Daily lake stage was simulated in the model by summing estimates of hydrologic-budget components. The Devils Lake hydrologic-budget components are precipitation on the lake surface, evaporation from the lake surface, runoff (consisting of overland flow to the lake and an intermittent stream flowing into the lake), and ground-water flow into and out of the lake.
The model was calibrated to measured lake stage for the period 1980-92. Simulated stage compares reasonably well with historical stage data for Devils Lake. The root mean square of the differences of simulated and measured daily lake stages for the period 1980-92 is 0.83 foot. Simulated lake stage is very sensitive to small changes in runoff and evaporation coefficients, and ground-water-flow rates used in the model.
The average model-calculated annual amounts of each hydrologic-budget component for the 1980-92 simulation period, in order of increasing volume, are evaporation (791 acre-feet), precipitation (973 acre-feet), runoff (1,107 acre-feet), and net ground-water flow, which is out of the lake (1,323 acre-feet).
Three mitigation plans were simulated. Mitigation plan 3, which includes the addition of water from a basin adjacent to the northeastern side of the Devils Lake Basin, allows for withdrawals of hypolimnetic water and maintaining lake stage closer to optimal levels than would result without mitigation.
In addition to an understanding of the hydrology, an analysis is needed to estimate changes in lake stage caused by proposed withdrawals of water from the lake and variations in hydrologic-budget components. A model can be used to test the current understanding of the hydrology and may be useful in assessing the effect of future-proposed mitigation plans on long-term lake stage.
A study to address the above needs began in October 1991. The study was conducted by the U.S. Geological Survey (USGS) in cooperation with the WDNR and the town of Baraboo.
Two previous studies (Wisconsin Department of Natural Resources, 1988; Dickrell, 1991) have addressed the hydrology of Devils Lake. During 1986-87, seepage meters were used to estimate nutrient loading from ground-water flow into the lake. The seepage-meter data indicated that the amount of ground-water inflow and the location of ground-water-inflow areas are highly variable (Wisconsin Department of Natural Resources, 1988). Dickrell (1991) estimated net ground-water flow by use of a hydrologic-budget approach. Hydrologic-budget components were estimated or measured, and net ground-water flow was calculated as a residual. Dickrell estimated net ground-water flow to be -5.1 x 106 and -4.4 x 106 ft3/month (approximately 0.01 ft/d net ground-water flow from the lake), respectively, for spring and fall 1988.
DeltaS = P + RO - E - GWout-in,
where
DeltaS is change in lake storage,
P is precipitation falling directly on the lake,
RO is stream inflow and overland runoff into the lake,
E is water evaporated from the lake surface, and
GWout-in is net ground-water flow.
The sum of the hydrologic-budget components determines the change in lake storage (DS). If the sum is positive there will be a corresponding increase in
lake storage which results in an increase in lake stage, area, and volume. The opposite results if the sum is negative. Regression equations, based on
the relationship between lake stage, area, and volume (table 1), were developed from a bathymetric map. The equations were used to calculate lake stage
from a calculated lake area or volume and to calculate lake area from a calculated lake stage (Appendix 1).
Historical climate data were obtained from nearby weather stations (Arlington and Baraboo, Wis.). Additional lake-stage, precipitation, and ground-water-level data from a gaging station installed during this study on the northeast shore of Devils Lake also were obtained. Historic measurements of lake stage (January 1980-July 16, 1991) were made by employees of Devils Lake State Park. In addition, eight piezometers were installed around the perimeter of the lake to determine the distribution of ground-water inflow and outflow areas.
Precipitation (P) data were obtained from the Baraboo weather station (January 1, 1980-July 16, 1991) and from the Devils Lake gaging station (July 17, 1991-September 30, 1992). The Baraboo weather station is approximately 3.25 mi north of Devils Lake.
============================================== Stage (feet above Area Volume sea level) (square feet) (cubic feet) ---------------------------------------------- 963 15,920,000 484,890,000 960 15,270,000 438,110,000 958 14,480,000 408,360,000 953 13,220,000 339,130,000 948 12,730,000 274,260,000 943 12,280,000 211,740,000 938 11,640,000 151,950,000 933 10,680,000 96,170,000 928 7,500,000 50,950,000 923 5,330,000 19,030,000 918 2,010,000 1,340,000 916 0 0 ==============================================
where DS is change in lake storage, P is precipitation, A is lake area, E is evaporation, GWnet is net ground-water flow, and DA is the lake drainage area. From these calculations (table 2), a single average runoff coefficient of 0.21 was estimated. The increase in lake volume resulting from a single storm was then calculated by relating the change in lake stage to lake volume.
Evaporation (E) from the lake surface was estimated by use of pan-evaporation data obtained from the Arlington weather station (not shown), approximately 20 mi southeast of Devils Lake. The pan-evaporation data for January 1, 1980 through September 30, 1992, were multiplied by a typical lake/pan coefficient for Wisconsin, to estimate evaporation from the lake surface (Chow, 1964, p. 11-9).
========================================================================
Net change in
lake volume
Period due to runoff Precipitation Runoff
(year/month/day) (cubic feet) (inches) coefficient
------------------------------------------------------------------------
92/05/11-92/05/12 770,000 0.81 0.12
92/04/08-92/04/11 1,690,000 .82 .27
92/04/15-92/04/16 1,070,000 .97 .15
91/11/16-91/11/17 1,380,000 .85 .54
91/11/22-91/11/23 1,230,000 1.09 .22
91/11/27-91/11/29 460,000 .72 .08
91/12/08-91/12/12 1,380,000 1.90 .10
---------------------
Average .21
========================================================================
Ground-water inflow and outflow areas were determined from eight piezometers installed around the northern and southern edges of the lake (fig. 2). Large pieces of quartzite covering the lake bottom along the eastern and western edges of the lake prevented installation of piezometers in these areas. Measured ground-water gradients indicate that Devils Lake is losing water to the ground-water system at five piezometer sites (fig. 2). A sixth piezometer was dry, possibly because an unsaturated zone was present beneath the lake at this site (the large negative gradient, -0.76, at an adjacent piezometer indicates this possibility). Some ground-water inflow occurs at the southwestern part of the lake. Similar results were also obtained from a piezometer survey by Dickrell (1991). Net ground-water flow, GW(out-in), is the sum of ground water flowing into the lake and lake water flowing into the ground. The piezometer survey indicates that the amount of lake water flowing into the ground is greater than the amount of ground water entering the lake. Net ground-water-flow rate was estimated by plotting evaporation and lake-stage recession rate during periods of no precipitation (fig. 3). The Y-axis intercept (zero evaporation) of the regression line is equal to net ground-water flow. The net ground-water-flow rate, equivalent to 0.007 ft of lake surface per day, was assumed to be the average net ground-water-flow rate (fig. 3). The stan- dard error of the intercept is 0.003 ft/d and the range of the intercept value for the 95-percent confidence interval is from 0.0002 to 0.01 ft/d. The significance level of the regression line is 0.01.
A series of "if statements" are executed in the program to determine the volume of runoff that occurred during the current day. Runoff for three periods, winter (December, January, and February), spring and fall (March, April, May, September, October, and November) and summer (June, July, and August) are treated differently. For winter, runoff volume is summed and added to the lake in two increments; one on February 15 and one on March 15. The volume of runoff on a daily basis is calculated by multiplying the daily water equivalent of snow or rainfall by lake-drainage area and then multiplying that product by the runoff coefficient. For spring and fall months, the volume of runoff for the current day is calculated by multiplying the current daily precipitation by lake-drainage area by the coefficient. Runoff for summer is calculated the same way as for spring and fall unless the current daily precipitation is less than 0.5 in. and there was no precipitation during the previous day. In this case, runoff is not calculated and is assumed to be zero for that day.
Finally, the lake stage for the start of the next day is calculated by adding precipitation and watershed runoff to, and subtracting evaporation and net ground-water flow from, the current daily lake stage. The next daily lake stage is then calculated in the model, starting with the calculation of a new lake storage volume.
Simulated stage compares reasonably well with historical stage data for Devils Lake (fig. 5). Long-term trends in the measured stage of Devils Lake indicate a gradual increase from 1980 to the end of 1986, a decrease to about 1988 and, then a fairly constant stage through 1992. These trends are also apparent in the simulated lake stage. The measured daily lake stage, however, does not always compare well with the simulated daily lake stage. Reasons for this are in all of the assumptions and necessary simplifications discussed in the previous section. The two most critical assumptions seem to be that precipitation measured at the Baraboo weather station is representative of precipitation measured at Devils Lake and that a single runoff coefficient of 0.21 can be used to calculate runoff to the lake.
An average runoff coefficient cannot be used to accurately calculate daily runoff because runoff is affected by antecedent moisture conditions and evapotranspiration, both of which can vary daily. Estimates of a runoff coefficient, calculated from data collected during this study, varied from 0.08 to 0.54. Furthermore, the ground-water-flow rate used is constant throughout model simulation. Figure 6 shows that the head difference between lake stage and the water table varies throughout the period of record, indicating that a variable ground-water-flow rate is probably necessary for accurate simulation of short-term net ground-water flow.
Sensitivity of the model was tested by changing values for runoff and evaporation coefficients, and ground-water-flow rate individually while keeping the other two values constant. It was found that the simulated lake stage is very sensitive even to small changes in all three of these values (fig. 7). A 10-percent change in any of these values causes a change of greater than 1 ft in the rms of the differences of simulated and measured lake stage.
A simulation was done in which precipitation data were restricted to data collected at Devils Lake (fig. 8), to further test the model calibration. The simulation covered the period July 17, 1991 through September 30, 1992. Trends in the simulated stage again compare well with the measured lake stage; however, discrepancies in daily fluctuations are large. These discrepancies cannot be attributed to errors in precipitation measurement because precipitation, except for the winter months, was recorded at the lake. Moreover, at least for the summer months, the discrepancies are probably not caused by model-simulation errors in evaporation or net ground-water flow from the lake. This is shown by comparison of the slopes of the simulated and measured lake-stage hydrographs, which are almost identical during extended recessions in July and August 1991 and May-August 1992. These nearly identical slopes indicate that the combined simulated rates of evaporation and net ground-water flow for this period are accurate. Comparison of the hydrograph slopes for the winter of 1991-92, however, indicates that the rate of simulated ground-water flow from the lake is greater than the measured rate. In addition, at times the average runoff coefficient of 0.21 is probably too high to simulate accurately storm events. For example, the major storm in September 1992 produced runoff too low to cause the simulated increase in stage (fig. 8).
The annual change in model-calculated lake volume compares well to the annual change in measured-lake volume for most years (table 3). The annual residual, defined as the yearly difference between the model-calculated and measured change in lake storage ranges from -389 to 250 acre-ft, and the mean is -13 acre-ft. The measured change in lake volume was calculated by relating the change in lake stage (the difference in lake stage at the beginning of the year and the end of the year) to lake volume.
The relative amounts of each hydrologic-budget component are also shown in table 3. The average model-calculated yearly amounts for the simulation period, 1980-92, in order of increasing volume are evaporation (791 acre-ft), precipitation (973 acre-ft), runoff (1,107 acre-ft), and ground-water flow out of the lake (1,323 acre-ft).
=========================================================================================================
Model calculated
----------------------------------- Measured
Net ground- Surface Change in change in
Water year1 Precipitation Evaporation water flow runoff volume volume Residual
---------------------------------------------------------------------------------------------------------
1980 864 720 1,307 929 -233 -292 59
1981 816 689 1,307 917 -263 0 -263
1982 1,053 687 1,306 1,261 322 359 -37
1983 938 707 1,319 1,075 -14 -11 -3
1984 1,290 836 1,328 1,468 594 415 179
1985 1,229 772 1,355 1,355 457 373 84
1986 837 858 1,359 941 -438 -348 -90
1987 912 1,071 1,345 1,027 -476 -440 -36
1988 817 781 1,319 965 -318 -493 175
1989 956 784 1,310 1,071 -69 320 -389
1990 875 797 1,309 998 -233 -483 250
1991 1,083 787 1,311 1,273 258 338 -80
-----------------------------------------------------------------------------------------------------
Average 973 791 1,323 1,107 -34 -22 -13
=========================================================================================================
Mitigation plan 1 is to withdraw phosphorus-rich water from near the lake bottom (hypolimnetic water) during late summer (Richard Lathrop, Wisconsin Department of Natural Resources, Bureau of Research, oral commun., 1991). Figure 9 shows the effects of removing water from the lake at two different rates during the months of September for 1980-92 and compares the results to the calibration stage for the same period. The hydrographs of simulated lake stage are shown as a result of withdrawing the volume of water occupied by the lowermost 3.28 and 6.56 ft of lake depth every September. The volumes of the lowermost 3.28 and 6.56 depths are 5,600,000 and 16,300,000 ft3, respectively. For withdrawal of the 3.28 ft and 6.56 ft volumes during September, water must be pumped continuously at rates of 972 and 2,800 gallons per minute, respectively. The lake stage is approximately 3.98 and 11.03 ft lower, respectively, than the simulated lake stage with no withdrawals (calibration stage) at the end of this period. If 95 percent of the 6.56 ft volume were returned, the simulated lake stage at the end of the 1980-92 period is approximately 0.5 ft lower than the simulated lake stage with no withdrawals.
Mitigation plan 2 is to withdraw water in September but only during years of high stages. The effects of removing the 6.56 ft volume during September 1986, a high-stage period, in relation to the calibration stage are shown in figure 10. The results indicate that even in 1992 (6 years after the 1986 withdrawal) lake stage is more than 1.0 ft lower than it would be had the withdrawal not occurred.
Both of these simulated mitigation plans would lower the lake stage to levels unacceptable for optimal recreation use. An optimal lake stage for recreation is assumed to be 963.75 ft above sea level, which is the approximate elevation of the end of asphalt closest to the lake at the boat landing.
A third mitigation plan is to maintain an optimal lake stage while withdrawing hypolimnetic water in August and September. Water withdrawn would be replaced with water from the intermittent stream in the watershed northeast of the Devils Lake watershed (fig. 1) by means of a control structure. The intermittent stream flows from east to west until it turns north, where it comes within about 200 ft of the northern side of Devils Lake. The drainage area of this watershed upstream of the proposed diversion point is 1.65 mi2. The Fortran program code was modified to account for available runoff that could be diverted from this basin to Devils Lake.
Mitigation plan 3 is simulated as follows: (1) Withdraw the 6.56 ft volume during August and September; (2) whenever lake stage is below a stage of 962.00, water is not withdrawn but is added from the northeastern basin; (3) add water from the northeastern basin equal to the amount withdrawn during August and September, if the lake stage is greater than 963.00; and (4) no water will be added if the stage is greater than 963.75 (the optimal lake stage) from the northeastern basin, but will still be withdrawn during August and September. In figure 11, the results of this simulation are shown for the period 1980-92, along with the calibration stage for the same period. In general, the effect of mitigation plan 3 is to increase lake stage during periods when lake stage would normally be low (that is, without the addition of water from the northeastern basin) and to de-crease lake stage when lake stage would normally be high (that is, without the withdrawal of hypolimnetic water). Mitigation plan 3 increases or decreases lake stage to maintain the optimal stage of 963.75.
Daily lake stage is simulated in the model by summing estimates of hydrologic-budget components. The Devils Lake hydrologic-budget components are precipitation on the lake surface, evaporation from the lake surface, runoff (consisting of overland flow to the lake and an intermittent stream flowing into the lake), and ground-water flow into and out of the lake. Amounts of precipitation and pan evaporation (multiplied by a coefficient) recorded at the nearby weather stations in Baraboo and Arlington, respectively, were used to estimate these components. A gaging station installed at the lake measured precipitation for July 1991 through September 30, 1992. A runoff coefficient was multiplied by daily precipitation and drainage area to estimate the runoff component. The method of calculating runoff volume was varied seasonally: the water equivalent of daily snowfall during the winter was summed and released to the lake on February 15 and March 15, and runoff was calculated for the summer months when precipitation exceeded 0.5 in. on the day of interest or when precipitation had fallen on the previous day. Net ground-water flow leaving the lake was calculated by multiplying the current lake area by a coefficient.
The model was calibrated by comparing model-calculated and measured lakes stages for the period 1980-92. The root mean square of the differences of simulated and measured daily lake stage for the period 1980-92 is 0.83 ft. Simulated lake stage is very sensitive to small changes in the evaporation and runoff coefficients and the ground-water-flow rate. A 10-percent change in the values of the coefficients or ground-water-flow rate causes the root mean square of the differences of simulated and measured lake stage to be greater than 1 ft.
The average model-calculated yearly volume for the simulation period 1980-92, in order of increasing volume, is evaporation (791 acre-ft), precipitation (973 acre-ft), runoff (1,107 acre-ft), and net ground-water flow which is out of the lake (1,323 acre-ft).
The model was used to simulate three possible mitigation plans to reduce the cycling of phosphorus in the lake. Mitigation plans 1 and 2 would lower lake stage to levels unacceptable for optimal recreation use. Mitigation plan 3, which includes the diversion of water from a nearby watershed to Devils Lake, allows for withdrawing hypolimnetic water and maintaining lake stage closer to optimal levels than would be possible without mitigation.