National Water-Quality Assessment Program

The Use of Process Models to Inform and 

Improve Statistical Models of Nitrate Occurrence, 

Great Miami River Basin, Southwestern Ohio

84°0'0"W

Stillwater River

Little Miami 

 River

Great Miami River

Mad River

Finite-difference grid

Great Miami

 River

Wolf Creek

Area excluded

No model solution

27 EXPLANATION

-9

Difference in probability of 

exceeding a concentration 

of 4 mg/L, in percent, from

500-meter circular buffers 

and modeled recharge areas 

No difference

Scientific Investigations Report 2012–5001

U.S. Department of the Interior

U.S. Geological Survey
Front-cover art: Photograph of grain bins on a Midwestern farm and a map of Miami River Basin, Ohio study area. 

Photograph by Kelly L. Warner, U.S. Geological Survey.

Back-cover art: National map of NAWQA basins.
The Use of Process Models to Inform and 

Improve Statistical Models of Nitrate 

Occurrence, Great Miami River Basin, 

Southwestern Ohio

By Donald A. Walter and J. Jeffrey Starn

National Water-Quality Assessment Program

Scientific Investigations Report 2012–5001

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2013

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Suggested citation:

Walter, D.A., and Starn, J.J., 2013, The use of process models to inform and improve statistical models of nitrate 

occurrence, Great Miami River Basin, southwestern Ohio: U.S. Geological Survey Scientific Investigations Report 

2012–5001, 75 p. (Also available at http://pubs.usgs.gov/sir/2012/5001.)
iii

Foreword

The U.S. Geological Survey (USGS) is committed to providing the Nation with reliable scientific 

information that helps to enhance and protect the overall quality of life and that facilitates 

effective management of water, biological, energy, and mineral resources (http://www.usgs.

gov/). Information on the Nation’s water resources is critical to ensuring long-term availability 

of water that is safe for drinking and recreation and is suitable for industry, irrigation, and fish 

and wildlife. Population growth and increasing demands for water make the availability of 

that water, measured in terms of quantity and quality, even more essential to the long-term 

sustainability of our communities and ecosystems.

The USGS implemented the National Water-Quality Assessment (NAWQA) Program in 1991 

to support national, regional, State, and local information needs and decisions related to 

water-quality management and policy (http://water.usgs.gov/nawqa). The NAWQA Program 

is designed to answer: What is the quality of our Nation’s streams and groundwater? How are 

conditions changing over time? How do natural features and human activities affect the quality 

of streams and groundwater, and where are those effects most pronounced? By combining 

information on water chemistry, physical characteristics, stream habitat, and aquatic life, 

the NAWQA Program aims to provide science-based insights for current and emerging water 

issues and priorities. From 1991 to 2001, the NAWQA Program completed interdisciplinary 

assessments and established a baseline understanding of water-quality conditions in 51 of the 

Nation’s river basins and aquifers, referred to as Study Units (http://water.usgs.gov/nawqa/

studies/study_units.html).

In the second decade of the Program (2001–2012), a major focus is on regional assessments 

of water-quality conditions and trends. These regional assessments are based on major river 

basins and principal aquifers, which encompass larger regions of the country than the Study 

Units. Regional assessments extend the findings in the Study Units by filling critical gaps in 

characterizing the quality of surface water and groundwater, and by determining water-quality 

status and trends at sites that have been consistently monitored for more than a decade. In 

addition, the regional assessments continue to build an understanding of how natural features 

and human activities affect water quality. Many of the regional assessments employ modeling 

and other scientific tools, developed on the basis of data collected at individual sites, to 

help extend knowledge of water quality to unmonitored, yet comparable areas within the 

regions. The models thereby enhance the value of our existing data and our understanding 

of the hydrologic system. In addition, the models are useful in evaluating various resourcemanagement scenarios and in predicting how our actions, such as reducing or managing 

nonpoint and point sources of contamination, land conversion, and altering flow and (or) 

pumping regimes, are likely to affect water conditions within a region.

Other activities planned during the second decade include continuing national syntheses 

of information on pesticides, volatile organic compounds (VOCs), nutrients, trace elements, 

and aquatic ecology; and continuing national topical studies on the fate of agricultural 

chemicals, effects of urbanization on stream ecosystems, bioaccumulation of mercury in 

stream ecosystems, effects of nutrient enrichment on stream ecosystems, and transport of 

contaminants to public-supply wells.
iv

The USGS aims to disseminate credible, timely, and relevant science information to address 

practical and effective water-resource management and strategies that protect and restore 

water quality. We hope this NAWQA publication will provide you with insights and information 

to meet your needs, and will foster increased citizen awareness and involvement in the 

protection and restoration of our Nation’s waters.

The USGS recognizes that a national assessment by a single program cannot address all 

water-resource issues of interest. External coordination at all levels is critical for cost-effective 

management, regulation, and conservation of our Nation’s water resources. The NAWQA 

Program, therefore, depends on advice and information from other agencies—Federal, 

State, regional, interstate, Tribal, and local—as well as nongovernmental organizations, 

industry, academia, and other stakeholder groups. Your assistance and suggestions are 

greatly appreciated.

William H. Werkheiser

USGS Associate Director for Water
v

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................2

Purpose and Scope ..............................................................................................................................4

Methods of Analysis.............................................................................................................................6

Statistical Models of Nitrate Occurrence ...............................................................................6

Numerical Model of the Great Miami River Basin ...............................................................10

Model Design and Calibration ........................................................................................11

Modifications to Regional Model of the Great Miami River Basin...........................11

Sources of Water and Advective Transport of Nitrate to Pumped Wells...........................................15

Factors Affecting Steady-State Contributing Recharge Areas to Pumped Wells ...................15

Pumping Rate .............................................................................................................................15

Hydrologic Setting .....................................................................................................................15

Well Depth...................................................................................................................................18

Duration of Pumping..................................................................................................................18

Advective Transport of Nitrate .........................................................................................................21

Traveltimes..................................................................................................................................22

Transport of Nitrate ...................................................................................................................25

Comparison of Statistical and Process Models .....................................................................................27

Predictions of Nitrate Occurrence by Statistical and Process Models at Selected Wells....28

Role of Pumping Rate and Well Depth ...................................................................................30

Estimates of Area-Weighted Spatial Variables ...........................................................30

Predictions of Nitrate-Exceedance Probabilities........................................................30

Role of Hydrologic Setting........................................................................................................33

Estimates of Area-Weighted Spatial Variables ...........................................................33

Predictions of Nitrate-Exceedance Probabilities........................................................33

Predictions of Nitrogen-Occurrence Probabilities for Public-Supply and 

Network Wells ..............................................................................................................36

Basin-Scale Comparisons of Statistical and Process Models ............................................................36

Comparison of Circular Buffers and Contributing Recharge Areas as the Assumed 

Sources of Water to Wells...................................................................................................39

Predictions of Nitrate Occurrence by Using Statistical Models ................................................43

Estimates of Area-Weighted Spatial Variables.....................................................................43

Predictions of Nitrate-Exceedance Probabilities.................................................................45

Role of Well Depth .....................................................................................................................47

Estimates of Depth to Water....................................................................................................51

Predictions of Nitrate Concentrations by Using Nitrogen Loading Rates.................................58

Comparison of Predictions of Nitrate Occurrences and Concentrations .................................58

Limitations of Analysis ................................................................................................................................62

Gradient and Boundary Effects on the Mass Balance of Recharge and Pumping .................64

Gradient and Boundary Effects on Estimated Spatial Variables and 

Statistical-Model Predictions .............................................................................................64

Summary........................................................................................................................................................70

References Cited..........................................................................................................................................74
vi

Figures

1. Map showing A, extent of the glacial aquifer system, texture of near-surface glacial 

deposits, the central framework area, and location of the study area within the upper 

part of the Great Miami River Basin and B, block diagrams of valley-fill and buriedvalley aquifers commonly found in the central framework area..........................................3

2. Map showing extent of buried-valley glacial aquifers within the study area near 

Dayton, Ohio, and extent of regional groundwater-flow-model grid, southwestern 

Ohio .................................................................................................................................................5

3. A, Graph showing sets of explanatory variables used in 10 logistic-regression models 

to describe nitrate occurrence in the glacial aquifer system and B, diagrams showing 

example of spatial variables within a 1,640-ft radius of a hypothetical well in the 

Great Miami River Basin, southwestern Ohio..........................................................................8

4. Maps showing spatial distribution of A, environmental variables—silt content, alfisol 

content, and depth to water—and B, nitrogen application rates in the modeled area 

of the Great Miami River Basin, southwestern Ohio ..............................................................9

5. Cross section showing generalized geology and groundwater-flow patterns in a 

buried-valley glacial aquifer typical of the upper part of the Great Miami River Basin, 

southwestern, Ohio ....................................................................................................................10

6. Map showing regional model grid, active model layers, extents of subregional 

models, simulated water table, and locations of production wells, upper part of the 

Great Miami River Basin, southwestern Ohio........................................................................12

7. Map showing hydrologic boundaries and hydraulic-conductivity and recharge 

parameter zones used to represent the regional aquifer system, upper part of the 

Great Miami River Basin, southwestern Ohio........................................................................13

8. Maps showing distribution of A, horizontal hydraulic conductivity values in layer 1 

and B, recharge from precipitation in the calibrated regional model of the Great 

Miami River Basin, southwestern Ohio...................................................................................14

9. Map showing the 1,640-ft-radius circular buffer and steady-state contributing 

recharge areas to well 1053 (a hypothetical well) for three pumping rates, 

southeastern part of the Great Miami River Basin, southwestern Ohio ...........................16

10. Map showing the 1,640-ft-radius circular buffer and steady-state contributing 

recharge areas for intermediate and low pumping rates in three hydrologic settings, 

southeastern part of the Great Miami River Basin, southwestern Ohio ...........................17

11. Map showing the 1,640-ft-radius circular buffer and steady-state contributing 

recharge areas to well 1053 for A, high and intermediate, and B, low and intermediate 

pumping rates for three well depths, southeastern part of the Great Miami River 

Basin, southwestern Ohio .........................................................................................................19

12. Diagram showing circular buffer of radius 1,640 feet, transient contributing recharge 

areas, steady-state recharge areas for intermediate pumping rates, and traveltimes 

under steady-state transport to well 1053 after A, 2 years and B, 20 years of pumping, 

southeastern part of the Great Miami River Basin, southwestern Ohio ...........................20

13. A, Diagram showing relation between well depth and traveltime; and B, map showing 

1,640-ft-radius circular buffers and steady-state traveltimes within contributing 

recharge areas to pumped wells in three hydrogeologic settings and areas of 

nitrogen application from fertilizer, southeastern part of the Great Miami River Basin, 

southwestern Ohio .....................................................................................................................23

14. Graph showing A, Steady-state traveltimes to pumped wells in three hydrologic 

settings; and B, transient traveltimes after different periods of pumping by well 1053, 

southeastern part of the Great Miami River Basin, southwestern Ohio ...........................24
vii

15. Graph showing steady-state traveltimes for well 1053 pumped at different depths 

and at different pumping rates, southeastern part of the Great Miami River Basin, 

southwestern Ohio .....................................................................................................................25

16. Graph showing time-varying nitrate concentrations for A, wells in three hydrologic 

settings and B, well 1053 for three well depths and low and intermediate pumping 

rates ..............................................................................................................................................26

17. Graph showing time-varying nitrate concentrations at well 1053 for different 

porosities and source-term characteristics under steady-state transport for 

intermediate pumping rates......................................................................................................28

18. Map showing circular buffer of radius 1,640-ft and equivalent contributing recharge 

area to well 1053 and spatial distribution of explanatory variables used to estimate 

nitrate occurrence, Great Miami River Basin, southwestern Ohio....................................29

19. Map showing circular buffer of radius 1,640-ft and equivalent contributing recharge 

areas to well 1053 for different combinations of well depths and pumping rates, and 

spatial distribution of explanatory variables used to estimate nitrate occurrence, 

Great Miami River Basin, southwestern Ohio........................................................................31

20. Graphs showing A, estimates of area-weighted explanatory variables and B, the 

probability of nitrate occurrence above threshold concentrations of 1 to 4 milligrams 

per liter as nitrogen by statistical models for the 1,640-ft circular buffer and different 

combinations of well depth and pumping rate at well 1053 ................................................32

21. Map showing circular buffer of radius 1,640-ft and steady-state contributing recharge 

areas to wells in three hydrologic settings for intermediate and low pumping rates, 

and spatial distribution of explanatory variables used to estimate nitrate occurrence, 

southeastern part of Great Miami River Basin, southwestern Ohio..................................34

22. Graphs showing A, estimates of area-weighted explanatory variables and 

B, predictions of nitrate occurrence above threshold concentrations from 1 to 

4 milligrams per liter as nitrogen by using statistical models for wells in three 

hydrologic settings for intermediate and low pumping rates .............................................35

23. Graphs showing A, estimates of area-weighted explanatory variables and 

B, predictions of nitrate occurrence above threshold concentrations of 1 to 

4 milligrams per liter as nitrogen determined by using contributing recharge areas 

and circular buffers for production wells, Great Miami River Basin, southwestern 

Ohio ...............................................................................................................................................37

24. Graphs showing A, estimates of area-weighted explanatory variables and 

B, predictions of nitrate occurrence above threshold concentrations of 1 and 

4 milligrams per liter as nitrogen determined by using recharge areas and circular 

buffers for network wells in the Great Miami River Basin, southwestern Ohio...............38

25. Map showing volumetric recharge rates through circular buffers for hypothetical 

wells uniformly spaced at 500-foot intervals across the Great Miami River Basin, 

southwestern Ohio .....................................................................................................................40

26. Map showing overlap between contributing recharge areas and circular buffers 

in three hydrologic settings, southeastern part of the Great Miami River Basin, 

southwestern Ohio .....................................................................................................................41

27. Map showing overlap between contributing recharge areas and circular buffers for 

20,920 hypothetical wells uniformly spaced at 500-foot intervals across the Great 

Miami River Basin, southwestern Ohio...................................................................................42

28. Histogram showing distribution of overlaps between contributing recharge areas 

and circular buffers for 20,920 hypothetical wells in the Great Miami River Basin, 

southwestern Ohio .....................................................................................................................43
viii

29. Graphs showing relation between contributing recharge areas and circular buffers 

for A, nitrogen-application rate and silt content, and B, alfisol content and depth to 

water.............................................................................................................................................44

30. Histograms showing differences in estimated area-weighted means for A, nitrogenapplication rate, B, silt content, C, alfisol content, and D, depth to water for 

contributing recharge areas and circular buffers ................................................................45

31. Maps showing differences in the area-weighted means of the A, nitrogenapplication rate and B, silt content estimated for contributing recharge areas and 

circular buffers, Great Miami River Basin, southwestern Ohio .........................................46

32. Maps showing probabilities of exceeding nitrate concentrations of A, 1 milligram 

per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted from logistic 

regression models on the basis of spatial variables for circular buffers, Great Miami 

River Basin, southwestern Ohio...............................................................................................48

33. Maps showing probabilities of exceeding nitrate concentrations of A, 1 milligram 

per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted by logistic 

regression models on the basis of spatial variables for contributing recharge areas, 

Great Miami River Basin, southwestern Ohio........................................................................49

34. Graph showing relation between probabilities of exceeding nitrate concentrations 

of 1 and 4 milligrams per liter as nitrogen on the basis of area-weighted mean 

explanatory variables for contributing recharge areas and circular buffers...................50

35. Histograms showing differences in predicted probabilities of exceeding 

concentrations of A, 1 milligram per liter as nitrogen; and B, 4 milligrams per liter as 

nitrogen on the basis of area-weighted mean explanatory variables for contributing 

recharge areas and circular buffers .......................................................................................51

36. Maps showing differences in the probabilities of nitrate concentrations exceeding 

A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted 

by logistic regression models on the basis of spatial variables for contributing 

recharge areas and circular buffers, Great Miami River Basin, southwestern Ohio .....52

37. Maps showing areas where probabilities of nitrate concentrations exceeding 

A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen were 

higher than the median probabilities predicted by logistic regression models 

for contributing recharge areas and circular buffers, Great Miami River Basin, 

southwestern Ohio .....................................................................................................................53

38. Map showing contributing recharge areas and the circular buffer to well 725 (a 

hypothetical well) for shallow and deep well screens, southeastern part of the 

Great Miami River Basin, southwestern Ohio........................................................................54

39. Graph showing relation between fractions of overlap between contributing recharge 

areas and corresponding circular buffers for wells pumped in shallow and deep 

parts of the glacial aquifer system, Great Miami River Basin, southwestern Ohio.........55

40. Graphs showing A, estimates of nitrogen-application rate and silt content, and 

B, probabilities of nitrate occurrence above threshold concentrations of 1 and 

4 milligrams per liter as N determined for contributing recharge areas and circular 

buffers for shallow and deep pumped wells, Great Miami River Basin, southwestern 

Ohio ...............................................................................................................................................56

41. Graphs showing relations between A, depths to water within circular buffers 

estimated from STATSGO data and calculated by a groundwater-flow model, and 

B, probability of nitrate concentrations exceeding 1 and 4 milligrams per liter as 

nitrogen on the basis of the two sets of estimated depths to water..................................57

42. Maps showing A, nitrate concentrations (as N) and B, concentration percentiles after 

50 years of steady-state transport from areas of nitrogen-fertilizer application to 

shallow pumped wells, Great Miami River Basin, southwestern Ohio..............................59
ix

Table

1. Summary of statistical models of nitrate occurrence in the glacial aquifer system 

for threshold concentrations of 1 to 10 milligrams per liter as nitrogen .............................7

43. Graph showing relations between simulated nitrate concentrations and areaweighted mean nitrogen-application rates within circular buffers and contributing 

recharge areas, Great Miami River Basin, southwestern Ohio..........................................60

44. Maps showing areas where nitrate concentrations are predicted to exceed 1 and 

4 milligrams per liter as nitrogen by a process model based on transport times of 

A, 50, and B, 10 years, Great Miami River Basin, southwestern Ohio ...............................61

45. Graph showing relationship between simulated nitrate concentrations for wells 

pumped in shallow and deep parts of the glacial aquifer, Great Miami River Basin, 

southwestern Ohio .....................................................................................................................62

46. Maps showing areas where the probability of exceeding A, 1 milligram per liter as 

nitrogen, and B, 4 milligrams per liter as nitrogen exceeds the basinwide median 

probability from statistical models and areas where simulated nitrate concentrations 

from a process model exceed the median for the basin, Great Miami River Basin, 

southwestern Ohio .....................................................................................................................63

47. Map showing contributing recharge areas that are examples of a gradient effect 

(well 1860) and a boundary effect (well 1629) in the northeastern part of the Great 

Miami River Basin, southwestern Ohio...................................................................................65

48. Graphs showing A, simulated ratio of pumping rates in shallow pumped wells to 

recharge rates estimated from contributing recharge areas and B, relative 

importance of gradient and boundary effects for shallow (layer 1) and deep (layer 3) 

pumped wells ..............................................................................................................................66

49. Map showing distribution of large gradient and boundary effects, defined as 

differences of more than 50 percent between recharge and pumping rates, Great 

Miami River Basin, southwestern Ohio...................................................................................67

50. Graph showing overlap between areas of circular buffers and contributing recharge 

areas for all wells in the basin and those wells with a mass balance between 

recharge and pumping rates within 10 percent, Great Miami River Basin, 

southwestern Ohio .....................................................................................................................68

51. Graph showing differences in A, estimates of area-weighted spatial variables, and 

B, probabilities of exceeding 1 and 4 milligrams per liter as nitrogen determined for 

contributing recharge areas and circular buffers for all hypothetical wells in the basin 

and a subset of wells with a mass balance between pumping and recharge rates 

within 10 percent, Great Miami River Basin, southwestern Ohio.......................................69
x

Conversion Factors, Datum, and Abbreviations

Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

Area

acre 0.4047 hectare (ha)

square mile (mi2) 259.0 hectare (ha)

Flow rate

foot per day (ft/d) 0.3048 meter per day (m/d)

million gallons per day (Mgal/d) 0.04381 cubic meter per second (m3/s)

Hydraulic conductivity

foot per day (ft/d) 0.3048 meter per day (m/d)

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NGVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Altitude, as used in this report, refers to distance above the vertical datum.

Concentrations of chemical constituents in water are given either in milligrams per liter (mg/L) 

or moles per liter (M/L).

Application rates are given in kilograms per year (kg/yr).

Abbreviations

NEU nitrogen uptake efficiency 

NAWQA National Water-Quality Assessment Program

NLCDE92 Enhanced National Land Cover Data 1992

STATSGO U.S. Department of Agriculture State Soil Geographic Database

USGS U.S. Geological Survey
The Use of Process Models to Inform and Improve 

Statistical Models of Nitrate Occurrence, Great Miami 

River Basin, Southwestern Ohio

By Donald A. Walter and J. Jeffrey Starn

Abstract

Statistical models of nitrate occurrence in the glacial 

aquifer system of the northern United States, developed by the 

U.S. Geological Survey, use observed relations between nitrate 

concentrations and sets of explanatory variables—representing 

well-construction, environmental, and source characteristics—

to predict the probability that nitrate, as nitrogen, will exceed a 

threshold concentration. However, the models do not explicitly 

account for the processes that control the transport of nitrogen 

from surface sources to a pumped well and use area-weighted 

mean spatial variables computed for a circular buffer around 

the well as a simplified source-area conceptualization. The use 

of models that explicitly represent physical-transport processes 

can inform and potentially improve these statistical models. 

Specifically, groundwater-flow models simulate advective 

transport—predominant in many surficial aquifers—and 

can contribute to the refinement of the statistical models by 

(1) providing for improved, physically based representations 

of a source area to a well, and (2) allowing for more detailed 

estimates of environmental variables.

A source area to a well, known as a contributing recharge 

area, represents the area at the water table that contributes 

recharge to a pumped well; a well pumped at a volumetric rate 

equal to the amount of recharge through a circular buffer will 

result in a contributing recharge area that is the same size as 

the buffer but has a shape that is a function of the hydrologic 

setting. These volume-equivalent contributing recharge areas 

will approximate circular buffers in areas of relatively flat 

hydraulic gradients, such as near groundwater divides, but 

in areas with steep hydraulic gradients will be elongated in 

the upgradient direction with respect to the corresponding 

circular buffers.

The degree to which contributing recharge areas 

estimated by process models, which simulate advective 

transport and therefore account for local hydrologic 

settings, would inform and improve the development of 

statistical models can be implicitly estimated by evaluating 

the differences between explanatory variables estimated 

from the contributing recharge areas and from the circular 

buffers used to develop existing statistical models. The 

larger the difference in estimated variables, the more likely 

that statistical models would be changed, and presumably 

improved, if explanatory variables estimated from contributing 

recharge areas were used in model development. Comparing 

model predictions from the two sets of estimated variables 

would further quantify—albeit implicitly—how an improved, 

physically based estimate of explanatory variables would be 

reflected in model predictions. Differences between the two 

sets of estimated explanatory variables and resultant model 

predictions vary spatially; greater differences are associated 

with areas of steep hydraulic gradients. A direct comparison, 

however, would require the development of a separate set 

of statistical models based on explanatory variables from 

contributing recharge areas.

In the Great Miami River Basin, a buried-valley glacial 

aquifer in southwestern Ohio, the agreement (measured as the 

coincident or overlapping area) between circular buffers and 

contributing recharge areas in a network of 21,774 uniformly 

spaced hypothetical wells ranged from less than 5 to more than 

95 percent and averaged 32 percent. In this study, source areas 

represented as circular buffers generally differed substantially 

from those represented as contributing recharge areas of 

equivalent size. This difference indicates that incorporating 

an improved understanding of groundwater processes through 

process models can inform statistical models of nitrate 

occurrence in glacial environments.

Area-weighted means of three environmental variables—

silt content, alfisol content, and depth to water from the U.S. 

Department of Agriculture State Soil Geographic (STATSGO) 

database—and one nitrogen-source variable (fertilizerapplication rate from county data mapped to agricultural land 

use within the Enhanced National Land Cover Data 1992 

(NLCDe92) database can vary substantially between circular 

buffers and volume-equivalent contributing recharge areas and 

among contributing recharge areas for different sets of well 

variables. The differences in estimated explanatory variables 

are a function of the same factors affecting the contributing 

recharge areas as well as the spatial resolution and local distribution of the underlying spatial data. As a result, differences 
2 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

in estimated variables between circular buffers and contributing recharge areas are complex and site specific as evidenced 

by differences in estimated variables for circular buffers and 

contributing recharge areas of existing public-supply and 

network wells in the Great Miami River Basin. Large differences in area-weighted mean environmental variables were 

observed at the basin scale, determined by using the network 

of uniformly spaced hypothetical wells; the differences have 

a spatial pattern that generally is similar to spatial patterns in 

the underlying STATSGO data. Generally, the largest differences were observed for area-weighted nitrogen-application 

rates from county and national land-use data; the basin-scale 

differences ranged from -1,600 (indicating a larger value for 

the area within the volume-equivalent contributing recharge 

area) to 1,900 kilograms per year; the range in the underlying 

spatial data was 0 to 2,200 kilograms per year. Silt content, 

alfisol content, and nitrogen-application rate are defined by 

the underlying spatial data and are external to the groundwater 

system; however, depth to water is an environmental variable that can be estimated in more detail and, presumably, in 

a more physically based manner by using a groundwater-flow 

model instead of the spatial data. Model-calculated depths to 

water within circular buffers in the Great Miami River Basin 

differed substantially from values derived from the spatial data 

and had a much larger range.

Differences in estimates of area-weighted spatial 

variables result in corresponding differences in predictions 

of nitrate occurrence in the aquifer. In addition to the 

factors affecting contributing recharge areas and estimated 

explanatory variables, differences in predictions also are a 

function of the specific set of explanatory variables used and 

the fitted slope coefficients in a given model. For models that 

predicted the probability of exceeding 1 and 4 milligrams per 

liter as nitrogen (mg/L as N), probabilities based on variables 

estimated from circular buffers and contributing recharge 

areas generally were correlated but differed significantly 

at the local and basin scale. The scale and distribution of 

prediction differences can be explained by the underlying 

differences in the estimated variables and the relative weight 

of the variables in the statistical models. Differences in 

predictions of exceeding 1 mg/L as N on the basis of only 

environmental variables generally were correlated with the 

underlying differences in STATSGO data, whereas differences 

in predictions of exceeding 4 mg/L as N were more spatially 

extensive because that model included environmental and 

nitrogen-source variables. Using depths to water from within 

circular buffers derived from the spatial data and depths 

to water within the circular buffers calculated from the 

groundwater-flow model, restricted to the same range, resulted 

in large differences in predicted probabilities. The differences 

in estimated explanatory variables between contributing 

recharge areas and circular buffers indicate that incorporation 

of physically based contributing recharge areas likely would 

result in a different set of explanatory variables and an 

improved set of statistical models.

The use of a groundwater-flow model to improve 

representations of source areas or to provide more detailed 

estimates of specific explanatory variables includes a number of limitations and technical considerations. Assumptions 

in these analyses are that (1) there is a state of mass balance 

between recharge and pumping, and (2) transport to a pumped 

well is under a steady-state flow field. Comparison of volumeequivalent contributing recharge areas under steady-state and 

transient transport conditions at a location in the southeastern part of the basin shows that the steady-state contributing 

recharge area is a reasonable approximation of the transient 

contributing recharge area after 10 to 20 years of pumping. 

The first assumption is a more important consideration for this 

analysis. A gradient effect refers to a condition where simulated pumping from a well is less than recharge through the 

corresponding contributing recharge area. This effect generally 

takes place in areas with steep hydraulic gradients, such as 

near discharge locations, and can be mitigated by using a finer 

model discretization. A boundary effect refers to a condition 

where the recharge rate through the contributing recharge 

area is less than the volumetric pumping rate. This effect 

indicates that other sources could be contributing water to the 

simulated well. In the Great Miami River Basin, large gradient effects, where the volumetric recharge rate is more than 

double the corresponding pumping rate, and large boundary 

effects, where the volumetric recharge rate is less than half 

the corresponding pumping rate, occurs in 5 and 14 percent 

of the modeled basin, respectively. The agreements between 

circular buffers and volume-equivalent contributing recharge 

areas, differences in estimated variables, and the effects on 

statistical-model predictions between the population of wells 

with balances between pumping and recharge rates within 

10 percent and the population of all wells were similar. These 

agreements indicated that process-model limitations did not 

affect the overall findings for the Great Miami River Basin; 

however, such a conclusion would be model specific, and 

prudent use of a process model needs to entail a limitations 

analysis and, if necessary, alterations to the model.

Introduction

Unconsolidated glacial sediments cover about 

722,000 square miles (mi2

) and portions of 25 states in the 

northern contiguous United States (fig. 1A). The glacial 

sediments, which consist of gravel, sand, silt, and clay and 

exceed 1,000 feet (ft) in thickness in some areas, compose an 

important regional surficial aquifer system. The glacial aquifer 

system, which is defined as all unconsolidated sediments 

overlying bedrock in glaciated terrain, was a source of water 

for about 41 million people in 2000; about 40 percent of the 

water was pumped from private wells, primarily in rural areas 

(Warner and Arnold, 2006). About half (51 percent) of the 

land area overlying the glacial aquifer system is agricultural 

land that receives large amounts of applied fertilizer, and wells 
Introduction 

3

70° 80° 90° 110°

45°35°

100° 120°

40°

0 200 MILES0 200 KILOMETERS

C A N A D A

ALASKA CANADA

WA

OR

MT

WY

CO

ND

SD

NE

KS

IA

MN

WI

IL

MO

IN

MI

KY

WV

VA

PA

NY

VT NH

ME

MA

RI

CT

NJ

DE

MD OH

Pacific Ocean

Lake Superior

LakeHuron

Lake Erie

Lake Ontario

AtlanticOcean

U N I T E D S T A T E S

Lake Michigan

Area of inset map

Area ofenlargement

Base from U.S. Geological Survey 1:2,000,000 Digital DataAlbers Equal-Area Conic projection, standard parallels 29° 30’ and 45° 30’, central meridian -96°, latitude of origin 23°

0 100 MILES

100 KILOMETERS 0 N

EXPLANATION

Texture of near-surface deposits of the glacial aquifer system

Coarse-grained sedimentFine-grained sediment (includes till and patchy Quaternary or organic-rich sediment)Undifferentiated or mixed fine- and coarse-grained sediment

Unclassified

Approximate extent of glacial aquifer systemWhite-Miami basin boundaryStudy area boundary

No Data are shown for Alaska AK

Study area

WV

OH

White-Miami  River Basin

Driftless area

Central framework area

Central framework area

Sand and gravelClay, silt, and very fine to fine sand

TillBedrock

Water

EXPLANATION

Valley-fill deposits Fracture

Buried

bedrock

valley

Buried

bedrock

valley

VALLEY-FILL AQUIFER BURIED-VALLEY AQUIFER B

A

Figure 1. A, Extent of the glacial aquifer system, texture of near-surface glacial deposits, the central framework area, and location of the study area within the upper part of the Great Miami River Basin and B, block diagrams of valley-fill and buried-valley aquifers commonly found in the central framework area. (Modified from Warner and Arnold, 2010).
4 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

in agricultural areas are susceptible to contamination from 

fertilizer-derived nitrate (DeSimone and others, 2009). Nitrate, 

which can adversely affect human health, is conservative 

in oxic environments and, under such conditions, is readily 

transported in groundwater to pumped wells (Hem, 1985).

Since 1991, the U.S. Geological Survey (USGS) has 

collected water-quality data for the Nation’s principal aquifers, 

including the glacial aquifer system, as part of the National 

Water-Quality Assessment (NAWQA) Program. DeSimone 

and others (2009) found that nitrate concentrations in private 

wells screened in the glacial aquifer system exceeded the U.S. 

Environmental Protection Agency Maximum Concentration 

Limit (MCL) of 10 milligrams per liter as nitrogen (mg/L 

as N) in about 3.8 percent of sampled wells. Concentrations 

exceeded the MCL in about 37.5 percent of private wells in 

agricultural areas; maximum concentrations were as high 

as 77 mg/L as N. Background concentrations of nitrate in 

the glacial aquifer system, where the aquifer is unaffected 

by anthropogenic sources, generally are less than 1 mg/L 

as N (Nolan and Hitt, 2003). Warner and Arnold (2010) 

identified relations between nitrate occurrence in the glacial 

aquifer system and a set of explanatory well-construction, 

environmental, and source variables and used the relations to 

develop statistical models of nitrate occurrence. The models 

predict the probability of nitrate occurrence above a set of 

threshold concentrations by using different combinations 

of well-construction and spatial variables; the latter set of 

variables was estimated by using a circular buffer around 

a well as a simplified representation of the source area to 

the well.

Although the statistical models may implicitly represent 

transport and attenuation processes through inclusion 

of environmental characteristics, they do not explicitly 

represent the physiochemical processes controlling nitrate 

concentrations in groundwater and do not account for many of 

the complexities of natural systems. Numerical process models 

simulating groundwater flow more explicitly represent areas 

contributing recharge to pumped wells and have the capacity 

to represent, to an extent, complexities of advective transport 

in natural systems.

In 2008, the USGS began a study to evaluate how 

incorporating an understanding of transport processes can 

inform or improve statistical models of nitrate occurrence. 

The area of investigation is within the upper part of the Great 

Miami River Basin in southwestern Ohio (fig. 2). The area 

is within the central framework region of the glacial aquifer 

system (fig. 1A), which is characterized by a predominance 

of buried-valley and valley-fill glacial aquifers that can have 

thick sequences of sand and gravel and high yields of water 

(fig. 1B) (Warner and Arnold, 2010). This basin, which is 

in the eastern part of the broader White-Miami River Basin 

encompassing parts of southern Ohio and Indiana (fig 1A), 

was chosen for study for a number of reasons: (1) the part of 

the basin used in this effort includes a mix of land uses typical 

of the broader glacial aquifer system; (2) environmental 

variables used in statistical models of nitrate occurrence had 

already been defined for the basin, and data from sampled 

wells in the basin were used in developing the models; and 

(3) a numerical groundwater-flow model of the area had been 

developed and maintained that can simulate the physical 

process of advective transport in the underlying glacial aquifer 

system (Sheets, 2007) (fig. 2).

Purpose and Scope

This report documents the capabilities of a process 

model, specifically a numerical groundwater-flow model, to 

inform and improve statistical models of nitrate occurrence in 

the glacial aquifer system. The capabilities of groundwaterflow models to represent the advective transport of nitrate as 

nitrogen are evaluated, including the ability to (1) provide an 

improved representation of the source area to a pumped well, 

and (2) predict time-varying concentrations of nitrate in a well 

arising from the process of advective transport. Results from 

process and statistical models are compared to evaluate how 

incorporation of a process-level understanding of groundwater 

flow and nitrate transport can inform the development and use 

of statistical models.

Factors affecting the size, shape, and location of 

simulated contributing recharge areas and differences 

between contributing recharge areas and circular buffers at 

specific locations are evaluated, including well depth and 

hydrologic setting. A range of pumping rates is evaluated— 

from rates consistent with public-supply wells to those 

consistent with domestic wells. The role of hydrologic setting 

on the distribution of steady-state traveltimes and transient 

contributing recharge areas is evaluated and related to steadystate contributing recharge areas. The use of a process model 

to estimate nitrate concentrations in pumped wells and the 

factors that can affect those predictions are presented.

Sets of environmental variables estimated from circular 

buffers, as used in the development of statistical models, 

are compared to those estimated from model-calculated 

contributing recharge areas. Comparisons are made for 

existing public-supply and NAWQA network wells as well 

as for hypothetical wells in different hydrologic settings and 

at different depths and pumping rates. The sets of estimated 

environmental variables are used to calculate and compare 

probabilities of nitrate occurrence above selected threshold 

concentrations predicted by using the statistical models 

developed by Warner and Arnold (2010). These comparisons 

are used to quantify the degree to which incorporation of 

groundwater-flow processes can affect predictions of nitrate 

occurrence in pumped wells. The same set of analyses is 

presented for a network of uniformly spaced hypothetical 

wells to evaluate differences between statistical and process 

models at the basin scale. The effect of hydrologic setting 

on physical representations of source areas and the potential 

effect of an improved representation of the source on statistical 

models across the basin are evaluated.
Introduction 5

DAYTON

GREENE COUNTY

CLARK COUNTY

PREBLE COUNTY MONTGOMERY COUNTY

MIAMI COUNTY

WARREN COUNTY BUTLER COUNTY CLINTON COUNTY

DARKE COUNTY

I-70

I-75

Little

Miami River

Great

Miami River

Wolf Creek

Stillwater

River

Mad River

84°32' 84°24' 84°16' 84°8' 84° 83°52'

39°28'

39°36"

39°44'

39°52'

40°

0 10 20 MILES

0 10 20 KILOMETERS

EXPLANATION

Valley fill or buried valley aquifer

Extent of active model area

Extent of model grid

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Wright-Patterson

Air Force Base

Oakwood

Figure 2. Extent of buried-valley glacial aquifers within the study area near Dayton, Ohio, and extent of regional 

groundwater-flow-model grid, southwestern Ohio. (Modified from Sheets, 2007).
6 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Limitations and technical considerations for use of 

process models to inform statistical models that are applicable 

to other basins are presented. Potential effects of processmodel design on physically based source areas and how those 

effects can, in turn, affect differences in statistical and processmodel predictions are evaluated. The evaluation of model 

limitations presents considerations for use in developing future 

process models or modifying existing process models to better 

facilitate their use in improving statistical models. It should 

be noted that the analyses presented in the report require 

existing statistical and process models. Although statistical 

models have been developed that are applicable to the entire 

glacial aquifer system, currently (2011), areas represented by 

numerical process models encompass a relatively small part of 

that system.

Methods of Analysis

This effort utilized statistical and process models 

developed for the glacial aquifer system as part of previous 

efforts by the USGS. Warner and Arnold (2010) developed 

statistical models—using linear and logistic regression—to 

relate nitrate concentrations and occurrence to sets of wellconstruction and environmental explanatory variables. Sheets 

(2007) modified an existing numerical groundwater-flow 

model of a glacial aquifer system near Dayton, Ohio, within 

the Great Miami River Basin; the model allows for the 

simulation of groundwater-flow processes and the advective 

transport of conservative solutes.

Statistical Models of Nitrate Occurrence

Warner and Arnold (2010) identified a number of 

correlations between the occurrence of nitrate and a number 

of well-construction and environmental variables; the study 

used water-quality data collected between 1991 and 2005 

from a network of 379 private wells, 41 public-supply wells, 

and 870 monitoring wells as part of the NAWQA Program. 

The correlations were used to develop statistical models 

of the probability of nitrate occurrence above 10 threshold 

concentrations: 1 to 10 mg/L as N. The models use logistic 

regression to estimate the probability that nitrate concentration 

in a well exceeds the threshold concentration. The general 

form of logistic regression is as follows:

P exp B BV B V B V

B BV B V B V

o n n

o n n

= + + + +

+ + + + +

( )

( )

1 1 2 2

1 1 2 2 1





/

, (1)

where

P is the probability of exceeding the threshold 

concentration,

Bo is the intercept,

Bn is the fitted slope parameter, and

Vn is the explanatory variable.

Three well-construction variables and four spatial 

variables were included, in differing combinations, in the 

logistic-regression models. The variables used in each of 

the 10 logistic-regression models developed by Warner and 

Arnold (2010) are summarized in table 1 and figure 3A. 

Well-construction variables include open-interval diameter, 

length of the open interval (the part of the well casing open 

to the aquifer), and depth to the top of the open interval; a 

fourth variable is an interaction term between depth to top of 

the open interval and open-interval diameter. Spatial variables 

include three environmental variables derived from the U.S. 

Department of Agriculture State Soil Geographic (STATSGO) 

database (Wolock, 1997): silt content, alfisol content, and 

depth to water. Alfisols, which cover about 25 percent of 

the glacial aquifer system, are clayey soils formed from 

weathering processes in humid areas with deciduous forest 

or mixed vegetative cover (Warner and Arnold, 2010). An 

additional spatial variable was the 10-year mean nitrogen 

fertilizer-application rate (1992–2001) estimated from 

county data and applied to agricultural land delineated in 

the Enhanced National Land Cover Data 1992 (NLCDe92) 

database (Nakagaki and others, 2007). The slope coefficients 

for explanatory variables and intercepts are summarized in 

table 1. Larger slope coefficients indicate that a variable has 

a larger effect on predicted probabilities than variables with 

smaller coefficients for a given unit of measure; a positive 

coefficient indicates that the probability increases as the 

variable increases, and a negative slope indicates that the 

probability decreases as the variable increases. The effect of a 

spatial variable on the predicted probability is the product of 

the slope coefficient and the value of the variable.

Environmental variables—silt content, alfisol content, 

and depth to water—were based on STATSGO data and 

had the same spatial variability; seven value fields were 

represented within the Great Miami River Basin (fig. 4A). Silt 

content ranged from 48.55 to 57.20 percent; alfisol content, 

where present, from 69 to 94 percent; and depth to water from 

2.48 to 4.88 ft. Application rates for nitrogen were estimated 

from county usage data applied to agricultural land delineated 

in the NCLDE92 database. The mean nitrogen-application 

rates (1992–2001) on agricultural land in the Great Miami 

River Basin ranged from 1,413 to 2,202 kg/yr (fig. 4B). The 

value of each spatial variable used in developing the models 

was determined by calculating the area-weighted mean for a 

circular area of radius 1,640 ft [500 meters (m)] around a well 

(fig. 3B); the circular buffer encompassed an area of about 

0.3 mi2.

The 7 variables included in the statistical models were 

a subset of 18 variables that were correlated with nitrate 

concentrations at a 95-percent confidence level (Warner and 

Arnold, 2010) by using Spearman’s rho rank-correlation 

coefficient (Helsel and Hirsch, 1992). The 18 variables were, 

in turn, a subset of 46 variables for which visual inspection 

of scatter plots suggested a linear correlation with nitrate 

concentrations; the variables included categories related 

to land use, watershed characteristics, soil properties, 
Introduction 

7

Table 1. Summary of statistical models of nitrate occurrence in the glacial aquifer system for threshold concentrations of 1 to 10 milligrams per liter as nitrogen (Warner and Arnold, 2010).[mg/L, milligrams per liter; N, nitrogen; >, greater than; kg/yr, kilograms per year; --, variable not used in model; logistic regression models used variable values for each private well in the glacial aquifer system; from Warner and Arnold, 2010]

Variable 

type Variable description Variable abbreviation

Nitrate concentration, in mg/L as N

Model parameters, slope coefficients, and intercepts

> 10 > 9 > 8 > 7 > 6 > 5 > 4 > 3 > 2 > 1

Intercept -2.1454 -1.0447 -1.396 -1.187 -4.7459 -6.8249 -5.3277 -4.6698 -5.1108 -3.9552Source 

variables

10-year mean (1992–2001) nitrogen 

applied to farms, in kg/yr

NFARM_10YR 0.000658 0.000474 0.00047 0.000422 0.000555 0.00042 0.000211 -- -- --

Transport 

variables

Depth to top of open interval, 

in inches

DEPTH_TOI_ -0.0123 -0.0104 -0.00812 -0.00793 -0.00563 -0.004 -0.00187 -0.00165 -0.00152 -0.012

Open interval diameter, in inches OI_CASE_DI -0.26 -0.2021 -0.1614 -0.1599 -0.0909 -0.05 0.0151 -- -- --Length of open interval, in inches OILENGTH_I -- -- -- -- -- -- -- 0.00111 0.000985 0.00309Silt in soil, in percent SILTAVE -- -- -- -- -- 0.039 0.0326 0.0481 0.0535 0.0356Depth to water, in feet WTDEPAVE -- -- -- -- 0.7818 0.8839 0.7355 0.6468 0.8013 0.7219

Attenuation 

variables

Alfisols, in percent ORDA -- -- -- -- -0.0211 -0.019 -0.0127 -0.0156 -0.0188 -0.0164Interaction term between depth to 

top and diameter of open interval

OI_CASE_D*DEPTH_

TOI_

0.00135 0.00103 0.000776 0.000772 0.000541 0.000367 -- -- -- --
8 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0

1

2

3

4

5

6

7

8

10 9 8 7 6 5 4 3 2 1

Nitrate concentration, in milligrams per liter as nitrogen

Number of variables

Interaction between depth to top of 

 open interval and open-interval 

 diameter

Length of open interval, in inches 

Mean percentage of silt in soil 

 within 1,640 feet of the well

Percentage of alfisols within 1,640 feet 

 of the well

Depth to saturated soil within 

 1,640 feet of the well, in feet

10-year mean (1992–2001) load of 

 nitrogen fertilizer applied to farms within 

 1,640 feet of the well, in kilograms 

 per year

Open-interval diameter, in inches

Depth to top of open interval, in inches

Variable used in the models

EXPLANATION

EXPLANATION

Nitrogen-application rate

2.845

4.177

4.835

Nitrogen-

 application load,

 in kilograms

 per year

0

1,413

1,977

Silt content

Depth to water Alfisol content

48.547

51.560

57.203

0.00

81.00 

85.00

Silt content,

 in percent

Depth to 

 water, in

 percent

Alfisol

 content,

 in percent

Average = 926.42 Average = 51.61

Average = 4.14 Average = 60.37

Well Well

Well Well

A

B

1,640 feet

1,640 feet 1,640 feet

1,640 feet

Figure 3. A, Sets of explanatory variables used in 10 logistic-regression models to describe nitrate occurrence in the glacial 

aquifer system (modified from Warner and Arnold, 2010) and B, example of spatial variables within a 1,640-ft radius of a 

hypothetical well in the Great Miami River Basin, southwestern Ohio.
Introduction 

9

84°10' 84°0' 84°20' 84°10' 84°0' 84°20'

0 3 6 MILES

EXPLANATION

STATSGO data

2.482.853.634.184.764.844.88

0

1,4131,8611,9772,202

48.5551.0951.4351.5652.0852.8857.20

 0.069.074.081.084.085.094.0

Depth to water, in feet

 Silt content, in percent

 Alfisol content, in percent

Nitrogen-application load, in kilograms per year

County data

A B

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84°

Active model area Active model area

Extent of model grid Extent of model grid

39°50'39°40'

0 6 KILOMETERS

Figure 4. Spatial distribution of A, environmental variables—silt content, alfisol content, and depth to water—and B, nitrogen application rates in the modeled area of the Great Miami River Basin, southwestern Ohio. 
10 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

population density, and waste disposal (Warner and Arnold, 

2010). The 7 variables explained the variability in nitrate 

concentrations at a significance level exceeding 0.05 as 

determined by a stepwise-forward-selection method (Warner 

and Arnold, 2010).

There are a number of assumptions inherent in 

statistical models of nitrate occurrence. The most important 

assumption is that the source of water to a pumped well is 

within a 1,640-ft-radius circular buffer around the well. In 

natural systems, the areas contributing recharge to a well 

are a function of the hydrologic setting (the magnitude and 

direction of hydraulic gradients), the depth of the well, and the 

pumping rate. It is likely that explanatory variables estimated 

for the area within the circular buffers would differ from those 

estimated for a physically based representation of the areas 

contributing recharge to pumped wells.

Numerical Model of the Great Miami River Basin

Advection refers to the movement of a solute with 

average groundwater flow and generally is the dominant 

component of transport in unconsolidated surficial aquifers, 

particularly in areas with permeable sediments and high 

recharge rates. Groundwater-flow models that simulate 

hydraulic gradients are effective for representing the advective 

transport of chemically conservative solutes, such as nitrate 

in oxic environments. Numerical groundwater-flow models 

generally use finite-difference methods to simulate heads and 

flows in a spatially discretized representation of an aquifer 

system. Model input parameters include intrinsic aquifer 

properties, hydraulic boundary conditions, and natural 

and anthropogenic stresses. Particle-tracking methods use 

simulated cell-by-cell flows to represent the movement of 

water within the aquifer, and can, in turn, be used to simulate 

contributing recharge areas and the advective transport of 

contaminants from surface sources to pumped wells.

A regional groundwater-flow model developed by 

Dumouchelle (1998) for part of the Great Miami River Basin 

was modified and recalibrated to simulate advective-transport 

paths and contributing recharge areas to public-supply wells 

(Sheets, 2007); detailed descriptions of the original and 

modified models are presented in Sheets (2007). The glacial 

aquifer system in the study area is a buried-valley aquifer in 

which glacial sediments fill bedrock valleys; the sediments, 

which are over 300 ft thick in some areas, generally are coarse 

grained with lenses of finer grained material (fig. 5) and can 

be productive sources of water (Sheets, 2007). Recharge 

from precipitation, which is about 40 in/yr in the basin, and 

flow from adjacent uplands—as streamflow or as shallow 

groundwater flow—are the sole sources of water to the 

aquifer; water leaves the system as streamflow or is withdrawn 

from wells (fig. 5). The part of the aquifer represented by the 

regional model also receives water from downvalley flow 

within the aquifer from the upgradient part of the aquifer 

and contributes water to downgradient parts of the aquifer. 

The aquifer is bounded laterally and below by relatively 

impermeable bedrock consisting of Ordovician and Silurian 

shale, limestone, and other types of carbonate rock (fig. 5).

The calibrated regional model of the Great Miami River 

Basin (Sheets, 2007) used MODFLOW–2000 (Harbaugh 

and others, 2000) to simulate hydrologic conditions in the 

glacial aquifer system and the inverse-calibration capabilities 

of MODFLOW–2000 (Hill and others, 2000) to calibrate the 

model to observed hydraulic conditions. In the current set of 

SW NE

275

100

185

Altitude, in meters

0 5,000 FEET

0 1,000 METERS

Wetland Tributary Great Miami

River

Production

Well

Production

well

Domestic

well

Recharge

Silurian

 carbonates

Sand and gravel

Clay till

Geology

EXPLANATION

SCHEMATIC ONLY, NOT TO SCALE

Recharge

Ordovician shale

 with limestone

Figure 5. Generalized geology and groundwater-flow patterns in a buried-valley glacial aquifer typical of the upper part of the 

Great Miami River Basin, southwestern, Ohio. (Modified from Sheets, 2007)
Introduction 11

analyses, the particle-tracking program MODPATH (Pollock, 

1994) was used to simulate the sources of water to pumped 

wells—existing and hypothetical—and the advective transport 

of nitrate for a number of hydrologic settings and pumping 

rates; the geoprocessing software MODTOOLS (Orzol, 1997) 

was used to convert model and particle-tracking output to 

georeferenced forms.

There are a number of assumptions inherent in steadystate numerical models of groundwater systems: (1) the 

hydraulic stresses (pumping and recharge) are constant over 

time, (2) aquifer properties are homogenous within model 

cells, and (3) calibrated aquifer properties and recharge 

reasonably match those of the real system. Although recharge 

varies over time, transport of contaminants often occurs over 

time scales long enough to make the assumption of steadystate conditions reasonable. Regional models, such as those 

of the Great Miami River Basin, address regional-scale 

questions that likely are not affected by variations of aquifer 

properties within model cells. The accuracy of calibrated 

aquifer properties can be assessed by comparing the match 

between measured and observed heads and flows, as discussed 

in Sheets (2007).

The regional model was originally developed to 

evaluate the source of water and transport of anthropogenic 

contaminants to public-supply wells, primarily in urbanized 

areas around Dayton, Ohio (Sheets, 2007). The analyses 

presented in this report represent a use of the model that 

differs from its original purpose; however, the results 

indicate that the model was appropriate for these analyses 

in most instances. Some model limitations of importance 

to these analyses, as discussed later in the report, likely are 

not of relevance to the analyses for which the model was 

originally designed.

Model Design and Calibration

The regional model finite-difference grid consists of 

230 rows, 370 columns, and 3 layers; has a uniform horizontal 

discretization of 500 ft; and is rotated 25 degrees (°) east 

of north (fig. 6). The active model area encompasses about 

254 mi2 and incorporates the buried-valley aquifer system 

near Dayton, Ohio (fig. 2). The simulated aquifer generally is 

thickest near the axis of buried bedrock valleys and thinnest 

in areas near bedrock and glacial till uplands and areas of 

bedrock highs within the aquifer; the latter are overlain by 

glacial till and soil. Simulated water-table altitudes range from 

over 1,000 ft near bedrock and glacial till uplands and along 

valley margins to about 600 ft where the Great Miami River 

exits the aquifer system (fig. 6). About 377 public-supply 

wells operate within the basin, generally near urban areas.

Hydraulic conductivity (HK1) in the top layer of the 

model and recharge (RCH) are grouped into four parameter 

zones within the aquifer system: buried-valley sediments 

(BV), bedrock/till uplands (UP), sediments marginal to 

uplands (EDGE), and sediments in the Oakwood area in the 

south-central part of the aquifer (OAK) (fig. 7). Multipliers 

were used to allow values in individual cells to vary within 

zones. Parameters also represent horizontal and vertical 

hydraulic conductivity in layers 2 and 3 and river leakances 

(RIV) in buried-valley and upland areas. A detailed discussion 

of the model parameterization is presented in Sheets (2007).

Formal sensitivity analyses and parameter estimation, 

the sensitivity analysis (SEN) and parameter estimation (PES) 

Processes in MODFLOW–2000 (Hill and others, 2000), 

respectively, were used to improve calibration of the model 

to observed hydrologic conditions (Sheets, 2007). Simulated 

equivalents at observation locations were sufficiently sensitive 

to allow for the estimation of six parameters: HK1_BV, 

HK2, RCH_BV, RCH_UP, RCH_EDGE, and RIV_BV; 

the remaining parameters used specified values. A detailed 

discussion of the inverse-calibration methods, sensitivity 

analyses, and model fit to observed hydrologic conditions is 

presented in Sheets (2007). Calibrated hydraulic conductivities 

in the top layer of the model ranged from less than 10 to more 

than 400 feet per day (ft/d) (fig. 8A), and recharge ranged from 

less than 6 to more than 12 inches per year (in/yr) (fig. 8B). 

Sediment porosity, which affects groundwater velocity and 

transport times, was uniform and specified as 0.20.

Modifications to Regional Model of the Great Miami 

River Basin

The current set of analyses used the regional model of 

the Great Miami River Basin documented in Sheets (2007). 

Model layering, parameter values, hydrologic boundaries, and 

natural recharge stresses were the same as in the documented 

model. Subregional models were developed for three areas 

in the southeastern part of the model (fig. 6) to allow for 

the representation of pumped wells with low withdrawal 

rates, such as are typical in domestic-supply wells. Spatial 

discretization in these areas was decreased from 500 to 10 ft, 

and the subregional models were linked to the regional model 

by using a specified-head boundary condition derived from 

regional-model-calculated heads; vertical discretization was 

the same as in the regional model. In each case, pumping 

stresses were consistent between the regional and subregional 

models, and simulated heads and hydrologic budgets for the 

subregional models were checked to make sure they were 

consistent with the coincident area within the regional model. 

The steady-state regional model was temporally 

discretized to simulate contributing recharge areas resulting 

from a transient pumping stress. The length of the transient 

stress period was 50 years divided into 300 uniform time 

steps. The transient stress period used the same constant 

recharge stresses as the regional model and a constant 

pumping stress, which resulted in a solution at the end of the 

transient period that was similar to that from the steady-state 

regional model. Model-calculated steady-state heads were 

used as an initial condition. Specific yield and storativity of 

the aquifer sediments were uniform and specified as 0.23 and 

0.0001, respectively.
12 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

39°50'

39°40'

0 3 6 MILES

Stillwater River

Little Miami River

Great Miami River

Mad River

Extent of model grid

Great Miami

 River

Wolf Creek

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Active model area

0 3 6 KILOMETERS

84°20' 84°10' 84°0'

750

750

700

900

850

800

900

750

850

800

750

750

Well 1053

Well 870

Well 1084

Well 1629

Well 1860

Well 725

EXPLANATION

Production well

Subregional model

Simulated head—elevation in feet (NGVD 88)

Layers 1–3 active

Layers 1–2 active

Layer 1 active

Hypothetical well and number

NAWQA network well

860

Well 1860

Figure 6. Regional model grid, active model layers, extents of subregional models, simulated water table, and locations 

of production wells, upper part of the Great Miami River Basin, southwestern Ohio. 
Introduction 13

Base from U.S. Geological Survey digital data, 1:24,000, 1983 0 3 6 MILES

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Active model area

0 3 6 KILOMETERS

Extent of model grid

EXPLANATION

Specified head

River

Drains

Specified flux

Hydraulic conductivity (HK1) and 

 recharge (RCH) parameter zones 

EDGE

BV

UP

OAK

Hydrologic boundaries 

84°20' 84°10' 84°0'

39°50'

39°40'

Figure 7. Hydrologic boundaries and hydraulic-conductivity and recharge parameter zones used to represent the 

regional aquifer system, upper part of the Great Miami River Basin, southwestern Ohio. 
14 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0 3

3

6 MILES

84°10' 84°0' 84°20'

39°50'39°40'

84°10' 84°0' 84°20'

EXPLANATION

Hydraulic conductivity, in feet per day

> 0 to 10> 10 to 20> 20 to 30> 30 to 50> 50 to 100

> 100 to 150> 150 to 200> 200 to 300> 300 to 400> 400

> 5.5 to 6.0> 6.0 to 7.0> 7.0 to 8.0> 8.0 to 9.0

> 9.0 to 10.0> 10.0 to 11.0> 11.0 to 12.0> 12.0 to 13.0

Active model area

Recharge, in inches per year

Active model area

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84° 0 6 KILOMETERS

Extent of model grid

Extent of model grid

A

B

Figure 8. Distribution of A, horizontal hydraulic conductivity values in layer 1 and B, recharge from precipitation in the calibrated regional model of the Great Miami River Basin, southwestern Ohio. 
Sources of Water and Advective Transport of Nitrate to Pumped Wells 15

Sources of Water and Advective 

Transport of Nitrate to Pumped Wells

The contributing recharge area to a pumped well refers 

to the area at the water table that contributes water to the 

well. Under steady-state conditions, there is a mass balance 

between water entering through the contributing recharge 

area and withdrawal from the well. In cases where recharge 

is the sole source of water to a pumped well—a common 

situation for surficial aquifers—the contributing recharge 

area is proportional to the pumping rate of the well, and the 

proportionality constant is the recharge rate. In some cases, 

sources of water to a well can include other sources, such 

as induced infiltration from nearby surface waters or, under 

transient conditions, storage in the aquifer.

An assumption implicit in statistical models of nitrate 

occurrence (Warner and Arnold, 2010) is that the source of 

water to a pumped well is within a radius of 1,640 ft (500 m) 

of the well (fig. 3B) and that area-weighted mean spatial 

variables for the area within that circular buffer approximate 

the mean values for the area of the land surface overlying the 

contributing recharge area to the well. Differences between 

a true (or modeled) contributing recharge area and a circular 

buffer around the well can result in differences in the estimates 

of area-weighted spatial variables. Process models that 

simulate groundwater flow within an aquifer can be used to 

estimate contributing recharge areas to wells and account for 

hydrologic conditions, including aquifer properties, hydrologic 

boundaries, and recharge and pumping stresses. A contributing 

recharge area to a well pumped at a rate equivalent to the 

volume of recharge through a circular buffer would be the 

same as the area of the circular buffer but would account for 

groundwater processes and better represent the actual source 

area to the well.

Factors Affecting Steady-State Contributing 

Recharge Areas to Pumped Wells

The source of water to a pumped well is a function of 

the pumping rate, hydrologic setting, depth, and duration of 

pumping of the well and characteristics of the aquifer system, 

including aquifer geometry, hydrologic boundaries, natural 

stresses, and sediment characteristics (Reilly and Pollock, 

1993; Barlow, 1994; Masterson and others, 1998; Walter 

and Whealan, 2003; Walter and others, 2004). The pumping 

rate determines the size of the contributing recharge area. 

The shape of the contributing recharge area is a function 

of hydraulic gradients, and the location of the contributing 

recharge area relative to the well is affected by the depth 

of the well screen. For a given pumping rate and screen 

elevation, local aquifer heterogeneities can result in additional 

complexities in the shape of the contributing recharge area.

Pumping Rate

Under steady-state conditions, there is a condition of 

mass balance between recharge and pumping, and therefore, 

for a given recharge rate, the size of the contributing recharge 

areas is proportional to the pumping rate and, for a given 

pumping rate, the size of the contributing recharge areas is 

inversely proportional to the recharge rate. Simulated steadystate contributing recharge areas for three pumping rates from 

a hypothetical well (well 1053) in the southeastern part of the 

basin (fig. 6) are shown in figure 9. The well is in an area with 

steep hydraulic gradients, and the contributing recharge area 

extends upgradient from the well towards a regional watertable divide (fig. 9).

In this report, a high pumping rate refers to a withdrawal 

equal to the mean withdrawal for public-supply wells in 

the basin (about 0.25 million gallons per day (Mgal/d)); 

an intermediate pumping rate refers to a rate equal to the 

recharge through the corresponding 1,640-ft-radius circular 

buffer (a rate of about 0.08 Mgal/d at this location); and a low 

pumping rate refers to a rate similar to the withdrawal rate of 

a domestic-supply well [about 350 gallons per day (gal/d)]. 

About 19 and 33 percent of steady-state contributing recharge 

areas for well 1053 under high and intermediate pumping 

rates, respectively, are within the 1,640-ft-radius circular 

buffer. The contributing recharge area under a low pumping 

rate is completely within the 1,640-ft-radius circular buffer but 

extends to near the edge (fig. 9).

Hydrologic Setting

Contributing recharge areas to pumped wells are also a 

function of the depth of the well screen and regional hydraulic 

gradients around the well. Wells 870 and 1084 are near a 

bedrock/glacial till upland where the aquifer is thin; well 

1084 is near a regional groundwater divide in an area with 

small, radial hydraulic gradients, and well 870 is in an area 

with steep hydraulic gradients (figs. 6 and 10). Well 1053 

is in an area of steep hydraulic gradients within a buried 

valley where the aquifer is thicker. At each location, the 

intermediate pumping rate and resulting contributing recharge 

areas are equivalent to the rate of recharge through and size 

of the corresponding circular buffer, so the effect of regional 

gradients can be quantified by the overlap of the contributing 

recharge area and the circular buffer—defined as the fraction 

of the recharge area that is inside the circular buffer. Near the 

groundwater divide (well 1084), where hydraulic gradients are 

relatively flat, the contributing recharge area is similar to the 

area of the circular buffer; about 87 percent of the contributing 

recharge area is inside the circular buffer (fig. 10). In areas 

with steep hydraulic gradients, there is less overlap between 

contributing recharge areas and circular buffers—about 

56 percent at well 870 and 33 percent at well 1053 (fig. 10).

Near the regional groundwater divide, where the 

contributing recharge areas are similar to the area of the 

circular buffer (well 1084), the entire area contributing 
16 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

84°3' 84°2' 84°1'

39°45'30"

39°45'

940

900

860

820

Well 1053

Well 1053

1,640 feet

Buried-valley aquifer

Direction of groundwater ow

0 2,000 METERS

0 2,000 FEET

EXPLANATION

Layer 1

Layers 1– 2

Layers 1–3

Circular buffer

 of radius

 1,640 feet 

Low pumping

 rate

Intermediate

 pumping rate

High pumping

 rate

Active model layers Pumped well

 and number 860

Water table—number 

 is altitude, in feet

 (NGVD 88)

Recharge area

Figure 9. The 1,640-ft-radius circular buffer and steady-state contributing recharge areas to well 1053 (a hypothetical well) for 

three pumping rates, southeastern part of the Great Miami River Basin, southwestern Ohio. 
Sources of Water and Advective Transport of Nitrate to Pumped Wells 17

84°4' 84°3' 84°2' 84°1'

39°47'

39°46'

39°45'

Buried-valley aquifer

Well 870

Well 1084

Well 1053

Well 1053

900

860

820

Direction of

groundwater

 ow

EXPLANATION

Well 1053

Well 1084

Well 870

Pumped well and

 number 880

Recharge Area

0 0.5 1 MILE

Recharge area for

 low pumping rate

Subregional model

Subregional model

Subregional

 model

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

0 0.5 1 KILOMETER

Layer 1

Layers 1– 2

Layers 1–3

Active model layers

Circular buffer of 

 radius 1,640 feet

Water table—number

 is altitude, in feet 

 (NGVD 88)

Figure 10. The 1,640-ft-radius circular buffer and steady-state contributing recharge areas for intermediate and low 

pumping rates in three hydrologic settings, southeastern part of the Great Miami River Basin, southwestern Ohio.
18 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

recharge under low-pumping conditions is within 500 ft of 

the well (fig. 10). Low-pumping contributing recharge areas 

in areas of steep hydraulic gradients at wells 870 and 1053 

extend 2,430 and 1,560 ft, respectively, upgradient from 

the well. About 33 percent of the low-pumping contributing 

recharge area at well 870 is outside of the 1,640-ft-radius 

circular buffer (fig. 10).

Well Depth

Wells with screens deeper in the aquifer generally 

withdraw water with longer flow paths and have contributing recharge areas that extend farther upgradient than wells 

screened in shallower parts of the aquifer (figs. 11A and 11B). 

Under intermediate pumping conditions, in which the pumping rate is equivalent to the recharge rate through the corresponding buffer zone, and the contributing recharge area is of 

a similar size, the contributing recharge areas to pumped wells 

screened deeper in the aquifer (altitudes of the boundaries of 

layer 2 are 785 and 680 ft, and 680 and 638 ft for layer 3), corresponding to screen tops 34 and 139 ft below the water table, 

respectively) extend farther upgradient than the contributing 

recharge areas screened in the shallowest part of the aquifer 

(altitudes of the boundaries of layer 1 are 819 and 785 ft, 

corresponding to screen tops at the water table). The amounts 

of overlap between the contributing recharge areas for wells 

in layers 2 and 3 and the circular buffer were about 20 and 

11 percent, respectively, compared to about 33 percent for a 

well screened in layer 1 (fig. 11A). For high pumping rates, 

contributing recharge areas for wells pumped in layers 1, 2, 

and 3, were essentially the same; the amounts of overlap with 

the circular buffer were 18.7, 17.6, and 16.6 percent, respectively (fig. 11A). At high rates, pumping changes regional gradients and creates strong gradients to the pumped well; under 

these conditions, well depth has a minimal effect.

The effect of well depth on a contributing recharge area 

is more pronounced for a low pumping rate (fig. 11B); wells 

pumped at low rates in deeper parts of the aquifer capture 

water from longer flow paths with recharge locations farther upgradient than for wells pumped in shallow parts of 

the aquifer because natural regional gradients are relatively 

unchanged as a result of pumping at low rates. The contributing recharge area to a pumped well near the water table (layer 

1) extends about 1,420 ft upgradient from the well and is completely within the 1,640-ft-radius circular buffer. Contributing 

recharge areas to pumped wells in layer 2 and 3 extend 4,610 

and 9,040 ft, respectively, from the well (fig. 11B). About 

5.8 percent of the contributing recharge area to a well pumped 

in layer 2 is within the circular buffer for low pumping rates. 

For a well pumped in the deepest part of the aquifer (layer 3)

with a screen altitude about 150 ft below the simulated water 

table, the contributing recharge area is outside the circular 

buffer; the nearest part of the contributing recharge area is 

about 2,910 ft from the edge of the circular buffer (fig. 11B). 

Because the mean depth of domestic-supply wells in the glacial aquifer system is 91 ft (Warner and Arnold, 2010), some 

wells in the NAWQA monitoring network might have source 

areas outside of the 1,640-ft-radius circular buffer. Although 

all or most of the contributing recharge areas to wells pumped 

at a low pumping rate in layers 2 and 3 are outside of the 

circular buffer, they are within the contributing recharge areas 

to the wells pumped at an intermediate rate (that is, a rate 

equivalent to recharge through the circular buffer). This result 

indicates that the use of equivalent contributing recharge areas 

(and wells pumped at a rate equal to the volume of recharge 

through the circular buffer) is an improved representation of a 

spatial buffer for wells pumped at low rates.

Duration of Pumping

The assumptions for a steady-state contributing recharge 

area are that recharge and pumping are constant over time and 

that the two stresses are in a state of mass balance. However, 

at the start of pumping, the pumped well receives water 

primarily from storage within the aquifer framework. As 

the flow system adjusts to the pumping stress, and hydraulic 

conditions approach steady state, a mass balance between 

recharge and pumping is approached. Whereas the system 

may adjust hydraulically within a relatively short time (days 

or weeks), the contributing recharge area is defined by the 

process of advective transport over time scales of multiple 

decades. This transient effect causes the contributing recharge 

area to become a function of the duration of pumping. The 

particle methodology used to estimate transient contributing 

recharge areas involves a phased analysis in which particles 

are tracked in the direction of groundwater flow to a pumped 

well under transient flow conditions for a specified period of 

pumping; the resulting particle field is then tracked in reverse 

(opposite the direction of flow) to recharge locations under 

steady-state nonpumping conditions to estimate the transient 

contributing recharge area.

The contributing recharge area to well 1053 after 2 years 

of pumping generally is within, but about 56.1 percent the 

size of, the steady-state contributing recharge area (fig. 12A). 

This result indicates that, initially, storage in the aquifer is 

a significant source of water to the well. After 10 years of 

pumping, the contributing recharge area is about 87 percent 

of the size of the steady-state contributing recharge area, 

and after 20 years of pumping, the contributing recharge 

area is essentially the same size and generally within the 

steady-state contributing recharge area (fig. 12B). Although 

conditions are site specific, these results indicate that steadystate contributing recharge areas likely are reasonable 

approximations of the true contributing recharge areas for 

wells in operation for more than 10 to 20 years in similar 

hydrologic settings. This conclusion is consistent with the 

results of analyses of transient contributing recharge areas 

presented in Masterson and others (2002) and Masterson 

and others (2004). The results also indicate that contributing 

recharge areas to relatively new wells may not be sufficiently 

represented by steady-state conditions. It should be noted 

that the upgradient parts of the 2- and 20-year contributing 
Sources of Water and Advective Transport of Nitrate to Pumped Wells 19

Well 1053

84°3' 84°2' 84°1'

39°45'30"

39°45'

940

900

860

820

Direction of groundwater ow Buried-valley aquifer

Well 1053

Well 1053

39°45'30"

39°45'

940

900

860

820

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Direction of groundwater ow Buried-valley aquifer

A

B

Low 860

pumping

rate

Intermediate

pumping

rate

Recharge Areas

EXPLANATION

Deep depth

Shallow depth

Intermediate depth

High

pumping

rate

0 1,200 2,400 FEET

0 300 METERS 600

Water table—number

is altitude, in feet 

(NGVD 88)

Pumped well and

number

Circular buffer of 

 radius 1,640 feet

Figure 11. The 1,640-ft-radius circular buffer and steady-state contributing recharge areas to well 1053 for 

A, high and intermediate, and B, low and intermediate pumping rates for three well depths, southeastern part 

of the Great Miami River Basin, southwestern Ohio.
20 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

84°3'0"W 84°2'0"W 84°1'0"W

940

900

860

820

940

900

860

820

39°45'30"

39°45'

39°45'30"

39°45'

A

B

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Direction of groundwater ow

Direction of groundwater ow

Well 1053

Well 1053

Well 1053

Buried-valley aquifer

Buried-valley aquifer

0 1,200 2,400 FEET

0 300 600 METERS

EXPLANATION

Layer 1

Layers 1–2

Layers 1–3

Steady-state

2-year steadystate traveltime

Active model layers

Pumped well 

and number

880

Steady-state recharge areas

2-year recharge area

20-year steadystate traveltime

20-year recharge area

 Transient recharge areas Water table—number

is altitude, in feet 

(NGVD 88)

Circular buffer of 

radius 1,640 feet

Figure 12. Circular buffer of radius 1,640 feet, transient contributing recharge areas, steady-state recharge areas 

for intermediate pumping rates, and traveltimes under steady-state transport to well 1053 after A, 2 years and 

B, 20 years of pumping, southeastern part of the Great Miami River Basin, southwestern Ohio.
Sources of Water and Advective Transport of Nitrate to Pumped Wells 21

recharge areas are outside of the steady-state contributing 

recharge area; traveltimes within the contributing recharge 

area can exceed 100 years and, as a result, most of the 

advective transport to the well from the upper part of the 

contributing recharge area after 20 years of pumping is 

under steady-state nonpumping conditions, which produce 

hydraulic gradients that differ from those produced by 

pumping conditions.

Advective Transport of Nitrate

Statistical models can predict the probability of a 

contaminant exceeding a threshold concentration by utilizing 

relations between observed concentrations and a set of defined 

explanatory variables, whereas process models can be used to 

predict solute concentrations by simulating the physiochemical 

processes controlling the transport of a solute, such as nitrate, 

from surficial sources to receptors, such as pumped wells. 

The one-dimensional transport of a solute in the principal 

direction of groundwater flow can be described by the 

generalized equation:

  C t v C  x D   C x   q t = − ( ) + L ( ) − 2 2 (2)

(Advection) (Dispersion) (Reactions),

where

C is the concentration of a solute (moles per 

liter (M/L3)),

t is the time (t),

x is the distance (L),

v is the groundwater velocity (L/t),

DL is the longitudinal dispersivity (L), and

Q is the concentration loss caused by 

reactions (M/L3).

The first term refers to advection, which is the movement 

of a mass of solute with average groundwater velocity, and 

the second term—dispersion—refers to the spreading of 

mass resulting from scale-dependent aquifer heterogeneity. 

Advection and dispersion are physical processes, and the 

relative importance of the two is a function of hydrogeologic 

setting; advective transport generally is the dominant 

component of transport in permeable surficial aquifers such as 

the glacial aquifer system. The third term refers to the effect 

of chemical reactions on solute concentrations. An important 

biologically mediated chemical reaction affecting nitrate 

concentrations in surficial aquifers is denitrification, which is 

the transformation of nitrate to nitrogen gas under anaerobic 

conditions. Nitrate generally is conservative and mobile under 

the oxic conditions prevalent in the glacial aquifer system, 

including the Great Miami River Basin (Sheets, 2007).

Groundwater-flow models simulate cell-by-cell 

flows within an aquifer; particle-tracking methods use this 

information to simulate the movement of water through the 

aquifer. Solute-transport models use the cell-by-cell flows 

to simulate physical transport processes—advection and 

dispersion—and can represent single-solute reactions such 

as sorption and decay. Reactive transport models use these 

same flows to represent advection and dispersion but can also 

simulate complex reactions among solutes, including chemical 

equilibrium and rate-dependent oxidation-reduction reactions 

like denitrification.

The advective transport of a conservative solute, such as 

nitrate in oxic conditions, to a well is a function of the spatial 

and temporal distribution of sources within the contributing 

recharge area to the well and traveltimes in the aquifer. 

Groundwater models can represent the advective transport 

of conservative solutes by using mass-weighted particle 

tracking—a method in which a mass of solute is assigned to 

particles, and the number and traveltimes of the particles are 

used to approximate time-varying loads or concentrations 

at critical receptors. Kauffman and others (2001) present 

a detailed analysis of the advective transport of nitrate in 

a coastal-plain aquifer in northern New Jersey by using 

mass-weighted particle tracking; the analysis presents the 

capabilities and limitations of using the method to represent 

the effect of transport processes and time-varying sources 

on nitrogen concentrations in pumped wells. Walter (2007) 

presents a comparison of mass-weighted particle tracking and 

solute-transport methods in estimating nitrogen loads and 

concentrations in wells and ecological receptors in a glacial 

aquifer on Cape Cod, Mass.

Even under the assumptions of no subsurface attenuation 

and limited dispersion, a number of considerations that 

relate to intrinsic aquifer properties and surface-source 

characteristics apply to the use of a groundwater-flow model 

to represent the advective transport of nitrate. Time-varying 

nitrate concentrations are affected by intrinsic aquifer 

properties, such as porosity and dispersivity. Groundwater 

velocities and therefore traveltimes are proportional to 

the porosity of aquifer sediments and, although differing 

porosities result in the same steady-state concentration, the 

temporal pattern of nitrate concentrations in a pumped well 

varies with porosity. Also, groundwater models and massweighted particle tracking do not account for the effects of 

dispersion arising from aquifer heterogeneities. Dispersion 

causes nitrate to arrive at pumping wells sooner than it would 

through advective transport alone. Steady-state concentrations 

arising from a continuous surface source are similar for 

simulations that both do and do not account for dispersion 

(Walter, 2007) because the largest component of dispersion is 

in the direction of groundwater flow (longitudinal). Surfacesource characteristics include source history and biologic and 

chemical processes that affect nitrate concentrations in water 

recharging the aquifer at the water table. Changes in land use 

and associated changes in surface sources of nitrate affect the 

source term used in models simulating nitrate transport and 

thus the estimates of time-varying nitrate concentrations in a 

pumped well.

Processes near the land surface and in unsaturated 

sediments above the water table, such as uptake by plants 
22 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

and denitrification, also affect the source terms used to 

estimate time-varying nitrate concentrations in pumped 

wells. Uptake of nitrogen from applied fertilizer varies by 

crop; for cereal crops, the average nitrogen uptake efficiency 

(NEU) is about 33 percent (Raun and Johnson, 1999) and 

can exceed 50 percent for corn (Bundy and Andraski, 2005). 

In addition to plant uptake, five additional processes—soil 

denitrification, volatilization, gaseous plant emissions, 

subsurface tile drains, and surface runoff—can cause 

additional near-surface losses of nitrogen (Raun and Johnson, 

1999). The remaining nitrogen that is not lost through these 

processes leaches into the ground and potentially enters the 

aquifer at the water table with recharge. Although nitrate in 

a glacial aquifer may be transported conservatively under 

oxic conditions, substantial losses of nitrogen—exceeding 

30 percent—can occur in association with reducing conditions 

in the unsaturated zone prior to recharge at the water table 

(Weiskel and Howes, 1992). Because these processes have 

an effect on the source term used to simulate the advective 

transport of nitrate to a pumped well, understanding those 

processes is critical. Although subsurface processes affecting 

nitrogen concentrations, such as dispersion and biochemical 

transformation, can be explicitly simulated by using solutetransport and reactive-transport models, surficial processes 

can be implicitly represented by altering source terms to 

reflect those processes. Some implicitly represented surficial 

processes are analogous to spatial variables of potential use in 

statistical models of nitrate occurrence.

Traveltimes

Groundwater age is an important consideration in 

evaluating the vulnerability of a well to contamination from 

surface sources of nitrogen; groundwater age has a control 

on the occurrence of nitrogen as well as the geochemical 

conditions that contribute to nitrogen contamination (Warner 

and Arnold, 2010). Traveltimes through an aquifer from 

recharge locations to pumped wells are a function of the 

depths of the flow paths; deeper flow paths generally are 

associated with longer traveltimes (fig. 13A). Steady-state 

traveltimes within a contributing recharge area, which are 

proportional to the porosity of aquifer sediments, can be 

defined by using particle tracking. Steady-state traveltimes 

in pumped wells in the Great Miami River Basin range from 

essentially instantaneous to more than 100 years; the longest 

traveltimes are for water recharging the upgradient parts of the 

contributing recharge areas, where flow paths are longer and 

deeper than for water recharging closer to the pumped well 

(fig. 13B). Maximum steady-state traveltimes to wells pumped 

in three hydrologic settings at a rate equivalent to the volume 

of recharge through the corresponding 1,640-ft-radius circular 

buffer exceeded 150 years, but median traveltimes to wells 

870, 1084, and 1053 were about 17.6, 24.8 and 36.6 years, 

respectively. The three wells have a similar distribution 

of traveltimes (fig. 14A); the longest traveltimes are from 

upgradient parts of the contributing recharge areas.

Steady-state traveltimes are valid after a well has been 

in operation, and the resulting hydraulic gradients have been 

established for a time longer than the longest traveltime within 

the contributing recharge area, which can exceed 200 years. 

In most cases, pumped wells in the glacial aquifer system 

have been in operation for traveltimes that are shorter than 

the maximum steady-state traveltimes within the contributing 

recharge area. At well 1053, the transient contributing recharge 

area approximates the steady-state contributing recharge area 

in less than 20 years (fig. 12B); however, transient traveltimes 

are longer than those under steady-state pumping conditions 

(fig. 14B). Steady-state traveltimes reflect steeper hydraulic 

gradients around the well caused by pumping and therefore 

are faster than under less steep, natural gradients. The median 

traveltimes within the contributing recharge area after 2, 10, 

and 20 years of operation were about 23, 25, and 41 years, 

respectively (fig. 14B). As a result, the distributions of 

traveltimes after 30, 40, and 50 years are similar; the median 

is about 50 years compared to a median of 36.6 years under 

steady-state conditions. Transient contributing recharge areas 

for a given period of pumping are much larger than for the 

equivalent steady-state traveltime band; for 2 and 20 years 

of operation, the corresponding 2- and 20-year steady-state 

contributing areas were 15.9 and 40 percent of the area of the 

transient contributing recharge area (fig. 12).

Steady-state traveltimes are a function of the depth of 

the pumped well and the pumping rate. At an intermediate 

pumping rate (equivalent to the rate of recharge through the 

corresponding circular buffer), traveltimes are longer for 

wells screened deeper in the aquifer because the wells capture 

water from longer flow paths. Median traveltimes for a well 

pumped in layers 1, 2, and 3 are 36.6, 50.3, and 58.2 years, 

respectively (fig. 15). The increased gradient causes the well 

to capture water from flow paths associated with a large range 

of traveltimes—from essentially instantaneous to more than 

150 years. The traveltimes for wells pumped in layers 1, 2, 

and 3 overlap over most of the ranges of traveltimes (fig. 15) 

because pumping alters regional hydraulic gradients and 

steepens gradients to the well; this result indicates that the 

recharge areas correspond to a similar range of traveltimes 

for wells pumped at different depths. Pumping has less effect 

on regional hydraulic gradients at low pumping rates, and 

traveltimes reflect natural regional age patterns. As a result, 

traveltimes have smaller ranges and less overlap between 

wells pumped from different depths (fig. 15); this result 

indicates that recharge areas for wells pumped at different 

depths correspond to different ranges of traveltimes. For 

a shallow pumped well (layer 1), traveltimes are less than 

35.2 years with a median of 14.9 years, compared to maximum 

and median traveltimes of 188.3 and 36.6 years, respectively, 

for an intermediate pumping rate. Median traveltimes for 

pumped wells in layers 2 and 3 are 93.5 and 196.9 years, 

respectively, for low pumping rates (fig. 15), larger than those 

of 50.3 and 58.3 for the intermediate pumping rate.
Sources of Water and Advective Transport of Nitrate to Pumped Wells 23

84°4' 84°3' 84°2' 84°1'

39°47'

39°46'

39°45'

Not to scale Depth below land surface, in feet NGVD 88

oldest

older

Land surface

Water table

Pumped wells

water

water

Direction of 

groundwater flow

young water

youngest water

A Nitrogen 

 input

920

880

840

B

Well 870

Well 1084

Well 1053

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Direction of 

groundwater flow

0 0.5 1 MILES

0 0.5 1 KILOMETERS

Well 1053

EXPLANATION

Layer 1

Layers 1–2

Layers 1–3

Active model layers

Pumped well 

and number

880

Water table—number

is altitude, in feet 

(NGVD 88) >0 to 20

>20 to 40

>40 to 60

>60 to 80

>80 to 100

>100

Traveltime, in years

Area of nitrogen

application 

(1,977 kilograms

per year)

Circular buffer of 

radius 1,640 feet

Figure 13. A, Relation between well depth and traveltime; and B, map showing 1,640-ft-radius circular buffers and steady-state 

traveltimes within contributing recharge areas to pumped wells in three hydrogeologic settings and areas of nitrogen application 

from fertilizer, southeastern part of the Great Miami River Basin, southwestern Ohio.
24 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0 100 200 300

 0

20

40

60

80

100

 A 

0 100 200 300

 0

20

40

60

80

100

Time, in years

Percentage of particles

B

Median

Median

EXPLANATION

Steady state

2-year recharge area

10-year recharge area

20-year recharge area

50-year recharge area

40-year recharge area

30-year recharge area

Transient

EXPLANATION

Well 870

Well 1084

Well 1053

Steady-state traveltime, in years

Figure 14. A, Steady-state traveltimes to pumped wells in three hydrologic settings; and B, transient traveltimes 

after different periods of pumping by well 1053, southeastern part of the Great Miami River Basin, southwestern 

Ohio.
Sources of Water and Advective Transport of Nitrate to Pumped Wells 25

0 200 200 300 400 500

 0

20

40

60

80

100

Time, in years

EXPLANATION

Layer 1

Layer 2

Layer 3

Intermediate 

 pumping rate

Low pumping rate

Layer 1

Layer 2

Layer 3

Median

Percentage of particles

Figure 15. Steady-state traveltimes for well 1053 pumped at different depths and at different pumping rates, 

southeastern part of the Great Miami River Basin, southwestern Ohio.

Transport of Nitrate

If nitrogen is transported conservatively in the form of 

nitrate, as under oxic conditions in the glacial aquifer system, 

groundwater-flow models and mass-weighted particle tracking 

can be used to simulate the advective transport of nitrate and 

estimate time-varying concentrations at a pumped well. The 

use of mass-weighted particle tracking entails assigning a 

mass of nitrate to a set of particles and tracking those particles 

forward to pumped wells. Adding the associated masses for 

different particle traveltimes can allow approximation of the 

time-varying load of nitrate to a well; the pumping rate of the 

well can be used to convert the loads to concentrations. The 

method requires a uniform spatial distribution of particles 

at the water table and a defined source term to determine 

mass weights. Nitrogen concentrations in a pumped well are 

a function of several factors, including traveltimes between 

recharge locations at the water table and the well and the 

distribution of sources of nitrogen at the land surface. 

Ten-year mean nitrogen-application rates (1992–2001) 

reported in Warner and Arnold (2010) were assembled 

from county usage data and applied to agricultural land 

identified in the NLCDe92 database. Application rates in 

the Great Miami River Basin, in kilograms per year (kg/yr) 

(fig. 4B), are average annual rates applied 500-meter landuse parcels. The nitrogen-application rate on agricultural 

land in the area around hypothetical wells 870, 1084, and 

1053 was 1,977 kg/yr (fig. 13B). It should be noted that 

statistical and process models are based on a steady-state 

application rate, which does not take into account complex, 

time-varying source terms. Agricultural land composed 

about 70.6, 20.3, and 47.0 percent of the total contributing 

recharge area to wells 870, 1084, and 1053, respectively 

(fig. 13B). Nitrate concentrations estimated under steadystate pumping conditions increase steadily until a steadystate concentration is reached after a transport time equal to 

the maximum traveltime within the contributing recharge 

area, which can exceed 100 years (fig. 13B). Under the 

assumptions of no attenuation of nitrogen prior to recharge 

and conservative transport, steady-state nitrate concentrations 

in wells 870, 1084, and 1053 were 34.6, 9.9, and 26.9 mg/L 

as N, respectively (fig. 16A). The concentrations are within 

the range reported for monitoring wells in the glacial aquifer 

system but exceed the 90th percentile (Warner and Arnold, 

2010). The steady-state concentrations were reached after 

65 to 70 years of transport and reflect the proportions of 

agricultural land and the distribution of traveltimes within the 

contributing recharge areas (fig. 13B).
26 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio Simulated nitrate concentration, in milligrams per liter as nitrogen

Steady-state concentration = 34.6 mg/L

Steady-state concentration = 26.9 mg/L

Steady-state concentration = 9.9 mg/L

0

10

20

30

40

 A 

0

10

20

30

Time, in years

Steady-state 

concentration

for low pumping

in layer 3 = 0 mg/L

 B 

0 10 20 30 40 50 60 70 80 90 100

EXPLANATION

Well 870

Well 1084

Well 1053

EXPLANATION

Layer 1

Layer 2

Layer 3

Intermediate 

 pumping rate

Low pumping rate

Layer 1

Layer 1

Layer 2

Transient—Intermediate

 pumping rate

Figure 16. Time-varying nitrate concentrations for A, wells in three hydrologic settings and B, well 1053 for three 

well depths and low and intermediate pumping rates.
Comparison of Statistical and Process Models 27

Estimated nitrate concentrations are affected by 

the same factors as the simulated contributing recharge 

areas and traveltimes, including well depth and pumping 

rates. At intermediate pumping rates, steady-state nitrate 

concentrations in well 1053 pumped in model layers 1, 2, 

and 3 were 26.9, 23.3, and 18.4 mg/L as N, respectively 

(fig. 16B). The contributing recharge areas to the well shift 

upgradient for pumping in successively deeper model layers 

(fig. 11), resulting in less agricultural source area within the 

contributing recharge areas. Steady-state concentrations under 

low pumping rates differ from those under higher pumping 

rates and are more affected by well depth (fig. 16B). Nitrate 

concentrations in a shallow well (layer 1) pumped at a low 

rate steadily increased to a concentration of 23.3 mg/L as N 

over the range of total traveltime—about 35 years—and likely 

may be near steady state (fig. 16B). Nitrate concentrations 

in a well pumped at a low rate deeper in the aquifer (layer 2) 

reached a steady-state concentration of 10.7 mg/L as N after 

about 86 years of transport (fig. 16B). Concentrations in a well 

pumped at low rate in the deepest part of the aquifer (layer 3) 

were zero over the range of transport times. Wells pumped at 

a low rate have less effect on regional gradients, and therefore 

regional hydraulic gradients and traveltimes have a larger 

effect on predictions of nitrate concentrations.

Most pumped wells are in operation for less time than 

the maximum steady-state traveltime within the contributing 

recharge area; as a result, concentrations from transient 

simulations of traveltimes (fig. 15) could differ from steadystate simulated concentrations. The method used to estimate 

transient contributing recharge areas results in a nonuniform 

particle field that makes assigning mass weights problematic. 

The transient contributing recharge areas after 2, 10, 20, 

30, 40, and 50 years were used to implicitly estimate the 

concentration in the pumped well at those times on the basis 

of the average nitrogen load for particles in the contributing 

recharge area and the total volume of recharge. Using the same 

method for steady-state contributing recharge areas confirmed 

that the results do match steady-state concentrations simulated 

by using mass-weighted concentrations. Nitrate concentrations 

under transient transport initially are higher and increase 

faster than those estimated under assumptions of steady-state 

transport but approach the same steady-state concentration 

after sufficiently long transport times (fig. 16B). The nitrate 

concentration in well 1053, pumped at an intermediate rate 

for 2 years in a shallow part of the aquifer, was 13.9 mg/L as 

N compared to 0 for the nitrate concentration estimated after 

2 years of steady-state transport. After 20 years, estimated 

concentrations under transient and steady-state flow conditions 

were 25.3 and 15.4 mg/L as N, respectively. The assumption 

of transient flow accounts for the fact that, initially, a pumped 

well may capture older water with nitrate transported under 

natural hydraulic gradients from recharge locations beneath 

existing sources, whereas the assumptions of steady-state flow 

are that the well has been in operation for a time equivalent 

to the maximum traveltime from the contributing recharge 

area and that there initially is no mass of nitrate in the aquifer. 

Changes in land use or the rate of nitrogen application 

onto agricultural land would affect estimated nitrogen 

concentrations; the effect would be a function of the timing of 

the resultant changes in the source term and the traveltimes to 

the pumped well.

Estimates of time-varying concentrations from 

groundwater-flow models representing advective transport are 

affected by a number of factors, including aquifer porosity 

and surface-source characteristics. Groundwater velocities 

and therefore traveltimes are proportional to the effective 

porosity of aquifer sediments, which generally range from 

0.2 to 0.4 for unconsolidated sediments (Freeze and Cherry, 

1979). Increasing porosity causes slower velocities and longer 

traveltimes. For well 1053, increasing the porosity from 0.2 

to 0.3 increases the amount of time required to reach the same 

steady-state concentration in a shallow pumped well from 

about 68 to about 100 years (fig. 17). The source term used 

to represent inputs of nitrate into the aquifer and the transport 

of nitrate to a pumped well also is affected by changes in the 

rate of application of nitrate at the land surface as well as the 

transformation and loss of nitrate between the land surface and 

the water table. If nitrogen in agricultural areas near well 1053 

were assumed to be applied for only 20 years, the steady-state 

concentration of 15.3 mg/L as N would be reached in 20 years; 

this concentration is less than the steady-state concentration of 

26.9 mg/L as N derived from an assumed continuous source 

over the entire period of steady-state pumping (fig. 17). Also, 

assuming a loss of 50 percent of nitrate prior to recharge to 

the water table from plant uptake and biochemical processes 

results in a steady-state concentration of 13.4 mg/L as N, 

which is half the concentration under the assumption of 

no loss (fig. 17). An assumed nitrogen loss of 50 percent 

or greater prior to recharge likely is more realistic than the 

assumption of conservative transport from land surface to the 

water table.

Comparison of Statistical and 

Process Models

Differences in predictions of nitrate occurrence by 

statistical and process models could arise from two sources. 

The first relates to differences in the assumed source of water 

to a pumped well—circular buffers (statistical models) or 

contributing recharge areas (process models)—that is used 

to estimate the values of the explanatory variables for which 

relations to nitrate occurrence have been defined. The second 

relates to the two different modeling approaches: predictions 

that utilize observed relations between nitrate occurrence and a 

set of explanatory variables (statistical models) and those that 

explicitly represent surface sources and transport processes 

(process models).
28 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Time, in years

0

10

20

30

Nitrate concentration, in milligrams per liter as N 

0 10 20 30 40 50 60 70 80 90 100

EXPLANATION

Land use less than 20 years

Porosity of 0.3

50-percent loss prior to recharge

 at the water table

Nitrate concentration, in milligrams

 per liter in well 1053

Porosity of 0.2, steady-state land

 use, no attenuation

Figure 17. Time-varying nitrate concentrations at well 1053 for different porosities and source-term characteristics under 

steady-state transport for intermediate pumping rates.

Predictions of Nitrate Occurrence by Statistical 

and Process Models at Selected Wells

The statistical models of nitrate occurrence presented in 

Warner and Arnold (2010) utilize relations between observed 

nitrate concentrations and a set of explanatory variables to 

predict the probability that nitrate concentrations at a well will 

exceed a given concentration. The set of variables includes 

environmental variables and a source variable defined as an 

area-weighted mean within 1,640 ft (500 m) of the pumped 

well. The assumption is that the circular buffer approximates 

the source of water to the well; this assumption does not 

account for local groundwater-flow conditions and can result 

in area-weighted explanatory variables and resultant model 

predictions that differ from those determined for the actual 

source area. The source of water to the well, or contributing 

recharge area, is a function of several factors, including pumping rate, screen depth, and hydrologic setting (figs. 10–12), 

and therefore area-weighted explanatory variables and model 

predictions would be affected by the same three factors.

The 1,640-ft-radius circular buffer around well 1053 

differs from the contributing recharge area to the well 

pumped at a rate equivalent to the volumetric rate of recharge 

through the circular buffer (fig. 18). The volume-equivalent 

contributing recharge area is similar in size but has a 

different shape, extends upgradient from the circular buffer, 

and results in different estimates of explanatory variables. 

The distribution of mean 10-year nitrogen-application rates 

estimated from county data and environmental variables 

(silt content, alfisol content, and depth to water) estimated 

from STATSGO data (figs. 4A and 4B) differ between 

the circular buffer and volume-equivalent contributing 

recharge area (fig. 18). As an example, the area-weighted 

mean nitrogen-application rate within the circular buffer 

and contributing recharge area were 928 and 779 kg/yr, 

respectively. The area-weighted silt content, alfisol content, 

and mean depth to water were 49.4 percent, 82.1 percent, 

and 4.64 ft, respectively, within the circular buffer and 

51.1 percent, 84.5 percent, and 4.27 ft respectively, within the 

volume-equivalent contributing recharge area. Differences in 
Comparison of Statistical and Process Models 29

84°03' 84°02' 84°01' 

39°

45'

30" 

39°

45' 

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

EXPLANATION

Pumped well and number 

Well 1053

Well 1053

Area of nitrogen-fertilizer application

(1,977 kilograms per year)

Area of no nitrogen fertilizer application—Alfisol content 

of 81 percent, silt content of 48.55 percent, and 

depth to water of 4.84 feet

Alfisol content of 85 percent, silt content of 51.56 percent, 

and depth to water of 4.18 feet

Circular buffer of radius of 1,640 feet

0 1,200 2,400 FEET

0 300 600 METERS

Nitrogen-application rate, in kilograms per year; 

Alfisol and silt content, in percent; and depth to 

water, in feet

O kg/yr, 85%, 51.56 %, 4.18 ft

0 kg/yr, 81%, 48.55%, 4.84 ft

0 kg/yr, 81%, 48.55%, 4.84 ft

1,997 kg/yr, 85%, 51.56%, 4.18 ft 

1,997 kg/yr, 85%, 51.56%, 4.18 ft

1,977 kg/yr, 81%, 48.55%, 4.84 ft

Recharge area

Circular buffer

Figure 18. Circular buffer of radius 1,640-ft and equivalent contributing recharge area to well 1053 and spatial distribution 

of explanatory variables used to estimate nitrate occurrence, Great Miami River Basin, southwestern Ohio. Ten-year mean 

nitrogen-application rate derived from county data, and environmental variables—alfisol content, silt content, and depth to 

water—derived from STATSGO data. 
30 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

estimated source and environmental variables between circular 

buffers and contributing recharge areas are affected by the 

same factors that affect the simulated contributing recharge 

area; these factors include pumping rate, screen depth, and 

hydrologic setting.

Role of Pumping Rate and Well Depth

The contributing recharge area to a well is affected by 

the rate at which the well is pumped and the depth of the 

well screen; the size of the contributing recharge area is 

proportional to the pumping rate, and wells screened deeper in 

the aquifer have contributing recharge areas that extend farther 

upgradient than shallow-screened wells. Generally, the effect 

of screen depth on the contributing recharge area is greater for 

wells pumped at low rates because natural regional hydraulic 

gradients are a more important control on the contributing 

recharge area (fig. 13). At a given location, the distributions 

of spatial variables and the area-weighted means of these 

variables in the contributing recharge areas differ for each 

combination of pumping rate and screen depth (fig. 19).

Estimates of Area-Weighted Spatial Variables

 Area-weighted mean variables at well 1053 vary for 

different simulated pumping rates from three screen depths 

(fig. 20A) corresponding to screen midpoints of about 

17, 86, and 162 ft below the water table. For a high pumping 

rate—equivalent to the mean pumping rate of public-supply 

wells in the basin—values of environmental variables (alfisol 

and silt content and depth to water) are similar for shallow, 

intermediate, and deep well screens (fig. 20A). The ranges for 

all three area-weighted environmental variables are less than 

0.5 percent of the averages for the three well-screen depths. 

These results are consistent with the similarity in simulated 

contributing recharge areas for a well pumped at a high rate 

at different depths (fig. 19). For an intermediate pumping 

rate—equivalent to the volume of recharge through the 

corresponding 1,640-ft-radius circular buffer—the ranges of 

the three area-weighted environmental variables for different 

well-screen depths were between 0.6 and 2 percent of the 

average. With an assumed low pumping rate of 350 gal/d, 

which is similar to the withdrawal rate from a domestic-supply 

well, ranges in area-weighted environmental variables for the 

three well-screen depths were between 2 and 7 percent of the 

average. The coefficients of variation—defined as the ratios 

of the variances to the means—for silt content, alfisol content, 

and depth to water were 0.007, 0.07, and 0.004, respectively, 

for the different combinations of pumping rates and screen 

depths. Estimates of the 10-year mean nitrogen-application 

rate (1992–2001) within contributing recharge areas varied 

more (fig. 20A) with a coefficient of variation of 321.8 for 

different combinations of pumping rates and screen depths. 

The ranges of area-weighted nitrogen-application rates for 

three different well-screen depths and high and intermediate 

pumping rates were about 1.6 and 33.7 percent, respectively, 

of the average value. Screen depth had a greater effect on 

area-weighted estimates of nitrogen-application rates at 

low pumping rates because natural gradients have a greater 

effect on flow patterns, and contributing recharge areas vary 

more with depth (fig. 19). The area-weighted mean nitrogenapplication rates for wells pumped in shallow, intermediate, 

and deep parts of the aquifer were 1,111; 525; and 0 kg/yr, 

respectively (fig. 20A). 

Area-weighted spatial variables for contributing recharge 

areas and different combinations of pumping rates and 

screen depths all differed from those for the corresponding 

1,640-ft-radius circular buffer around well 1053 (fig. 20A). 

Area-weighted alfisol and silt content were lower and depth to 

water higher within the circular buffer than in the contributing 

recharge areas for all combinations of pumping rates and 

screen depths. The mean nitrogen-application rate within the 

circular buffer was in the range of variation of values from 

different combinations of pumping rates and screen depths 

(fig. 20A).

Predictions of Nitrate-Exceedance Probabilities

The effects of variations in area-weighted spatial 

variables on predictions of nitrate occurrence by logistic 

regression are functions of the explanatory variables used 

in a model, the scale of variability of the area-weighted 

explanatory variables, and the weight of the explanatory 

variables in the logistic regression models indicated by 

the slope coefficient (Bn) in equation 1. In addition to 

the explanatory variables estimated from spatial data, 

the statistical models include well-construction variables 

(fig. 3A): depth to and length of the well screen and the 

diameter of the well. In this analysis, a well diameter of 7 in., 

a screen length of 7.8 ft—corresponding to the medians for 

private-supply wells used to develop the statistical models 

(Warner and Arnold, 2010)—and a well depth of 10 ft were 

assumed; the well depth of 10 ft is similar to the median of 

the simulated depth to water of 7 ft in the basin and consistent 

with the depths of wells screened in shallow parts of the 

aquifer (layer 1). For purposes of comparison, consistent 

well-construction variables are used so that differences in 

predicted probabilities are a function of only differences in 

area-weighted explanatory variables for simulated source 

areas. The statistical models (Warner and Arnold, 2010) were 

developed based on an assumed 1,640-ft-radius circular buffer 

as the source area to the well; these models include spatial 

variables estimated in the same manner. Although spatial 

variables estimated from contributing recharge areas were 

used in this analysis, the results provide insight into the degree 

to which improving representation of the source area would 

affect the statistical model development and performance.

The probabilities of exceeding threshold concentrations 

of 1, 2, 3, and 4 mg/L as N in well 1053 were predicted to 

be 49, 41, 33, and 21 percent, respectively, by using median 

values of the area-weighted spatial variables for the circular 

buffer in the logistic regression models presented in Warner 
Comparison of Statistical and Process Models 31

84°3' 84°2' 84°1'

39°

46'

39°

45'

0 1,200 2,400 FEET

0 300 600 METERS

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

EXPLANATION

Well 1053

Well 1053

Low

pumping

rate,

370 gallons

per day

Intermediate

pumping

rate,

0.08 million gallons

per day

High

pumping

rate,

0.25 million gallons

per day

Shallow depth

Intermediate depth

Deep depth

Circular buffer of 

radius 1,640 feet

Pumped well and number

Area of nitrogen-fertilizer 

application

(1,977 kilograms per year)

Area of no nitrogen fertilizer 

application—Alfisol 

content of 81 percent, 

silt content of 48.55, 

and depth to water 

of 4.84 feet

Alfisol content 

of 85 percent, silt content 

of 51.56, and depth to water 

of 4.18 feet

Recharge Areas

Figure 19. Circular buffer of radius 1,640-ft and equivalent contributing recharge areas to well 1053 for different 

combinations of well depths and pumping rates, and spatial distribution of explanatory variables used to estimate nitrate 

occurrence, Great Miami River Basin, southwestern Ohio. Ten-year mean nitrogen-application rate derived from county data, 

and environmental variables—alfisol content, silt content, and depth to water—derived from STATSGO data.
32 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio Alfisol content, in percent Nitrogen-application rate, in kilograms per year

82

83

84

85

86

Depth to water, in feet

49

50

51

52

Silt content, in percent

0

200

400

600

800

1,000

1,200

4.0

4.2

4.4

4.6

4.8

Alfisol

content

Nitrogenapplication rateSilt Depth to water

Spatial variable

5.0

1 2 3 4

0.0

0.1

0.2

0.3

0.4

0.5

Modelled threshold concentration, in milligrams per liter

Probability of exceeding threshold concentration

A

B

Intermediate

Low

Shallow

Deep

Intermediate High

EXPLANATION

Circular buffer of radius 1,640 feet

Pumping rate Depth

Figure 20. A, Estimates of area-weighted explanatory variables and B, the probability of nitrate occurrence above 

threshold concentrations of 1 to 4 milligrams per liter (mg/L) as nitrogen by statistical models for the 1,640-ft circular buffer 

and different combinations of well depth and pumping rate at well 1053.
Comparison of Statistical and Process Models 33

and Arnold (2010) (fig. 20B). Logistic regression models 

for predicting exceedances of 1, 2, and 3 mg/L as N do not 

include the 10-year nitrogen-application rate as an explanatory 

variable. The predicted probabilities based on the circular 

buffer around the well were higher than predicted probabilities 

based on area-weighted statistical variables estimated from 

contributing recharge areas for different combinations of 

pumping rates and screen depths (fig. 20B). The highest 

variability in probabilities for different screen depths was 

at low pumping rates. The coefficients of variation for the 

probabilities of nitrate exceeding 1, 2, 3, and 4 mg/L as N 

were less than 0.003. The exceedance probabilities varied 

less than the values of the area-weighted spatial variables 

(figs. 20A and B) because the probabilities are functions 

of both spatial and well-construction variables, which only 

partially control the variability in predictions of nitrate 

occurrence.

Role of Hydrologic Setting

The importance of well location on estimates of areaweighted spatial variables for contributing recharge areas 

arises from two sources: (1) the hydrologic setting of 

the pumped well, and (2) the areal distribution of spatial 

characteristics upgradient from the well. The hydrologic 

setting of a well affects the shape of the contributing recharge 

area and therefore the areal distribution of characteristics 

defined by land-use or soil data layers upgradient from 

the well. The interaction of hydrologic setting and the 

distribution of spatial variables results in unique estimates 

of area-weighted spatial variables for a given set of well 

characteristics (location, pumping rate, and depth).

Wells in three hydrologic settings—shallow and deep 

saturated sediments in areas of relatively steep hydraulic 

gradients (well 870 and 1053) and shallow saturated sediments 

near a groundwater divide (well 1084)—are associated with 

differing areal distributions of spatial characteristics within 

circular buffers and contributing recharge areas (fig. 21). 

When a well near a groundwater divide is pumped at a rate 

equivalent to recharge through the corresponding circular 

buffer, the recharge area approximates the circular buffer, 

and the two likely would have similar spatial characteristics. 

Contributing recharge areas to similar pumped wells in areas 

of steep hydraulic gradients extend upgradient from the well, 

beyond the 1,640-ft-radius circular buffer (fig. 21), and likely 

have spatial characteristics that differ from the corresponding 

circular buffers.

Estimates of Area-Weighted Spatial Variables

Estimates of area-weighted spatial variables differ for 

wells pumped in the three different locations and hydrologic 

settings at intermediate and low pumping rates (fig. 22A). The 

circular buffers and contributing recharge areas for wells 870 

and 1084 are within the same areas of defined environmental 

characteristics from the STATSGO database (silt content, 

alfisol content, and depth to water) (fig. 21), so there is no 

variability among estimates of area-weighted mean values 

for these two areas (fig. 22A). Contributing recharge areas 

and the circular buffer for well 1053 encompass two different 

regions of defined environmental characteristics resulting in 

some variability in estimates of area-weighted mean values 

(fig. 22A). 

The circular buffers and contributing recharge areas of 

all three wells encompass two regions of defined 10-year 

mean nitrogen-application rates. The coefficients of variation 

of estimated area-weighted mean nitrogen-application rates 

for wells 870, 1084, and 1053 were 292.5, 243.3, and 29.5, 

respectively. Well 870 had the most variable and highest 

mean area-weighted nitrogen-application rate (1,363.0 kg/yr) 

(fig. 22A). The area-weighted mean nitrogen-application rate 

ranged from 715.6 kg/yr for the circular buffer to 1,977 kg/yr 

for the contributing recharge area at low pumping rates. About 

36 percent of the circular buffer overlaps an area with a mean 

nitrogen-application rate of 1,977 kg/yr—the remaining 

part of the circular buffer overlaps an area with no nitrogen 

application—whereas the entire contributing recharge area to 

the well for low pumping rates is within the area of nitrogen 

application (fig. 21). Conversely, the contributing recharge 

area to well 1084 for low pumping rates is almost completely 

within an area with no nitrogen application, and the circular 

buffer overlaps an area with nitrogen application (fig. 21).

Predictions of Nitrate-Exceedance Probabilities

Probabilities of exceeding nitrate concentrations 

of 1, 2, 3, and 4 mg/L as N at the three wells by using 

circular buffers and contributing recharge areas for low and 

intermediate pumping rates are a function of the estimated 

spatial variables (fig. 22B). Probability predictions were the 

same for the circular buffers and the contributing recharge 

areas for low and intermediate pumping rates for wells 870 

and 1084 (fig. 22B) because the three logistic regression 

models for nitrate exceedances of 1, 2, and 3 mg/L as N used 

the same estimated values of the three environmental spatial 

variables for both the circular buffers and the contributing 

recharge areas (fig. 22A). Estimated spatial variables for the 

contributing recharge areas and the circular buffer for well 

1053 and the resulting predictions of nitrate exceedances as N 

are more variable (fig. 22B), but the coefficients of variation 

of the predictions for the three models are less than 0.003. 

The logistic regression model for nitrate exceedance of 

4 mg/L as N includes the mean 10-year nitrogen-application 

rate as an explanatory variable, which is the variable with 

the largest coefficient of variation for circular buffers and 

contributing recharge areas for the three wells (fig. 22A). 

The predicted probabilities of exceeding 4 mg/L as N 

range from 14 to 21 percent (fig. 22B); the coefficient of 

variation of all the predictions of exceeding 4 mg/L as N 

was about 0.004. The three well locations include a range 

of spatial variables, particularly nitrogen application rate 

(0 to 1,977 kg/yr). Probabilities of nitrate exceeding a given 
34 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

84°04' 84°03' 84°02' 84°01'

39°46'

39°45'

900

860

820

Well 870

Well 1084

Well 1084

Well 1053

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Direction of

groundwater ow

EXPLANATION

Pumped well and number

Recharge area, low pumping

Circular buffer of radius 1,640 feet 

Water table, number is altitude, in feet (NGVD 88)

Area of nitrogen-fertilizer application

 (1,977 kilograms per year)

Area of no nitrogen fertilizer and STATSGO data—Alfisol 

content of 81 percent, silt content of 48.55 percent, and depth

to water of 4.84 feet

STATSGO data—Alfisol content of 85 percent, silt content of 51.56 percent, 

and depth to water of 4.18 feet

940

860

0 0.5 1 MILES

0 0.5 1 KILOMETERS

Nitrogen-application rate, in kilograms per year; 

Alfisol and silt content, in percent; and depth 

to water, in feet

0 kg/yr, 85%, 51.56%, 4.18 ft

0 kg/yr, 81%, 48.55%, 4.84 ft

0 kg/yr, 81%, 48.55%, 4.84 ft

1,977 kg/yr, 85%, 51.56%, 4.18 ft

1,977 kg/yr, 85%, 51.56%, 4.18 ft

1,977 kg/yr, 81%, 48.55%, 4.84 ft

Recharge area

Circular buffer

Figure 21. Circular buffer of radius 1,640-ft and steady-state contributing recharge areas to wells in three hydrologic 

settings for intermediate and low pumping rates, and spatial distribution of explanatory variables used to estimate 

nitrate occurrence, southeastern part of Great Miami River Basin, southwestern Ohio.
Comparison of Statistical and Process Models 35

Alfisol

content

Mean 10-year

nitrogen

application rate

Silt Depth to water

0

500

1,000

1,500

2,000

Spatial variable

Nitrogen-application rate, in kilograms per year

4.0

4.2

4.4

4.6

4.8

5.0

49

50

51

52

Silt content, in percent

1 2 3 4

0.0

0.1

0.2

0.3

0.4

0.5

Modeled threshold concentration, in milligrams per liter 

Probability of exceeding threshold concentration

A

B

Alfisol content, in percent

82

83

84

85

86

Depth to water, in feet

Recharge area,

intermediate

pumping rate

Circular buffer

Recharge area,

low pumping

rate

#1053 #1084 #870

EXPLANATION

Well

Figure 22. A, Estimates of area-weighted explanatory variables and B, predictions of nitrate occurrence above 

threshold concentrations from 1 to 4 milligrams per liter as nitrogen by using statistical models for wells in three 

hydrologic settings for intermediate and low pumping rates.
36 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

threshold concentration generally showed less variability 

than the estimated spatial variables because the predicted 

probabilities are functions of the same sets of wellconstruction variables.

Predictions of Nitrogen-Occurrence Probabilities 

for Public-Supply and Network Wells

A total of 309 pumping centers—each with 1 or more 

wells—are represented in the regional model of the Great 

Miami River Basin (Sheets, 2007); these wells withdraw 

about 18.4 Mgal/d from the glacial aquifer, primarily around 

the city of Dayton, Ohio (figs. 2 and 6). Spatial variables 

estimated for 1,640-ft-radius circular buffers for these 

pumping centers differed to varying degrees from estimates 

for corresponding steady-state contributing recharge areas to 

these centers; adjusted R2 values ranged from 0.27 to 0.44. 

The highest correlations were for mean nitrogen-application 

rate (R2=0.41) and silt content (R2=0.44) (fig. 23A). Spatial 

variables for circular buffers estimated from the STATSGO 

database—silt content, alfisol content, and depth to water—

were statistically different than those for contributing recharge 

areas. Comparisons of estimated variables for circular buffers 

and contributing recharge areas by a paired t-test (Helsel and 

Hirsch, 1992) showed that silt content was lower (p was less 

than 0.001), and alfisol content and depth to water were higher 

(p was less than 0.001), within circular buffers (fig. 23A), 

Nitrogen-application rates estimated for circular buffers and 

contributing recharge areas were not statistically different.

The logistic-regression models presented in Warner and 

Arnold (2010) and spatial variables estimated for circular 

buffers and corresponding contributing recharge areas 

were used to predict the probability of exceeding nitrate 

concentrations of 1 to 4 mg/L as N; well-construction data 

were used to determine nonspatial explanatory variables. 

Predictions of the probability of exceeding 1 to 4 mg/L as N 

generally were correlated for the two sets of spatial-variable 

estimates (fig. 23B). Adjusted R2

 values for probabilities 

of exceeding 1 to 4 mg/L as N were 82 and 94 percent, 

respectively. Models of nitrate exceedance of 1 to 4 mg/L 

as N are useful for comparison because (1) the 1 mg/L as N 

includes only environmental variables from the STATSGO 

database, and (2) 4 mg/L as N is the lowest threshold 

concentration that includes all spatial variables—all of 

the environmental variables and nitrogen-application rate. 

Predictions of exceeding 1 mg/L as N generally were greater 

for the set of spatial variables estimated from the circular 

buffers than those estimated from the contributing recharge 

areas (fig. 23B). This result is consistent with differences 

observed in area-weighted mean values estimated from the 

STATSGO database. The slope coefficients of silt content, 

alfisol content, and depth to water for the prediction model 

for 1 mg/L as N are 0.0356, -0.0164, and 0.7219, respectively 

(table 1), indicating that lower silt content and higher depth 

to water contribute to higher probabilities of exceedance. The 

large slope coefficient for depth to water indicates that the 

probabilities predicted for the circular buffers (fig. 23B) could 

be higher because of the higher area-weighted depth to water 

within the circular buffers. 

A network of 1,290 wells was used to evaluate water 

quality in the glacial aquifer system (DeSimone and others, 

2009); the network included public- and private-supply wells 

and monitoring wells used to develop statistical models of 

nitrate occurrence (Warner and Arnold, 2010). A total of 

55 network wells were within the modeled area of the Great 

Miami River Basin (fig. 6A). Spatial data for 1,640-ft-radius 

circular buffers and corresponding contributing recharge areas 

for network wells—at a pumping rate equivalent to the volume 

of recharge through the circular buffer—show different trends 

than do spatial data from public-supply wells: silt content 

generally is higher (p is less than 0.001) inside circular 

buffers than in the corresponding contributing recharge 

areas (fig. 24A), and alfisol content and depth to water are 

lower. The probabilities of exceeding 1 and 4 mg/L as N in 

water from wells inside circular buffers and corresponding 

contributing recharge areas are highly correlated with adjusted 

R2 values of 0.80 and 0.88, respectively (fig. 24B), similar 

to predictions for public-supply wells. However, predictions 

of exceeding 1 and 4 mg/L as N generally are higher (p less 

than 0.001) by using spatial variables for circular buffers 

than for contributing recharge areas (fig. 24B). The results 

show that, although depth to water generally is lower within 

circular buffers, predicted probabilities of exceeding 1 mg/L 

as N are higher. The results from individual wells show 

that large differences in alfisol content contributed to the 

higher predicted probabilities. The fact that different trends 

in estimated values of spatial variables but similar trends in 

predicted probabilities were observed in the two sets of wells 

underscores the hypothesis that physical processes likely do 

not underlie the observed trends. One possible reason could 

be the chosen locations of the public-supply wells, such as the 

siting of wells in areas of sandy soils or in areas near rivers, 

where hydraulic gradients are steeper. The results show that 

representing source areas as circular buffers or contributing 

recharge areas can greatly affect estimates of spatial 

variables and probability predictions, but that the effects are 

site specific.

Basin-Scale Comparisons of Statistical 

and Process Models

Differences in estimates of area-weighted spatial 

variables and the resultant predictions of nitrate occurrence 

by using circular buffers and contributing recharge areas are 

functions of several factors, including (1) hydrologic setting, 

which in turn is a function of aquifer geometry and properties 

and hydraulic boundaries and gradients; and (2) areal 

differences in the values of explanatory variables derived from 

spatial data (fig. 4), which can affect the resultant predictions 
Basin-Scale Comparisons of Statistical and Process Models 37

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Probability of exceeding threshold concentration from 1,640-foot circular buffer

Probability of exceeding threshold concentration from recharge area

1 mg/L

EXPLANATION

4 mg/L

EXPLANATION

Nitrogen-application rate from 1,640-foot circular buffer, in kilograms per year

Silt content from recharge area, in percent

Nitrogen-application rate from recharge area, in kilograms per year

Silt content from 1,640-foot circular buffer, in percent

R2=0.82

R2

=0.94

0 400 800 1,200 1,600 2,000

0

400

800

1,200

1,600

2,000

48 50 52 54 56 58

48

50

52

54

56

58

A

B

Line of equality

Line of equality

Nitrogen-application 

rate 

Silt concentration 

Figure 23. A, Estimates of area-weighted explanatory variables and B, predictions of nitrate occurrence 

above threshold concentrations of 1 to 4 milligrams per liter as nitrogen determined by using contributing 

recharge areas and circular buffers for production wells, Great Miami River Basin, southwestern Ohio.
38 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Nitrogen-application rate from 1,640-foot circular buffer, in kilograms per year 

Silt content from recharge area, in percent

Nitrogen-application rate from recharge area, in kilograms per year

Silt content from 1,640-foot circular buffer, in percent

Nitrogen-application rate

EXPLANATION

Silt content

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 400 800 1,200 1,600 2,000

0

400

800

1,200

1,600

2,000

48 50 52 54 56 58

48

50

52

54

56

58

R2=0.80

R2

=0.88

A

B

Probability of exceeding threshold concentration from 1,640-foot circular buffer

Probability of exceeding threshold concentration from recharge area

Line of equality

Line of equality

1 mg/L

EXPLANATION

4 mg/L

Figure 24. A, Estimates of area-weighted explanatory variables and B, predictions of nitrate occurrence 

above threshold concentrations of 1 and 4 milligrams per liter as nitrogen determined by using recharge areas 

and circular buffers for network wells in the Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 39

of nitrate occurrence. Contributing recharge areas simulated 

by process models account for regional groundwater-flow 

conditions and better represent spatial characteristics in the 

source area to a well than do circular buffers.

The effect of hydrologic setting on the correlations 

between estimates of spatial variables and predictions of 

nitrogen occurrence in contributing recharge areas and 

circular buffers were evaluated at the basin scale within the 

Great Miami River Basin. Estimates of spatial variables 

and predictions of nitrate occurrence were made for 21,774 

hypothetical wells spaced at regular intervals of 500 ft. Wells 

within 1,640 ft of the edge of the active model area were 

excluded from the analysis so that the spatial variables for the 

circular buffers, like the simulated contributing recharge areas, 

would represent only the active modeled area (fig. 4).

At each location, the area-weighted means of explanatory 

spatial variables—mean 10-year nitrogen application, silt 

content, alfisol content, depth to water, and volumetric rate 

of recharge —were estimated for a 1,640-ft-radius circular 

buffer around the well. Volumetric rates of recharge through 

the 1,640-ft-radius circular buffers ranged from 10,550 to 

23,589 ft3/d (about 0.08 to 0.18 Mgal/d) (fig. 25) and were 

spatially correlated with modeled recharge to the aquifer 

(fig. 8B). A well was then pumped in layer 1 of the model at a 

rate equivalent to the volume of recharge through the circular 

buffer. In cases where the model went dry in layer 1, the well 

was pumped in successive layers until the layer around the 

well remained saturated. In cases where no solution could be 

obtained, the well was discarded. A total of 854 simulations 

either went dry or did not yield a solution; this outcome was 

common where saturated sediments were thin, such as near 

bedrock uplands. Only the hypothetical well of interest was 

pumped during each simulation so that natural gradients were 

represented to the extent possible; pumping at rates typical 

of the recharge rates through corresponding circular buffers 

generally did not alter natural gradients. The relative areal 

overlap—defined as the ratio of the portion of the contributing 

recharge area within the circular buffer to the total area of the 

contributing recharge area—was determined for each well site.

For each of the 20,920 successful model runs, the 

contributing recharge area to the well and area-weighted 

means of the spatial explanatory variables were determined. 

The sets of estimated spatial variables from contributing 

recharge areas and circular buffers were used to predict 

probabilities of nitrate exceeding threshold concentrations 

of 1 to 10 mg/L as N by using the logistic regression models 

in Warner and Arnold (2010). Well-construction variables 

were the same in all predictions, so that differences in the 

probabilities were functions only of differences in spatial 

variables which, in turn, are functions of the differences 

between circular buffers and volume-equivalent contributing 

recharge areas. A well diameter of 7 in. and screen length 

of 7.8 ft were used in the statistical models, consistent with 

the mean values for private-supply wells in the NAWQA 

monitoring network (Warner and Arnold, 2010); a screen 

depth of 10 ft—consistent with the screen depth for a shallow 

pumped well, but less than the mean for private-supply wells 

(about 91 ft)—was also used. Assuming a shallower well 

depth would result in larger probabilities of nitrate exceedance 

for a given set of spatial variables, because well depth has a 

negative slope coefficient and is inversely related to predicted 

probabilities (table 1). However, on the basis of results given 

earlier in this report, trends in differences between estimates 

of spatial variables and resultant predictions for probabilities 

estimated for circular buffers and contributing recharge areas 

would be similar. In addition to estimates of area-weighted 

spatial variables and predictions of nitrate occurrence, massweighted particle tracking was used to estimate time-varying 

nitrate concentrations arising from fertilizer application on 

agricultural areas within the contributing recharge area to each 

of the 20,920 successfully modeled wells.

Comparison of Circular Buffers and Contributing 

Recharge Areas as the Assumed Sources of 

Water to Wells

The potential effects of hydrologic setting on estimates 

of spatial variables for a source area to a pumped well can 

be quantified by using the overlap between a circular buffer 

around a well and the simulated contributing recharge area to 

the well pumped at a rate equivalent to the volumetric rate of 

recharge through the circular buffer. In areas where hydraulic 

gradients are nearly flat, such as near groundwater divides, 

contributing recharge areas are nearly circular and generally 

overlap the circular buffer, whereas wells pumped in regions 

with steeper regional hydraulic gradients have elongated 

contributing recharge areas that overlap less of the circular 

buffer (figs. 21, 26). At three locations in the southeastern part 

of the basin, the amount of overlap ranged from 85.6 percent 

in well 1084, which was near a regional groundwater divide, 

to 32.7 percent in well 1053, which was in an area of steep 

hydraulic gradients (fig. 26).

The amount of overlap between circular buffers and 

corresponding contributing recharge areas in the basin ranged 

from 2 to 99 percent (fig. 27). Amounts of overlap generally 

were larger near upland margins, near groundwater divides 

and away from discharge areas, although basin-scale spatial 

trends were variable. The mean and median overlaps were 28 

and 32 percent, respectively (fig. 28). The results indicate that, 

at a basin scale, contributing recharge areas differ substantially 

from circular buffers, and large differences in estimated spatial 

variables and resulting predictions are likely.
40 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Recharge through 1,640-foot circular buffers, 

in cubic feet per day

24,000

10,000

EXPLANATION

Area not modeled

84°20' 84°10' 84°

39°

50'

39°

40'

Great Miami River

Little Miami River

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

0 3 6 MILES

0 6 KILOMETERS

Extent of model grid

Figure 25. Volumetric recharge rates through circular buffers for hypothetical wells uniformly spaced at 500-foot intervals 

across the Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 41

Overlap = 54.8 percent

Overlap = 85.6 percent

Overlap = 32.7 percent

Buried-valley aquifer

84°04' 84°03' 84°02' 84°01'

39°47'

39°46'

39°45'

EXPLANATION

Pumped well and number 

Well 1084

Well 1053

Well 870

Recharge area Overlap between recharge areas

and circular buffers

Well 1084

Well 1053

Well 870

880 No pumping

Pumping in well 1053

Layer 1

Layers 1−2

Layers 1−3

Active model layers

920

900

820

860

Well 870

Well 1084

Well 1053

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Circular buffer of radius 1,640 feet

Water table—number is altitude, in feet 

(NGVD 88)

0 0.5 1 MILE

0 0.5 1 KILOMETER

Well 870

Figure 26. Overlap between contributing recharge areas and circular buffers in three hydrologic settings, southeastern part 

of the Great Miami River Basin, southwestern Ohio.
42 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Stillwater River

Little Miami River

Great Miami River

Mad River

Great Miami

River

Wolf Creek

84°20' 84°10' 84°

39°50'

39°40'

EXPLANATION

No modeled solution

Area not modeled

Overlap between recharge areas and 

1,640-foot circular buffers, in percent

100

0

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

0 1.5 3 6 MILES

0 6 KILOMETERS

Extent of model grid

Figure 27. Overlap between contributing recharge areas and circular buffers for 20,920 hypothetical wells uniformly spaced 

at 500-foot intervals across the Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 43

0 10 20 30 40 50 60 70 80 90 100

Overlap between circular buffers and recharge areas, in percent

200

400

600

800

Number of wells

0

N = 20,920

Median=32

Mean=28

Figure 28. Distribution of overlaps between contributing recharge areas and circular buffers for 20,920 hypothetical wells 

in the Great Miami River Basin, southwestern Ohio.

Predictions of Nitrate Occurrence by Using 

Statistical Models

Values of spatial variables were obtained from two 

sources: soil characteristics from the STATSGO database 

(Wolock, 1997) and nitrogen-application rates from county 

data mapped to areas of agricultural data in the NLCDe92 

database. Differences in the area-weighted means of spatial 

variables between circular buffers and contributing recharge 

areas, as indicated by the generally small overlap between 

the two at the basin scale (figs. 27 and 28), would cause 

differences in the resultant predictions of nitrate occurrence by 

statistical models.

Estimates of Area-Weighted Spatial Variables

Area-weighted mean nitrogen-application rates inside 

circular buffers and contributing recharge areas generally 

are correlated (R2

=0.71, p less than 0.01) with no statistical 

difference between the two sets of estimates (fig. 29A). 

Area-weighted means of STATSGO data—silt content, alfisol 

content, and depth to water—also generally were correlated 

(R2

 values between 0.58 and 0.62, p less than 0.01) (fig. 29B). 

Silt content was statistically higher, and alfisol content and 

depth to water were statistically lower (p less than 0.001), 

within circular buffers than in contributing recharge areas; 

however, there likely are no physical processes underlying 

those differences. Differences in area-weighted mean spatial 

variables between circular buffers and contributing recharge 

areas are functions of the overlap between the buffers and 

contributing recharge areas (fig. 27) and the variability of 

the two sets of spatial data from which the area-weighted 

means were derived (fig. 4). Differences in area-weighted 

mean nitrogen-application rates between circular buffers and 

contributing recharge areas, determined by subtracting the 

value estimated for the contributing recharge area from the 

value estimated for the circular buffer, ranged from about 

-1,560 kg/yr (indicating that the value for the circular buffer 

was less than the value for the contributing recharge area) 

to 1,880 kg/yr. The mean and median differences were 7.8 

and 0.0 kg/yr, respectively (fig. 30A). Area-weighted means 
44 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0 500 1,000 1,500 2,000

0

500

1,000

1,500

2,000

48 50 52 54 56 58

48

50

52

54

56

58

Nitrogen-application rate from 1,640-ft circular buffer, in kilograms per year

Silt content from recharge area, in percent

Nitrogen-application rate from recharge area, in kilograms per year

Silt content from 1,640-ft circular buffer, in percent

0 20 40 60 80 100

0

20

40

60

80

100

2

3

4

5

2

Depth to water from 1,640-ft circular buffer, in feet

Alfisol content from recharge area, in percent

Alfisol content from 1,640-ft circular buffer, in percent

Depth to water

EXPLANATION

Alfisol content

Depth to water from 1,640-ft recharge area, in feet

A

B

N=20,920

N=20,920

Line of equality

Line of equality

3 4 5

Nitrogen-application rate

EXPLANATION

Silt content

Figure 29. Relation between contributing recharge areas and circular buffers for A, nitrogenapplication rate and silt content, and B, alfisol content and depth to water.
Basin-Scale Comparisons of Statistical and Process Models 45

A

C D

B

-1,200 -800 -400 0 400 800 1,200

0

5

10

15

20

25

-8 -6 -4 -2

0

10

20

30

40

50

0 8642

Difference in alfisol content, in percent Difference in depth to water, in feet

Difference in nitrogen application rate, in kilograms per year Difference in silt content, in percent

Frequency, in percent

Frequency, in percent

Frequency, in percent

Frequency, in percent

-60 -40 -20

0

10

20

30

40

50

0 20 40 60

N = 20,920

N = 20,920 N = 20,920

N = 20,920

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

10

20

30

40

50

Mean = 7.8

Median = 0.0

Mean = 0.24

Median = 0.0

Mean = -6.0

Median = 0.0

Mean = -0.05

Median = 0.0

Figure 30. Differences in estimated area-weighted means for A, nitrogen-application rate, B, silt content, C, alfisol content, and D,

depth to water for contributing recharge areas and circular buffers. A positive difference indicates that the prediction for circular 

buffers was larger.

from circular buffers and contributing recharge areas were not 

statistically different. The estimated mean nitrogen-application 

rates were equal at 4,209 locations, representing about 

20.4 percent of the total. 

Area-weighted means for spatial variables estimated 

from the STATSGO database—silt content, alfisol content, 

and depth to water—were the same for 6,915 locations, about 

33.6 percent of the total. Differences ranged from -7.5 to 

48.9 percent for silt content, from -81.8 to 81.4 percent for 

alfisol content, and from -1.68 to 4.77 ft for depth to water. 

The mean differences, defined as the value for the recharge 

area subtracted from the value for the circular buffer, for silt 

content, alfisol content, and depth to water were 0.24 percent, 

-6.0 percent, and -0.05 ft, respectively (figs. 30B–30D). Silt 

content was statistically higher, and alfisol content and depth 

to water statistically lower, inside circular buffers than inside 

contributing recharge areas (p less than 0.001). The spatial 

distribution of differences in estimates of spatial variables 

for circular buffers and contributing recharge areas (fig. 30) 

reflects trends in the two sets of underlying spatial data 

(fig. 4). Differences in nitrogen-application rates were largest 

and more variable in the northeastern and southeastern parts 

of the basin (fig. 31A), generally corresponding to agricultural 

areas with the largest rates of fertilizer application (fig. 4B). 

Differences in silt content between circular buffers and contributing recharge areas (fig. 31B) generally were largest and 

more variable in areas where the STATSGO data are more 

variable, such as near major streams in the center of the basin 

(fig. 4A).

Predictions of Nitrate-Exceedance Probabilities

Nitrate exceedance probabilities are functions of the 

scales of the differences and the weights of the variables 

in the logistic regression model, as indicated by the slope 
46 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio No modeled solution Area not modeled Difference in nitrogen-application rate, in kilograms per year, from county data 1,900 EXPLANATION -1,600 Difference in silt content, in percent, from STATSGO data 49 -8 No difference

Stillwater River

Little Miami River

Great Miami River

Mad River

Great Miami

River

Wolf Creek

Stillwater River

Great Miami River

Mad River

Great Miami

River

Wolf Creek

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84°

0 3 6 MILES

0 6 KILOMETERS

Extent of model grid

Extent of model grid

Little Miami River

84°10' 84°20' 84°

A

B 84°10' 84°20' 84°

39°50'39°40'

Figure 31. Differences in the area-weighted means of the A, nitrogen-application rate and B, silt content estimated for contributing recharge areas and circular buffers, Great Miami River Basin, southwestern Ohio. 
Basin-Scale Comparisons of Statistical and Process Models 47

coefficients (table 1). When calculated for circular buffers, the 

probability of exceeding a nitrate concentration of 1 mg/L as 

N ranged from 20.1 to 57.0 percent, and the mean and median 

were both 47.0. The highest probabilities generally were near 

major streams in the center of the basin and in the southeastern 

part of the basin (fig. 32A). Data for water in some wells 

indicated that high predicted probabilities for the center of 

the basin were related to the absence of alfisol soils, whereas 

high probabilities for upland areas were related to large depths 

to water (fig. 4A). The predicted probability of exceeding 

4 mg/L as N ranged from 5.4 to 30.4 percent, and the mean 

and median were 19.5 and 18.7, respectively. The largest areas 

of high probability were in the southeastern and northeastern 

parts of the basin (fig. 32B); the logistic regression model 

for predicting nitrate exceeding 4 mg/L as N included the 

nitrogen-application rate as an explanatory variable, and 

the high predicted probabilities for the northeastern part 

of the basin were related to areas of high rates of fertilizer 

application (fig. 4B). Probabilities predicted on the basis of 

spatial variables estimated for contributing recharge areas 

(figs. 33A and 33B) showed trends that generally were 

similar to trends in probabilities estimated for circular buffers 

(figs. 33A and 33B). The mean and median of probabilities 

of nitrate exceeding 1 mg/L as N predicted on the basis of 

variables estimated from contributing recharge areas were 

45.2 and 42.7 percent, respectively; the mean and median 

of probabilities of exceeding 4 mg/L as N were 18.7 and 

18.3 percent, respectively. 

Probabilities predicted on the basis of spatial variables 

estimated for circular buffers and contributing recharge areas 

generally were well correlated with adjusted R2

 values of 

0.69 and 0.68 for the predictions of concentrations exceeding 

1 and 4 mg/L as N, respectively (fig. 34). Predictions based 

on variables estimated for circular buffers were statistically 

higher than were predictions for contributing recharge 

areas (p less than 0.001) for both 1 and 4 mg/L as N; this 

result reflects statistical differences in the estimated spatial 

variables. Statistical differences were likely not caused by any 

underlying physical processes and thus could differ in other 

basins. Differences in predicted probabilities of exceeding 

1 mg/L as N on the basis of spatial variables from circular 

buffers and contributing recharge areas ranged from -17.6 

to 48.7 percent; a positive value indicated higher predicted 

probabilities for circular buffers. The mean and median 

values were 1.8 and 0.2 percent, respectively (fig. 35A). For 

predicted probabilities of exceeding 4 mg/L as N, differences 

ranged from -9.2 to 21.8 percent and had a mean and median 

of 0.8 and 0.4 percent, respectively (fig. 35B). Predictions for 

exceeding concentrations of 1 and 4 mg/L as N were identical 

for circular buffers and contributing recharge areas in 35.4 

and 10.2 percent of the hypothetical wells, respectively. The 

number of locations for the 4 mg/L threshold was lower than 

the number of locations for the 1 mg/L threshold because the 

use of nitrogen-application rate as an explanatory variable 

in the logistic regression model for the 4 mg/L threshold 

resulted in more potential for differences in predictions from 

contributing recharge areas and circular buffers (table 1). 

Although predictions generally were correlated, differences 

for some areas were large (figs. 32 and 33), and using spatial 

variables for contributing recharge areas can significantly 

affect statistical model predictions.

Spatial trends in differences among predictions generally 

reflect spatial patterns in the differences among the spatial 

variables estimated for circular buffers and contributing 

recharge areas. The largest differences among predictions 

of exceeding concentrations of 1 mg/L as N generally were 

near major streams near the center of the basin (fig. 36A), 

for which differences in environmental variables from the 

STATSGO database were large (fig. 31B). Areas for which 

predictions for circular buffers and contributing recharge areas 

were the same—about 35 percent of the modeled basin—

generally corresponded to large contiguous areas for which 

environmental variables from the STATSGO database were 

similar (fig. 4A). Differences in predictions of exceeding 

4 mg/L as N were more widespread; predictions were the 

same for circular buffers and contributing recharge areas in 

only about 10 percent of the modeled basin (fig. 36B). Areas 

where differences for exceeding 4 mg/L as N were larger than 

differences for exceeding 1 mg/L as N generally corresponded 

to areas where differences in the nitrogen-application rate 

were larger, such as agricultural areas in the southeastern and 

northeastern parts of the basin (fig. 4B). Despite the large 

differences in predictions caused by differences in spatial 

variables, the spatial patterns were similar. Areas where the 

probability of nitrate concentrations exceeding the threshold 

was higher than the median probability predicted for circular 

buffers generally coincided with the areas similarly predicted 

for contributing recharge areas (figs. 37A and 37B). The 

areas where predicted probabilities of exceeding 1 mg/L as 

N were greater than the median values—about 47 percent for 

circular buffers and about 43 percent for contributing recharge 

areas—overlapped by 93.0 percent (fig. 37A). For predicted 

probabilities of exceeding 4 mg/L as N, areas greater than 

the medians of about 19 and 18 percent for circular buffers 

and contributing recharge areas overlapped by 88.6 percent 

(fig. 37B). These differences do not reflect an underlying 

nonrandom process, however, and patterns in differences 

among spatial variables for this area would not be transferable 

to other basins.

Role of Well Depth

Contributing recharge areas to wells pumped near 

the water table partially overlap the circular buffers to 

those wells; however, in many cases, circular buffers and 

contributing recharge areas to deeper wells do not overlap 

because contributing recharge areas to wells pumped deeper 

in the aquifer are likely to extend farther upgradient than 

contributing recharge areas to shallower wells (fig. 11). 

As an example, the contributing recharge area to well 725, 

pumped in the shallow part of the aquifer, is partially inside 

(about 13 percent) and upgradient from the corresponding 
48 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio Probability of exceeding nitrate concentrations, in percent Area not modeled 57 EXPLANATION 20 1 mg/L 30

5

4 mg/L

Stillwater River

Little Miami River

Great Miami River

Mad River

Wolf Creek

Stillwater River

Little Miami River

Great Miami River

Mad River

Great Miami

River

Wolf Creek

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84°

Extent of model grid

Extent of model grid

Great Miami

River

84°10' 84°20' 84°

A

B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 32. Probabilities of exceeding nitrate concentrations of A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted from logistic regression models on the basis of spatial variables for circular buffers, Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 49 No modeled solution Area not modeled Probability of exceeding nitrate concentrations, in percent 57 EXPLANATION 2 1 mg/L 171 4 mg/L

Stillwater River

Little Miami River

Great Miami River Mad River

Great Miami

River

Wolf Creek

Stillwater River

Little Miami River

Great Miami River Mad River

Wolf Creek

Great Miami

River

Extent of model grid Extent of model grid

84°10' 84°20' 84° A B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84°

Figure 33. Probabilities of exceeding nitrate concentrations of A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted by logistic regression models on the basis of spatial variables for contributing recharge areas, Great Miami River Basin, southwestern Ohio.
50 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1 mg/L

EXPLANATION

4 mg/L

Probability of exceeding threshold concentration of nitrate from 1,640-foot circular buffer

Probability of exceeding threshold concentration of nitrate from recharge area

R2=0.69

R2

=0.68

Line of equality

Figure 34. Relation between probabilities of exceeding nitrate concentrations of 1 and 4 milligrams per liter as nitrogen on 

the basis of area-weighted mean explanatory variables for contributing recharge areas and circular buffers.

circular buffer, whereas the contributing recharge area to 

a well pumped deep in the aquifer (layer 3) at the same 

location is far upgradient (fig. 38). The closest edge of the 

contributing recharge area to the deep well is about 15,000 ft 

from the edge of the 1,640-ft radius circular buffer (fig. 38). 

The fraction of overlap between contributing recharge areas 

and circular buffers is generally lower for deep wells than for 

shallow wells (fig. 39). The mean and median of the fraction 

of overlap between circular buffers and contributing recharge 

areas to deep wells were 0.24 and 0.27, respectively; the 

contributing recharge areas to about 63.1 percent of wells 

pumped from deep parts of the aquifer did not overlap any 

of the corresponding circular buffers. For deep wells where 

the contributing recharge area partly overlapped the circular 

buffer, the fractions of overlap for shallow and deep wells 

were positively correlated (fig. 39).

Differences among contributing recharge areas to 

wells pumped at different depths indicate that differences 

in estimates of spatial variables and resultant predictions 

of nitrate occurrence that arise from differences in circular 

buffers and contributing recharge areas are greater for wells 

pumped in deep than for those pumped in shallow parts of the 

aquifer, and that estimated spatial variables for contributing 

recharge areas are likely to be different for wells pumped at 

different depths. Estimates of nitrogen-application rates for 

contributing recharge areas to shallow wells (layer 1) and deep 

wells (layer 3) generally were correlated (R2

=0.64) (fig. 40A), 

and the two sets of estimated variables were not statistically 

different; however, there is a large amount of scatter in the 

data arising from large differences in contributing recharge 

areas from shallow and deep wells at individual locations 

(fig. 40A). The correlation is less for estimates of spatial 

variables such as silt content from STATSGO data between 

contributing recharge areas from shallow and deep pumped 

wells (fig. 40A). Correlations of silt and alfisol content were 

statistically higher, and depth to water statistically lower, in 

contributing recharge areas to deep wells than contributing 

recharge areas to shallow wells (p less than 0.001). No 

underlying physical processes likely caused these differences. 

Probabilities of nitrate concentrations exceeding 1 and 

4 mg/L as N in shallow and deep wells at a consistent depth 

of 10 ft (to identify trends in the predictions arising only 
Basin-Scale Comparisons of Statistical and Process Models 51

-30 -20 -10 0 10 20 30

0

10

20

30

40

50

-30 -20 -10 0 10 20 30

A B

Median=0.2

Mean=1.8

Mean=0.8

Median=0.4

Difference in predicted probabilities of exceeding threshold nitrate concentration, in percent

Fraction of total, in percent

N = 20,920

N = 20,920

Figure 35. Differences in predicted probabilities of exceeding concentrations of A, 1 milligram per liter as nitrogen; and 

B, 4 milligrams per liter as nitrogen on the basis of area-weighted mean explanatory variables for contributing recharge 

areas and circular buffers. A positive difference indicates that the prediction for circular buffers was larger.

from differences in estimated spatial variables) generally 

were correlated (fig. 40B). The probabilities of exceeding 1 

and 4 mg/L as N predicted for contributing recharge areas 

to shallow wells (layer 1) are statistically higher than those 

predicted for contributing recharge areas to deep wells 

(layer 3), likely reflecting statistical differences in spatial 

variables estimated from the STATSGO database. Based 

on a well depth of 91 ft, which is the mean depth of private 

wells in the NAWQA monitoring network and is common for 

wells pumped in layer 3 of the model, predicted probabilities 

of exceeding 1 and 4 mg/L as N are much lower than those 

predicted on the basis of an assumed well depth of 10 ft. The 

negative slope coefficients of the well-depth explanatory 

variables in the logistic regression models (table 1) indicate 

that predicted probabilities decrease with increasing depth 

to the top of the well screen. The variability in predicted 

probabilities arising from differences in estimated spatial 

variables for contributing recharge areas to shallow and deep 

wells is substantially smaller for larger assumed well-screen 

depths (fig. 40B). This result indicates that although 

differences in estimated spatial variables are larger for deep 

wells, the use of well-screen depths consistent with deeper 

wells lessens the effects of those differences on predictions of 

nitrate occurrence.

Estimates of Depth to Water

Values of the spatial variables included as explanatory 

variables in the logistic regression models are based on 

national databases and generally are coarse at the local scale. 

Nitrogen application rates are calculated by applying total 

fertilizer use, based on county-specific data, to agricultural 

parcels in the NLCDe92 database; these data can provide 

a source term for statistical and process models of nitrate 

occurrence or transport. Soil characteristics that are based 

on STATSGO data—silt content and alfisol content—can 

inform process and statistical models regarding the potential 
52 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio No modeled solution Area not modeled 1 mg/L 27 EXPLANATION-9 Difference in probability of exceeding nitrate concentration, in percent, from circular buffers and recharge areas 48-17 No difference 4 mg/L

Stillwater River

Little Miami River

Great Miami River

Mad River

Great Miami

 River

Wolf Creek

Stillwater River

Little Miami River

Great Miami River

Mad River

Great Miami

 River

Wolf Creek

Extent of model grid

Base from U.S. Geological Survey digital data, 1:24,000, 1983Albers Equal Area projectionStandard parallels 37°50' and 41°10', central meridian -84°

Extent of model grid

84°10' 84°20' 84°

A

B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 36. Differences in the probabilities of nitrate concentrations exceeding A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen predicted by logistic regression models on the basis of spatial variables for contributing recharge areas and circular buffers, Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 53 Area where probability of nitrate concentrations exceeding Area not modeled No model solution 1 mg/L is greater than median predicted probability EXPLANATION Area where probability of nitrate concentrations exceeding Circular buffer, median of 47 percent Recharge area, median of 43 percent 4 mg/L is greater than median predicted probability Circular buffer, median of 19 percent Recharge area, median of 18 percent Base from U.S. Geological Survey digital data, 1:24,000, 1983 Albers Equal Area projection Standard parallels 37°50' and 41°10', central meridian -84° Extent of model grid Extent of model grid

84°10' 84°20' 84° A B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 37. Areas where probabilities of nitrate concentrations exceeding A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen were higher than the median probabilities predicted by logistic regression models for contributing recharge areas and circular buffers, Great Miami River Basin, southwestern Ohio.
54 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Well 725 Mad River

Buried-valley aquifer

Well 725

Pumped well and number

Shallow (layer 1)

Deep (layer 3)

EXPLANATION

Layer 1

Layers 1−2

Layers 1−3

Active model layers

880

Recharge areas

860

820

820

900

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Direction of 

groundwater ow

Circular buffer of radius 1,640 feet 

Water table, number is altitude, in feet

(NGVD 88)

0 0.5 1 MILES

0 0.5 1 KILOMETERS

84°07' 84°06' 84°05' 84°04'

39°47'

39°46'

39°45'

Figure 38. Contributing recharge areas and the circular buffer to well 725 (a hypothetical well) for shallow 

and deep well screens, southeastern part of the Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 55

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Fractions of overlap between recharge area and corresponding

1,640-foot circular buffer, well pumped in layer 1 

Fractions between recharge area and corresponding 1,640-foot 

circular buffer, well pumped in layer 3 

Line of equality

Figure 39. Relation between fractions of overlap between contributing recharge areas and corresponding circular buffers for 

wells pumped in shallow and deep parts of the glacial aquifer system, Great Miami River Basin, southwestern Ohio.

for nitrogen attenuation, but are external to the models. 

Unlike other spatial variables used in the logistic-regression 

models, depth to water (saturates soil), derived from the 

STATSGO database, represents the depth from land surface to 

the water table and is a function of the local flow system; the 

variable, therefore, can be calculated from a process model 

that simulates hydrologic processes. Depth to water has large 

slope coefficients in the statistical models of nitrate occurrence 

above six of the ten threshold concentrations (1 to 10 mg/L as 

N) (table 1) (Warner and Arnold, 2010) and therefore has an 

important control on predictions of nitrate occurrence.

Depth to water in the STATSGO database ranged from 

2.48 to 4.88 ft (fig. 4A); area-weighted mean values for the 

1,640-ft-radius circular buffers to the 21,774 hypothetical 

wells had the same range (fig. 41A). Area-weighted mean 

estimates of depth to water determined from the groundwaterflow model, representing the distance between the land 

surface and the calculated water-table altitude, ranged from 

0.02 to 78 ft—a much larger range than area-weighted means 

determined from the STATSGO database (fig. 41A). Modelcalculated depths to water better represent local hydrologic 

conditions (determined from aquifer properties, hydrologic 

boundaries, and recharge conditions) and, if the model is well 

posed, likely represent more accurate and detailed estimates of 

the explanatory variable used in statistical models.

For the range of depths to water in the STATSGO database (2.48 to 4.88 ft), the probabilities of exceeding 1 and 

4 mg/L as N predicted by using the subset of model-calculated 

depths to water and depths to water from the STATSGO 

database differed substantially (fig. 14B). The two sets of 

predictions were not correlated and statistically different 

(p less than 0.001). As an example, for a prediction of 0.5 for 

an exceedance of 1 mg/L as N on the basis of STATSGO data, 

the predicted model-calculated depths to water ranged from 

0.18 to 0.73 (fig. 41B). The large difference between probabilities predicted on the basis of depths to water estimated 

from the STATSGO database and probabilities calculated by 

the groundwater-flow model indicates that the more detailed, 

site-specific estimates from the process (groundwaterflow) model could inform or improve statistical models of 

nitrate occurrence.
56 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

Probability of exceeding threshold concentration for wells pumped in layer 1

Probability of exceeding threshold concentration

for wells pumped in layer 3

A

B

Nitrogen-application rate from recharge areas, in kilograms per year, for wells pumped in layer 1

Silt content from recharge areas, in percent, for wells pumped in layer 1

Nitrogen-application rate from recharge areas, in kilograms per year,

for wells pumped in layer 3

Silt content from recharge areas, in percent, 

for wells pumped in layer 3

0.0 0.2 0.4 0.6

0.0

0.2

0.4

0.6

1 mg/L

EXPLANATION

4 mg/L

Depth to water of 10 feet for shallow wells 

Depth to water of 91 feet for deep wells 

1 mg/L

4 mg/L

0 500 1,000 1,500 2,000

0

500

1,000

1,500

2,000

48 50 52 54 56 58

48

50

52

54

56

58

Nitrogen-application rate

EXPLANATION

Silt content

Line of equality

Figure 40. A, Estimates of nitrogen-application rate and silt content, and B, probabilities of nitrate occurrence 

above threshold concentrations of 1 and 4 milligrams per liter as N determined for contributing recharge areas 

and circular buffers for shallow and deep pumped wells, Great Miami River Basin, southwestern Ohio.
Basin-Scale Comparisons of Statistical and Process Models 57

0

20

40

60

80

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Depth to water within circular buffers, in feet, from STATSGO database

Depth to water within circular buffers, in feet, calculated

by groundwater-flow model

Probability of exceeding threshold concentration of nitrate on

the basis of depth to water from STATSGO database

Probability of exceeding threshold concentration of nitrate by using

depth to water calculated by groundwater-flow model

1 mg/L

EXPLANATION

4 mg/L

Range of values 

from STATSGO 

database

4.18

2.85

4.84

4.88

3.63

4.76

2.48

A

B

2

Line of equality

3 4 5

Figure 41. Relations between A, depths to water within circular buffers estimated from STATSGO data 

and calculated by a groundwater-flow model, and B, probability of nitrate concentrations exceeding 1 and 

4 milligrams per liter as nitrogen on the basis of the two sets of estimated depths to water.
58 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

The use of a process model of saturated flow to calculate 

depths to water has two important limitations: (1) the inability 

to represent perched water tables, which could cause lower 

depths to water, particularly for areas with clayey soils 

(such as alfisols), and (2) uncertainties associated with the 

flow model that can result in simulated water-table altitudes 

above the land surface and thus negative depths to water. 

Calculations of negative depths to water were excluded from 

the analysis.

Predictions of Nitrate Concentrations by Using 

Nitrogen Loading Rates

Process models that simulate sources, flow, and transport 

can be used to estimate concentrations of a solute from surface 

sources in an aquifer system. For conservative species, such 

as nitrate in oxic environments within the glacial aquifer 

system, process models that simulate only groundwater flow 

can use mass-weighted particle tracking to implicitly represent 

the advective transport of nitrate from surface sources, 

such as areas of fertilizer application, to pumped wells. 

The method predicts nitrogen concentrations by tracking 

water—represented as a series of particles—from recharge 

locations at the water table to a pumped well, analogous to 

the recharge area, and assigning to each particle a mass of 

nitrogen estimated from the associated source area at that 

location. Accountings of cumulative mass and transport 

times are used to estimate the time-varying concentration in 

the well. The analyses are presented as a way of comparing 

process-model predictions that explicitly represent physical 

transport processes with statistical-model predictions of the 

probability of exceeding a threshold concentration from the 

set of estimated explanatory variables. The analyses are based 

on the spatial distribution of nitrogen application that was 

used to develop the statistical models (fig. 4); the distributions 

of nitrogen application at the local scale and nitrogen from 

other sources are not represented. Also, assumptions regarding 

nitrate attenuation in the unsaturated zones and in the aquifer 

are inherent in the analyses.

Under the assumed nitrogen-application rates from 

county data (fig. 4B), a nitrogen uptake of 50 percent (Bundy 

and Andraski, 2005), and losses of 30 percent to other 

biologic and geochemical processes near land surface and 

in the unsaturated zone (Weiskel and others, 1992), nitrate 

concentrations determined from mass-weighted particle 

tracking to 21,774 hypothetical pumped wells ranged 

from 0 to 12 mg/L as N after 50 years of transport through 

shallow parts of the Great Miami River Basin (fig. 42A). The 

median value was 2 mg/L as N, consistent with the median 

for monitoring wells in the NAWQA well network (Warner 

and Arnold, 2010). Nitrate concentrations arising from the 

application of fertilizer were spatially correlated with the 

distribution of nitrogen application in the basin; nitrate 

concentrations exceeding the median value of 2 mg/L as N 

(higher than the 50th percentile) were in the northeastern, 

southeastern, and southwestern parts of the aquifer (fig. 42B) 

beneath the three areas of highest fertilizer-application rates 

on agricultural land (fig. 4B). It should be noted that the 

concentrations account only for fertilizer application and not 

other sources such as septic systems in residential areas.

Simulations of nitrate concentrations by a groundwaterflow model and mass-weighted particle tracking use the 

nitrogen-application rate (fig. 4B) as the only source input 

and, as a result, the steady-state concentrations of nitrate in 

pumped wells are highly correlated with area-weighted mean 

nitrogen-application rates within contributing recharge areas 

(R2

=0.90) (fig. 43). The proportionality constant between the 

area-weighted mean nitrogen-application rate and steady-state 

nitrogen concentration is a function of the volume of recharge 

through the contributing recharge area; the scatter in the data 

arises from the spatially variable recharge volumes in the basin 

(fig. 25). The nitrate concentration also is correlated with the 

area-weighted mean nitrogen-application rate within circular 

buffers (R2

=0.65) (fig. 43) because area-weighted means from 

circular buffers and contributing recharge areas are correlated 

(R2

=0.71) (fig. 29A).

Nitrogen concentrations predicted by a groundwaterflow model are a function of simulated transport times 

because concentrations change over time until a steady-state 

concentration is reached (fig. 16A). After an assumed transport 

time of 50 years, simulated concentrations exceed 1 and 

4 mg/L in about 57.6 and 28.4 percent of the basin (fig. 44A). 

After an advective transport time of 10 years, simulated 

concentrations exceed 1 mg/L as N in about 44.7 percent of 

the basin and 4 mg/L as N in about 5.6 percent of the basin 

(fig. 44B). Simulated concentrations also are a function of 

depth consistent with observations of the relations between 

nitrate concentrations, groundwater age, and well depth 

(Warner and Arnold, 2010). Simulated nitrate concentrations 

in water from the same source generally are lower for wells 

pumped from deep parts of the aquifer (layer 3) than those 

pumped from shallow parts (layer 1) (fig. 45); longer transport 

times for water withdrawn from greater depths in the aquifer 

result in lower concentrations than for water withdrawn from 

shallower depths after 50 years of transport. The large amount 

of scatter results from large differences among sources of 

water and associated nitrogen inputs to wells pumped at 

different depths (fig. 38).

Comparison of Predictions of Nitrate 

Occurrences and Concentrations

Logistic regression models use a set of explanatory 

variables that include combinations of spatial variables 

describing soil characteristics affecting nitrogen attenuation 

and sources of nitrogen in agricultural areas; the result 

is a prediction of the probability that concentrations at a 

location will exceed a specific threshold concentration. The 

use of a process model to simulate nitrogen transport, such 

as can be done by a groundwater-flow model coupled with 
Basin-Scale Comparisons of Statistical and Process Models 59 Nitrate concentration, in milligrams per liter as N 120 EXPLANATION No modeled solution Area not modeled 0 to 20 >20 to 30 >30 to 40 >40 to 50 >50 to 60 >60 to 70 >70 to 80 >80 to 90 >90 to 100 Percentiles of nitrate concentration Extent of model grid Extent of model grid Base from U.S. Geological Survey digital data, 1:24,000, 1983 Albers Equal Area projection Standard parallels 37°50' and 41°10', central meridian -84°

84°10' 84°20' 84° A B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 42. A, Nitrate concentrations (as N) and B, concentration percentiles after 50 years of steady-state transport from areas of nitrogen-fertilizer application to shallow pumped wells, Great Miami River Basin, southwestern Ohio.
60 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0 500 1,000 1,500 2,000

0

 2

 4

 6

 8

10

12

Nitrogen-application rate, in kilograms per year

Simulated nitrate concentration, in milligrams per liter as nitrogen

1,640-foot circular buffer

EXPLANATION

Recharge area

Figure 43. Relations between simulated nitrate concentrations and area-weighted mean nitrogen-application rates within 

circular buffers and contributing recharge areas, Great Miami River Basin, southwestern Ohio.

mass-weighted particle tracking, produces an estimate of 

a concentration in a pumped well that incorporates source 

terms and advective-transport processes; attenuation variables 

affecting nitrogen prior to recharge to the aquifer are not 

explicitly included in the process model but can be included 

implicitly as correction factors to the source term. The two 

differing methodologies and sets of included information 

complicate a direct comparison between the two.

One possible comparison is to use a process model to 

develop a prediction more analogous to statistical-model 

predictions. A process model can identify areas where 

the predicted concentration exceeds a given threshold 

concentration. These areas can be compared to areas where 

the statistical models predict that concentrations are likely to 

exceed that same threshold concentration (corresponding to 

predicted probabilities exceeding 0.5). However, uncertainties 

associated with nitrogen sources and attenuation processes 

explicitly represented in the process model would limit the 

utility of this kind of comparison.

The basin-scale patterns of model-prediction values can 

be used to show how differing sets of variables can affect the 

agreement between statistical and process models; this more 

limited comparison can show if the two methods qualitatively 

agree on where predictions of nitrogen concentration and 

occurrence are greater than a common measure. The patterns 

can be related by identifying areas where a given prediction 

is greater or less than the basin-wide median for that type 

of prediction; in this approach, attenuation processes do not 

affect results under the reasonable assumption that those 

processes, though unknown, are spatially consistent. The parts 

of the basin where simulated nitrate concentrations (from a 

process model) after 50 years of transport exceed the median 

value of 2 mg/L as N for the basin are in agricultural areas 

in the northeastern, southeastern, and southwestern parts of 

the basin (fig. 42B). Areas where probabilities of exceeding 

1 mg/L as N, estimated on the basis of contributing recharge 

areas, are greater than the median value of 0.427 are primarily 

near the center of the basin and overlap about 42 percent of 

the area where nitrate concentrations (predicted by a process 

model) exceed the median for the basin (fig. 46A); the two 

have the same total areas, because areas greater and less 

than median values are equal in size. The two areas did not 
Basin-Scale Comparisons of Statistical and Process Models 61 1 > 4 to 4 No modeled solution < 1 EXPLANATION Nitrate concentration, in milligrams per liter < 1 1 > 4 to 4 50 years of transport 10 years of transport Extent of model grid Base from U.S. Geological Survey digital data, 1:24,000, 1983 Albers Equal Area projection Standard parallels 37°50' and 41°10', central meridian -84° Extent of model grid

84°10' 84°20' 84° A B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 44. Areas where nitrate concentrations are predicted to exceed 1 and 4 milligrams per liter as nitrogen by a process model based on transport times of A, 50, and B, 10 years, Great Miami River Basin, southwestern Ohio.
62 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0 2 8 10

0

2

4

6

8

10

4 6

Nitrate concentration after 50 years of pumping in layer 1, in milligrams per liter as nitrogen

Nitrate concentration after 50 years of pumping in layer 3,

in milligrams per liter as nitrogen

Line of equality

Figure 45. Relationship between simulated nitrate concentrations for wells pumped in shallow and deep parts of the glacial 

aquifer, Great Miami River Basin, southwestern Ohio.

generally overlap each other because the explanatory variables 

for the logistic regression model used to predict probabilities 

of exceeding 1 mg/L as N did not include a nitrogen-source 

variable, whereas the process model uses only a nitrogensource term to estimate concentrations. 

Predictions of nitrate occurrence outside of agricultural 

areas could be more consistent with the process model if the 

latter incorporated nitrogen inputs from other sources, such 

as septic systems. A nitrogen-source variable is not included 

in models of threshold concentrations less than 4 mg/L as N 

because observed concentrations in network wells could not be 

correlated with defined sources to an extent that would warrant 

inclusion in the statistical models (Warner and Arnold, 2010). 

Areas where probabilities of nitrate concentrations exceeding 

4 mg/L as N are greater than the median probability for the 

basin are more in agreement with areas where concentrations 

simulated by a process model also exceed the median for 

the basin—the two overlap by about 62 percent (fig. 46B). 

The percentage of overlap is higher because both the logistic 

regression and process models incorporate a nitrogen-source 

variable. Areas where the agreement improves by inclusion of 

the nitrogen-application rate as an explanatory variable are the 

southwestern and northeastern parts of the basin (fig. 46B).

Both methods used in this study are associated with 

uncertainties. Statistical models have uncertainties associated 

with the degree of fit and an inability to represent potential 

variables and processes that may affect predictions. In this 

analysis, the process models had uncertainties associated with 

model parameters as well as with sensitivity to factors for 

which there is little information, such as attenuation processes 

prior to recharge. As an example, the lack of agreement 

between statistical and process models in estimating the extent 

of concentrations exceeding 4 mg/L as N in the basin could be 

the result of either the inaccuracy of the statistical models or 

an incorrect estimate of the amount of attenuation taking place 

prior to recharge.

Limitations of Analysis

An assumption in this steady-state modeling analysis is 

that there is a mass balance between recharge and pumping 

rates so that the volume of water entering the aquifer through 
Basin-Scale Comparisons of Statistical and Process Models 63 Area excluded EXPLANATION Area in which prediction exceeds basinwide median No modeled solution Probability of exceeding 1 milligram per liter, from recharge area Nitrate concentration Probability of exceeding 4 milligrams per liter, from recharge area Finite-difference grid Extent of model grid Extent of model grid Base from U.S. Geological Survey digital data, 1:24,000, 1983 Albers Equal Area projection Standard parallels 37°50' and 41°10', central meridian -84°

84°10' 84°20' 84° A B 84°10' 84°20' 84°

39°50'39°40'

0 3 6 MILES

0 6 KILOMETERS

Figure 46. Areas where the probability of exceeding A, 1 milligram per liter as nitrogen, and B, 4 milligrams per liter as nitrogen exceeds the basinwide median probability from statistical models and areas where simulated nitrate concentrations from a process model exceed the median for the basin, Great Miami River Basin, southwestern Ohio.
64 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

the contributing recharge area is the same as the volume 

being withdrawn from the pumped well. Factors that affect 

the mass balance between recharge and pumping could affect 

comparisons of estimates of area-weighted mean variables 

and resultant predictions of nitrate-exceedance probabilities. 

Two factors that can affect this balance are hydraulic gradient 

effects and boundary effects.

Gradient effects are largest in areas of steep hydraulic 

gradients and are caused by the use of a finite-difference 

model in the analysis; if model cells are sufficiently large, 

so that flow through the cell containing the simulated well 

is larger than the pumping rate, the well is a weak sink and 

the contributing recharge area—which is the contributing 

recharge area to the model cell—will be larger than the actual 

contributing recharge area to the well. Although areas of 

steep gradients are areas where possible weak sinks need to 

be considered, these areas also are where a process model 

most likely can improve estimates of spatial variables over 

estimates based on a circular buffer. As an example, the 

contributing recharge area to well 1860 is 2.17 times the size 

of the circular buffer around the well—under equilibrium 

conditions, the buffer and the contributing recharge area 

should be the same size because the well is pumped at a rate 

equivalent to recharge through the buffer (fig. 47). This well is 

upgradient from a hydrologic boundary representing a major 

stream in an area of steep hydraulic gradients. 

A boundary effect refers to the influence of a nearby 

hydrologic boundary on a contributing recharge area and 

may represent a real hydrologic condition. The result of a 

boundary effect is that the model cell containing the pumped 

well receives water from other sources in addition to recharge, 

usually represented in a model as a specified-head or headdependent flux boundary; the result is a contributing recharge 

area smaller than the actual contributing recharge area. As an 

example, the contributing recharge area to well 1629, which 

is near a specified-head boundary along the edge of the active 

model domain, is about 30 percent the size of the circular 

buffer (fig. 47), indicating that most of the flow through the 

cell is from the specified-head boundary.

Gradient and Boundary Effects on the Mass 

Balance of Recharge and Pumping

The degree to which gradient and boundary effects 

could affect comparisons of estimated spatial variables and 

resultant predictions is site specific and would be expected to 

vary spatially. The ratio of the volumetric rate of pumping to 

the volumetric rate of flow through the contributing recharge 

area can be used to quantify the degree to which gradient and 

boundary effects could affect the balance between pumping 

and recharge. The mean and median of the ratio of pumping 

to recharge for shallow wells in the basin were 0.98 and 

0.96, respectively (fig. 48A); pumping and recharge were 

within 10 percent for pumped wells in about 50 percent of 

the basin and within 20 percent in about 65 percent of the 

basin, indicating that, generally, there was a reasonable mass 

balance between recharge and pumping across most of the 

modeled basin. Groundwater fluxes in porous, unconsolidated 

aquifers, such as glacial aquifers, increase exponentially with 

depth, and therefore the relative importance of gradient and 

boundary effects are a function of well depth. At locations 

in which wells were pumped in shallow (layer 1) and deep 

(layer 3) parts of the aquifer, the ratios of pumping to recharge 

were statistically higher in deep wells (p less than 0.001) with 

means of 0.97 (shallow) and 1.04 (deep), respectively. The 

comparison indicates that deeper wells generally were more 

affected by hydrologic boundaries (indicated by a ratio of 

pumping to estimated recharge greater than 1), but the relative 

importance of gradient and boundary effects was variable 

(fig. 48B).

Large gradient effects, defined as a condition where 

pumping is less than 50 percent of recharge, occur in pumped 

wells in about 5 percent of the basin, and large boundary 

effects, defined as a condition where recharge is less than 

50 percent of pumping, occur in about 14 percent of the 

basin (fig. 49). Wells with large gradient effects generally are 

near discharge boundaries representing major surface-water 

features. Wells with large boundary effects generally are near 

specified-head boundaries along the edge of the active model 

domain or downgradient from head-dependent flux boundaries 

where simulated surface-water features are contributing water 

to the simulated well (fig. 49).

Gradient and Boundary Effects on 

Estimated Spatial Variables and 

Statistical-Model Predictions

Steep hydraulic gradients and hydrologic boundaries 

can affect the mass balance between pumping and recharge 

and, therefore, estimates of spatial variables and resultant 

predictions of the probability of exceeding threshold 

concentrations. The amount of overlap between contributing 

recharge areas and circular buffers was statistically higher 

(p less than 0.001) for the population of wells with mass 

balances within 10 percent than for the population of all 

wells in the basin; gradient and boundary effects likely would 

cause less agreement between contributing recharge areas and 

buffers. The mean and median values of the amount of overlap 

for the two populations were similar: median overlaps of 0.32 

and 0.28 and mean overlaps of 0.34 and 0.33 for all wells 

and wells with mass balances within 10 percent, respectively 

(fig. 50).

The nitrogen-application rates within contributing 

recharge areas of the population of wells with mass balances 

within 10 percent and the population of all wells in the basin 

were not statistically different; the relation between areaweighted mean nitrogen-application rates within contributing 

recharge areas and circular buffers shows a similar degree 

of variability for both populations (fig. 51A). There was 

no statistical difference in silt content—estimated from 
Limitations of Analysis 65

84°21' 84°20' 84°19'

39°42'

39°41'

780

760

740

Well 1629

Well 1860

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

Pumped well and number

860

Specified head

Head-dependent flux boundary

Near-boundary area excluded

from analysis

Well 1860

Recharge area

EXPLANATION

Well 1629

Boundary conditions

Circular buffer with radius of 1,640 feet

Water table—number is altitude, 

 in feet (NGVD 88)

Well 1860

0 0.5 1 MILES

0 0.5 1 KILOMETERS

Figure 47. Contributing recharge areas that are examples of a gradient effect (well 1860) and a boundary effect (well 1629) 

in the northeastern part of the Great Miami River Basin, southwestern Ohio.
66 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Boundary effect: Layer 1

Gradient effect: Layer 3

Gradient effect: Layer 1

Gradient effect: Layer 3

Boundary effect: Layer 1

Boundary effect: Layer 3

Gradient effect: Layer 1

Boundary effect: Layer 3

+/- 30 percent

+/- 20 percent

+/- 10 percent

Ratio of pumping to recharge rates estimated from recharge areas

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0

500

1,000

1,500

2,000

2,500

Gradient effect Boundary effect

Number of wells

A

B

Median=0.96

Mean=0.98

N=20,920

Ratio of pumping to recharge rates estimated from recharge areas in layer 1

Ratio of pumping to recharge rates estimated from recharge areas in layer 3

Figure 48. A, Simulated ratio of pumping rates in shallow pumped wells to recharge rates estimated 

from contributing recharge areas and B, relative importance of gradient and boundary effects for shallow 

(layer 1) and deep (layer 3) pumped wells.
Limitations of Analysis 67

Base from U.S. Geological Survey digital data, 1:24,000, 1983

Albers Equal Area projection

Standard parallels 37°50' and 41°10', central meridian -84°

No modeled solution

Head-dependent flux boundary

Specified head

Boundary effects

EXPLANATION

Recharge rates less than 50 percent of pumping rates

Gradient effects

Pumping rates less than 50 percent of recharge rates

Boundary conditions

Area excluded from analysis

Pumped well and number

Well 1860

Extent of model grid

Well 1629

Well 1860

84°20' 84°10' 84°

39°50'

39°40'

0 3 6 MILES

0 3 6 KILOMETERS

Figure 49. Distribution of large gradient and boundary effects, defined as differences of more than 50 percent between 

recharge and pumping rates, Great Miami River Basin, southwestern Ohio.
68 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

0

200

400

600

800

EXPLANATION

Overlap between areas of recharge and circular buffers, in percent

All wells

Mass balance between pumping and recharge rates within 10 percent

Overlap between areas of circular buffers and recharge, in percent

Number of wells

0 10 20 30 40 50 60 70 80 90 100

Mean = 0.33

Mean = 0.34

Median = 0.32

Median = 0.28

Figure 50. Overlap between areas of circular buffers and contributing recharge areas for all wells in the basin and those 

wells with a mass balance between recharge and pumping rates within 10 percent, Great Miami River Basin, southwestern 

Ohio.
Limitations of Analysis 69

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

4

EXPLANATION

1

Concentrations in all wells,

in milligrams per liter

1

4

Concentrations in wells with

 mass balance between recharge

 and pumping within 10 percent,

 in milligrams per liter

0 500 1,000 1,500 2,000

0

500

1,000

1,500

2,000

Probability of exceeding threshold concentration from 1,640-foot circular buffer

Probability of exceeding threshold concentration from recharge area

Nitrogen application rate from 1,640-foot circular buffer, in kilograms per year

Nitrogen application rate from recharge area, in kilograms per year

EXPLANATION

All wells

Mass balance between pumping 

and recharge within 10 percent

Line of equality

A

B

Figure 51. Differences in A, estimates of area-weighted spatial variables, and B, probabilities of exceeding 1 and 4 milligrams per 

liter as nitrogen determined for contributing recharge areas and circular buffers for all hypothetical wells in the basin and a subset 

of wells with a mass balance between pumping and recharge rates within 10 percent, Great Miami River Basin, southwestern Ohio.
70 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

STATSGO data—between the two populations. Alfisol 

content and depth to water were statistically lower (p less 

than 0.001) in wells with mass balances between recharge and 

pumping within 10 percent than in the population of all wells. 

Predictions of the probabilities of exceeding 1 and 4 mg/L as 

N show similar variability between the population of wells 

with mass balances between recharge and pumping within 

10 percent and the population of all wells (fig. 51B); however, 

predicted probabilities for wells with a mass balance within 

10 percent were statistically higher (p less than 0.001) than 

for the population of all wells; these differences likely reflect 

differences in estimates of the spatial data.

The results show that gradient and boundary effects can 

affect comparisons of statistical and process models arising 

from different representations of the source of water to a 

pumped well; however, the differences are small, and general 

trends and conclusions are not affected. In the regional model 

of the Great Miami River Basin, gradient and boundary effects 

(greater than 10 percent) occur in about half the model cells, 

but effects on estimates of spatial variables and predictions 

of nitrate occurrence at the basin scale are small. Gradient 

effects can be minimized by using models with sufficiently 

small discretization to allow simulated pumping rates to be 

essentially the only sink within the cell. Boundary effects 

can be minimized by using, to the extent possible, boundary 

conditions that do not contribute water to the aquifer; the 

utility of this approach depends on the system being simulated. 

Boundary effects may represent real processes, and if there are 

additional sources of water, the use of statistical models that 

are based on circular buffers or contributing recharge areas 

may not be appropriate.

Summary

Unconsolidated glacial sediments consisting of gravel, 

sand, silt, and clay in the northern conterminous United States 

compose an important regional aquifer system that, in 2000, 

supplied water to about 41 million people. About 40 percent 

of the water was pumped from private wells, primarily in rural 

areas where agriculture is the predominant land use. Wells in 

these areas are vulnerable to contamination from nitrogen, 

primarily in the form of nitrate, from fertilizer application on 

agricultural land. The USGS, through the NAWQA Program, 

has characterized the distribution of nitrogen in the glacial 

aquifer system and used logistic regression based on observed 

relations between nitrogen concentrations and sets of wellconstruction and spatial characteristics to develop statistical 

models of nitrate occurrence. A set of 10 models describes the 

probability of exceeding concentrations of 1 to 10 milligrams 

per liter as nitrogen (mg/L as N) by using a set of explanatory 

variables—well-construction variables (well depth, diameter, 

and screen length) and spatial variables. The latter set consists 

of soil characteristics (silt content and alfisol content) and 

depth to water from the U.S. Department of Agriculture 

State Soil Geographic (STATSGO) database and nitrogenapplication rates from county data mapped to NLCDe92 

agricultural land use. The four explanatory variables used in 

the statistical models represent area-weighted means computed 

for a 1,640-foot-radius circular buffer around the well.

Statistical models may implicitly represent transport 

and attenuation process through inclusion of environmental 

characteristics but do not explicitly account for complexities 

in natural systems. By contrast, process models can explicitly 

represent physical and chemical processes by using numerical 

methods and have the capacity to represent, to an extent, those 

complexities. However, the use of process models to represent 

nitrogen transport requires detailed knowledge of nitrogen 

sources, attenuation processes from land surface to the point 

of interest in the aquifer, and the parameters that control 

transport. Process models can inform and improve statistical 

models primarily by providing for physically based source 

areas to pumped wells to account for advective-transport 

processes; enhanced information on nitrogen sources and 

nitrogen-attenuation processes that can affect source terms 

would allow further improvement of statistical models of 

nitrogen occurrence. To evaluate the potential use of process 

models to inform statistical models, the two methods were 

compared at existing and hypothetical wells at a local scale in 

different hydrologic settings in the Great Miami River Basin, 

southwestern Ohio, and at a basin scale by using a network 

of hypothetical wells uniformly spaced at 500-foot intervals 

across the basin.

Contributing recharge areas, which are a function of 

local hydrologic setting, were used to improve estimates of 

explanatory variables over estimates based on the simplified 

circular buffers used in statistical models. Differences in 

estimated explanatory variables were used to evaluate 

how incorporating an enhanced understanding of physical 

transport could improve statistical models. The larger the 

difference, the more likely that statistical models would be 

improved if contributing recharge areas were used to estimate 

explanatory variables. Comparing model predictions from 

the two sets of variables would provide an evaluation of the 

degree to which incorporation of physically based estimates of 

explanatory variables would affect the statistical models. The 

set of explanatory variables based on contributing recharge 

areas, however, was not used to develop the models; a direct 

comparison would require the use of predictions from separate 

sets of statistical models developed from the two sets of 

explanatory variables.

Steady-state contributing recharge areas, which represent 

a mass balance between recharge and pumping rates, are more 

accurate representations than circular buffers of the areas at 

the water table that contribute recharge to a pumped well. If a 

well is pumped at a rate equivalent to the volume of recharge 

through the circular buffer, the contributing recharge area to 

the well—referred to as a volume-equivalent contributing 

recharge area—is the same size as the circular buffer (about 

0.3 square miles for a radius of 1,640 feet). A number of 

factors affect the size and shapes of contributing recharge 
Summary 71

areas and the extent to which they differ from circular 

buffers, including pumping rate, well depth, hydrologic 

setting, and length of pumping. Source areas represented 

as circular buffers differ substantially from source areas 

represented as contributing recharge areas of equivalent size, 

and therefore spatial variables estimated for the assumed 

source areas and the resultant predictions of nitrate would 

also differ. The differences generally are larger in areas of 

steep hydraulic gradients where differences in circular buffers 

and contributing recharge areas are larger. Incorporating an 

improved understanding of groundwater processes through 

process models can inform statistical models of nitrate 

occurrence in glacial environments.

• Calculations of steady-state contributing recharge 

areas were based on the assumptions that hydraulic 

stresses are constant over time and that there is a mass 

balance between pumping and recharge rates; these 

assumptions are likely valid only after the well has 

been in operation longer than the maximum steadystate traveltime within the contributing recharge area, 

in some cases longer than 100 years. Most wells 

typically are in operation for much less time than is 

necessary to establish a true steady-state transport 

condition. Comparisons of steady-state contributing 

recharge areas with transient contributing recharge 

areas to a well after different durations of pumping 

showed that hydrologic conditions simulated for 

the two types of contributing recharge areas were 

in reasonable agreement after about 10 to 20 years 

of simulated pumping, and that the steady-state 

assumption underlying recharge delineation was valid 

in most wells.

• Near groundwater divides, where hydraulic gradients 

are small, volume-equivalent steady-state contributing 

recharge areas were similar to circular buffers, whereas 

contributing recharge areas extended upgradient from 

wells in areas with steep hydraulic gradients and 

differed greatly in shape from circular buffers. At three 

hypothetical wells in the same area but in different 

hydrologic (gradient) settings in the southeastern 

part of the Great Miami River Basin, the amount 

of overlap between contributing recharge areas and 

circular buffers ranged from about 87 percent near 

a groundwater divide to about 33 percent where 

hydraulic gradients were steeper. 

• Contributing recharge areas to wells pumped in 

shallow parts of an aquifer were more similar to the 

areas of circular buffers than the contributing recharge 

areas to wells pumped in deeper parts of the aquifer; 

the importance of well depth on the agreement between 

contributing recharge areas and circular buffers is a 

function of pumping rate. In an area of steep hydraulic 

gradients, contributing recharge areas to shallow and 

deep wells pumped at a high rate (equal to the mean 

pumping rate of 0.25 million gallons per day for 

public-supply wells in the basin), were similar; the 

amount of overlap differed by about 2 percent. For a 

low pumping rate of 350 gallons per day (generally 

consistent with a domestic well), there was a large 

difference in contributing recharge areas to shallow 

and deep wells. The contributing recharge area to a 

shallow well pumped near the water table in an area 

of steep hydraulic gradients was completely within 

the circular buffer, whereas the contributing recharge 

area to a deeper well was more than 2,000 feet outside 

of the circular buffer. This difference reflects the 

strong effect of the natural hydraulic gradients on the 

contributing recharge areas at low pumping rates.

• The mean depth of domestic wells in the NAWQA 

sampling network is 91 feet, similar to the deep well 

simulated in the model. This well depth coupled 

with the low pumping rates typical of domestic wells 

suggests that contributing recharge areas to these 

wells could be outside of the circular buffer. However, 

the contributing recharge areas to wells pumped at 

different depths at a low pumping rate are within 

the corresponding volume-equivalent contributing 

recharge areas. These results suggest that the use of 

volume-equivalent contributing recharge areas not only 

better represents groundwater-flow processes, but also 

more likely includes the actual source areas to wells 

pumped at low rates regardless of depth.

Differences in contributing recharge areas result in 

differences in area-weighted means for the four explanatory 

spatial variables—STATSGO data (silt content, alfisol content, 

and depth to water) and nitrogen-application rate. These 

differences, in turn, affect predicted probabilities of nitrate 

occurrence. Estimates of area-weighted mean spatial variables 

depend on the same factors that affect the contributing 

recharge areas to a well—hydrologic setting, well depth, and 

pumping rate.

• Differences in estimates of area-weighted means for 

spatial variables varied among three hypothetical wells 

in different hydrologic settings in the southeastern part 

of the basin; area-weighted means based on STATSGO 

data were the same for a well near a groundwater 

divide, where the contributing recharge area is similar 

to the circular buffer, but differed substantially in 

an area with steep hydraulic gradients. The largest 

variability was for the nitrogen-application rate, which 

was determined from the Enhanced National Land 

Cover Data 1992 (NLCDE92) data.

• For wells pumped at low rates in an area of steep 

hydraulic gradients, well depth had the largest effect 

on estimated spatial variables, reflecting the large 

differences in contributing recharge areas. As an 

example, area-weighted mean nitrogen-application 

rates in contributing recharge areas to shallow and 
72 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

deep wells pumped at a high rate were 731 and 

743 kilograms per year (kg/yr), respectively, but 

1,111 and 0 kg/yr, respectively, for wells pumped at a 

low rate.

• The differences in estimated spatial variables resulted 

in differences in the predicted probabilities of nitrate 

occurrence, which varied less than the estimated 

variables because variability in the spatial variable 

only partly explained variability in the predictions with 

the remainder being a function of well-constriction 

variables. Differences also were a function of the 

set of explanatory variables included in a given 

statistical model. As an example, predictions of 

nitrate exceeding 1, 2, and 3 mg/L as N were the same 

for three hypothetical wells in different hydrologic 

settings because the models included only explanatory 

variables from STATSGO data, whereas predictions 

of exceeding 4 mg/L as N differed because nitrogenapplication rate, which varied the most, was included 

as an explanatory variable in the model.

• Estimates of area-weighted means for circular buffers 

and contributing recharge areas differed substantially 

between public-supply and NAWQA network wells 

in the Great Miami River Basin. Spatial variables for 

the buffers and the contributing areas were weakly 

correlated with large differences between the two sets 

of estimates. Predictions of nitrate exceeding 1 and 

4 mg/L as N were more correlated, but there were large 

differences among individual wells.

The overlap between contributing recharge areas and 

circular buffers is highly site specific and varies greatly across 

the basin with hydrologic setting. Differences in estimates of 

area-weighted means are a function of not only hydrologic 

setting but also spatial variability in the data used to estimate 

the area-weighted means. These differences also result in 

basin-scale differences in the predicted probabilities of 

nitrate exceeding threshold concentrations. The differences in 

estimated explanatory variables and resultant statistical-model 

predictions between contributing recharge areas and circular 

buffers indicate that the incorporation of physically based 

contributing recharge areas likely would result in a different 

set of explanatory variables and an improved set of statistical 

models. The greatest improvement in the statistical models 

likely would be in areas with steep hydraulic gradients.

• In the Great Miami River Basin, the amount of overlap 

between circular buffers and volume-equivalent 

contributing recharge areas ranged from less than 

5 to more than 95 percent and generally varied 

with location relative to groundwater divides and 

discharge locations. The mean and median amounts 

of overlap between contributing recharge areas and 

circular buffers for the basin were 34 and 32 percent, 

respectively, indicating that there are substantial 

differences in the two representations of source areas 

to wells.

• The general lack of agreement between estimates of 

area-weighted mean explanatory variables for circular 

buffers and volume-equivalent contributing recharge 

areas resulted in large differences in these estimates 

across the basin. Differences in estimated variables 

derived from STATSGO data (silt content, alfisol 

content, and depth to water) were correlated with the 

spatial variability of that dataset; the largest differences 

were in the central part of the basin. The largest 

variability was for estimates of nitrogen-application 

rates. The differences ranged from 1,880 kg/yr (a 

higher mean within circular buffers) to -1,560 kg/yr 

and generally were highest in the northeastern and 

southeastern parts of the basin—generally correlated 

with the NLCDe92 data from which the estimates 

were derived. 

• The predicted probabilities of nitrate exceeding 1 and 

4 mg/L as N on the basis of spatial variables estimated 

for circular buffers and contributing recharge areas 

differed in about 65 and 90 percent of the basin; 

spatial patterns in predictions were related to spatial 

differences in estimates of spatial variables and 

were significant in some areas (exceeding 0.45 and 

0.25, respectively).

• Soil characteristics (silt content and alfisol content) and 

nitrogen-application rate were external to the models, 

and information about these variables was limited to 

the underlying spatial data. However, depth to water 

is a function of local groundwater conditions, and a 

groundwater-flow model was considered able to inform 

statistical models by providing for more physically 

based estimates of depth to water. Depth to water from 

the STATSGO database ranged from 2.48 to 4.88 feet 

in the Great Miami River Basin, whereas depth to 

water calculated by the model ranged from about 

0 to 78 feet. Based on area-weighted means for circular 

buffers and only model-calculated depths to water 

inside the circular buffer that were within the range 

of the STATSGO data, predictions of concentrations 

exceeding 1 and 4 mg/L as N differed greatly. This 

result suggests that use of a physically based depth 

to water could greatly change the development of 

statistical models.

• Contributing recharge areas to wells pumped in deeper 

parts of the aquifer overlapped less with circular 

buffers than did contributing recharge areas to shallow 
Summary 73

wells. The mean and median amounts of overlap for 

deep wells were 0.24 and 0.27. Contributing recharge 

areas to most (about 65 percent) deep wells did not 

overlap any part of the corresponding circular buffers 

and, depending on hydrologic setting, were far from 

the buffers. The greater difference between circular 

buffers and contributing recharge areas to deep wells 

resulted in larger differences in estimates of spatial 

variables and resulting model predictions.

Groundwater-flow models coupled with mass-weighted 

particle tracking can simulate nitrate concentrations in 

groundwater transported by advection from surface sources. 

With an assumed a crop uptake of 50 percent and additional 

losses of 30 percent(consistent with published values) and 

the nitrogen-application rates used in the statistical models, 

the median simulated nitrate concentration in the Great 

Miami River Basin was similar to that measured in the 

nationwide glacial aquifer system—about 2 mg/L as N. Direct 

comparisons between results from statistical and process 

models cannot be made because statistical models predict 

the probability of exceeding a given concentration and not 

concentration values. The two methods generally are in poor 

agreement, but the respective uncertainties and limitations 

of the two methods would make the comparison difficult 

to analyze.

• The parts of the basin where probabilities of exceeding 

1 mg/L as N exceeded the basinwide median prediction 

(about 0.43) overlapped about 42 percent of the parts 

of the basin where simulated nitrate concentrations 

exceeded the median nitrate concentration. For a 

concentration of 4 mg/L as N, the amount of overlap 

between areas where both statistical-model predictions 

and process-model predictions exceeded the basinwide 

median prediction, and simulated nitrate concentrations 

exceeded the median simulated nitrate concentration, 

was about 62 percent. 

• The improved agreement resulted from the statistical 

model describing nitrate occurrence above 1 mg/L as 

N not including nitrogen-application rate as a variable, 

which was the only control on the distribution of 

simulated concentrations from the process model. The 

statistical model describing nitrate occurrence above 

4 mg/L as N did include the nitrogen-source variable, 

resulting in more agreement in agricultural areas. 

A model-calculated contributing recharge area better 

represents the source area to a well and is more likely to 

include the actual source area to a well with a low pumping 

rate; however, the use of a process model has limitations. An 

implicit assumption in using a contributing recharge area to 

define spatial variables that can affect water quality in a well is 

that pumping and recharge rates are in a state of mass balance. 

Two possible factors that can affect the utility of a finitedifference model are gradient effects and boundary effects, 

both of which can affect simulated contributing recharge 

areas and, in turn, estimates of spatial variables and resultant 

model predictions.

• Gradient effects refer to a condition where pumping 

within a model cell is less than total flow through the 

cell and generally occur in areas with large hydraulic 

gradients. If the pumping rate is equivalent to the 

volume of recharge through the circular buffer, the 

result is a contributing recharge area larger than the 

circular buffer. Gradient effects are a function of model 

discretization and, in the model of the Great Miami 

River Basin, pumping was less than 50 percent of 

flow through the contributing recharge area in about 

5 percent of the basin; these areas were generally 

upgradient from discharge locations. Gradient effects 

can be minimized by using a finer model discretization.

• Boundary effects refer to a condition where there are 

other sources of water to a cell containing a pumped 

well, resulting in a contributing recharge area that 

is smaller than it would be if pumping and recharge 

were in mass balance. Boundary effects occur near 

simulated hydrologic features and may represent 

real hydrologic processes, in which case the chosen 

model representation of the source area either as a 

circular buffer or contributing recharge area may not 

be appropriate. In about 14 percent of the area of the 

Great Miami River Basin, recharge was less than 

50 percent of the total rate of inflow to wells; these 

boundary effects were in areas generally near the 

margins of the simulated aquifer and near simulated 

surface waters. Wells pumped in deep parts of the 

aquifer generally were more likely to be associated 

with boundary effects.

• Mass balances between pumping and recharge in 

the Great Miami River Basin are within 20 percent 

in about 65 percent and within 10 percent in about 

half of the model cells. The amount of overlap for 

the population of wells with mass balances within 

10 percent were similar to the rates for the population 

of all wells. Estimates of spatial variables and 

predictions of nitrate occurrence exceeding 1 and 

4 mg/L as N for the population of wells with a mass 

balance within 10 percent and the population of all 

wells generally were similar, indicating that boundary 

and gradient effects occur within the modeled area 

but generally do not affect analytical results. The 

importance of gradient and boundary effects is model 

specific and needs to be evaluated to see if changes in 

the model are warranted.
74 Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Ohio

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areas of public-supply wells in simple and complex flow 

systems, Cape Cod, Massachusetts: U.S. Geological Survey 

Water Supply Paper 2434, 66 p.

Bundy, L.G., and Andraski, T.W., 2005, Recovery of fertilizer 

nitrogen in crop residues and cover crops on an irrigated 

sandy soil: Soil Science Society of America Journal, v. 69, 

p. 640–648.

DeSimone, L.A., Hamilton, P.A., and Gilliom, R.J., 2009, 

Quality of water from domestic wells in principal aquifers 

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Walter and Starn—Use of Process Models to Inform and Improve Statistical Models of Nitrate Occurrence, Great Miami River Basin, Southwestern Ohio—SIR 2012–5001

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