimated by comparison with the curves for Big Dry Creek near Van Norman (U.S. Geological Survey station number 6-1310) and Sand Creek near Jordan (6-1307). These two gaged streams were selected because they are located within the same low-precipitation area as the ungaged streams being studied, and because Big dry Creek is larger than any of the ungaged streams while Sand Creek is smaller than any of them. Consequently, since the other basin characteristics are not excessively dissimilar (see Table 1), the dimensionless curves of the ungaged streams are likely to be between the curves of the two gaged streams. The estimated dimensionless curves are shown in Figure 2. The corresponding dimensioned flow-duration curves are shown in figure 3

One year of streamflow record (1975 water year) was available for Sunday Creek below the confluence of North Sunday Creek and South Sunday Creek. However, that record was not used to generate a flow-duration curve for Sunday Creek, because the nature of the precipitation distribution in the 1975 water year was such that the direct runoff in the Sunday Creek drainage was probably abnormally high compared to the direct runoff in nearby basins with long periods of record. This condition is indicated by the following facts: (1) about fifty-nine percent of the total annual flow for the 1975 water year in the Sunday Creek drainage was direct runoff from storms that occurred between May 5 and May 8, and (2) precipitation during this period was much greater at Miles City, near the mouth of Sunday Creek, than elsewhere in southeastern Montana. This extra precipitation at Miles City would be expected to cause a flow-duration curve estimated from correlating Sunday Creek streamflow data with nearby streams to be much too high on the graph. Such poor correlation of short-tern streamflow in adjacent intermittent streams can occur in southeastern Montana because a large part of the annual discharge of an intermittent stream for a particular year may be generated by one or two large rainstorms, and the intensity of these rainstorms varies considerably within short distances. Consequently, the runoff relationship between an index basin and other basins is not likely to be consistent from year to year, and any specific year's relationship may be considerably different from the long-term relationship. Sunday Creek was treated as an ungaged stream so that only long periods of record would be used in estimating its long-term flow-duration curve. The flow-duration curve thus produced is shown in Figure 3.

Subsequent to phase I of this investigation, computer-calculated low-flow frequency data for the Little Powder River was obtained from the Water Resources Division of the U.S. Geological Survey, Helena, Montana. This computer-calculated data allowed better low-flow frequency curves to be prepared, and these curves are presented in Figure 4 of the present report.

Low-flow frequency curves.

Two points that should be considered in interpreting the flow-duration curves (Figure 3) and the low-flow frequency curves (Figure 4) are (1) the period of the streamflow records does not include the drought of the 1930's and (2) unknown differences in the amount of underflow in the alluvium of the valleys of the ungaged streams might produce considerable error in the extreme low-flow ends of their estimated flow-duration curves. When the curves are used, a suitable margin of safety should be introduced.


Flow-duration curves were prepared for Great Porcupine Creek, Little Porcupine Creek, North Sunday Creek, South Sunday Creek and Sunday Creek. These are all located in an area of low annual precipitation north of the Yellowstone River in the Hysham-Miles City area. The streams are ungaged, except Sunday Creek, which has a one-year streamflow record. Big Dry Creek near Van Norman and Sand Creek near Jordan are gaged streams in the same low-precipitation area, and they were selected as index streams for estimating the flow-duration curves for the ungaged streams. The flow-duration curves are probably fairly good, but a suitable margin of safety should be used when applying the curves to practical problems.


Butler, S. S. (1967): Free-aquifer ground-water depletion hydrographs; Jour. Irrigation and Drainage Div., Am. Soc. Civil Engr., Vol. 93, No. Ir1, p. 65-81.

Earth Science Services, Inc. (1976): Preliminary investigation of practical methods for estimating low-flow frequencies of streams in southeastern Montana; Report to Yellowstone-Tongue APO, Broadus, Montana, 11 p.

Riggs, H. C. (1965): Estimating probability distributions of drought flows; Water and Sewage Works, Vol. 112, No. 5, p. 153-157.

Searcy, J. K. (1959): Flow-duration curves; U.S. Geological Survey Water Supply Paper 1542-A.

Southard, A. R. (1973): Soils of Montana; Montana Agricultural Experiment Station Bulletin 621.

Thomas, D. M. and M. A. Benson (1970): Generalization of streaflow characteristics from drainage basin characteristics; U.S. Geological Survey Water Supply Paper 1975, 55 p.

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.