Estimating Ground Water Recharge from Topography, Hydrogedogy, and Land Cover by Douglas 5. Cherkauer' and Sajjad A. Ansari*

Abstract Proper management of ground water resources requires knowledge of the rates and spatial distribution of

recharge to aquifers. This information is needed at scales ranging from that of individual communities to regional. This paper presents a methodology to calculate recharge from readily available ground surface information without long-term monitoring. The method is viewed as providing a reasonable, but conservative, first approximation of recharge, which can then be fine-tuned with other methods as time permits.

Stream baseflow was measured as a surrogate for recharge in small watersheds in southeastern Wisconsin. It is equated to recharge (R) and then normalized to observed annual precipitation (P). Regression analysis was con- strained by requiring that the independent and dependent variables be dimensionally consistent. It shows that R/P is controlled by three dimensionless ratios: (1) infiltrating to overland water flux, ( 2 ) vertical to lateral distance water must travel, and (3) percentage of land cover in the natural state. The individual watershed properties that comprise these ratios are now commonly available in GIS data bases.

The empirical relationship for predicting R/P developed for the study watersheds is shown to be statistically viable and is then tested outside the study area and against other methods of calculating recharge. The method pro- duces values that agree with baseflow separation from streamflow hydrographs (to within 15% to 20%), ground water budget analysis (4%), well hydrograph analysis ( 12%), and a distributed-parameter watershed model calibrated to total streamflow (18%). It has also reproduced the temporal variation over 5 yr observed at a well site with an aver- age error < 12%.

Introduction As human populations continue to grow, the demands

placed on ground water resources will accelerate. As an example, 70% of the more than five million residents of Wisconsin rely on ground water as their sole source of water supply. These increasing demands create a need to define accurately the spatial distribution of recharge across large areas from readily available information so communities can examine not only the extent of their resource, but also how their use of it is interconnected with their neighbors.

To date, much of the effort to quantify recharge has been directed at semiarid areas (De Vries and Simmers 2002). There are many examples, however, where bur-

'Department of Geosciences, University of Wisconsin-Mil- waukee, P.O. Box 413, Milwaukee, WI 53201 ; aquadoc@uwm.edu

2Department of Geosciences, University of Wisconsin-Mil- waukee, P.O. Box 413, Milwaukee, WI 53201

Received March 2002, accepted November 2003. Copyright 0 2005 by the National Ground Water Association.

geoning population in humid areas relying on ground water has caused water levels to decline significantly. Clearly, demand in these areas exceeds recharge.

The availability of recharge cannot be factored into the planning process unless it has been quantified. To be useful to resource manager, recharge definition must satisfy four criteria. (1) Focus on the influx to saturated systems. ( 2 ) Accurately define that influx down to the community scale (several km2). (3) Rely exclusively on readily available data. (4) Be readily applicable across the regional scale ( lo2 to lo4 h2). The majority of methods developed to date do not meet all these criteria. The scale issues present the biggest hurdle. Well hydrograph analysis (Ketchum et al. 2000) works at small scales (a single well) but is difficult to extend to larger areas without extensive monitoring sys- tems. Stream hydrograph separation (Mau and Winter 1997) and ground water budget analyses are well suited to the scale of a watershed where external divides limit lateral fluxes and simplify the process. It is difficult, however, to resolve either measure to areas smaller than a gauged watershed. Geochemical tracers in the saturated zone are

102 Vol. 43, No. I-GROUND WATER-January-February 2005 (pages 102-1 12)

also best suited to measuring average recharge rates across broad areas; finer resolution is uncommon (Scanlon et al. 2002).

Measurement of infiltration at the ground surface can determine recharge to the saturated zone only if the effects of evapotranspiration losses and transient storage in the unsaturated zone can be defined. Unfortunately, the uncer- tainty attached to calculation of water budget terms in the unsaturated zone makes the resultant recharges question- able. They are also difficult to extrapolate to large scales.

Recent work by Braun et al. (2003) estimates recharge potential by combining mapped distributions of the poten- tial for surface runoff, percolation, and evapotranspiration. This work uses geographic information systems (GIS) inputs to generate high resolution recharge potentials across an entire county. However, it does not actually quan- tify recharge rates.

Recharge from ground water models is controlled by the hydraulic properties and boundary conditions built into the model and is therefore nonunique. Scanlon et al. (2002) point out that the reliability of recharge in models depends on the accuracy of hydraulic conductivity inputs. In addi- tion, Tortomasi (2003) has shown that the amount of recharge necessary to calibrate a flow model varies directly with the total conductance of the surface water bodies sim- ulated. Thus for a model with fixed hydraulic conductivi- ties, the amount of recharge is entirely dependent on the resolution of the surface water bodies included. Flux targets only resolve this dilemma if they are available for most of the surface water bodies, a rare situation. Hence flow mod- els may appear to provide high resolution recharge mea- sures over large areas, but the accuracy of their values is unknown.

Cherkauer (2003) has shown that a distributed-param- eter model can produce recharge rates which are accurate at the watershed scale and which can be resolved to finer scale. Such models can only be calibrated on gauged water- sheds, although their results may be extendable to nearby ungauged areas. The model used, Precipitation-Runoff Modeling System (Leavesley et al. 1983), is extremely data intensive and its operation is probably beyond the capabil- ity of ground water using communities. More complete reviews of all these recharge methods are presented in Scanlon et al. (2002) and Cherkauer (2003).

What is needed then is a relatively simple, accurate measure of ground water recharge in humid areas that relies on readily available information. It needs to be able to read- ily provide communities with a solid estimate of the total recharge to their aquifer and its spatial distribution. The lat- ter would allow communities to identify their primary recharge areas and allow them to plan for the protection of recharge quantity and quality.

Objectives This paper presents the results of a project which was

undertaken to fill the need previously described. Specifi- cally, it had three objectives: 1. Obtain a reliable measure of the actual ground water

recharge rate for a representative area in a humid cli- mate.

2.

3.

Determine the independent factors that control the spa- tial distribution of that recharge. Test the relations between recharge and independent factors to determine the reliability of the method and whether it can be extrapolated beyond the study area.

If at all possible, the independent factors used in the study should be readily measurable, so that the results of the study can be used as widely as possible.

Methods This paper uses the baseflow separation from total

stream discharge to obtain a measure of ground water recharge. It will then establish the relationship that exists between the rate of recharge and controlling physical prop- erties of watersheds. Baseflow separation has been shown to be a viable means of estimating recharge (Arnold and Allen 1999). It uses the stream drainage system to integrate a hydrologic response over the entire drainage basin, thus reducing the problem of site specificity.

Baseflow separation involves developing a simple ground water budget equation for a small watershed:

(1)

where I is infiltration to the system, GW, is ground water influx to the watershed through aquifers, Qhf is ground water discharge to stream baseflow, GW,,, is ground water efflux from the watershed through aquifers, ET is evapo- transpiration losses from the watershed, NP is net pumpage of ground water by humans into or out of the watershed, and AS/t is the rate of change of ground water storage with respect to time.

If watersheds can be selected where GW, = GW,,, = NP = ASIt = 0, and if recharge is defined as net ground- water recharge (I - ET), then Equation 1 reduces to

Recharge = Net recharge = Qhf = Stream baseflow (2)

In order to meet these conditions, a watershed should have coincident surface and ground water divides, no human transfers of water across those divides, and ground water storages that do not change significantly from year to year. Hydrograph separation inherently assumes that direct surface runoff and ground water discharge are the only components of streamflow. Thus watersheds with signifi- cant surface storage locations (lakes, wetlands) must also be avoided. Using watersheds smaller than most U.S. Geo- logical Survey (USGS) gauging sites allows small-scale resolution of recharge and its controls, but requires field observations to produce a stream hydrograph.

I + GW, = Qbf + GW,,, + ET + NP + AS/t

Conceptualization and Procedures The interpretation of flows in these small watersheds is

focused on their primary, perennial channel. For simplicity, the watersheds are conceived of as rectangular solids with the horizontal length being the length of the main channel (L,) and width being drainage a r e a , , which is also twice the length of surface flow (L,) (Figure 1). Each individual watershed is then considered to be internally homogeneous, with each physical property (elevation, slope, etc.) repre- sented as the mean value measured across the entire area.

D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12 103

1_

T Dw

---------- Water tmtble

Stream

Figure 1. Conceptualized geometric dimensions of study watersheds. Half of the watershed of drainage area A, is shown; other half is mirror image. L, is length of main chan- nel. L, , length of overland flow, is A& L,), S is average sur- face slope toward stream, and D, is the average depth to the water table.

Within the watersheds, precipitation falls on the ground surface and flows toward the channel. Along the path of flow, water is distributed between that which enters the stream as runoff and that which becomes net recharge and enters the stream as baseflow. The physical properties of the watershed then control this partitioning. The entire drainage area is conceived of as recharge area, and all ground water discharge occurs at the stream.

This conceptual model is highly simplified. It does not account for heterogeneity of properties or the presence of ground water discharge zones other than the stream. It does not incorporate the effects of antecedent soil moisture on net recharge. It also assumes that precipitation is uniformly distributed across the watershed and that it occurs with uni- form intensity. The test of whether these simplifications are appropriate is the viability of the relation of recharge to physical properties.

Study Site Selection Southern Washington County, Wisconsin, was chosen

as the primary focus of the study (Figure 2). It is traversed by a glacial interlobate moraine, which provides sufficient local topographic relief that surface and ground water drainage divides generally coincide. It is underlain by a shallow aquifer system consisting of both sand and gravel in the glacial deposits and Silurian dolomite. Beneath that, a shale serves as the base to the shallow aquifer system.

The area drained by five adjacent streams was subdi- vided into 10 contiguous watersheds to be monitored for this study (Figure 2b). Several sets of these watersheds are nested along the same stream, which allows determination of how baseflow changes downstream. These watersheds were treated as serially linked, independent land areas. The topography and other controlling factors were measured for the drainage area contributing to the stream between the station of interest and the next one upstream. Similarly, the streamflows (both total and baseflow) were calculated as the gain from the upstream site to the one of interest.

A

7

I r

Washington County

HF r - - - ' O R F

-

Figure 2. Location map of study watersheds and other sites. (a) Regional setting. RF, GT, IF, and ML are Richfield, Ger- mantown, Hartford and Milwaukee precipitation sites. F1, F2, CC, MT, and KK are Fox above Waukesha, Fox at Wauke- sha, Cedar Creek, Menomonee at Wauwatosa, and Kinnick- innic gauging stations, respectively. MQ is the Mequon water budget site. Dashed rectangle is the primary study area. (b) Drainage system with gauging sites in the study area. CE, CO, M, W, and WB are Cedar, Coney, Menornonee, Willow, and West Branch, respectively. For multiple sites on the same stream, the watersheds are the areas contributing to the stream between two gauging points.

Comparison of water table maps (Young and Batten 1980; SEWRPC 2002) with water levels from well con- struction reports and topographic maps shows no signifi- cant temporal change in ground water levels and coincidence of the surface and subsurface divides in these watersheds. Virtually all residents in these watersheds have wells in the shallow aquifer and treat waste water via sep- tic systems. Hence there is no significant human transfer of water into or out of the area.

These 10 watersheds are in an area where land use is primarily agricultural, residential, or low-density commer- cial or industrial. In order to extend the recharge analysis to more urban land uses, an additional watershed, the Kin- nickinnic (Figure 2a) in Milwaukee County, was added to the list of study watersheds. It is monitored continuously for streamflow by the USGS and also meets the selection criteria reasonably well.

Precipitation and Streamflow Monitoring For the 10 contiguous watersheds in Washington

County, precipitation from National Oceanic and Atmos- pheric Administration (NOAA) stations at Hartford and

104 D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12

Germantown and a private site maintained for the study (Figure 2) were used. For the Kinnickinnic River site, data from the NOAA site at Mitchell Airport in Milwaukee were used. Values for the precipitation in each study watershed were interpolated from the recording stations using the Thiessen polygon method (Fetter 1994).

Monitoring sites were selected in accordance with the criteria presented in Buchanon and Somers (1969). Stream levels were measured manually on a staff gauge between one and three times each week for the period June 1997 through June 1999. Flow at the monitoring site was gauged using a pygmy Price meter and following USGS gauging standards (Buchanon and Somers 1969). The full gaugings were done once every week to three weeks as needed to establish a consistent and reproducible rating curve or curves for the site.

Using the rating curves, discharge was obtained for each monitoring date and plotted against time as a stream hydrograph. It should be noted that these hydrographs are not the result of continuous recording; discharges have been interpolated between actual monitoring dates. The monitoring missed peak flows, but this is not a serious shortcoming, because the study is focused on defining the recession of low flows. Manual monitoring was necessi- tated by the studys budget and the fact that the gauging sites were all on private property.

Baseflow was separated from total streamflow, using a manual method presented by Linsley et al. (1982). It was developed for single events and was modified here for use with an entire years hydrograph (Figure 3 ) . A persistent baseflow recession rate (slope of the semilogarithmic hydrograph) is identified by looking at the distribution of observed discharge minima. This recession rate is then maintained throughout the year. Recessions are extended from tc after a streamflow peak until t, after the next peak, with t, = where Ad is the drainage area in square miles (Linsley et al. 1982). The area under the separation line (Figure 3 ) is the total baseflow for the year.

Szilagyi ( 1999) demonstrated that the baseflow reces- sion curve may not be semilog linear for aquifers with very small vertical thicknesses. For conditions appropriate to the study watersheds the semilog linear recession is conserva- tive compared to Szilagyis approach, because it underesti- mates ground water discharges early in the response to a recharge event.

The study watersheds meet the criteria required for the establishment of Equation 2. Therefore, the baseflow at each site is equated to the ground water recharge that has occurred in each watershed draining to the monitoring site. From this point on, the baseflow per unit drainage area from each site will be referred to as recharge rate or recharge (dimensions L/T).

Measurement o f Watershed Properties The amount of recharge that occurs within a watershed

intuitively should be dependent upon precipitation, topogra- phy, geology, and land cover. Determination of the topo- graphic, hydrogeologic, and land use information was initially done manually because GIS coverage was not read- ily available. Within each watersheds drainage area, aver- age hillslope and the length of the main channel were

measured from 1 :24,000 topographic maps. Hillslope was measured as the average of 15 to 20 randomly selected lines within each watershed. The accuracy of all these parameters is constrained by the 10-foot contour interval of the maps.

The rate at which ground water is recharged is related to the conductivity of the surface sediments and the thick- ness of the unsaturated zone. The vertical hydraulic con- ductivity (K,,) was calculated from US. Soil Conservation Service county maps and measurements. A $ for each soil in the watershed was calculated from measured values for the soils horizons. These were then combined into a weighted mean conductivity based on the soils areal extent (Ansari 1999).

Water table contour maps and isopach maps of the thickness of the unsaturated zone were generated from the water levels reported on well construction reports. Land cover (from 1995 maps at I :24,000 scale) was lumped into five categories: natural land (woodlands, wetlands, parks, golf courses); agricultural; developed (residential, commer- cial, industrial); extractive (quarry and aggregate opera- tions); and open water.

Data Analysis A link between recharge and independent control para-

meters was developed using multiple regression on the first year of record at the study watersheds. The process was subject to two constraints. The independent variables had to be intuitively reasonable within the conceptual model, and they had to be dimensionally consistent with the dependent variable. The acceptance criterion used was that the regres- sion equation had to explain at least 85% of the variation in recharge observed in the study watersheds.

Three levels of testing were used to confirm the rela- tions veracity. First, the relation was tested against a sec- ond data set, the recharge observed in the study watersheds for a second year of record. Second, the relation was used to calculate the expected recharge rates at sites outside the set of study watersheds. Comparison to observed values determines if the relation can be extrapolated outside the study watersheds. Finally, the accuracy of the relation was tested by comparing its recharge rates to those developed by other methods for the same locations.

For the external testing, recharge and the independent variables were obtained at six additional locations. Four are USGS gauging sites on larger watersheds around the out- side of the 10 contiguous sites. They are the Menomonee River at the Wauwatosa gauging site, the Fox River stations both above and at Waukesha, and the Cedar Creek station above Cedarburg (Figure 2). For multiple gauging sites on a given stream, the watersheds were treated as the drainage areas contributing flow between gauging stations, and base- flow was the amount gained between those stations. The topographic, hydrogeologic, and land use conditions of these test areas were initially obtained manually as earlier described for the study watersheds. Subsequently, GIS cov- erage (30 m pixel resolution) became available and some parameters were remeasured for comparison. Precipitation was obtained from additional NOAA sites which provide the coverage for these watersheds.

Three other methods were used to calculate recharge in and around the study area. Cherkauer and Bacon (1978)

D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12 105

used a water budget to calculate recharge in Mequon, Wis- consin, east of the study watersheds (Figure 2). Recharge has also been calculated on 63 subwatersheds in the area (Cherkauer 2003), using the distributed parameter model PRMS (Leavesley et al. 1983). The empirical relation developed in this paper was used to calculate a long-term recharge for the same subwatersheds. In addition, annual recharge was measured from changes in water level in the well at the Richfield site (Figure 2).

I I , , . \ : . , , . I : , , 2 . . . , I . j

Relation o f Recharge to Watershed Properties

Characteristics o f the Study Watersheds The hydrogeologic, topographic, and land cover prop-

erties of the study watersheds are presented in Table 1. Drainage areas range from 3 up to 48.7 km2, although all but three of the sites are < 14 km2. Soil conductivities range from 0.7 to 4.1 m/d, with a mean value of 1.5 m/d. For the study watersheds in Washington County, the dominant land use is agriculture (Table l), although the total amount of development ranges up to 38%. Inclusion of the Kinnickin- nic River watershed (88% developed) provides an urban end member.

The western watersheds are in a glacial moraine and have steep hillslopes and channel gradients. Those toward the northeast (all West Branch, Menomonee, Cedar 1) lie below the moraine and have much flatter topography (Table 1). As a general rule, depths to the water table vary with the local relief. They are much greater in the north and west watersheds in Washington County (Table 1 ).

The range of conditions observable within the study area is representative of most of southeastern Wisconsin. Coupled with the proximity among the study watersheds, this makes them an ideal set for examining the spatial vari- ability of recharge.

Precipitation and Baseflow Annual precipitation varies across the study water-

sheds (Table 2). To eliminate the effect of spatially varied precipitation on recharge, the latter was normalized to total annual precipitation and is expressed as a ratio (recharge per unit precipitation, or UP) for the remainder of this paper. This value is the percentage of the annual precipita- tion in a watershed that becomes ground water recharge and then makes its way to a stream as baseflow.

Three constraints were imposed on the manual base- flow separation. First, a baseflow recession rate (slope of the baseflow line in Figure 3) was defined visually from the spring and early summer baseflows. Next, flow events that deviate greatly from the recession line in periods when the ground is not frozen were assumed to indicate recharge

1EM1 r I

~

Table 1 Properties of the Study Watersheds

Hydrogeology Land Cover Topography

Ad Ks Dw N D Ag Lc S Watershed (km2) ( d d ) (m) (%I (%I (%) (km) ( d m )

Cedar 1 Cedar 2 Cedar 3 Coney Kinnickinnic Menomonee West Branch 1 West Branch 2 West Branch 3 Willow 1 Willow 2 Uncertainty #

10.9 3.0

13.2 25.0 48.7 36.5 3.9 6.6 2.9 4.6 9.0

f 5%

1.2 11.3 1.1 10.7 0.55 15.8 2.4 32.6 0.33 12.2 0.91 10.1 0.91 11.3 0.91 9.1 4.1 14.0 2.7 9.8 0.73 9.4

f 15% f 20%

12.9 15.0 71.7 8.0 25.8 65.9

22.2 19.3 57.5 26.9 6.2 66.3 12.3 87.7 0 26.7 13.5 59.0 9.6 9.8 80.3

11.5 14.0 13.9 15.2 38.3 45.6 22.8 30.0 45.9 17.4 26.3 55.6

* 15% f 15% f 15%

2.9 2.0 4.2

12.0 7.9

11.0 1.5 4.0 2.2 3.7 6.1

f 5%

.040

.042 ,056 ,082

,0095 .029 ,029 .038 .044 ,068 ,044

* 15% ~~~

Ad =drainage area: S =average hillslope; N = natural; D =developed; Ag = agricultural; Lc = length of main channel: Ks = effective soil conductivity; Dw = depth to water table

106 D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12

Table 2 Precipitation and Normalized Recharge in the Study Watersheds

Year 11 Year I Precip. R/P obs R/P calc Difference Precip R/P obs R/P calc Difference

Watershed (cm) (cdcm) (cdcm) (%) (cm) (cdcm) (cdcm) (%)

Cedar 1 Cedar 1 Cedar 2 Cedar 3 Coney Kinnickinnic Menomonee West Branch 1 West Branch 2 West Branch 3 Willow 1 Willow 2 Mean

104.0 104.0 101.2 95.6 89.7 101.5 106.7 103.1 101.8 98.9 101.2 101.6 100.5

0.142 0.144 0.142 0.144 0.059 0.066 0.090 0.070 0.102 0.105 0.1 16 0.113 0.176 0.186 0.1 14 0.145 0.101 0.097 0.251 0.239 0.168 .134

(0.175) (0.065) 0.132 0.130

+1.4 +1.4 -1 1.4 -22.1 +2.4 -2.6 +6.0 -26.5 -3.7 -4.7 -20.1

(-62.7) 10.1*

93.5 0.104 0.130 93.5 0.104 0.130 93.4 0.050 0.061 93.3 0.091 0.068 87.3 0.100 0.102 94.5 0.114 0.106 93.5 0.124 0.163 93.5 (0.067) (0.100) 93.4 0.081 0.089 93.4 0.243 0.226 93.4 0.156 0.124 93.4 (0.127) (0.060) 93.0 0.113 0.119

.+24.6 +24.6 +22.2 -25.1 +1.9 -7.3

+31.0 (+49.4)

-10.1 -6.9

-20.7 (-52.9)

19.9*

Year I is June 13, 1997 to June 12, 1998 for precipitation. Baseflow measurements are lagged one week behind. Year I1 data are for the same periods in 1998 to 1999 Values in ( ) are anomalies discussed in text and not included in means. *Means in difference columns are of the absolute values of differences. obs = observed; calc = calculated with Equation 3

(additions to baseflow). Finally, it was assumed that no recharge would occur during the period when the ground surface was frozen (based on observations by the first author at Richfield). This means that the rise in streamflow between days 155 and 245 (Figure 3) is assumed to be entirely the result of surface runoff, with no additional input from baseflow. This final constraint is conservative. While some recharge may occur during the frozen period, hydro- graph separation would greatly overestimate any baseflow stemming from it. The frozen ground period observed at Richfield was assumed valid across the entire study area.

Normalized recharge varies spatially across the study area by a factor of 5 (Table 2). This variability, especially in light of the fact that most of the watersheds are geo- graphically contiguous, demonstrates that recharge is con- trolled by spatially variable factors. In addition, observed precipitation varies < 10% about the mean, indicating that it is factors other than precipitation which cause the spatial changes in recharge rates.

Factors Contributing to the Spatial Variability o f Recharge Regression analysis commonly results in relations

which are not dimensionally consistent. In this study, UP, a dimensionless parameter, did not correlate well with indi- vidual watershed properties that had dissimilar dimensions. It showed a much stronger correlation when it was regressed against dimensionless parameters. A variety of ratios of representative lengths, areas, volumes, and fluxes were tested. R/P was correlated with the three best in a mul- tiple regression. The result is

IUP = 0.0085(Kv/ (S -4.18{Dw/Lf} + 0.0025 (N} + 0.022 (3)

where UP is normalized annual recharge (recharge per unit precipitation in cdcm) ; & is effective vertical soil con- ductivity ( d d ) ; S is average hillslope in a watershed ( d m ) ; Dw is average depth to the water table (m); Lf is length of flow to the main channel, which is the drainage area42 X channel length) (km); D is the portion of devel- oped land in the watershed (as a %); and N is the portion of natural land cover (as a %).

Both the second and third terms of the equation are intuitively reasonable. Normalized recharge varies directly with the amount of natural land cover (woodlands, wet- lands, and parks) and the distance water travels overland to reach the main channel (L, ). It varies inversely with depth to the water table (D,,,). Net recharge is enhanced by a longer length of overland flow because water has a greater opportunity to infiltrate before it reaches the stream. It is reduced by a greater depth to the water table because there is more opportunity for water to be lost to evapotranspira- tion before it reaches the saturated zone.

The first term in Equation 3 is intended to be the ratio of representations of the vertical flux of water through the soil to the horizontal flux across the ground surface. Nei- ther has been measured, so both are estimated within the framework of the conceptual model.

The vertical flux occurs when the surface soil is satu- rated and under a vertical hydraulic gradient of unity. Then,

9 v = c 1 & (4) where qv is the vertical flux (L/T), and C, is a dimensionless proportionality coefficient. The denominator of the first term of Equation 3 represents the flux of runoff. In a system where gravity and friction are the dominant stresses, that flux should vary directly with surface slope and inversely with ground surface roughness:

D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12 107

q h = (c, s>/ ( 5 1 where qh is the horizontal flux (discharge per unit area) (LiT), S is ground surface slope (L/L), F is surface rough- ness (L), and C, is a proportionality coefficient with dimen- sions (T-I).

One major factor that affects a watershed's roughness is the extent of land surface development (D). As D increases, the construction of ditches, storm sewers, and paved surfaces all increase runoff rates by effectively reducing the watershed's roughness. Therefore, this esti- mate of inverse roughness has been made:

F = C, D-' (6)

The exponent x was determined empirically to be 0.3, the value which optimizes the correlation of runoff flux with UP. C, is a proportionality coefficient with dimensions

Taking the ratio of qv to qh and substituting Equation 6 produces (C Ks)/(S Do.,) for the first dimensionless term in Equation 3 , where the proportionality coefficient C has the dimensions (T/L). It indicates that recharge should vary directly with soil conductivity and inversely with surface slope and human development, all intuitively reasonable.

The three ratios in braces in Equation 3 are the inde- pendent variables in this multiple regression analysis. They exhibit no correlation among themselves. Based on t-tests, there was far less than an 80% probability that any nonran- dom relation existed among them. An F-test on Equation 3 (with three independent variables and six degrees of free- dom) shows that there is a > 95% probability that the rela- tion is meaningful. In addition, a stepwise regression test shows that the correlation coefficient increases from 0.66 to 0.82 to 0.91 as each independent ratio is added in the order they appear in Equation 3 . This indicates that each term contributes to the overall statistical significance of the rela- tion and that they are presented in Equation 3 in descend- ing order of their relative importance.

(L-1).

When Equation 3 is used to calculate R/P for the study watersheds in year I, it produces values which have an average error of 10% (Table 2) and a correlation coefficient of 0.91. Thus it is able to explain 91% of the variation observed in normalized recharge, except for one watershed (Figure 4). The anomalous gauge site is located down- stream from a major highway which lies directly atop shal- low bedrock. It appears that this unique combination. of conditions is acting as a partial barrier to ground water flow and forcing an inordinate amount into the stream. The result is higher than expected baseflow. The site was plot- ted (Figure 4), but has been excluded from both the devel- opment of Equation 3 and from the calculation of the correlation coefficient and errors.

Equation 3 is thus derived from 10 observation points. Although that is a relatively small number, each point required the equivalent of 80 to 100 person-hours to obtain. Budget and logistics precluded the expansion of the local set. Other nearby USGS stations were all so large that the effects of specific watershed characteristics would not be resolvable. The validity of the data set will be determined by the testing of the relation in the next section.

Each of the parameters in Equation 3 has an inherent uncertainty associated with its measurement (Table 3). When combined, they propagate a composite error for the calculated R/P o f f 37%, which is shown as representative vertical bars in Figure 3. Similarly, both the baseflow and precipitation measures have indeterminate errors (esti- mated as k 10% and & 5%, respectively). These propagate a total error on the observed R/P of -+ 1 1 % (horizontal error bars, Figure 3).

The sensitivity of FVP to the uncertainty of each water- shed property in Equation 3 was tested (Table 3). Effects of individual properties on the calculated R/P range from 1 % to 11%. When the values are all modified in the direction they would increase or discharge recharge, they produce a composite change of k 32%. This is smaller than the prop- agated value because some of the factors counteract one

Table 3 Sensivitity of Calculated Rm to Input Uncertainty

Independent Manual Measurements GIS Measurements Parameter Uncertaintys A(R/P)b Rel.Sens.c Uncertaintyd A(WP)' Rel.Sens.c

Ks S D Ad Lc Dw N Composi tef

f 15% f 15% f 20% < f 5 % < + 5 % f 20% f 15% f 37%

-8.9% +0.60 +10.5% -0.70 +3.0% -0.20 -0.8% +0.16 +0.8% -0.16 +3.1% -0.16 -5.8% +0.40 f 32%

.t 15% + 30% + 30% f 5% * 5% -35% +30% f 65%

-8.9% +Oh0 -13.8% -0.46

-0.8% +0.16 +0.8% -0.16 +5.5% -0.16 +11.7% +0.40

-4.5% .GO. 15

+5 yo

aUncertainty = indeterminate error bChange in calculated R/p caused by reduction of parameter the amount of listed uncertainty ERelative sensitivity = [(Wpc-WPb)lWpb]/[(IPc-1Pb)nPb] where b = base, c = changed, IP = independent parameter, R = recharge dBias in CIS relative to manual: + indicates CIS consistently > manual; - is < manual; + is no bias Thange in R/P with change in parameter in direction of bias. For +, parameter is reduced. 'Composite values for uncertainty calculated as propagation of indeterminate error. Those for ARK' calculated using Equation 3 with parameter set in direction of bias

~ ~ ~ ~

108 D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1. 102-1 12

another. A reasonable estimate of the total uncertainty asso- ciated with the watershed properties would be -r- 30%. Only the aforementioned outlier falls outside this envelope in Figure 4. The relative sensitivities (Table 3) indicate that the three most important watershed inputs for Equation 3 are surface slope, soil conductivity, and the natural land cover.

Work with Equation 3 has shown that it may calculate extremely high recharge in areas where natural land cover remains on gently sloping surfaces with permeable soils and shallow water tables, attributes common to large wet- lands, for example. To be conservative, the relation should not be applied where combinations of recharge enhancing conditions occur that are outside the dominant ranges tested in this study. Therefore, the following limits on input Val- ues are recommended: K, 52.7 m/d; N 5 30%; S 2 0.03; D, 2 9.1 m; and D 2 5%.

The relation may also calculate negative normalized recharge when the second term of the equation is very large. Negative recharge is impossible within the context of the conceptual model presented and should be interpreted as 0. It arises in heavily urbanized areas with low perme- ability soils and deep water tables. Both these constraints have been applied in the tests which follow.

Testing the Recharge Relationship When applied to the second year of observations at the

study watersheds, Equation 3 calculates values for R/P with an average absolute difference of 20% from observed (Table 2). It explains 82% of the observed spatial variation, and all but two of the watersheds fall within the f. 30% error enve- lope. One of these is the same anomaly from year I (Willow 2). The other (West Branch 1) is the result of very small baseflow influx between stations in year 11. The change in total streamflow between WB2 and WB1 is less than the estimated 2 10% resolution of stream gauging, so the base- flow change for this site could not be resolved accurately.

Figure 4. Comparison of observed and predicted normalized recharge for year I in the study watersheds. Calculations are with Equation 3. Correlation coefficient calculated without outlier at lower right is 0.91. Values and prediction errors are in Table 2. Composite indeterminate error shown at repre- sentative sites. Error envelope (k 30%) is developed in text.

Use in Areas Outside the Study Watersheds To determine if the relation is valid outside the set of

study watersheds, measures of recharge from multiple methods were obtained at six nearby sites. Hydrograph sep- aration was used at four USGS gauging sites. For multiple gauging sites on the same stream, the calculations were again done for the incremental drainage area between two adjacent sites. Properties for these sites were available from GIS (Wisconsin Department of Natural Resources 2000). When compared to manual measures, GIS consistently overestimates slope and land cover percentage?, while underestimating depth to the water table (Table 3). In fact, the average surface slope calculated by ArcView is often more than twice that measured manually, largely because the two methods measure different things. The manual process is designed to obtain a general watershed slope toward the main channel, while ArcView is calculating the slope in each 30 m pixel and averaging those values. .Many of the latter are actually oriented away from the stream and therefore do not show up in the manual measure. Dividing the GIS value by two makes it more representative of the slope measure sought, although it still exceeded the manual value.

When the difference between the manual and GIS Val- ues is propagated through Equation 3, the result is an inde- terminate error of -t 65% (Table 3). However, the GIS values have distinct biases relative to the manual and they tend to counteract in Equation 3. When the uncertainties are entered in the direction of the bias, they produce a compos- ite error of just f 5% (Table 3). In other words, R/P calcu- lated with manual or GIS data agree to within 5% and therefore can be considered equivalent.

Normalized recharge for each independent gauging site was calculated using Equation 3 for the 2 yr of the study and compared very well to the observed value (Fig- ure 6). Equation 3 accounts for 8 1 % of the observed varia- tion and reproduces individual values to within r 16%.

Comparison to Other Methods o f Recharge Estimation For further testing, recharge was obtained using meth-

ods other than hydrograph separation. A water budget analysis was conducted for a part of Mequon, Wisconsin (Cherkauer and Bacon 1978), outside the study area. Equa- tion 3 was used with 1978 precipitation and land cover and produced a normalized recharge within 3.1% of the budget value (Figure 6, Table 4). Recharge was also calculated fr6m the well hydrograph at the Richfield site (Figure 2). The 5 yr average normalized recharge is 0.109, and Equa- tion 3 calculates 0.100 cm/cm, a difference of 9% (Fig- ure 6).

As another test, the precipitation runoff modeling sys- tem (PRMS) was calibrated in seven watersheds in south- eastern Wisconsin (Cherkauer 2004). This distributed- parameter model takes climatic input (temperature and pre- cipitation) to a watershed and then distributes the incoming water among evapotranspiration, interception, runoff, and infiltration. The model combines surface runoff, interflow, and baseflow to produce a simulated stream hydrograph, which can then be calibrated to the observed hydrograph in a gauged watershed. The process, explained in Cherkauer (2004), allows ground water recharge to be calculated from

D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1 : 102-1 12 109

0.25

0.2

'z1 0.15 s! - c 5 0.1 s -

0.05

0 0 0.05 0.1 0.15 0.2 0.25

Observed RIP

Figure 5. Comparison of observed and predicted normalized recharge for year I1 in the study watersheds. Calculations are with Equation 3. Correlation coefficient calculated without outlier at lower right is 0.83. Error bars and envelope are as in Figure 4.

Table 4 Summary of Errors Observed in Testing

Recharge Calculation Relation

Method Number of Avg. Absolute

Observations Error (%)

Hydrograph separation - internal sites - Yr I1

Hydrograph separation -

9

all external sites 8

Hydrograph separation - external sites - Yr I1 4

Well hydrograph 5

Water budget 1

Distributed parameter model (PRMS) 63

19.9

15.5

19.4

11.8

3.1

17.5

the model as the direct inflow to the saturated zone. This has been done for 63 subwatersheds within the seven gauged watersheds. Six of these subwatersheds overlap with study watersheds but 'do not coincide identically. The remaining 57 are entirely outside the study watersheds. The method used 30 yr average daily precipitation and temper- ature to generate a 30 yr average recharge rate.

Equation 3 was used to calculate normalized recharge, which was then converted to recharge using the 30 yr aver- age precipitation at each site. Recharge rates in the study vary spatially by a factor of five, while precipitation varies only 10% about the mean. The conversion of R/P to recharge thus retains essentially all of the inherent variabil- ity and allows recharge converted from R P to be compared to that from methods which estimate recharge directly. Equation 3 accounts for 68% of the spatial variability gen- erated by PRMS (Figure 7a) and predicts recharge to within f 18% (Table 4). Both sets of recharges show the same overall range of values but slightly different frequency dis- tributions (Figure 7b). The resulting mean recharge from

PRMS is - 22% higher than that for the empirical method, indicating that Equation 3 is conservative.

The empirical relation has the advantage over PRMS in that it can be readily applied to broader areas. It also cal- culates recharges that are accurate to within f 20%. Fur- thermore, PRMS can only be used directly on gauged

Figure 6. Comparison of observed and predicted normalized recharge at independent test areas. Calculations are with Equation 3. Correlation coefficient is 0.81. Prediction errors are in Table 5. Solid symbols are values from USGS gauging stations calculated using baseflow separation. Water budget and well methods are defined in text.

Figure 7. Comparison of recharge rates calculated with Equation 3 and PRMS. (a) Scatter plot for 63 subwatersheds in seven calibrated watersheds. Correlation coefficient is 0.68. (b) Frequency histograms of recharge calculated with Equa- tion 3 and PRMS.

110 D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12

watersheds, although Cherkauer (2004) suggests it can be extrapolated to ungauged areas near multiple calibrated watersheds.

Comparison to Other Time Periods It has already been demonstrated that Equation 3 can

successfully reproduce a second year of normalized recharge observed in the study watersheds. Similarly, it has also been applied on the test sites outside the study water- sheds for the second year of the study. The comparison between observed and predicted normalized recharge in year I1 (Figure 6) indicates an average difference from observed of k 19% (Table 4). Thus the empirical relation reproduces to within -c 19% of the normalized recharge obtained from baseflow separation for different years.

Recharge rates were calculated through time from well hydrograph variations in a domestic well at the Richfield site (Figure 2). Water level was monitored weekly for > 5 yr at times when the well had recovered from pumping. Annual recharge was calculated from

R = (AH) n (9)

where R is recharge rate (cdyr) , AH is change in head between the annual minimum and maximum (crdyr), and n is effective porosity (%). The porosity of the sand and gravel aquifer in which the well is finished is estimated to be 17%. Equation 3 was used to calculate the expected nor- malized recharge for each year at the same site, using on- site precipitation measurements. Conversion of the normalized recharge to annual recharge allows comparison with Equation 3 results (Figure 8) . For both calculations, the annual period used was started on December 1, roughly the start of the period of frozen ground. Again, the compar- ison is excellent, reproducing observations to within .t 12% (Table 4).

Summary and Conclusions In areas with humid climate, the management of

ground water has not received as much attention as in arid areas. Yet overuse is not uncommon, and demands in some humid areas are already known to exceed replenishment via recharge. The need exists, therefore, for a method that can provide a reasonable measure of recharge at scales ranging from individual communities to entire political regions. Ideally it should be based on readily available information and be relatively simple to apply without requiring data from long monitoring programs. In the authors view, any uncertainties in the method should be conservative (biased toward lower recharge) to prevent overstating the availabil- ity of ground water resources. Most existing methods do not achieve the full set of criteria, limiting their immediate usefulness in this ground water management issue. By the time sufficient data are collected to use water budgets or tracers or other procedures, political and economic deci- sions about development will have been made, perhaps with insufficient ground water information,

This paper presents a methodology that allows a more rapid determination of recharge. An empirical relation between normalized recharge and readily available cli- matic, topographic, hydrogeologic, and land measures has

Figure 8. (a) Comparison of recharge rates calculated with Equation 3 and from a well hydrograph. (b) Well hydrograph from Richfield site in Figure 2.

been developed for small watersheds in the glaciated terrain of southeastern Wisconsin. The analysis has been con- strained by forcing the independent variables in multiple regression to have the same dimensions as the dependent variable. It shows that the dependent normalized recharge varies directly with the ratio of infiltration flux to overland runoff flux and with the ratio of natural land cover to total surface area. It also varies inversely with the ratio of repre- sentative vertical and lateral travel distances in a watershed.

With few exceptions, the input needed to determine all these independent ratios is now available in CIS data bases in many locations.

The relation (Equation 3 ) was derived for 10 small watersheds based on one year of intensive observations. The small watersheds allow resolution of how specific water- shed attributes relate to recharge. The results show that recharge in the study watersheds ranges from 3% to 25% of precipitation (Table 2), depending on local conditions.

The relation was subjected to a variety of tests. Despite the relatively small observation set, the relation was shown to be statistically significant. It also reproduces both the spatial and temporal variation of observed recharge (expressed as either recharge per unit precipitation or recharge rate) in areas outside the study watersheds and for times other than the study period all within about f 20%. The method produces values that agree with those obtained from hydrograph separation (to within 20%), water budget calculation (4%), well hydrograph analysis (12%), and a

D.S. Cherkauer, S.A. Ansari GROUND WATER 43, no. 1: 102-1 12 11 1

calibrated, distributed-parameter model (1 8%). It was shown to work in southeastern Wisconsin for watersheds between 3 and 300 km2 in area. Based on one years record, it reproduced measures of recharge in a second year in the study area, for 5 yr of record on a well hydrograph, from watershed models calibrated to 30 yr average streamflows, and with a budget analysis done for a period of 20 yr prior to the study. The method works with either manually or digitally measured watershed properties.

The method presented cannot and should not be viewed as providing an absolute value of recharge, even within the study area. It is not presented as an alternative to more direct procedures that have been developed to obtain recharge. Rather, it provides a quick way to calculate a rea- sonably accurate estimate of recharge from commonly available information. It can provide a community with a first estimate of its overall recharge rates (and thus a total inflow to the ground water system). It also provides a way to identify areas with high recharge rates where develop- ment should be scrutinized to minimize impact on the quan- tity and quality of recharge.

The method has significant advantages over other methods in providing a conservative first approximation of recharge. It can be readily applied in ungauged watersheds, where baseflow separation is not possible. It is much sim- pler to apply than distributed parameter or other surface hydrologic models. Finally, it can be applied across an entire region, but with a resolution down to the scale of 3 to 5 km2. As an outgrowth of this last point, the method was used successfully to generate estimates of the magnitude and spatial variability of recharge rates to be used as input to ground water flow models (Feinstein et al. 2004; Cherkauer and Carlson 2002). These regional calculations show considerable similarity to the recharge potentials developed by Braun et al. (2003) for the county immedi- ately south of the study area.

The method remains to be tested outside southeastern Wisconsin. The dimensionless ratios of flux, travel dis- tance, and area developed for Equation 3 have intuitively reasonable relations to recharge. It is the authors belief that these same ratios may function as controlling parameters on recharge outside the study area, but that the values of the regression coefficients will change. These could be deter- mined elsewhere, however, by using baseflows separated from stream hydrographs at USGS gauging stations. Imposing the dimensionless, independent parameters on the relation would allow use of larger scale gauging stations for this purpose.

Acknowledgments This work was funded by the University of Wisconsin

System Ground Water Research Coordinating Council and by the Source Water Protection Program of the Wisconsin Department of Natural Resources. We thank Kenneth Bradbury, Peter Cook, and some anonymous reviewers for their critical assessment and suggestions, which resulted in substantive improvement of the manuscript.

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17-30.

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