WATER RESOURCES BULLETIN
VOL. 13. NO. 2 AMERICAN WATIR RLSOURCLS ASSOCIATION APRIL 1977
STATISTICAL ANALYSIS OF THE IMPACT OF
"GROUND WATER PUMIPAGE ON LOW-FLOW HYDROLOGY'
SC. '. Fetter, Jr.
ABSTRACT Ground-water pumpage withdrew 57 cubic feet per second from aquifers beneath
Sthe Vahara River Basin in 1970. Forty-si\ cubic feet per second were exported by the diversion
of treated wastewater from the drainage basin.
I he lo\ -flow hydrology of the upper Yahara River has been impacted by this diversion. .
Prior to 1959, the \wastewater was discharged into the ri\er, au.menting the baseflow during
losw-flow periods. As much as 85" of streamflow was due to eftfhent discharge. In 1959 the
wastewater wa-, transferred from the river basin. Tlie result was a dec rease of about one-third in
mean annuiil streanfltow. and a decrease of more than 50 in the 70 and 70,10. Repression
anal\ sis show cd tlhe annual 7-day low-flow and 60-day h m-tlow have a statistically significant
correlation with mean annual flow. Using predictions of future mean annual discharge of the .
rixer with increasin.L interbasin transfers, it is shown that by 1 91 i there is a significant proba-
bility that in some \ cars the (O6-day low-flow in the river \\ ill be /ero. .- ,.-.
(KIV'Y T1 R1S: low-flow hydrology: repression analysis: low-tow predict ions.)
The cil\ of Madison. Wisconsin. is situated on a series of lakes which are part of the
S Yahara River system. Tlie streams, lakes and ground water form interdependent parts of a
S regional hydiologic system, and changes induced in one part of the system may affect
other system parameters. .
Ground water is pumped from a number of deep wells in the drainage basin. In 1970 ;
an average of 57 cutbic feet per second was withdrawn from the deep aquifers. Of this
amountt. 11 cubic feet per second was used consumptively or seeped back into the
ground. lea ing -4o cubic feet per second of pumped ground water passing to the waste-
water tIeattument plant.
As a part of a water-quality improvement program for the lower Mladison lakes, an
effluentt-divecsio. scheme was impletnmented in D)ecember, 1958. Wastewater which had
been discharged into tlhe uttpper Yahara Riser via Lake Waubesa was diverted to Badfish
Creek. a tributary of the lower Yahara River. It is the purpose of this paper to evaluate
1Paper \o. o, i, h :-C'tr sor'\( r''.s H /kt'tui. Discussions are open until lDecemnber 1. 1t77.
"- \-,osi;ate P'ros:,,, i' < ,;l.,-\ t:isersit \\ iscottsin .Ost kosti. O S(.kosh. \\'isconsin 54901.1
SStatistical Analysis of the Impact of Ground Water Pumpage on Low-Flow Hydrology 317
S all of the variation, there is a probability that actual values of Q7 will not be the
calculated values. Confidence intervals can be established for the regression line for differ-
K ent probability levels (Draper and Smith. 1967, p. 17-24). The 95% confidence band for
the regression was determined and plotted on Figure 4. For this problem we are especially
interested in the X-intercept of the regression line. This is the value of the mean-annual
"0 flow where the value of the seven-day low flow is calculated as zero. The regression
equation yields a value of 85.4 cubic feet per second for the X-intercept. The 95%
Confidence band indicates the Q7 low-flow at a Q365 of 85.4 would fall in the range of 0
S to 18 c.f.s.
Sixty-Da)y Low Flows.
A similar regression analysis was made for the sixty-day low flow for the period April 1,
_0 0 1959. through March 31. 1974. The regression equation is:
>a 60 .583 Q365 37.9
< The regression line is given on Figure 5. The regression is also statistically significant at
sevean-ay low o ul bes e period. n
the 0.01 level with an F value of 24.4 The value of R2 was .65" so that the regression
Accounts for 65.2 of the variation in the data. This is greater than the value for the
CJter pumpag -tdesewager dischares o; l '
\ wseven-day low flow, as would be expected since the sixty-day period represents a greater
portion of the year than a seven-day period. The value of the X-intercept is 65.2 c.f.s. The
95m% confidence limit for c 7 is 0 to 26 c.f.s. at a value of 65.2 c.f.s. for Q 365
FREQUENCY OF MEAN ANNUAL FLOW
0 In determining the mean annual flow there are several time periods which could be
Y utilized. They include the entire period of record, 1931-1974, the post-diversion period,
S1959-1974, and the pre-diversion period. 1931-1959. Since the regression analysis was
,. =,made for the post-diversion period it might be desirable to determine the frequency for
th mean annual flow for the same period.
S However, it has been shown through the double-mass curve that the post-diversion
data of mean annual flow are biased since water was diverted from the basin. It would
z seem desirable to avoid these data. and only analyze the pre-diversion data. 1931-1959.
"Mean annual flows during this period were probably not significantly affected by ground
water pumpage and sewage discharges.
It should be noted that by using the data from 1959-1974 for the regression analysis
0 and 1931-1959 for the frequency analysis, the results may not be totally compatible since
different hydrologic periods are involved. However, the low flows in the pre-diversion
"period were a'fectcd by low- low augmentation with effluent, and the mean-annual flows
S in the post-diversion period were affected by eftlient diversions from the basin.
The frequency of the mean-annual discharge for the Yahara River for the period April
1, 1931, through March 31, 1q59, was determined. During this period median-anntial
S streamfiow was 143 c.f.s. and the mean-annual streamflow was 149.6 c.f.s. with a stan-
dard deviation of 40.69, and a probable deviation of 26.35. If there is a normal distribu-
tion of strcainflow. the median should qoual the mean and one probable deviation on
Statistical Analysis of the Impact of Ground Water Pumpage on Low-1 low Hydrology 319
either side of the mean should include 50% of the streamflows (Leopold, 1969). For the
period of record of the Yahara River 13 of 28 values, or 47%, lie in this range. Likewise,
82% of the data should fall within two probable deviations of the mean, which is approxi-
-l mately the case for the period 193 1-1959 (actual value 79%).
The frequency distribution of the data plot as a straight line on probability paper
indicating the data are approximately normally distributed. This has been done on
o Figures 6 and 7. A linear-probability curve was established by the mean streamflow
(plotted at 50% probability), the streamflow one probable deviation greater than the
mean (plotted at 75% probability) and the streamflow one probable deviation less than
the mean (plotted at 25% probability).
c FUTURE LOW-FLOW PROBABILITIES
o Larger diversions of sewage effluent from. the upper Yahara drainage basin will lower
S. the mean-annual discharge. This will not be a one-to-one relationship as growing pumping
cones in the future will induce additional ground water inflow to the basin. McLeod
(1975b) using a two-layer digital ground water model calculated the diversion ot water
S, t -from the drainage basin of the upper Yahara River due to ground water going to con-
,4 1 sumptive uses and sewage diversions. These computed values are reproduced in Table 1.
| TABLE 1. Source of Ground Water Diverted to Wells in Madison Area
Year From paper Yahara River Basin From Adjacent River Basins
1970 52 c.f.s. 5 C.f.s.
S"| .15 c.f.s.
_4 4 I o o 5 2 8 C:f.S f> "^ \
< 1980 67 c.f.s. 8 C.f.s.
| 1990 81 c.f.s. 12 c.f.s.
S0 < .
As the water is diverted from the river, it represents a reduction in annual discharge.
o \ These diversions will reduce the annual flow by the specified amount. ,
\\-.If'one assumes that the April 1931 to March 1959 frequency distribution is represent.-
SIm tive, and that diversions do not change the slope of the line, then a family of trequencV ,:
ci \r -c es for varying rates of diversion can be established. This was done on Fi,,ues 6 and 7
by lowering the linear-frequency curve by an amount equal to varying rates ot divet:ion.
Lines for 56. 67 and 81 c.f.s. diversions are given. .
SBy combining each of the regression curves with the family of tequency curves, n
^ assessment of the probability of future low flows may be made for both a seven-day and a
S ercn-ID .- /.(t- I .. k.. .
A heavy dashed line is drawn on Figure 5 at 85 c.f.s. mean-annual flow, which is t -,h,
X-intercept of the seven-day low flow regression line.
______ ^ ,,.r,.,....____X-^^ ^&W^. ______ --