Prepared in cooperation with the 

Michigan Department of Natural Resources

Ground-Water-Withdrawal Component 

of the Michigan Water-Withdrawal Screening Tool

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2009–5003 

Ground-Water-Withdrawal Component 

of the Michigan Water-Withdrawal 

Screening Tool 

By Howard W. Reeves, David A. Hamilton, Paul W. Seelbach, and 

A. Jeremiah Asher 

Prepared in cooperation with the 

Michigan Department of Natural Resources

Scientific Investigations Report 2009–5003 

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2009

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Reeves, H.W., Hamilton, D.A., Seelbach, P.W., and Asher, A.J., 2009, Ground-water-withdrawal component of the 

Michigan water-withdrawal screening tool: U.S. Geological Survey Scientific Investigations Report 2009–5003, 36 p. 
iii

Contents

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

Introduction.....................................................................................................................................................1

Assessment Process and Screening Tool ........................................................................................1

Streamflow Depletion by Pumping Wells .........................................................................................2

Purpose and Scope ..............................................................................................................................2

Study Approach.....................................................................................................................................2

Ground-Water-Withdrawal Component of Screening Tool ....................................................................3

Input of Constant and Time-Varying Pumping Rates ......................................................................4

Aquifer Properties Required for the Water-Withdrawal Screening Tool ....................................4

Definition of Valley Segments....................................................................................................5

Assignment of Bedrock Aquifer Properties to Valley Segments .........................................7

Definition of Great Lake Shoreline Catchments .....................................................................8

Distribution of Streamflow Depletion Among Valley Segments in 

Neighboring Catchments .......................................................................................................8

Demonstration of Analytical Model and Distribution Approach............................................................9

Limitations of Testing..........................................................................................................................12

Summary of the Ground-Water Component of the Water-Withdrawal Screening Tool ..................12

Example Screening and Confirmation of Water-Withdrawal Screening-Tool Results ....................13

References Cited..........................................................................................................................................15

Appendix 1. Background and Literature Review on Streamflow Depletion Modeling.................17

Appendix 2. Distribution of Withdrawal Between Neighboring Watersheds ................................27 

Figures

1. Graph showing reported aquifer storage coefficients for wells completed in 

glacial deposits from aquifer tests reviewed by the State of Michigan as a 

function of depth; arithmetic, geometric, and harmonic means also are shown..............6

2. Map showing four bedrock categories used in the water-withdrawal screening 

tool for Michigan...........................................................................................................................7

3. Graph showing recharge and pumping rates used in the base simulations with 

the Kalamazoo regional ground-water-flow model ................................................................9

4–6. Maps showing:

4. A grid of points used to compare the analytical model and different 

distribution methods to MODFLOW results from the Kalamazoo regional 

ground-water-flow model.................................................................................................10

5. Estimated streamflow depletion from valley segment 3544 as a percentage 

of pumping rate after 5 years of pumping. A, results from a sequential 

introduction of test wells in the MODFLOW ground-water-flow model for 

Kalamazoo County, Mich. B, results from analytical model with inverse 

distance to stream distribution of streamflow depletion............................................11

6. Example test point, valley-segment catchment containing the point, and 

neighboring valley-segment catchments ......................................................................13
iv

1–1 – 2–1. Diagrams showing:

1–1. Alternate conceptual models for streamflow depletion by a pumping 

well. A, Fully penetrating stream with no streambed resistance. 

B, Fully penetrating stream with streambed resistance. C, Partially 

penetrating stream with streambed resistance. D, Partially 

penetrating stream in semiconfining layer with pumping from 

underlying semiconfined aquifer.............................................................................17

1–2. Alternate sources of water for capture by a well near a stream. 

A, Stream in an aquifer that overlies a regional aquifer with potentially 

different hydraulic properties. B, Stream in an aquifer system where 

pumping may interact with plants and change evapotranspiration 

losses from the system..............................................................................................21

2–1. Pumping well between two parallel streams with notation used in an 

equation for steady-state streamflow depletion...................................................27

2–2 – 2–5. Maps showing:

2–2. A grid of points used to compare the analytical model and different 

distribution methods to MODFLOW results from the Kalamazoo 

regional ground-water-flow model..........................................................................31

2–3. Estimated streamflow depletion from valley segment 3544 resulting 

from a sequential introduction of test wells in the MODFLOW groundwater-flow model for Kalamazoo County, Mich., as a percentage of 

pumping rate after 5 years of pumping...................................................................32

2–4. Estimated streamflow depletion from valley segment 3544 resulting 

from a sequential application of the analytical model to test wells with 

no weighting approach between adjacent valley segments as a 

percentage of pumping rate after 5 years of pumping.........................................33

2–5. Estimated streamflow depletion from valley segment 3544 as a 

percentage of pumping rate after 5 years of pumping. A, results from 

a sequential application MODFLOW. Results from sequential application 

of the analytical model with B, the inverse distance to the stream, 

C, inverse distance to the stream squared, and D, natural neighbor 

distribution methods ..................................................................................................35 

Tables

1. Bedrock categories and response by water-withdrawal screening tool to 

proposed bedrock well in each category.................................................................................8

2. Compilation of estimates for an example test point using the Michigan waterwithdrawal screening tool and the set of batch programs used in screening-tool 

development and testing ...........................................................................................................14 
v

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

meter (m) 3.281 foot (ft)

mile (mi) 1.609 kilometer (km)

kilometer (km) 0.6214 mile (mi)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Flow rate

gallon per minute (gal/min) 3.785 liter per minute (L/min)

gallon per day (gal/d) 4.3 x 10-8 cubic meter per second (m3

/s)

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

/s)

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

Conversion Factors and Datum
This page is intentionally blank.
Abstract 

A water-withdrawal assessment process and Internetbased screening tool have been developed to evaluate proposed new or increased high-capacity water withdrawals 

in Michigan. Michigan legislation defines high capacity 

withdrawals as those capable of removing an average of 

100,000 gallons per day for a consecutive 30-day period. This 

report describes the ground-water component of the screening tool, provides background information used to develop 

the screening tool, and documents how this component of the 

screening tool is implemented. The screening tool is based 

on application of an analytical model to estimate streamflow 

depletion by a proposed pumping well. The screening tool 

is designed to evaluate intermittent pumping, to account for 

the dynamics of stream-aquifer interaction, and to apportion streamflow depletion among neighboring streams. The 

tool is to be used for an initial screening of a proposed new 

or increased high-capacity withdrawal in order to identify 

withdrawals that may cause adverse resource impacts. The 

screening tool is not intended to be a site-specific design tool. 

Results of an example application of the screening tool in Kalamazoo County, Mich., are compared to streamflow depletion 

estimated by use of a regional ground-water-flow model to 

demonstrate its performance.

Introduction

The water-withdrawal assessment process mandated 

by State of Michigan Public Act 34 of 2006 (2006 PA 34) 

(Michigan State Legislature, 2006a) was proposed to “assist 

in determining whether the proposed [water] withdrawal may 

cause an adverse impact to the waters of the state or to the 

water-dependent natural resources of the state.” The waterwithdrawal assessment process was developed under the 

auspices of the Michigan Ground Water Conservation Advisory Council (GWCAC), which was formed though Public 

Act 148 of 2003 (2003 PA 148) (Michigan State Legislature, 

2003) and revised by 2006 PA 34. In response to 2006 PA 34, 

the U.S. Geological Survey (USGS) entered into a cooperative 

agreement in 2006 with the Michigan Department of Natural 

Resources; the Michigan Department of Environmental Quality; the Institute for Fisheries Research, University of Michigan; and the Institute for Water Research, Michigan State 

University, to assist in developing the technical aspects of the 

water-withdrawal assessment process.

The 2006 PA 34 legislation presumed that, for the first 

2 years of the act, wells deeper than 150 ft or further away 

than 1⁄4-mi from trout streams did not create an adverse 

resource impact on a trout stream (Michigan State Legislature, 

2006a). Wells within 1⁄4-mi of a trout stream and less than 

150 ft deep may have posed an adverse resource impact on the 

stream, but the legislation did not specify how this potential 

should be analyzed. This presumption expired in 2008, and the 

legislature replaced it with the water-withdrawal assessment 

process including an Internet-based screening tool (Public 

Act 185 of 2008 (2008 PA 184), Michigan State Legislature, 

2008a). The water-withdrawal process identifies withdrawals 

likely to cause an adverse environmental impact on the waters 

of the State by assessing whether the withdrawal will affect 

the ability of a stream to support the characteristic fish population at a site (State of Michigan Ground Water Conservation 

Advisory Council, 2007). 

Assessment Process and Screening Tool

The legislature and GWCAC envisioned a screening 

tool as part of the water-withdrawal assessment process. In 

this report, the term “assessment process” refers to the entire 

process used to assess a new or increased high-capacity 

withdrawal from surface water or ground water, including 

initial screening, site-specific review, and agency procedures. 

High capacity withdrawals are defined through legislation as 

those capable of removing an average of 100,000 gal/d for a 

Ground-Water-Withdrawal Component of the Michigan 

Water-Withdrawal Screening Tool 

By Howard W. Reeves1, David A. Hamilton2, Paul W. Seelbach3, and A. Jeremiah Asher4 

1

 U.S. Geological Survey.

2

 Michigan Department of Environmental Quality, Land and Water 

Management Division.

3

 Michigan Department of Natural Resources, Institute for Fisheries 

Research.

4

 Michigan State University, Institute for Water Research.
2 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

consecutive 30-day period (as defined in section 32701 of Public Act 33 of 2006 (2006 PA 33), Michigan State Legislature, 

2006b). The screening tool is an Internet-accessible program 

used for an initial screening of a proposed new or increased 

high-capacity withdrawal. This screening tool identifies withdrawals that are not likely to cause an adverse resource impact 

upon any type of stream in the State and allows users proposing such withdrawals to proceed with the withdrawal after 

registering the use. Withdrawals identified as having potential 

to cause an adverse resource impact are expected to require 

additional agency review to determine their disposition (State 

of Michigan Ground Water Conservation Advisory Council, 

2007). This tool is designed to focus regulatory resources on 

withdrawals that have greater potential to cause environmental 

harm. The screening tool uses a statewide database of aquifer 

types and properties in conjunction with a robust analytical 

equation to identify withdrawals not likely to cause an adverse 

environmental impact. 

In this report, the component of the screening tool used 

to estimate the amount of streamflow reduction caused by a 

given ground-water withdrawal is presented and discussed. 

The complete screening tool, including ground-water and surface-water withdrawals and the decision rules used to evaluate 

withdrawals, will be documented in a separate report.

Streamflow Depletion by Pumping Wells

The key to understanding the potential impact of a pumping well on streamflow is to recognize the source of water to 

wells as described, for example, by Theis (1940), Bredehoeft 

and others (1982), Sophocleous (1997), Alley and others 

(1999), and Bredehoeft (2002). When a well is pumped, water 

is removed initially from storage in the aquifer, and the potentiometric level, or head, near the well is reduced. Once the 

head near the well is reduced, then flow is induced towards the 

well. Water is removed from storage, and the potentiometric 

level declines away from the pumping well, thereby creating 

a cone of depression around the well. This cone of depression 

continues to expand until the pumping can be balanced by 

(a) an increase in recharge to the system, induced by the lowering of the potentiometric level in the aquifer; (b) a decrease 

in the discharge from the system, resulting from the lowered 

potentiometric level in the system and decreased gradients in 

discharge areas; or (c) a combination of increased recharge 

and decreased discharge (Bredehoeft, 2002). The sum of 

increased recharge and decreased discharge is called capture (Lohman and others, 1972). If the well cannot capture 

enough water to balance pumping, water levels will continue 

to decline until pumping cannot continue at the initial rate. 

The capture of a pumping well does not depend on the initial 

recharge rate; rather, it depends on the change in recharge 

induced by the pumping well (Bredehoeft and others, 1982; 

Bredehoeft, 2002).

Ground-water pumping can capture streamflow in two 

ways for an initially gaining stream: (1) decreased discharge 

from the aquifer to the stream because of the lowered potentiometric gradient, and (2) induced infiltration from the stream 

to the aquifer because of a reversal in potentiometric gradient. 

In the second instance, the flow is from the stream toward the 

well, and the affected segment of the stream becomes a losing segment. The source of water to a pumping well changes 

with time from water released from storage to water captured 

because of decreased discharge or increased recharge, and this 

shift depends on the geometry of the system and aquifer properties (Bredehoeft, 2002). Appendix 1 offers an analysis of the 

various ways that the interaction between a pumping well and 

a stream have been quantified.

Purpose and Scope

The purpose of this report is to document the groundwater-withdrawal component of the Michigan water-withdrawal screening tool. In practice, the impact of wells on 

streams may not be limited to the 1⁄4-mi buffer and 150 ft 

cutoff depth used in the original legislation so the groundwater component of the statewide assessment tool is designed 

to estimate streamflow depletion resulting from pumping a 

well without imposing arbitrary spatial constraints. The final 

screening tool also is designed to evaluate intermittent pumping and to account for the dynamics of stream-aquifer interaction so that ground-water withdrawals are not considered to 

be immediate from the stream at the pumping rate. Finally, 

the screening tool was designed to estimate the depletion from 

neighboring streams, not just from the stream closest to the 

proposed well.

In this report, the methods used to evaluate the impact of 

a new or increased ground-water withdrawal upon streamflow 

are described. Types of data required for the evaluation and the 

sources of data are provided. The distribution of streamflow 

depletion between neighboring streams is discussed in detail 

because the method used for this distribution was developed 

in this project. The screening tool was tested by comparison 

of results from it to streamflow depletion estimates generated 

with a numerical ground-water-flow model of Kalamazoo 

County, Mich. (Luukkonen and others, 2004), and these results 

are presented and discussed. 

Study Approach

The ground-water component of the water-withdrawal 

screening tool was designed to account for the removal of 

water from storage and the resulting delay in the impact of 

new or increased pumping on streamflow. As a conservative approach, the only sources of water to the well in the 

screening tool are storage and streamflow depletion. For 
Ground-Water-Withdrawal Component of Screening Tool 3

the screening tool, recharge to the aquifer is assumed to be 

unaffected by the addition of a high-capacity well. Several 

other potential sources of water to the well are omitted in this 

approach, including decreased loss of ground water to vegetation capable of using shallow ground water, and changes in 

water exchange between adjacent aquifers through semiconfining layers. The pumping well also is assumed to not significantly affect the water level in the stream. Because stream 

elevation and recharge are assumed to be unaffected by the 

additional pumping, natural seasonal variation in the system 

(including variations in stream elevation and recharge) does 

not influence the estimation of streamflow depletion.

As stated by Bredehoeft (2002) and described in appendix 1, the only way to assess capture by a pumping well is by 

using either analytical or numerical approaches to model the 

system. In general, analytical models are applicable for simple 

geometries and uniform aquifer properties, whereas numerical models allow for more complicated systems. For simple 

systems, both approaches yield similar estimates. For more 

complicated systems, analytical solutions may not capture 

the complexity of the system. Numerical models can describe 

a more complicated system, but these models require more 

extensive site-specific data and analysis. The goal of this project was to devise a procedure applicable for statewide screening; therefore, an analytical model was selected to estimate 

streamflow depletion by a pumping well. Use of an analytical 

model yields a screening tool that is sufficiently accurate but 

requires minimal input data and user training. 

Ground-Water-Withdrawal Component 

of Screening Tool

Upon review of the literature related to modeling 

streamflow depletion by pumping wells, the analytical model 

selected for the ground-water component of the water-withdrawal screening tool is an equation derived by Hunt (1999). 

This analytical model describes streamflow depletion by a 

pumping well for a partially penetrating stream in an infinite 

aquifer with streambed resistance between the stream and the 

aquifer. This analytical model is appropriate for Michigan 

streams, which typically do not fully penetrate the aquifers 

used for water supply, and it is sufficiently simple for statewide screening. This analytical model was implemented in a 

Fortran computer code (Reeves, 2008). In the final implementation of the screening tool, an Internet-accessible version of 

the analytical model was required. The essential elements of 

the analytical model were programmed in VBScript (Microsoft 

Corporation, 2007) for implementation in the Internet version. 

This report includes verification that the VBScript version of 

the analytical model yields solutions similar to those from the 

Fortran version documented by Reeves (2008).

To illustrate the aquifer properties required by the analytical model, the form of the model is a useful summary. Hunt 

(1999) derives the analytical model as 

Q Q Sd

Tt

t

ST

d

T

s w =





































− +

















erfc

exp

2

2

4

4 2

l l 



 +





































erfc l2 2

4 4

t

ST

Sd

Tt

, (1)

where

Qs is the rate of streamflow depletion (length 

cubed per unit time),

Qw is the pumping rate (length cubed per unit 

time),

erfc() is the complementary error function 

(dimensionless),

exp() is the exponential function (dimensionless),

d is the distance from the well to the stream 

(length),

S is the storage coefficient, or storativity, of the 

aquifer (dimensionless),

T is the transmissivity of the aquifer (length 

squared per unit time), 

t is the time from the start of pumping, and

λ is the streambed conductance term (length per 

unit time).

The major assumptions used to derive equation 1 (Hunt, 

1999) are the following:

· Horizontal flow dominates any potential vertical 

flow; the Dupuit assumption is valid.

· The aquifer is homogeneous and isotropic and has 

constant saturated thickness.

· The aquifer is either confined or, if unconfined, 

change in hydraulic head in the aquifer is small 

compared to the saturated thickness.

· The stream is straight, infinitely long, and remains 

in hydraulic connection with the aquifer. 

· The pumping does not change the stage of the 

stream.

· Recharge to the system is unchanged by pumping.

· The streambed may offer resistance to groundwater flow. 

· There is no streambank storage.

· The pumping rate is constant.

· The aquifer extends to infinity. 
4 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Examination of equation 1 reveals that streamflow depletion depends on aquifer and streambed properties, the distance 

from the well to the stream, and time. In this model, streamflow depletion is described by a Darcy expression describing 

the flux between the stream and the aquifer (Hunt, 1999). The 

aquifer is assumed to remain in hydraulic contact with the 

stream, which means that the pumping well does not cause 

the hydraulic head in the aquifer to be lower than the streambed. See appendix 1 for more discussion of this assumption. 

To examine the sensitivity of streamflow depletion results on 

variations of input parameters, see Hunt (1999) and Reeves 

(2008).

The aquifer is assumed to be infinite, and, therefore, no 

additional data regarding the aquifer geometry are required 

by the model beyond the distance between the well and the 

stream. The remaining model input includes transmissivity, 

streambed conductance, storage coefficient, pumping rate, 

and time desired for the evaluation. As described in Reeves 

(2008), the analytical model was implemented in the computer 

code STRMDEPL08, which was used for development of the 

screening tool, with the option to simulate time-varying pumping rates.

Input of Constant and Time-Varying Pumping 

Rates

The impact of time-varying pumping is estimated in the 

screening tool because removal of water from storage may 

reduce the streamflow depletion. The streamflow depletion 

caused by time-varying pumping may be much lower than the 

pumping rate, and the estimated depletion caused by timevarying pumping often continues in time even while the pump 

is off. For analysis of streamflow depletion resulting from 

a constant pumping rate, the user must input the pumping 

rate. The duration of pumping in the screening tool is set to 

1,825 days (5 years). For time-varying pumping, the user must 

not only provide the pumping rate but also a pumping schedule. Appendix 1 provides more discussion regarding the affect 

of time-varying pumping rates.

The interface to the screening tool prompts the user for 

more information if the user indicates a ground-water withdrawal with time-varying pumping. The user must enter the 

months when the well will be pumped, the well capacity, 

the days per week that the well will be pumped during those 

months, and the hours per day pumped. If the hours pumped 

per day are less than 24, the well capacity is prorated to yield a 

daily pumping rate. The days per week and months of pumping are then evaluated to determine the number of times that 

the pumping rate is changed. A superposition in time technique 

is used to account for time-varying pumping in the screening 

tool. Assuming that only streamflow depletion caused by a 

new or increased use is of interest, the equation for superposition used to evaluate time-varying pumping may be written as 

Q t Q t R t s i k

k

i

k k ( ) ( ) ( ) = =

∑∆ ∆

1

, (2)

where

Qs(ti) is the streamflow depletion at time interval i 

(length cubed per unit time),

DQk(tk) is the change in pumping rate during the 

interval k (length cubed per unit time),

R(Dt

k

) is the ratio of streamflow depletion to 

pumping rate given by equation 1 for time 

interval k (length cubed per unit time), 

t

i is the length of time from the beginning of the 

pumping analysis to the time of interest,

Dt

k is ti

 – tk, the difference in time between the 

time of interest and the time when the 

pumping rate changed for interval k, 

i is the number of times when the pumping rate 

changes (dimensionless), and

k is the time interval number (dimensionless).

Streamflow depletion is estimated for every time when 

the pumping rate changes and at the end of the 5-year evaluation period. The maximum streamflow depletion from each of 

these estimations is identified in the screening tool and used to 

evaluate the potential impact of the proposed pumping.

Aquifer Properties Required for the WaterWithdrawal Screening Tool

The analytical model used for the water-withdrawal 

screening tool, equation 1, requires aquifer transmissivity, streambed conductance, and aquifer storage coefficient. 

One of the assumptions is that the aquifer is homogeneous. 

Aquifers in Michigan, however, are typically heterogeneous. 

For the screening-level evaluation, an estimated transmissivity and storage coefficient were assigned to watersheds and 

used in the screening tool. A safety factor was assigned in the 

screening tool to allow for more site-specific evaluation of 

withdrawals that may lead to adverse environmental impact. 

The procedures used to assign aquifer properties to watersheds 

for the water-withdrawal screening tool are presented in this 

section.

For the statewide water-withdrawal screening tool, the 

most consistent source of aquifer properties was assembled 

recently into the Michigan Ground Water Inventory and Map 

(GWIM) database (Michigan Department of Environmental 

Quality, 2005). The GWIM database includes estimates for 

aquifer transmissivity mapped to a 1-km by 1-km (3,281 ft 

by 3,281 ft) grid across the State for glacial deposits and 

bedrock aquifers. Constant values for aquifer storage coefficient were assumed in the GWIM analysis for illustration of 

potential well-to-well impacts. These values were different for 

the glacial deposits and bedrock aquifers, but the values for 

both types were indicative of leaky (semiconfined) systems. 
Ground-Water-Withdrawal Component of Screening Tool 5

The range of storage coefficients observed in aquifer tests in 

Michigan is discussed later in this report. The GWIM database 

information was used to assign transmissivity, streambed conductance, and storage coefficient in the screening tool.

Definition of Valley Segments

The surface-water spatial framework used for the groundwater component of the water-withdrawal screening tool was 

built on the classification of Michigan stream arcs into ecological valley segments (Seelbach and others, 2006; Brenden 

and others, 2008; Zorn and others, 2009). Each valley segment 

is a contiguous length of stream with relatively homogeneous 

hydraulic, hydrologic, and ecologic characteristics. The catchment area, which is the surface watershed drainage area, for 

each valley segment was defined by combining the surface 

catchments of the approximately 30,000 individual stream arcs 

in the 1:100,000-scale National Hydrography Dataset (NHD) 

as processed in the Great Lakes Regional Aquatic Gap project 

(Morrison and others, 2003; Brenden and others, 2006). The 

final framework consists of approximately 10,000 valley segments. The reduction in streamflow caused by a ground-water 

withdrawal is assigned to the downstream point of the valleysegment catchment. To apply the water-withdrawal screening 

tool, aquifer properties were required for each unique valley 

segment.

Assignment of Aquifer Properties for Glacial 

Deposits to Valley Segments

Three aquifer properties are required by the analytical 

model: transmissivity, streambed conductance, and storage 

coefficient. Values for these properties were estimated from 

information in the GWIM database (Michigan Department of 

Environmental Quality, 2005). In this database, transmissivity of glacial deposits is mapped to a grid for the entire State 

of Michigan. The median transmissivity for the grid cells 

within each valley segment catchment is assigned to the valley 

segment and used in the screening model. The median was 

selected as the representative statistic instead of the arithmetic 

mean because the median is less influenced by extremely low 

or high transmissivity estimates in the database and therefore may be the more reasonable statistic for screening-level 

estimation.

The streambed conductance used in equation 1 may be 

written as 

l = ( ) K w

b

v , (3)

where

Kv is the vertical hydraulic conductivity of the 

aquifer (length per unit time),

w is the stream width (length), and 

b is the distance from the bottom of the stream 

to the top of the well screen or open 

interval of a bedrock well (length).

The distance from the bottom of the stream to the top 

of the well screen or open interval, b, is provided as the well 

depth by the user. For the statewide screening, the stream 

width was estimated by use of a regression equation developed 

to relate stream width to drainage area of a stream segment 

(T.G. Zorn, Michigan Department of Natural Resources, written commun., 2007):

w = da

−

3 28 10 0 522358 1 6093 2

0 18786

. * ( ^(( . * log( * . ^ ))

. )) , (4)

where

da is the drainage area of the catchment (square 

miles) and

w is width (feet).

An estimate of the vertical hydraulic conductivity of the aquifer was computed by dividing the median transmissivity by the 

average thickness of aquifer material for each valley-segment 

catchment. This thickness was estimated by interpolating the 

thickness used for the wells in the GWIM transmissivity-estimation procedure to a 3,281 ft by 3,281 ft grid over the State 

and then computing the mean estimated thickness for the grid 

cells within each valley-segment catchment. To avoid dividing 

by zero in areas where glacial deposits are thin, a minimum 

value of 5 ft was assigned. 

Equation 2 allows the streambed conductance used in 

the screening tool to vary with the depth to the top of the 

well screen entered by the user. A factor of 1/10 is included 

in the estimated streambed conductance to account for the 

anticipated anisotropy of aquifers in Michigan. This factor 

is reasonable for aquifers without clay units and probably 

underestimates the anisotropy in areas where clay layers 

are present (Todd and Mays, 2005). Anisotropy in hydraulic 

conductivity describes the tendency for the aquifer material to 

have less resistance to flow in the horizontal direction than the 

vertical direction primarily because of layering of the system: 

underestimation of this factor is conservative in the estimate 

of streamflow depletion by a pumping well. Because the user 

may request site-specific review of a proposed withdrawal if 

the screening tool identifies it as having greater potential for 

causing adverse environmental impact, this conservative factor 

was appropriate. The final expression for streambed conductance used in the screening tool is

l = (( / ') / T B w )

b

10 , (5)

where

B’ is the estimated mean thickness of glacial 

deposits for the valley-segment catchment 

(length) and

b is the input value of depth to the top of the 

well screen or open interval of the well 

(length).
6 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

The final aquifer property is storage coefficient. In the 

analysis for the GWIM, constant values for storage coefficient were used: 0.0016 for glacial deposits and 0.0004 for 

bedrock aquifers. These values were determined by use of the 

geometric means of reported storage coefficients from aquifer 

tests reviewed by the State. For the screening tool, the value 

for the glacial deposits was believed to be too small, resulting 

in overprediction of the streamflow depletion from wells in 

unconfined or leaky aquifers. The storage coefficients reported 

for aquifer tests in glacial deposits vary over four orders of 

magnitude and do not correlate well with location in the State, 

surficial geology, or depth. Relation of storage coefficient 

with depth is illustrated in figure 1. On the statewide scale, it 

has not been possible to reliably identify areas of Michigan 

dominated by confined, leaky, or unconfined conditions. Use 

of the geometric mean, or a smaller value typical of a confined 

aquifer, may lead to a potentially contradictory situation where 

the assumption is made in the screening tool that the aquifer 

behaves as confined but also in good hydraulic contact with 

headwater streams. Conversely, assuming that an unconfined 

specific yield represents statewide conditions is not conservative, especially for time-varying pumping estimates. To give 

a conservative estimate that is consistent with the observed 

uncertainty in the estimated storage coefficients across the 

state, a constant value of 0.01, which is representative of a 

leaky aquifer, was used in the screening tool.

Figure 1. Reported aquifer storage coefficients for wells completed in glacial deposits from aquifer tests reviewed by the State of 

Michigan as a function of depth; arithmetic, geometric, and harmonic means also are shown. Shaded zones show typical ranges of 

specific yield and storage coefficient (for a 100 foot aquifer) for unconsolidated materials (Domenico and Schwartz, 1990). 

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.0 100.0 200.0 300.0 400.0 500.0 600.0

STORAGE COEFFICIENT, DIMENSIONLESS

DEPTH, IN FEET

Value from aquifer test

Geometric mean

Harmonic mean

Arithmetic mean

Specific yield,

unconfined aquifers

Storage coefficient,

leaky aquifers

Storage coefficient,

confined aquifers
Ground-Water-Withdrawal Component of Screening Tool 7

Figure 2. Four bedrock categories used in the water-withdrawal screening tool for Michigan. 

0 50 100 MILES

0 50 100 KILOMETERS

EXPLANATION

Bedrock geology

-1 Bedrock aquifer in limited hydraulic

 connection but may be saline

0 Bedrock typically not used for

 high-capacity wells

1 Bedrock aquifer in limited hydraulic

 connection with streams

2 Bedrock aquifer may be in hydraulic

 connection with streams

Bedrock geology from Michigan Department of Environmental Quality, 1987

88°W

84°W

46°N

42°N

Assignment of Bedrock Aquifer Properties to 

Valley Segments

Bedrock aquifers are used in parts of Michigan, and in 

some areas of the State these bedrock aquifers are separated 

from overlying streams by thick glacial deposits containing 

layers of material with low hydraulic conductivity, such as silt 

or clay. In these areas, the hydraulic connection between these 

aquifers and nearby streams is limited. In other areas, there 

may be a greater hydraulic connection between the bedrock 

aquifer and overlying streams. Saline water may be present 

in some areas of the major bedrock aquifers, and it may occur 

at depth under a freshwater zone or through the entire thickness of the bedrock aquifer. Finally, in some areas of the State, 

bedrock aquifers are not suitable for high-capacity wells. To 

account for these different potential conditions in the screening tool, the bedrock aquifers were grouped into four categories (fig. 2). The categories and response given by the waterwithdrawal screening tool are summarized in table 1.
8 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Definition of Great Lake Shoreline Catchments

Many small valley-segment catchments were defined 

adjacent to one of the Great Lakes. If the valley segment in a 

catchment adjacent to a Great Lake did not contain a stream or 

was smaller than approximately 3 to 6 mi2, then the valleysegment catchment was designated as a shoreline catchment. 

No limitation on depletion was set from these shoreline catchments because pumping was assumed to come from the lake 

and not adversely impact a stream. Entities proposing withdrawals greater than 2 Mgal/d, however, are required to obtain 

a water-withdrawal permit (Public Act 180 of 2008 (2008 PA 

180), Michigan State Legislature, 2008b), and the potential 

ecological impact of such withdrawals may be assessed in the 

permitting process.

Distribution of Streamflow Depletion Among 

Valley Segments in Neighboring Catchments

Unlike direct surface-water withdrawal, ground-water 

withdrawals are thought to potentially affect valley segments 

in neighboring catchments. The analytical solution, however, 

estimates streamflow depletion only between the well and a 

single stream. Nine different methods to apportion streamflow 

depletion among neighboring valley segments were evaluated. 

Details regarding this evalution are presented in appendix 2. 

The simplest approach is to estimate streamflow depletion 

only for the valley segment draining the catchment containing the well. This simple approach, however, performed the 

worst compared to the results from a numerical ground-waterflow model. On the basis of the analysis, an inverse distance 

method was selected for the screening tool because (1) it 

produces a reasonable overall pattern of streamflow depletion 

compared to a numerical ground-water-flow model, (2) it is 

the most straightforward to implement in the Internet-based 

screening tool, and (3) it has some theoretical basis in steadystate analysis (Wilson, 1993). The weighting used for this 

distribution method may be written as 

f d

d

i

i

j n j

=

=

∑

1

1

1,

, (6)

where 

f

i is the fraction of the captured water attributed 

to valley segment i, 

n is the number of adjacent valley-segment 

catchments, and 

Table 1. Bedrock categories and response by water-withdrawal screening tool to proposed bedrock well in each category. 

Bedrock 

category Description of bedrock hydraulic conditions Response from water-withdrawal screening tool

-1 Bedrock aquifers typically in only limited hydraulic 

connection with streams but in areas where 

ground water in the bedrock aquifer is potentially 

saline based on a regional assessment (Westjohn 

and Weaver, 1996). Note that in some areas, saline 

water may occur at depth below a freshwater zone, 

and in other areas, saline water may be present 

throughout the entire bedrock thickness.

The proposed well passes the screen and the user is informed to 

register the use as required by statues. Because in some areas 

identified as saline (Westjohn and Weaver, 1996) bedrock wells 

may produce potable water and saline water becomes an issue only 

for deeper bedrock wells, this flag is not used in the current version 

(January 2008) of the screening tool. The information is retained in 

the underlying database in case this information is desired in future 

versions. The user is warned in all uses of the screening tool that the 

tool is not intended as a design tool and that the evaluation does not 

guarantee ground-water quantity or quality.

0 Bedrock typically not used for high-capacity wells 

because the dominant bedrock unit in the area is 

not a productive aquifer. For example, this 

category is used for areas in the state underlain 

by the Coldwater Shale.

The user is informed through a pop-up message that bedrock aquifers 

in this area are generally not used for high-capacity wells. The 

screening tool uses aquifer properties from the glacial deposits to 

evaluate the potential streamflow depletion for the proposed well.

1 Bedrock aquifers typically in only limited hydraulic 

connection with streams, especially smaller 

streams.

The proposed well passes the screen, and the user is informed to 

register the use as required by statues.

2 Bedrock aquifer may have hydraulic connection 

with streams.

The proposed withdrawal is evaluated by use of bedrock properties 

from the Ground-Water Inventory and Map database that were 

processed in the same way as the properties for the glacial deposits. 

The streambed conductance between the bedrock aquifer and the 

stream is estimated by use of the properties of the glacial deposits 

overlying the bedrock. The assigned storage coefficient is 0.0004, 

as used in the Ground-Water Inventory and Map.
Demonstration of Analytical Model and Distribution Approach 9

di is the distance from the proposed well to the 

closest point on the valley segment within 

catchment i. 

Demonstration of Analytical Model and 

Distribution Approach

The ground-water-flow model for the area around Kalamazoo County, Mich. (Luukkonen and others, 2004) was used 

to test the nine distribution methods discussed in appendix 2 

and to demonstrate the performance of the analytical model 

used in the screening tool. The ground-water-flow model is a 

finite-difference (MODFLOW) simulation model with 6 layers, 154 rows, and 162 columns. The smallest grid spacing 

in the model is 660 ft by 660 ft. Valley-segment catchments 

in the interior of the model that have this finest grid spacing 

were used in the methods testing. The model used for the test 

case includes approximately 90 existing wells, and streamflow 

depletion caused by the addition of a new well to different 

parts of the model is examined. To illustrate the applicability 

of the analytical model and distribution approach to field problems, a transient simulation with seasonally varying recharge 

and pumping rates was used as the test case. Luukkonen 

and others (2004) presented the results for a 9-year transient 

simulation; but, in this work, the length of the simulation 

was reduced to 5 years to reduce computer run time (fig. 3). 

The simulation illustrates seasonal pumping and recharge. 

This simulation is used in the demonstration to illustrate that 

the analytical model given by equation 1 is not affected by 

time-varying areal recharge as long as the areal recharge is 

independent of the imposed pumping. Because areal recharge 

in the numerical model is specified and does not depend on the 

pumping imposed on the system, the numerical model is consistent with the assumptions made in the analytical model. For 

many cases in Michigan, pumping will not induce increased 

recharge, and this assumption is valid at least on a screening 

level.

A method to illustrate streamflow capture by pumping 

wells (Leake and Reeves, 2008) was used to examine the 

spatial distribution of streamflow capture from the valley 

segments. In this method, hypothetical wells are added to the 

MODFLOW model sequentially on a grid of cells (fig. 4). 

Figure 3. Recharge and pumping rates used in the base simulations with the Kalamazoo regional ground-water-flow model (Luukkonen 

and others, 2004). 

-2.0E+07

0.0E+00

2.0E+07

4.0E+07

6.0E+07

8.0E+07

1.0E+08

1.2E+08

0 500 1,000 1,500 2,000

RATE, IN CUBIC FEET PER DAY

TIME, IN DAYS

Wells

Recharge

EXPLANATION
10 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Figure 4. A grid of points used to compare the analytical model and different distribution 

methods to MODFLOW results from the Kalamazoo regional ground-water-flow model 

(Luukkonen and others, 2004). 

EXPLANATION

Michigan valley-segment catchments

Finite-difference grid used in Kalamazoo 

regional ground-water-flow model

Michigan valley segments

Points used for sequential introduction of test wells

42°16'

42°12'

85°34' 85°30'

0 0.5 1 MILE

0 0.5 1 KILOMETER

Michigan

Finite-difference grid

Kalamazoo, Mich.

Base watershed and streams from Brenden and others, 2006
Demonstration of Analytical Model and Distribution Approach 11

One well is added to a specified cell, and the new simulation is 

computed. The results from this new simulation are subtracted 

from base-model results to determine the capture and change 

in storage caused by the new well. The process is repeated 

with one new well at a time, generating a grid of results that 

may be mapped. The fluxes to individual river cells representing the valley segments from the statewide data framework 

were computed. These fluxes were recorded by use of a river 

observation file (Hill and others, 2000) to designate the finitedifference cells assigned as river cells to the appropriate valley 

segments. The river flux output from the system with the new 

well for each designated valley segment was subtracted from 

the valley-segment flux computed with the original model to 

yield the streamflow depletion due to the new well as simulated with the MODFLOW model. Values for streamflow 

depletion as a percentage of pumping rate for the grid of wells 

shown in figure 4, which were placed in layer 3 of the model, 

are shown in figure 5. The two methods, MODFLOW and the 

analytical model, produce similar results for wells introduced 

into the test valley-segment catchment. The maximum depletion for wells introduced near the valley segment within the 

catchment is between 70 and 80 percent, and the minimum 

depletion near the boundary of the catchment is less than 

10 percent (See appendix 2 for more detailed discussion of the 

methods to distribute streamflow depletion among neighboring 

valley segments).

Figure 5. Estimated streamflow depletion from valley segment 3544 as a percentage of pumping rate after 5 years of pumping. 

A, results from a sequential introduction of test wells in the MODFLOW ground-water-flow model for Kalamazoo County, Mich. 

B, results from analytical model with inverse distance to stream distribution of streamflow depletion.

A MODFLOW B INVERSE DISTANCE TO STREAM

0 1 MILE

0 1 KILOMETER

EXPLANATION

Streamflow depletion, in percent

6.6 - 10

10.1 - 20

20.1 - 30

30.1 - 40

40.1 - 50

50.1 - 60

60.1 - 70

70.1 - 80

80.1 - 90

90.1 - 100

Base catchments from Brenden and others, 2006

Michigan valley

segment catchment 3544

Kalamazoo, Mich.

Michigan
12 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Limitations of Testing

The MODFLOW simulation model for the Kalamazoo 

area provides a convenient way to test the analytical model 

and distribution methods. The ground-water-flow model, 

however, is a regional simplification of the flow system, and 

neither the MODFLOW model nor the analytical model may 

accurately estimate the true impact of a specific well on a 

specific stream in the area. Site-specific investigation would be 

required for such an estimate. The MODFLOW model resolution of the stream network and well distribution also affects 

the estimates. The smallest grid size in the ground-waterflow model is 660 ft by 660 ft, and this becomes the shortest 

distance between the well and the stream that can be tested 

with the MODFLOW model. If a test point is placed in the 

same cell as a valley segment, the effective distance between 

the well and the stream is zero in the MODFLOW model. For 

the test grid (fig. 4), no wells were placed in valley-segment 

cells. The distances used for the analytical model are from 

geoprocessing of the data-framework files. These distances are 

more accurate than those used in the MODFLOW model, and 

some mismatch between MODFLOW results and the analytical estimates is expected because of these distance differences. 

In addition, the aquifer characteristics used in the MODFLOW 

model are different from those used in the analytical model. 

The transmissivity, storage coefficient, and streamflow-conductance values used in the analytical model in the test were 

estimated from the statewide databases, as described earlier 

in this report. The values used in the MODFLOW model, 

however, were the calibrated values from Luukkonen and 

others (2004). Different input values were used to more fully 

test the approach for the screening tool described previously. 

The reasonable match between the MODFLOW results and 

the analytical-model results supports the use of the analytical model, the use of the estimated aquifer properties derived 

from the GWIM database, and the use of the inverse-distance 

weighting scheme to distribute streamflow depletion among 

neighboring valley-segment catchments. 

The aquifer and surface-water network in Kalamazoo 

County is not necessarily representative of conditions across 

all of Michigan. In addition to this quantitative test, a series 

of qualitative ad hoc tests were performed by applying the 

analytical model to locations of high-capacity wells in every 

county of the state. The ad hoc tests relied on the experience 

of the testing team to assess whether the results appeared to 

be reasonable, and no major problems were identified through 

this testing. 

Summary of the Ground-Water 

Component of the Water-Withdrawal 

Screening Tool

The water-withdrawal screening tool was designed to be 

accessed through the Internet. This access required the integration of several technologies. The screening tool essentially 

implements the analytical model and inverse-distance weighting distribution method described in the previous sections 

but with different computer software than used in the testing 

and development. A brief list of steps may help illustrate the 

ground-water component of the screening tool. The overarching technology is an ArcIMS Web site (Environmental 

Systems Research Institute, Inc. (ESRI), 2004) that provides a 

map interface to the screening tool for the user. The following 

is the sequence of steps for a typical screening-tool session:

1. When the user enters the location for a new 

withdrawal, a server running ArcView with Avenue Scripts (for example, Razavi and Warwick, 

1999) determines the valley segment containing 

the proposed well. 

2. The user specifies whether the proposed well is 

in glacial deposits or bedrock. If bedrock, then 

the bedrock type for the valley-segment catchment is identified.

3. A GIS data file is accessed to gather the aquifer properties assigned to the valley-segment 

catchment and to determine whether the valleysegment catchment is identified as a Great Lake 

shoreline catchment.

4. The neighboring valley segments are identified.

5. The distances from the proposed well to the 

nearest valley segment in each of the valleysegment catchments are computed. 

6. VBScripts are used to run the Hunt (1999) analytical model (equation 1) for each of the valley 

segments using the aquifer properties for the 

valley-segment catchment containing the well 

and the distances computed in step 5. If the user 

specifies time-varying pumping, then superposition (equation 2) is used to compute the maximum streamflow depletion during the 5-year 

evaluation period. For steady pumping, the solution is evaluated after 5 years of pumping.

7. VBScripts are used to distribute the streamflow 

depletion between the neighboring valley segments using inverse distance weighting (equation 6).
Example Screening and Confirmation of Water-Withdrawal Screening-Tool Results 13

8. Additional scripts are used to collect this 

information, compare the results to streamflow 

estimates, and apply screening rules to provide 

feedback to the user. 

Example Screening and Confirmation 

of Water-Withdrawal Screening-Tool 

Results

The tests of the analytical model presented in this report 

were generated by use of Python scripts and the Fortran-based 

STRMDEPL08 program (Reeves, 2008). The STRMDEPL08 

program was used to evaluate the Hunt (1999) analytical 

model (equation 1). Examples were run in this manner as 

batch tests for the technical subcommittee of the GWCAC. 

Because the STRMDEPL08 program has been documented by 

Reeves (2008), it serves as a standard for comparison of the 

Internet-based water-withdrawal screening tool. In this final 

section, results from the water-withdrawal screening tool using 

the sequence of ArcIMS, VBScript, and Avenue scripts listed 

above are shown to match results from STRMDEPL08 with 

inverse-distance weighting applied to distribute withdrawal 

among neighboring valley segments. 

For this demonstration, an example in the northern part 

of the Lower Peninsula of Michigan (fig. 6) was evaluated 

with the water-withdrawal screening tool. The latitude and 

Figure 6. Example test point, valley-segment catchment containing the point, and 

neighboring valley-segment catchments.

-84°50' -84°40' -84°30' -84°20'

45°10'

45°0'

44°50'

44°40'

EXPLANATION

Test point

Valley segments

Valley-segment catchment with test point

Neighboring valley-segment catchments

0 6,000 Meters 12,000

0 25,000 Feet 50,000
14 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Table 2. Compilation of estimates for an example test point using the Michigan water-withdrawal screening tool and the set of 

batch programs used in screening-tool development and testing. 

[The number of digits used does not signify precision of estimates but is reported to show the difference between the two sets of calculations] 

Valley 

segment 

identification

Distance to 

example well, 

feet

Removal from 

valley segment, 

percent

Analytical removal, 

gallons per minute

Estimated removal 

from valley segment, 

gallons per minute

Screening 

tool Batch Screening tool Batch Screening tool Batch Screening tool Batch

8 14,802 14,798.9 9.89 9.89 52.02 52.02 5.15 5.15

9 12,609.2 12,609.6 11.61 11.61 54.30 54.30 6.31 6.30

11 15,750.5 15,745.0 9.30 9.30 51.05 51.03 4.75 4.74

27 22,567.6 22,562.4 6.49 6.49 44.24 44.25 2.87 2.87

9741 27,565.2 27,561.4 5.31 5.31 39.52 39.49 2.10 2.10

10532 33,059.5 33,052.5 4.43 4.43 34.62 34.60 1.53 1.53

11967 14,846.3 14,844.0 9.86 9.86 51.98 51.97 5.13 5.13

12515 17,042.5 17,033.9 8.59 8.59 49.73 49.73 4.27 4.27

12573 11,959.5 11,960.4 12.24 12.24 54.98 54.98 6.73 6.73

12941 19,070.8 19,063.4 7.68 7.68 47.69 47.71 3.66 3.66

13925 10,028.9 10,030.6 14.60 14.59 57.01 57.00 8.32 8.32

longitude of the point were input to the Python scripts used for 

the screening-tool development. The point, the valley-segment 

catchment containing the point, the neighboring valleysegment catchments, and the neighboring valley segments 

identified by the Internet-based suite of programs and the 

Python scripts were identical. The percent removal from each 

of the valley segments and the total removal given a well at 

a depth of 80 ft and a continuous pumping rate of 70 gal/min 

generated by the STRMDEPL08 program (Reeves, 2008) and 

the water-withdrawal screening tool were nearly identical 

(table 2). The differences between the STRMDEPL08 and 

water-withdrawal screening tool may be attributed to slight 

differences in the estimated distances between the proposed 

well and valley segments, most likely caused by round-off in 

the latitudes and longitudes used to locate the point. 
References Cited 15

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3512.

Zorn, T.G., Seelbach, P.W., Rutherford, E.S., Wills, Todd, 

Cheng, Su-Ting, and Wiley, M.J., 2009, A landscape-scale 

habitat suitability model to evaluate impacts of groundwater 

withdrawals on fish communities in Michigan streams: Ann 

Arbor, Mich., Michigan Department of Natural Resources 

Fisheries Research Report 2089, 46 p.
Appendix 1 17

Streamflow depletion has been modeled with various 

analytical and numerical models. The conceptual model underlying each approach may be used to distinguish and classify 

the analytical and numerical models. Five conceptual models 

and extensions are discussed to provide an overview of the 

relevant literature on streamflow depletion by wells. 

Fully Penetrating Stream With No Streambed 

Resistance

The first conceptual model discussed in this report was 

used to derive a solution by Theis (1941) that was later recast 

in the now more familiar form by Glover and Balmer (1954). 

This model is for a stream that fully penetrates the aquifer. 

There is no resistance to flow offered by the streambed. The 

domain is homogeneous and extends to infinity away from 

the stream. The stream is infinite, and the interaction between 

the pumped aquifer and the entire stream length is considered. 

There is no flow across the bottom of the aquifer. Flow is 

horizontal (Dupuit approximation) and, if the aquifer is unconfined, then drawdown in the aquifer is small compared to the 

saturated thickness such that the system may be modeled with 

constant transmissivity (fig. 1–1A). Glover (1974) presents a 

solution to the problem that provides an estimate of streamflow depletion for an arbitrary length of the stream. 

Appendix 1. Background and Literature Review on Streamflow Depletion 

Modeling 

Figure 1–1. Alternate 

conceptual models for 

streamflow depletion by 

a pumping well. (A, Fully 

penetrating stream with 

no streambed resistance. 

B, Fully penetrating stream 

with streambed resistance. 

C, Partially penetrating stream 

with streambed resistance. 

D, Partially penetrating stream 

in semiconfining layer with 

pumping from underlying 

semiconfined aquifer. In the 

figure, d is the distance from 

the well to the stream, Qw is 

the pumping rate from the 

well, b’ is the thickness of the 

streambed, B’ is the distance 

from the land surface to the 

top of the leaky aquifer, B’’ is 

the distance from the bottom 

of the stream to the top of the 

leaky aquifer, and b is the width 

of the stream (Modified from 

Reeves, 2008)). 

C D

A B

d

b’ d

d

Qw

B’ B’’

d

b Qw

Qw Qw
18 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

The Theis (1941) and Glover and Balmer (1954) solutions are transient but consider only a constant pumping rate. 

Jenkins (1968) and Wallace and others (1990) describe how 

to use superposition to extend the Glover and Balmer (1954) 

solution for intermittent or cyclic pumping rates. Jenkins 

(1968) introduces the concept of the streamflow depletion 

factor (sdf) that is used by several subsequent authors (for 

example, Burns, 1983). Jenkins (1968) focuses on the use of 

superposition to solve intermittent pumping problems and 

demonstrates that streamflow depletion continues for some 

time even after pumping has been stopped. Wallace and others (1990) examine the time required for the system to attain 

a dynamic steady state and the conditions under which a 

cycle-average pumping rate could be used in the estimation 

of streamflow depletion by cyclic pumping. Under certain 

conditions, discussed in the paper, the use of the cycle-average 

pumping rate to analyze streamflow depletion severely underestimates the peak depletion rates; therefore, an analysis with 

the true cyclic pumping rates is preferable. Barlow (2000) 

wrote a computer program to compute the Glover and Balmer 

(1954) solution for intermittent pumping, using the techniques 

described by Jenkins (1968). 

In contrast to the transient solutions, Newsom and Wilson 

(1988) and Wilson (1993) present steady-state solutions to 

this problem. For all the solutions discussed for this conceptual model, the total streamflow depletion is the pumping rate 

because there are no other sources of water to allow the system 

to reach steady state other than capture from the stream. The 

Glover (1974) solution and the Newsom and Wilson (1988) or 

Wilson (1993) solutions can be used to estimate the amount 

of streamflow depletion in a prescribed length of the stream 

reach. As this length becomes large, the streamflow depletion 

from the specified stream reach approaches the pumping rate. 

The papers by Newsom and Wilson (1988) and Wilson (1993) 

illustrate the importance of regional ground-water flow on the 

solution. The analysis in the papers also distinguishes between 

streamflow capture by interception of ground water that 

would have been discharged from the aquifer as base flow and 

streamflow capture by induction of stream water into the aquifer. Wilson (1993) also provides analysis for a well between 

two streams with vertical recharge and for an aquifer with both 

a stream and a barrier boundary. 

Distinguishing between capture from decreased base flow 

to the stream and induced infiltration from the stream may 

be important for water-quality analysis if the water-quality 

characteristics of the stream and the aquifer are different. The 

steady-state solutions may be used to analyze the infiltration 

rate of poor-quality surface water to a supply well or the discharge rate of contaminated ground water to a stream despite 

the installation of an extraction well. The critical pumping rate 

required to induce infiltration from the stream is shown to be a 

function of the angle between the ambient flow direction and 

the stream (Newsom and Wilson, 1988). If the ambient flow 

rate is perpendicular to the streambed, which is the typical 

situation, the critical flow rate is calculated as follows (Newsom and Wilson, 1988; Wilson, 1993): 

Q d q c = a p , (1–1)

where

Qc is the critical pumping rate, d is the distance 

between the well and the stream, and 

	 qa is the ambient flow rate of ground water to the 

stream per unit length of stream. 

This last term is the specific discharge from the aquifer to the 

stream integrated by the thickness of the aquifer, and it has 

units of length squared per unit time (Newsom and Wilson, 

1988). Newsom and Wilson (1988) present results that show 

that this critical pumping rate increases by approximately 

10 percent at an approximate angle between the ambient flow 

and the streambed of 29 degrees. The critical pumping rate is 

increased for angles between 0 and approximately 55 degrees 

and decreases between 55 degrees and 90 degrees. For flow 

parallel to the streambed (90-degree angle between ambient 

flow and the streambed), the critical pumping rate is approximately 40 percent of the value given above.

Chen (2003) combines transient analysis and the questions posed by Newsom and Wilson (1988) and Wilson (1993) 

to present a transient investigation of induced infiltration from 

the stream and decreased base flow to the stream. In particular, Chen (2003) derives the critical time required to induce 

infiltration from the stream as 

t d S

T q d Q c

a

= −

2

4 ln( / ) p

, (1-2)

where 

t

c is the critical time when infiltration from the 

stream to the aquifer is induced, 

T is the aquifer transmissivity (length squared 

per unit time), 

S is the aquifer storage coefficient 

(dimensionless), and 

Q is the pumping rate of the well (length cubed 

per unit time). 

Chen (2003) also discusses the residual impact of pumping on streamflow after pumping has ceased. As noted by other 

transient analysis (for example, Jenkins, 1968; Wallace and 

others, 1990), streamflow depletion continues after pumping has stopped until, under this conceptual model, the total 

volume removed as streamflow depletion is the same as the 

volume of water removed from the aquifer by the intermittent 

pumping. 

Fully Penetrating Stream With Streambed 

Resistance

Hantush (1965) presented the solution for the second conceptual model that includes streambed resistance as illustrated 

in figure 1–1B. The resistance in the streambed is described by 
Appendix 1 19

a streambed leakance or retardation coefficient, L = (Kb’/K’), 

where K is the hydraulic conductivity of the aquifer, K’ is the 

hydraulic conductivity of the streambed, and b’ is the thickness of the streambed1

. The streambed leakance term causes 

the interaction between the pumping well and the stream to 

be delayed. As discussed for the previous conceptual model, 

because there are no other sources of water for the well to 

capture, the streamflow depletion is the pumping rate at steady 

state. The Hantush (1965) solution is a transient solution for 

a constant pumping rate, and the superposition techniques 

described by Jenkins (1968) can be applied to this solution. 

Barlow (2000) includes the Hantush (1965) solution in the 

computer program cited above as an input option selected by 

the user. 

Many solutions that consider streambed resistance also 

incorporate partial penetration of the streambed into the aquifer, and these solutions will be discussed in the next section. 

Three papers that evaluated analytical solutions and identified shortcomings with various conceptual models are now 

reviewed to emphasize the difference between the conceptual 

models and associated analytical solutions. Spaulding and 

Khaleel (1991) and Sophocleous and others (1995) analyzed 

the analytical solutions by Glover and Balmer (1954), Jacob 

(1950) (discussed in the next section), Hantush (1965), and 

Glover (1974) and provide results illustrating the importance 

of streambed resistance and partial penetration of the streambed into the aquifer. See figure 1–1C for an illustration of a 

partially penetrating streambed. Spaulding and Khaleel (1991) 

used a finite-element ground-water-flow model from the 

literature and Sophocleous and others (1995) used the finitedifference-based model MODFLOW (Harbaugh and others, 

2000) to test the various analytical solutions for different 

conceptual models. Both papers conclude that streambed resistance and partial penetration of the streambed into the aquifer 

are important to the solution and that conceptual models that 

do not account for these factors may severely overestimate 

the streamflow depletion rate at a given time. Conrad and 

Beljin (1996) also used MODFLOW to investigate various 

assumptions inherent in the Wilson (1993) solution. These 

authors also investigated the impact of streambed resistance 

and partial penetration by the stream. In addition, they present 

tests exploring the impact of vertical anisotropy in the aquifer 

and of considering typical variations in stream levels on the 

simulation of the aquifer system. All of these papers show that 

partial penetration of the streambed into the aquifer and incorporation of streambed resistance are important features that 

should be included in the analysis of streamflow depletion.

Partially Penetrating Stream With Streambed 

Resistance 

Hunt (1999) generalizes the problem more by considering a partially penetrating stream in a fully confined or 

1

 Streambed resistance in the paper by Hunt (1999) is L, but Hantush (1965) 

denotes this coefficient by a.

unconfined aquifer. The rest of the assumptions used for the 

previous cases still hold: horizontal flow following the Dupuit 

approximation, small changes in saturated thickness for the 

unconfined case, and a homogeneous aquifer. In this case, the 

aquifer has infinite extent and the stream width and depth are 

small compared to the aquifer thickness, such that the stream 

may be modeled as a line (zero width). Hunt (1999) includes 

the resistance through a streambed using Darcy’s Law, Q = 

l (H-h), where l is a streambed-conductance term that must 

incorporate the hydraulic conductivity of the streambed, 

geometry of the streambed, and streambed thickness; H is the 

elevation of the water level in the stream; and h is the head 

in the aquifer at the streambed. As discussed by Hunt (1999), 

and emphasized by Rushton (1999), the head in the aquifer is 

assumed to remain in the streambed in this solution. (Solutions 

considering the case where the head in the aquifer drops below 

the streambed are discussed later.) This solution considers the 

impact of pumping on the head in the aquifer on the opposite 

side of the stream from the pumping well (fig. 1–1C). The 

solution yields the time-dependent response of the aquifer 

and for streamflow depletion to steady pumping from a well. 

Hunt (1999) shows how the solution presented approaches the 

Glover and Balmer (1954) solution as l approaches infinity and the Hantush (1965) solution if l is replaced by 2T/L. 

Hunt (1999) points out that Hantush (1965) suggested that the 

distance from the stream to the well, d, in the solution may 

be varied to account for partial penetration of the stream (see 

also Jacob, 1950; and Spaulding and Khaleel, 1991) and that 

this suggestion appears to be incorrect. Evidently, the Hantush 

(1965) solution for the semi-infinite domain and fully penetrating stream may be used for the infinite domain and partially 

penetrating stream by appropriate correction to the retardation 

coefficient, L, using L = 2T/l. 

The notion that partial penetration of the streambed into 

the aquifer may be modeled by varying the distance from the 

well to the stream in the solution is not unique to Hantush 

(1965). Jacob (1950) suggested that the distance from the well 

to the stream may be varied in the Glover and Balmer (1954) 

solution to account for both streambed resistance and partial 

penetration of the streambed (Spaulding and Khaleel, 1991). 

To facilitate comparison of the analytical solutions, Spaulding 

and Khaleel (1991) used a three-dimensional analytical solution to derive a set of curves defining the effective distance 

from the well to the stream for different degrees of partial 

penetration for use in either the Glover and Balmer (1954) 

or Hantush (1965) solutions. Darama (2001) used this set of 

curves along with techniques from Wallace and others (1990) 

to examine cyclic pumping in the case of partially penetrating streambed with streambed resistance. Darama (2001) 

echoed the conclusions of Spaulding and Khaleel (1991) and 

Sophocleous and others (1995) that partial penetration of the 

streambed and streambed resistance are important factors in 

the solution that should be considered in the conceptual model.

Singh (2003) presents an alternate solution for a partially 

penetrating stream by adding an equivalent resistance length 

to the Hantush (1965) retardation coefficient to account for 
20 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

partial penetration. In this way the solution addresses both 

streambed resistance and partial penetration. An expression to approximate the additional resistance length that 

depends on the thickness of the aquifer and the stream width 

is provided. Singh (2003) also discusses the use of superposition to account for intermittent pumping. The geometry of 

the conceptual model is not discussed by Singh (2003), and 

the Hunt (1999) solution is not cited. Because of the relation between the Hantush (1965) and Hunt (1999) solutions, 

and because the Singh (2003) solution is an extension of the 

Hantush (1965) solution, there must be a relation between the 

Singh (2003) and Hunt (1999) solutions. This relation implies 

that the Singh (2003) solution could be used for the conceptual 

model shown in figures 1–1B or 1–1C. An analysis to relate 

the Hantush (1965) retardation coefficient, the streambed-conductance term used by Hunt (1999), and the resistance length 

added due to partial penetration in the Singh (2003) would be 

interesting but beyond the scope of this review.

Another approach to this conceptual model was offered 

by Zlotnik and Huang (1999), who derived an analytical 

solution by breaking the problem domain into two regions. 

The first region is the aquifer below the partially penetrating 

stream, and the second region is the adjacent semi-infinite 

aquifer. The solution is derived by simultaneously solving the 

equations for the two regions that are coupled by the boundary 

condition at the interface between the two regions. The solution is given as a Laplace Transform and is awkward to evaluate. A simplified version of the solution in which the storage 

coefficient in the region below the streambed is ignored yields 

a simpler solution that has a similar form to many of the other 

analytical solutions discussed. One advantage of this solution 

is that the stream has finite width in contrast to the solution 

presented by Hunt (1999) that models the stream as a line.

Butler and others (2001) extend the earlier work by 

Zlotnik and Huang (1999) and compare results of this analytical model to results from the solution presented by Hunt 

(1999) and to numerical results generated from MODFLOW. 

This analysis again emphasizes the importance of considering 

both partially penetrating streambed and streambed resistance 

when simulating the interaction between a pumping well and 

a stream. The authors noted that the model by Hunt (1999), 

which does not account for stream width, yields essentially the 

same result as a finite-width solution if the well is more than 

five stream-widths away from the stream. Comparison with 

the numerical model allowed the evaluation of the assumption 

that the aquifer is infinite in extent away from the stream. This 

analysis suggested that the aquifer width has to be hundreds 

of stream widths in extent before the analytical solution yields 

similar results as the numerical solution.

A solution for a finite-width stream also was given by 

Fox and others (2002). The nature of the solution and an 

alternate explanation for convergence problems identified by 

Fox and others (2002) is presented in the discussion by Hunt 

(2004). The difference between the finite-width solution (Fox 

and others, 2002) and line-width solution (Hunt, 1999) was 

investigated for a range of conditions. The difference depends 

on the distance from the well to the stream and the simulation time. When the well is more than 50 stream widths from 

the stream, the difference between the two solutions is less 

than 1 percent. In contrast to the findings of Butler and others 

(2001), the lower limit reported by Fox and others (2002) for 

agreement between the solutions is 25 stream widths. When 

the well is closer than 15 stream widths away from the stream, 

the maximum error between the solutions under the worst-case 

conditions is stated to be as great as 10 percent.

Di Matteo and Dragoni (2005a, 2005b) used MODFLOW 

to investigate partial penetration of the streambed into the 

aquifer and the impact of regional gradients on the solution. 

The importance of regional gradients for fully penetrating 

streams also was illustrated by Newsom and Wilson (1988), 

Wilson (1993), and Chen (2003). Di Matteo and Dragoni 

(2005a) present an interesting approach wherein MODFLOW 

was used to generate streamflow-depletion estimates, and then 

these estimates were fit by a nonlinear function in terms of the 

aquifer parameters and stream geometry. This empirical relation was then tested for a variety of conditions to illustrate the 

importance of anisotropy, partial penetration of the streambed, 

and the overlap between the bottom of the streambed and top 

of the well casing. This is the only paper that explicitly considers this overlap in the analysis.

Steady-state solutions to the partially penetrating streambed are presented by Ernst (1979) and Bakker and Anderson 

(2003). Ernst (1979) uses image-well theory to derive expressions for drawdown and streamflow depletion, but this paper 

appears to be largely forgotten because it is not often cited in 

the literature. Bakker and Anderson (2003) also use imagewell theory and allow for a hydraulic gradient in the stream 

and for ambient flow in the aquifer at an arbitrary angle to the 

streambed. The solution is very interesting in that it illustrates 

the impact of pumping a well on the opposite side of the 

stream as solved by Hunt (1999) and schematically illustrated 

in figure 1–1C. In the special case of no hydraulic gradient 

in either the stream or in the aquifer (no ambient flow), the 

solution reduces to a steady-state evaluation of the solution 

by Hunt (1999). The Bakker and Anderson (2003) solution is 

similar in concept to that of Newsom and Wilson (1988) and 

Wilson (1993) except that it considers partial penetration of 

the streambed. The partial penetration of the streambed into 

the aquifer complicates matters and leads to a solution written 

in terms of complex potentials. As in the case of the earlier 

steady-state solutions, this solution is useful to delineate 

ultimate capture zones for wells and to distinguish between 

captured base flow and induced infiltration from the stream to 

the aquifer.

Partially Penetrating Stream With Drawdown in 

the Aquifer Below the Streambed

Several papers in the literature have discussed the importance of unsaturated flow in the analysis of the interaction of a 

stream and an aquifer when the head in the aquifer falls below 
Appendix 1 21

the streambed and the stream becomes disconnected from the 

water table (perched). In all of the solutions presented earlier 

in this review, the head in the aquifer must remain within 

the streambed. In cases where the resistance to flow in either 

the aquifer or the streambed is large enough, the head in the 

aquifer may drop below the streambed in response to pumping. Classically, when the head in the aquifer falls below the 

streambed, the rate of streamflow depletion becomes constant 

and independent of the pumping rate. The rate of streamflow 

depletion depends on the water level in the stream, the elevation of the streambed, and the hydraulic characteristics of the 

streambed (as modeled in MODFLOW, for example (Harbaugh and others, 2000); see also the discussion by Rushton, 

1999). Soil properties under unsaturated flow conditions, 

however, lead to more complicated behavior, and the rate of 

streamflow depletion also depends on the degree of saturation 

of the aquifer material below the streambed, the saturationcapillary pressure characteristics of the aquifer material, and 

the distance between the streambed and the water table.

Peterson and Zhang (2000) use a numerical model for 

unsaturated flow to demonstrate that the capillary pressure 

that develops below a perched stream may become an important consideration to the estimate of streamflow depletion. 

Solutions that use the classic approach described above may 

significantly underestimate the streamflow-depletion rate 

because they do not consider the flow from the stream to the 

aquifer caused by capillary-pressure differences and described 

by unsaturated flow theory. 

Osman and Bruen (2002) recognized this problem and 

proposed a modified algorithm for losing streams for MODFLOW that includes a term to account for unsaturated flow. 

These authors compared the results of the modified MODFLOW model to a full unsaturated-flow model. They report 

that the modified approach performed much better than the 

classic MODFLOW approach compared to the unsaturatedflow model for the water-table configuration under a perched 

stream and for the estimated streamflow depletion. Bruen and 

Osman (2004) used Monte Carlo experiments to investigate 

the importance of unsaturated flow below perched streams and 

the heterogeneity of the underlying aquifer. They report that 

explicit consideration of aquifer heterogeneity compared to 

modeling with average properties increases in importance in 

the estimate of streamflow depletion when the water table is 

well below the streambed.

Fox and Durnford (2003) modify the Hunt (1999) 

solution to allow part of the stream to become perched. The 

segment of the stream that is perched delivers water to the 

unsaturated zone at a rate determined by the unsaturated 

properties of the aquifer, as described by the previous papers. 

Empirical testing allowed Fox and Durnford (2003) to generate an analytical-solution procedure. The authors showed 

that if the stream becomes perched, the drawdown is greater 

and streamflow depletion is less than that predicted by use of 

the Hunt (1999) solution, which does not allow the stream to 

become perched. 

Streamflow Depletion in the Presence of Other 

Water Sources

The only water source for capture by the well to attain 

a steady state for all of the solutions presented thus far is the 

stream. The capture is either captured discharge from the aquifer that would have reached the stream or induced infiltration 

from the stream to the aquifer. The differences between the 

solutions are the rate that the system reaches steady state and 

the configuration of drawdown in the aquifer, such as whether 

the solution predicts drawdown on the side of the stream opposite from the pumping well. Other sources of water, however, 

may exist in some practical situations, and the estimate that 

ultimately streamflow depletion equals the pumping rate will 

not hold in these situations. Several papers present and analyze 

various other water sources (figs. 1–1D and 1–2). 

B

A

d

Qw

d

Qw

EXPLANATION

Qw - pumping rate

d - distance from 

 well to stream

Figure 1–2. Alternate sources of water for capture by a well 

near a stream. A, Stream in an aquifer that overlies a regional 

aquifer with potentially different hydraulic properties. B, Stream 

in an aquifer system where pumping may interact with plants and 

change evapotranspiration losses from the system. 
22 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Hunt (2003a) presents an analysis for a stream in an 

semiconfining unit that overlies a leaky aquifer with a well 

(fig. 1–1D). Water may be released from storage from both the 

leaky aquifer and the semiconfining unit. The water released 

from storage from the semiconfining unit is the additional 

water source in this solution. The solution exhibits delayedyield behavior: after an initial fairly rapid response to pumping, usually considered the result of the expansion of water 

and compression of the matrix, the response curve flattens 

for a time and then sharpens once again. The flattening of 

the response curve in this case is a result of the additional 

water source yielding water to the well without requiring 

additional drawdown in the aquifer. It is the only analytical 

solution to pumping a well near a stream that can fit this type 

of response by an aquifer. After long periods of time, as the 

system approaches a steady state, and the streamflow depletion will again equal the pumping rate. Release of water from 

storage from the semiconfining unit further delays the steady 

state such that the streamflow depletion may be less than the 

pumping rate for practical problems and time frames. In this 

conceptual model, the water in storage in the semiconfining 

unit is not recharged; therefore, the semiconfining unit cannot 

provide a long-term source of water to allow the system to 

reach steady state. This solution is used by several authors 

to analyze field data that exhibit delayed yield (Hunt, 2003b; 

Kollet and Zlotnik, 2003, 2005; Fox, 2004; and Lough, 2005).

Zlotnik (2004) solves a conceptual model that provides 

for a long-term source of water and generates a solution where 

the streamflow depletion rate is not necessarily the pumping rate at steady state. The conceptual model describes a 

stream in a shallow aquifer that overlies a deeper regional 

aquifer (fig. 1–2A). The shallow and regional aquifers may be 

separated by a confining or semiconfining unit. The regional 

aquifer presumably receives recharge that is independent of 

the stream and shallow aquifer and is modeled as maintaining 

a constant peizometric head. The conceptual model used for 

this analytical solution does not consider streambed resistance 

or partial penetration of the streambed into the shallow aquifer. 

The key result is that the streamflow depletion rate is dependent on the hydraulic properties of the shallow aquifer and the 

unit separating the shallow and regional aquifers. The paper 

also presents analysis of different boundary conditions for the 

shallow aquifer and summarizes test cases illustrating this type 

of behavior in the field. 

Two papers investigated in interaction between pumping 

a well near a stream and changes in evapotranspiration losses 

from the aquifer (fig. 1–2B). Darama (2004) used analytical 

models, and Chen and Shu (2006) used the numerical model 

MODFLOW. Both of these papers illustrate how ground 

water may be captured by a reduction of evapotranspiration 

loss as the head in the aquifer is lowered by pumping. This 

evapotranspiration capture decreases the streamflow capture 

required for the system to reach steady state. Chen and Shu 

(2006) summarize two important points regarding this situation that depend on whether the issue is streamflow depletion 

in response to pumping or streamflow recovery in response 

to reductions in pumping. The first point is that the water 

captured by a reduction in evapotranspiration rate causes 

the streamflow depletion to be less than the pumping rate. 

The second point is that if pumping rates are reduced to help 

maintain streamflow, then the potential increase in evapotranspiration rates as the head in the aquifer is increased causes the 

increase in streamflow to be less than the reduction in pumping rate.

Application and Analysis of StreamflowDepletion Models

Several articles and reports have been written that apply 

or analyze streamflow-depletion models. For this review, they 

are classified into two areas: application of the techniques to 

field problems and use of streamflow-depletion models for 

water-resources management. Most of the papers already 

cited provide application of the methods and discussion of the 

results. The papers in this section use the previous solutions or 

numerical methods that have already been summarized.

Application to Field Problems

Christensen (2000) demonstrated how the Hunt (1999) 

solution could be used to estimate streamflow-depletion 

parameters from aquifer-test data. This paper provides a 

detailed derivation of the inverse method and a discussion 

of the uncertainties associated with this type of analysis. 

The paper does not give an example of an application of 

the techniques to field data but refers to work that was later 

published by Nyholm and others (2002, 2003). Nyholm and 

others (2002) analyze two aquifer tests for wells located 60 m 

(200 ft) from a stream in a glacial outwash plain in Denmark. 

As discussed, the authors state that the conditions required 

to use the analytical solution by Hunt (1999) do not hold for 

these particular aquifer tests and use MODFLOW to simulate 

the tests. The inverse modeling code MODFLOWP (Hill, 

1992) was used in this case (although MODFLOW–2000 

should be able to produce similar results). The paper is notable 

in the analysis of the MODFLOWP information and related 

statistical tools and in the detailed discussion of this analysis. 

More detail regarding the tests and analysis of the test results, 

including the analysis of application of the Hunt (1999) solution, is offered by Nyholm and others (2003).

Hunt and others (2001) present an analysis of a field 

experiment that estimated the aquifer transmissivity and 

storage coefficient and the streambed-conductance factor by 

application of the Hunt (1999) solution to field data. (See also 

the discussion of the paper by Kollet and Zlotnik, 2002.) Hunt 

and others (2001) demonstrate the techniques used to estimate 

the aquifer and streambed properties.

Hunt (2003b) presents a short paper comparing application his 1999 model and his 2003 model to data from a field 

experiment. The brief paper demonstrates that the newer solution exhibiting delayed yield fits the field data better than the 
Appendix 1 23

earlier solution, which does not include semiconfined conditions (Hunt, 1999). The newer solution fits over the entire 

time of the test and yields reasonable values for the aquifer 

characteristics.

Kollet and Zlotnik (2003) describe a detailed aquifer test 

near a stream with clusters of piezometers between a pumping well and a stream. They analyzed the data with both the 

Hunt (1999) and the Butler and others (2001) approaches. The 

authors argue that Butler and others’ approach (2001) allows 

for a better fit of model parameters to the measured data 

because this model was better able to capture different hydrogeology at the site. Lough (2005) offers a different analysis 

of the data using the Hunt (2003a) model for a leaky aquifer 

exhibiting delayed yield and states that this approach better fits 

the entire response at the observation wells than does the latetime portion used by Kollet and Zlotnik (2003). Kollet and 

Zlonik (2005) reply that the aquifer is not leaky and suggest 

that the Hunt (1999; 2003a) analytical solutions do not fit the 

experimental data well because the assumption of horizontal 

flow (Dupuit assumption) is violated near the stream, as shown 

in the observation well data.

Chen and Chen (2003a) present another case study 

applying the Hunt (1999) solution to field data to estimate 

streamflow depletion parameters. The authors also present an 

ad hoc sensitivity analysis of the Hunt (1999) solution and an 

evaluation of the reliability of estimates for hypothetical cases. 

This paper received two comments in the literature stating that 

it did not recognize earlier work by Christensen (2000) or the 

application by Nyholm and others (2002). (See Christensen, 

2005; Chen and Chen, 2005a.) Kollet (2005) argues that the 

Hunt (1999) solution was inappropriate for this field test and 

that Chen and Chen (2003a) did not adequately discuss the differences between their results and previously published work 

(see Chen and Chen, 2005b).

Fox (2004) presents another case study and compares the 

result of fitting the Hunt (1999), Butler and others (2001), Fox 

and others (2002), and Hunt (2003a) solutions to field data. In 

this case, the data again show a delayed-yield response, and 

the solution by Hunt (2003a) is claimed to fit the drawdown 

data the best (other fits are not shown). All of the methods 

were stated to adequately fit the late-time data obtained in the 

field, and the transmissivity values reported using each of the 

solutions were reasonable.

Use of Streamflow-Depletion Models for WaterResources Management

The STRMDEPL program was first documented in a 

report describing the application of a precipitation-runoff 

model for the impact of water withdrawals on streamflow by 

Zarriello and Ries (2000). This implementation of the analytical solutions for streamflow depletion from wells was subsequently used in several other USGS studies in New England. 

Representative of these are two other precipitation-runoff 

models (Zarriello and Bent, 2004; Barbaro and Zarriello, 

2007) and a detailed water-use and water-availability analysis (Barlow, 2003). In all four of these studies, STRMDEPL 

(Barlow, 2000) was used to estimate daily streamflow depletion based on daily pumping records from municipal wells. 

Streamflow depletion is delayed in time and attenuated 

compared with pumping records, and the analytical solution 

allowed the precipitation-runoff models to be appropriately 

calibrated to the pumping records. 

A series of papers analyzed the proportion of streamflow 

depletion from induced infiltration from the stream to the 

aquifer and from captured discharge that would have reached 

the stream as base flow under different pumping or aquifer 

conditions (Chen, 2001; Chen and Yin, 2001; Chen and Shu, 

2002; Chen and Chen, 2003b; and Chen and Yin, 2004). These 

papers, in addition to those by Newsom and Wilson (1988), 

Wilson (1993), and Bakker and Anderson (2003), may be used 

to assess water-quality issues arising from induced infiltration 

from the stream to an aquifer. 

 Chen and Yin (2001) and Chen and Shu (2002) use 

MODFLOW to study induced infiltration of stream water to 

an aquifer stressed by seasonally pumped wells. Chen and Shu 

(2002) emphasize streamflow depletion and the analysis of 

the depletion that continues after pumping is stopped. Chen 

(2001) uses particle tracking and a simple analytical model 

(Theis solution with an image well) to examine the capture 

zone of a well. Chen and Chen (2003b) use MODFLOW and 

particle tracking to assess the impact of anisotropy on the 

induced infiltration of stream water. Chen and Yin (2004) use 

the Hunt (1999) solution and apply the concepts illustrated by 

Wilson (1993) to identify the reach of stream where induced 

infiltration from the stream to the aquifer occurs. The ratio of 

capture by induced infiltration to total capture is presented for 

various conditions, and comparison of the analytical model 

results to numerical results generated with MODFLOW are 

presented. As discussed, distinguishing between induced infiltration and base-flow reduction is more important in waterquality analysis than in streamflow-depletion analysis because 

the total streamflow depletion is the same regardless of the 

partitioning between these mechanisms.

The Environment Agency, which manages water 

resources in England and Wales, evaluated potential ways to 

assess ground-water pumping on streamflow. The “Impact 

of Groundwater Abstractions on River Flows (IGARF I)” 

program was discussed by Kirk and Herbert (2002) and in 

technical reports by the Environment Agency (for example, 

Parkin and others, 2002). 

Case studies examining whether reductions in pumping 

will change river flows form the core of a paper by Rushton (2002). The major message of this paper is that natural 

systems are more complicated than those described by most 

models and that careful hydrogeologic investigation and data 

collection are required to estimate whether ground-water 

pumping affects river flow at a given site. Rushton (2002) 

shows how the timing of recharge events, high and low flows, 

and seasonal changes in ground-water levels determine the 

response of the river to reductions in pumping when the river 
24 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

is at low flow. In some cases, Rushton argues, the pumping 

wells are virtually disconnected from the river during low flow 

and reductions in pumping may have only a marginal benefit 

on river flows.

References Cited in Appendix 1

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near a stream with a leaky bed: Ground Water, v. 41, no. 6, 

p. 833–840. 

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model for the Blackstone River Basin, Massachusetts and 

Rhode Island: U.S. Geological Survey Scientific Investigations Report 2006–5213, 71 p.

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Lower Blackstone River Basin, northern Rhode Island and 

south-central Massachusetts, 1995–99: U.S. Geological Survey Water-Resources Investigations Report 03–4190, 75 p.

Barlow, P.M., 2000, Documentation of computer program 

STRMDEPL—A program to calculate streamflow depletion 

by wells using analytical solutions, in Zarriello, P.J., and 

Ries, K.G., III, A precipitation‐runoff model for analysis 

of the effects of water withdrawals on streamflow, Ipswich River Basin, Massachusetts: U.S. Geological Survey 

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stream‐aquifer seepage to spatial variability of the saturated 

hydraulic conductivity of the aquifer: Journal of Hydrology, 

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ground‐water and surface‐water management alternatives in 

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vicinity of a partially penetrating stream: Ground Water, 

v. 39, no. 5, p. 651–659.

Chen, Xunhong, 2001, Migration of induced‐infiltrated stream 

water into nearby aquifers due to seasonal ground water 

withdrawal: Ground Water, v. 39, no. 5, p. 721–728.

Chen, Xunhong, 2003, Analysis of pumping‐induced 

stream‐aquifer interactions for gaining streams: Journal of 

Hydrology, v. 275, no. 1–2, p. 1–11.

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determination of streambed leakance and aquifer hydraulic 

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284.

Chen, Xunhong, and Chen, Xi, 2003b, Effects of aquifer 

anisotropy on the migration of infiltrated stream water to 

a pumping well: Journal of Hydrologic Engineering, v. 8, 

no. 5, p. 287–293.

Chen, Xunhong, and Chen, Xi, 2005a, Reply to comment by 

Steen Christensen on “Sensitivity analysis and determination of streambed leakance and aquifer hydraulic properties”: Journal of Hydrology, v. 303, no. 1–4, p. 322–327.

Chen, Xunhong, and Chen, Xi, 2005b, Reply to comments by 

Stefan Kollet on “Sensitivity analysis and determination 

of streambed leakance and aquifer hydraulic properties”: 

Journal of Hydrology, v. 303, no. 1–4, p. 331–333.

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interactions—Evaluation of depletion volume and residual 

effects from ground water pumping: Ground Water, v. 40, 

no. 3, p. 284–290.

Chen, Xunhong, and Shu, Longcang, 2006, Groundwater 

evapotranspiration captured by seasonally pumped wells 

in river valleys: Journal of Hydrology, v. 318, no. 1–4, 

p. 334–347.

Chen, Xunhong, and Yin, Yanfeng, 2001, Streamflow depletion—Modeling of reduced baseflow and induced stream 

infiltration from seasonally pumped wells: Journal of 

the American Water Resources Association, v. 37, no. 1, 

p. 185–195.

Chen, Xunhong, and Yin, Yanfeng, 2004, Semianalytical solutions for stream depletion in partially penetrating streams: 

Ground Water, v. 42, no. 1, p. 92–96.

Christensen, Steen, 2000, On the estimation of stream flow 

depletion parameters by drawdown analysis: Ground Water, 

v. 38, no. 5, p. 726–734.

Christensen, Steen, 2005, Comment on “Sensitivity analysis and determination of streambed leakance and aquifer 

hydraulic properties,” by X. Chen and X. Chen: Journal of 

Hydrology, v. 303, no. 1–4, p. 316–321.

Conrad, L.P., and Beljin, M.S., 1996, Evaluation of an induced 

infiltration model as applied to glacial aquifer systems: 

Water Resources Bulletin, v. 32, no. 6, p. 1209–1220.

Darama, Yakup, 2001, An analytical solution for stream depletion by cyclic pumping of wells near streams with semipervious beds: Ground Water, v. 39, no. 1, p. 79–86.

Darama, Yakup, 2004, Stream depletion by a pumping well 

including the effects of nonlinear variation of captured 

evaporation from a phreatic aquifer: Hydrological Sciences 

Journal, v. 49, no. 3, p. 443–460.
Appendix 1 25

Di Matteo, Lucio, and Dragoni, Walter, 2005a, Empirical 

relationships for estimating stream depletion by a well 

pumping near a gaining stream: Ground Water, v. 43, no. 2, 

p. 242–249.

Di Matteo, Lucio, and Dragoni, Walter, 2005b, Erratum—

Empirical relationships for estimating stream depletion by 

a well pumping near a gaining stream, Ground Water, 2005, 

v. 43, no. 2, p. 242–249: Ground Water, v. 43, no. 6, p. 772.

Ernst, L.F., 1979, Groundwater flow to a deep well near a 

rectilinear channel: Journal of Hydrology, v. 42, no. 1–2, 

p. 129–146.

Fox, G.A., 2004, Evaluation of a stream aquifer analysis 

test using analytical solutions and field data: Journal of 

the American Water Resources Association, v. 40, no. 3, 

p. 755–763.

Fox, G.A., DuChateau, Paul, and Durnford, D.S., 2002, Analytical model for aquifer response incorporating distributed 

stream leakage: Ground Water, v. 40, no. 4, p. 378–384.

Fox, G.A., and Durnford, D.S., 2003, Unsaturated hyporheic 

zone flow in stream/aquifer conjunctive systems: Advances 

in Water Resources, v. 26, no. 9, p. 989–1000.

Glover, R.E., 1974, Transient ground water hydraulics: Fort 

Collins, Colo., Colorado State University, Dept. of Civil 

Engineering, 413 p., as cited in Sophocleous, Marios; 

Koussis, Antonis; Martin, J.L.; and Perkins, S.P., 1995, 

Evaluation of simplified stream aquifer depletion models 

for water rights administration: Ground Water, v. 33, no. 4, 

p. 579–599.

Glover, R.E., and Balmer, G.G., 1954, River depletion resulting from pumping a well near a river: Transactions, American Geophysical Union, v. 35, no. 3, p. 468–470.

Hantush, M.S., 1965, Wells near streams with semipervious beds: Journal of Geophysical Research, v. 70, no. 12, 

p. 2829–2838.

Harbaugh, A.W., Banta, E.R., Hill, M.C., and McDonald, 

M.G., 2000, MODFLOW–2000, the U.S. Geological Survey modular ground‐water model—User guide to modularization concepts and the ground‐water flow process: 

U.S. Geological Survey Open‐File Report 00–92, 121 p.

Hill, M.C., 1992, A computer program (MODFLOWP) for 

estimating parameters of a transient, three‐dimensional, 

ground‐water flow model using nonlinear regression: 

U.S. Geological Survey Open‐File Report 91–484, 358 p.

Hunt, Bruce, 1999, Unsteady stream depletion from ground 

water pumping: Ground Water, v. 37, no. 1, p. 98–102.

Hunt, Bruce, 2003a, Unsteady stream depletion when pumping 

from semiconfined aquifer: Journal of Hydrologic Engineering, v. 8, no. 1, p. 12–19.

Hunt, Bruce, 2003b, Field‐data analysis for stream depletion: 

Journal of Hydrologic Engineering, v. 8, no. 4, p. 222–225.

Hunt, Bruce, 2004, Discussion of “Analytical model for 

aquifer response incorporating distributed stream leakage,” 

by Garey A. Fox, Paul DuChateau, and Deana S. Durnford, 

Ground Water, 2002, v. 40, no. 4, p. 378–384: Ground 

Water, v. 42, no. 1, p. 139–140; author reply, p. 140.

Hunt, Bruce, Weir, Julian, and Clausen, Bente, 2001, A stream 

depletion field experiment: Ground Water, v. 39, no. 2, 

p. 283–289.

Jacob, C.E., 1950, Flow of groundwater, in Rouse, H., ed., 

Engineering hydraulics: New York, John Wiley, as cited in 

Spalding, C.P., and Khaleel, Raziuddin, 1991, An evaluation 

of analytical solutions to estimate drawdowns and stream 

depletions by wells: Water Resources Research, v. 27, no. 4, 

p. 597– 609.

Jenkins, C.T., 1968, Computation of rate and volume of stream 

depletion by wells: U.S. Geological Survey Techniques of 

Water‐Resources Investigations, book 4, chap. D1, 17 p.

Kirk, Stuart, and Herbert, A.W., 2002, Assessing the impact of 

groundwater abstractions on river flows, in Hiscock, K.M., 

Rivett, M.O., and Davison, R.M., eds., Sustainable groundwater development: London, Geological Society, Special 

Publication 193, p. 211–233. 

Kollet, Stefan, 2005, Comment on “Sensitivity analysis and 

determination of streambed leakance and aquifer hydraulic properties” by Xunhong Chen and Xi Chen, Journal of 

Hydrology, 2003, v. 284, no. 1–4, p. 270–284: Journal of 

Hydrology, v. 303, no. 1–4, p. 328–330.

Kollet, S.J., and Zlotnik, V.A., 2002, Discussion of “A stream 

depletion field experiment,” by Bruce Hunt, Julian Weir, 

and Bente Clausen, Ground Water, 2001, v. 39, no. 2, 

p. 283–289: Ground Water, v. 40, no. 4, p. 448–449; author 

reply, p. 449.

Kollet, S.J., and Zlotnik, V.A., 2003, Stream depletion 

predictions using pumping test data from a heterogeneous 

stream‐aquifer system (a case study from the Great Plains, 

USA): Journal of Hydrology, v. 281, no. 1–2, p. 96–114.

Kollet, S.J., and Zlotnik, V.A., 2005, Reply to comment by 

H. Lough, Department of Civil Engineering, University 

of Canterbury, Christchurch, New Zealand, on the paper 

“Stream depletion predictions using pumping test data from 

a heterogeneous stream‐aquifer system (a case study from 

the Great Plains, USA)” by S.J. Kollet and V.A. Zlotnik, 

Journal of Hydrology, v. 281, no. 1–2, p. 96–114: Journal of 

Hydrology, v. 313, no. 3–4, p. 149–152.
26 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Lough, Hilary, 2005, Comment on paper “Stream depletion 

predictions using pumping test data from a heterogeneous 

stream‐aquifer system (a case study from the Great Plains, 

USA)” by Stefan J. Kollet and Vitaly A. Zlotnik, Journal of 

Hydrology, v. 281, no. 1–2, p. 96–114: Journal of Hydrology, v. 313, no. 3–4, p. 153–157.

Newsom, J.M., and Wilson, J.L., 1988, Flow of ground water 

to a well near a stream—Effect of ambient ground–water 

flow direction: Ground Water, v. 26, no. 6, p. 703–711.

Nyholm, Thomas, Christensen, Steen, and Rasmussen, K.R., 

2002, Flow depletion in a small stream caused by ground 

water abstraction from wells: Ground Water, v. 40, no. 4, 

p. 425–437.

Nyholm, Thomas, Rasmussen, K.R., and Christensen, Steen, 

2003, Estimation of stream flow depletion and uncertainty 

from discharge measurements in a small alluvial stream: 

Journal of Hydrology, v. 274, no. 1–4, p. 129–144.

Osman, Y.Z., and Bruen, M.P., 2002, Modelling stream‐aquifer 

seepage in an alluvial aquifer—An improved losing‐stream 

package for MODFLOW: Journal of Hydrology, v. 264, 

no. 1–4, p. 69–86.

Parkin, G., Birkinshaw, S., Rao, Z., Murray, M., and Younger, 

P.L., 2002, Impact of groundwater abstractions on river 

flows—Phase 2—A numerical modelling approach to the 

estimation of impact (IGARF II): Environment Agency 

R&D User Manual W6‐046/M, 30 p.

Peterson, D.M., and Zhang, H., 2000, Simulation of stream 

losses affected by the vadose zone, in Bentley, L.R., Sykes, 

J.F., Brebbia, C.A., Gray, W.G., and Pinder, G.F., eds., 

Computational methods in water resources—Volume 2— 

Computational methods, surface water systems and hydrology, Proceedings of the 13th International Conference 

on Computational Methods in Water Resources, Calgary, 

Alberta, Canada, June 2000: Rotterdam, A.A. Balkema, 

p. 1131–1138. 

Reeves, H.W., 2008, STRMDEPL08—An Extended Version 

of STRMDEPL with Additional Analytical Solutions to 

Calculate Streamflow Depletion by Nearby Pumping Wells: 

U.S. Geological Survey Open-File Report 2008–1166, 22 p.

Rushton, Ken, 1999, Discussion of “Unsteady stream depletion from ground water pumping” by B. Hunt, Ground 

Water, 1999, v. 37, no. 1, p. 98–102: Ground Water, v. 37, 

no. 6, p. 805.

Rushton, K.R., 2002, Will reductions in groundwater abstractions improve low river flows?, in Hiscock, K.M., Rivett, 

M.O., and Davison, R.M., eds., Sustainable groundwater 

development: London, Geological Society, Special Publication 193, p. 199–210. 

Singh, S.K., 2003, Flow depletion of semipervious streams 

due to pumping: Journal of Irrigation and Drainage Engineering, v. 129, no. 6, p. 449–453.

Sophocleous, Marios, Koussis, Antonis, Martin, J.L., and 

Perkins, S.P., 1995, Evaluation of simplified stream‐aquifer 

depletion models for water rights administration: Ground 

Water, v. 33, no. 4, p. 579–588.

Spalding, C.P., and Khaleel, Raziuddin, 1991, An evaluation 

of analytical solutions to estimate drawdowns and stream 

depletions by wells: Water Resources Research, v. 27, no. 4, 

p. 597–609.

Theis, C.V., 1941, The effect of a well on the flow of a nearby 

stream: Transactions, American Geophysical Union, v. 3, 

p. 734–738.

Wallace, R.B., Darama, Yakup, and Annable, M.D., 1990, 

Stream depletion by cyclic pumping of wells: Water 

Resources Research, v. 26, no. 6, p. 1263–1270.

Wilson, J.L., 1993, Induced infiltration in aquifers with ambient flow: Water Resources Research, v. 29, no. 10, p. 3503–

3512.

Zarriello, P.J., and Bent, G.C., 2004, A precipitation-runoff 

model for the analysis of the effects of water withdrawals 

and land-use change on streamflow in the UsquepaugQueen River Basin, Rhode Island: U.S. Geological Survey 

Scientific Investigations Report 2004–5139, 75 p.

Zarriello, P.J., and Ries, K.G., III, 2000, A precipitation‐runoff 

model for analysis of the effects of water withdrawals on 

streamflow, Ipswich River Basin, Massachusetts: U.S. Geological Survey Water‐Resources Investigations Report 

00–4029, 99 p.

Zlotnik, V.A., 2004, A concept of maximum stream depletion 

rate for leaky aquifers in alluvial valleys: Water Resources 

Research, v. 40, no. 6, p. W065071–W065079.

Zlotnik, V.A., and Huang, Huihua, 1999, Effect of shallow 

penetration and streambed sediments on aquifer response to 

stream stage fluctuations (analytical model): Ground Water, 

v. 37, no. 4, p. 599–605. 
Appendix 2 27

Appendix 2. Distribution of Withdrawal Between Neighboring Watersheds 

A major aspect of the screening tool is the distribution of 

pumping between several neighboring valley segments. The 

simplest approach to estimate the impact of a proposed well 

on the stream network is to treat the withdrawal as a surfacewater withdrawal and assume that the streamflow depletion 

due to pumping is immediate in time and entirely on the valley 

segment that drains the catchment containing the well. Use of 

the Hunt (1999) analytical model relaxes one of these assumptions and accounts for the delay in streamflow depletion due 

to removal of water from storage in the aquifer. All of the 

removal may be assigned to the valley segment in the catchment containing the well with no removal from other valley 

segments, but this approach is not physically realistic because 

a real well may capture water from adjacent valley segments. 

Because of the potentially complex geometry of the stream 

network, numerical ground-water-flow modeling would be 

required to rigorously estimate the capture from each valley 

segment. Keeping with the screening nature of the analysis; 

however, an approach to use an analytical solution was sought.

Wilson (1993) presented an analysis of streamflow depletion by pumping wells considering ambient flow to the stream. 

This analysis focused on steady-state solutions of induced 

infiltration from the stream to identify whether a well induced 

flow from the stream to the aquifer or whether the stream 

remained a gaining stream. In the latter case, pumping lowers 

the potentiometric level in the aquifer and reduces discharge to 

the stream but does not induce flow from the stream. This situation also may be described as capturing recharge that would 

have reached the stream in the absence of pumping. A solution 

is presented for a well between two parallel gaining streams 

(fig. 2–1). 

The solution (Wilson, 1993) can be written as 

Q x q Q

x

L

d

L

s a = − + w

− 









































 − 2

1

1 ' cos

cosh ' cos

p

p p

cosh ' cos p p x

L

d

L























− 











































































, (2–1)

where

d is the distance from the well to one of two 

streams (length), 

x’ is the distance along the stream from the point 

opposite the pumping well to the end of 

the zone of induced infiltration from the 

stream (length),

cosh () is the hyperbolic cosine function 

(dimensionless),

	 qa is the ambient flow between the two streams 

per unit length of stream (length squared 

per unit time), and 

L is the distance between the two parallel 

streams (length).

Points xs’ and xs are known as the stagnation points, and 

induced infiltration from the stream occurs along the distance 

-x’ to x’ (fig. 2–1).

Figure 2–1. Pumping well between two parallel streams with 

notation used in the Wilson (1993) equation for steady-state 

streamflow depletion. 

Well

L

• d

x'

-x'

Stream 1 Stream 2

• xs'

•

xs

The ambient flow per unit length of stream, qa, may be 

written as 

q

NL

L a = + −2

Φ Φ 2 1 , (2–2)

where

N is the recharge rate (length per unit time),

Fi is the state variable (length cubed per unit 

time) at stream i defined by,

Φi =Thi for a confined aquifer, or

Φ = Khi

2 for an unconfined aquifer,

T is the transmissivity of the aquifer (length 

squared per unit time), and

K is the hydraulic conductivity of the aquifer 

(length per unit time).
28 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

The stagnation point, xs, is found by solving 

Q

Lq

x

L w

a

S

2

2

2 =























 cosh cos( ) −

sin( )

π δ

δ , (2–3)

where

δ π = d

2L

. (2–4)

Details regarding the results from this solution are given 

by Wilson (1993). In summary, at low pumping rates, the well 

does not induce flow from either stream, and all of the well 

capture at steady state is from intercepted recharge. Determination of the percentage of this capture from recharge as 

diminishment of streamflow for each stream can be determined only through numerical ground-water-flow modeling 

(Wilson, 1993). As the pumping rate is increased, the well 

induces flow from the nearer stream and then from the second 

stream. At even higher pumping rates, as defined by the 

distance between the streams and the ambient flow between 

the streams controlled by recharge and aquifer conditions 

as shown in equation 2–3, the source of water to the well is 

dominated by induced flow from the two streams. At these 

higher pumping rates, the proportion of water induced by the 

pumping well from each stream depends only on the inverse of 

the distance from the well to each stream (Wilson, 1993). This 

last observation is used to guide the analysis for application of 

the analytical solution to multiple streams. 

Distribution Methods Tested 

Several methods based on either distances between a proposed well and the streams in the network or on the areas of 

the surface-water catchments associated with a proposed well, 

as approximations to the ground-water catchments, were tested 

to determine the most appropriate method to distribute water 

withdrawals between adjacent catchments in the screening 

tool. The methods distribute streamflow capture among adjacent or neighboring catchments defined as the valley-segment 

catchments that touch the boundary of the valley-segment 

catchment containing the proposed pumping well. Nine distribution methods were tested, and each method was computed 

by means of Python scripts that use geoprocessing commands 

(Tucker, 2005; van Rossum and Drake, 2006). The methods 

were inverse distance from valley segment, inverse distance 

from valley segment squared, inverse distance from catchment 

center, inverse distance from catchment center squared, transmissivity weighted, transmissivity weighted squared, natural 

neighbor, buffer around well, and no weighting. No weighting 

is the solution when all the streamflow depletion is assigned to 

the valley segment associated with the catchment containing 

the well.

Inverse Distance From Valley Segment

One distribution method, which is a direct extension of 

the analysis by Wilson (1993), is to assume that the distribution is directly proportional to the inverse distance from the 

pumping well to each stream. The contribution of any valleysegment catchment, i, is described by

f d

d

i

i

m

j

m j n

=

∑=

1

1

1,

, (2–5)

where 

f

i is the fraction of the pumping attributed to 

valley segment i, 

m is a factor to adjust weighting,

n is the number of adjacent valley-segment 

catchments, and 

di is the distance from the proposed well to the 

closest point on the valley segment within 

catchment i. 

To allow for close points to have greater influence on 

the contribution percentages, the factor m was included in 

the algorithm. Two methods of inverse distance weighting 

were tested: linear, m=1; and squared, m=2. Inverse distance 

squared weighting gives greater contribution to close valley 

segments compared to valley segments further from the proposed well than the linear inverse distance weighting.

Inverse Distance From Catchment Center

The philosophy of the Michigan statewide valleysegment framework is to group short arcs identified in the 

NHD dataset by hydrology and ecology into larger units. 

These larger units are the valley segments. Often, catchments 

associated with the valley segments contain several arcs for 

the same stream or arcs from different similar streams that join 

within the catchment. A second weighting scheme is based 

on this philosophy in that it uses the distance from the well to 

the adjacent valley-segment catchment centers to compute the 

contribution for each valley segment: 

f l

l

i

i

m

j

m j n

=

∑=

1

1

1,

, (2–6)

where

l

i is the distance from the proposed well to the 

center of the valley-segment catchment i. 

As in the previous case, inverse distance squared weighting also is implemented.
Appendix 2 29

Transmissivity Weighted

The final distance-based weighting scheme uses the 

approach for computing the equivalent vertical hydraulic 

conductivity in a layered system. In this case, the transmissivity assigned to each valley-segment catchment also is used 

to weight the contribution from the catchment in combination 

with the inverse distance between the pumping well and the 

valley segment. Valley-segment catchments with low transmissivity will contribute less than adjacent catchments with 

higher transmissivity. The contribution for valley-segment 

catchment i is given by 

f

T

d

T

d

i

i

i

m

j

j

m j n

=

∑=1,

, (2–7)

where 

Ti is the transmissivity assigned to valleysegment catchment i. 

Again, inverse distance squared weighting is implemented for the testing.

Buffer Around Well

In contrast to the distance-based weighting schemes, 

two area-based schemes were developed. These methods 

were tested as options that may reproduce the interception of 

recharge by the pumping well better than the distance-based 

weightings. The first uses a circular buffer around the well to 

identify the potentially contributing valley-segment catchments. The fraction of the buffer area attributed to each catchment is used to determine the contribution for each valleysegment catchment. For this method, the size of the buffer is 

important. If the buffer is very small, then all the weight is 

assigned to the valley-segment catchment containing the well 

unless the well is quite close to the catchment boundary. If 

the buffer is too large, too many neighboring valley segments 

are assigned a fraction of the streamflow depletion, and the 

contribution from the catchment containing the well may be 

too low. To be consistent with the statewide screening, and to 

allow the hydrogeology of the setting to influence the estimate, 

a simple well drawdown computation with ad hoc values for 

drawdown, time, pumping, and storage coefficient was done 

to determine the buffer radius used in the computations. The 

Theis solution was used: 

s Q

T

W u u Sr

Tt = = 4 4

2

p ( ) , . (2–8)

The equation was solved by bisection for each valleysegment catchment to determine the radius, r, for a drawdown, 

s, of 0.1 ft after 5 years of pumping, t, at 70 gal/min, Q, given 

the transmissivity, T, assigned to the valley-segment catchment in the screening tool, and a storage coefficient, S, of 0.1.

The contribution for each valley-segment catchment is 

given by 

f a

a i

i

j j n

= ∑=1,

, (2-9)

where 

n is the number of adjacent valley-segment 

catchments, and 

ai is the area of the buffer enclosed by valleysegment catchment i. 

Natural Neighbor 

The final method uses natural neighbor weighting to 

compute the contribution from the adjacent valley-segment 

catchments. Natural neighbor weighting uses Theissen polygons to define the natural neighbors of the proposed well and 

to compute areal weighting for the well (Sibson, 1980, 1981). 

The points defining the Theissen polygons are the closest 

points from the well to the valley segments that are used in 

the analytical solution. Natural neighbor weighting is computed by first generating a set of Theissen polygons for these 

points. A second set of Theissen polygons is then created with 

the closest valley-segment points and the well location. The 

overlap between these two sets of polygons is used to estimate 

the weighting assigned to each of the valley-segment points. 

If the new polygon generated around the well does not overlap 

with some of the original Theissen polygons, then the weight 

assigned to the points in these polygons is zero. If the well 

falls exactly on one of the valley-segment points, which means 

that the well is placed on a valley segment, the polygon generated in the second step exactly overlaps one polygon from the 

first set and the weight assigned to this point is 1. The contribution for each valley-segment catchment is given by 

f a

a i

i

j j n

= ∑=1,

, (2–10)

where 

ai is the area of influence determined by means 

of the overlapping areas of the Thiessen 

polygons for each of the natural neighbors.

The analytical model by Hunt (1999) was used to compute the time-dependent streamflow depletion due to a pumping well. This model provides the fraction of the pumping rate 

attributed to removal from storage and to streamflow capture. 

The pumping from a proposed well is distributed between 

adjacent valley segments by estimating a contribution for each 

valley-segment catchment. These contributions are assumed to 

be valid for the entire pumping time for the well. To estimate 

streamflow depletion for each of the valley segments, the 

analytical solution is computed for the well and each valley 

segment independently. These solutions are multiplied by a 

fraction representing the contribution to the pumping well for 
30 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

each valley-segment catchment. The sum of the contributions 

from each valley segment must equal 1 so that the sum of the 

streamflow depletions for all the contributing valley-segment 

catchments is equal to the pumping rate as the system reaches 

steady state. To compute the analytical solution for each valley 

segment, the distance between the closest point on the valley 

segment and the proposed well is determined. This distance 

is used along with the aquifer properties of the catchment 

containing the well to compute the streamflow depletion with 

time. 

Testing Procedure

The ground-water-flow model for the area around Kalamazoo County, Mich. (Luukkonen and others, 2004) was used to 

test the distribution methods. The ground-water-flow model is 

a finite-difference (MODFLOW) simulation model with 6 layers, 154 rows, and 162 columns. MODFLOW is documented 

by Harbaugh and others (2000). The smallest grid spacing 

in the model is 660 ft by 660 ft. Valley-segment catchments 

in the interior of the model that have this finest grid spacing 

were used in the methods testing. To illustrate the applicability 

of the analytical model to field problems, a transient simulation with seasonally varying recharge and pumping rates was 

used as the test case. Luukkonen and others (2004) presented 

the results for a 9-year transient simulation; but, in this work, 

the length of the simulation was reduced to 5 years to reduce 

computer run time.

A method to illustrate streamflow capture by pumping 

wells (Leake and Reeves, 2008) was used to examine the 

spatial distribution of streamflow capture from the valley 

segments. In this method, hypothetical wells are added to the 

MODFLOW model sequentially on a grid of cells (fig. 2–2). 

One well is added to a specified cell, and the new simulation is 

computed. The results from this new simulation are subtracted 

from base-model results to determine the capture and change 

in storage caused by the new well. The process is repeated 

with one new well at a time, generating a grid of results that 

may be mapped. The fluxes to individual river cells representing the valley segments from the statewide data framework 

were computed. These fluxes were recorded by use of a river 

observation file (Hill and others, 2000) to designate the finitedifference cells assigned as river cells to the appropriate valley 

segments. The river flux output from the system with the new 

well for each designated valley segment was subtracted from 

the valley-segment flux computed with the original model to 

yield the streamflow depletion due to the new well as simulated with the MODFLOW model. Values for streamflow 

depletion as a percentage of pumping rate for the grid of 

wells shown in figure 2–2 placed in layer 3 of the model were 

interpolated by means of inverse distance weighing to produce 

the map shown in figure 2–3. These streamflow depletions and 

the spatial pattern produced through this technique are used 

to access the distribution schemes applied with the analytical 

model.

The analytical model was solved for the same grid of 

points as the MODFLOW model illustrated in figure 2–2. 

Python scripts for the distribution methods used geoprocessing commands (Tucker, 2005; van Rossum and Drake, 2006) 

to identify the catchment containing the well, to identify the 

neighboring catchments, and to compute the different weightings. The analytical solution was computed for the well and 

each of the nearby valley segments sequentially by use of 

the transmissivity assigned to the valley-segment catchment 

containing the test point, which for these trials was always 

valley-segment catchment 3544. The storage coefficient and 

streamflow-conductance terms for this catchment from the 

water-withdrawal screening tool also were used. 

The streamflow depletion for each valley segment and 

each distribution method was estimated by multiplying the 

analytical solution for that valley segment-test point combination by the computed percent contribution for the valleysegment catchment. The resulting streamflow depletions were 

compared to the depletions computed with the MODFLOW 

model through the river observation file (Hill and others, 

2000) for each valley segment. The results from the groundwater-flow model and the analytical solutions after 5 years 

were compared to ensure that differences in the solutions 

resulting from differences in the storage coefficient used in the 

screening tool and the MODFLOW model do not dominate the 

analysis. To avoid problems with differences between small 

estimates of streamflow depletion biasing the analysis, the 

difference between the MODFLOW results and the analytical model was computed only if either the MODFLOW or the 

analytical model estimated a streamflow depletion of 5 percent 

of the pumping rate or greater. 

The simplest approach for the screening procedure 

would be the no-weighting approach. The capture caused 

by a proposed well would be assigned to the valley segment 

in the catchment containing the well. In essence, use of this 

approach assumes that the ground-water divides are coincident with surface-water divides and that the divides do not 

change in response to pumping. Because ground-water divides 

will respond to pumping, this should be a poor assumption. 

Testing reveals that streamflow depletion was overestimated 

in this approach compared to the MODFLOW model results 

(figs. 2–3 and 2–4). The minimum streamflow depletion estimated by way of this approach is approximately 63 percent, 

and the estimated streamflow depletion is greater than 80 percent for much of the valley-segment catchment. These results 

do not compare well with results generated using the MODFLOW model. For the MODFLOW model, the maximum 

estimated streamflow depletion for the valley segment was 

less than 80 percent for most of the catchment and it was less 

than 20 percent for parts of the valley-segment catchment near 

the most distant surface-water divides defining the catchment. 

On the basis of this comparison, the no-weighting approach—

although computationally the most straightforward method—

was rejected as an option for the screening model.
Appendix 2 31

EXPLANATION

Michigan valley-segment catchments

Finite-difference grid used in Kalamazoo 

regional ground-water-flow model

Michigan valley segments

Points used for sequential introduction of test wells

42°16'

42°12'

85°34' 85°30'

0 0.5 1 MILE

0 0.5 1 KILOMETER

Michigan

Finite-difference grid

Kalamazoo, Mich.

Base watershed and streams from Brenden and others, 2006

Figure 2–2. A grid of points used to compare the analytical model and different distribution 

methods to MODFLOW results from the Kalamazoo regional ground-water-flow model. 
32 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

Figure 2–3. Estimated streamflow depletion from valley segment 3544 resulting from a sequential 

introduction of test wells in the MODFLOW ground-water-flow model for Kalamazoo County, Mich., 

as a percentage of pumping rate after 5 years of pumping.

0 1 MILE

0 1 KILOMETER

EXPLANATION

Valley-segment catchment 3544

Valley segment 3544

Streamflow depletion, in percent

6.8 - 10

10.1 - 20

20.1 - 30

30.1 - 40

40.1 - 50

50.1 - 60

60.1 - 70

70.1 - 80

80.1 - 90

90.1 - 100

Base watershed and streams from Brenden and others, 2006

Michigan

Michigan valley

segment catchment 3544

Kalamazoo, Mich.

-
Appendix 2 33

0 1 MILE

0 1 KILOMETER

Base watershed and streams from Brenden and others, 2006

EXPLANATION

Valley-segment catchment 3544

Valley segment 3544

Streamflow depletion, in percent

6.8 - 10

10.1 - 20

20.1 - 30

30.1 - 40

40.1 - 50

50.1 - 60

60.1 - 70

70.1 - 80

80.1 - 90

90.1 - 100

Michigan

Michigan valley

segment catchment 3544

Kalamazoo, Mich.

-

Figure 2–4. Estimated streamflow depletion from valley segment 3544 resulting from a sequential 

application of the analytical model to test wells with no weighting approach between adjacent 

valley segments as a percentage of pumping rate after 5 years of pumping.
34 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

The distribution approaches that matched the MODFLOW results the best were the inverse distance to valley 

segment, inverse distance to valley segment squared, and 

natural neighbor weightings (fig. 2–5). The inverse distance to 

valley segment squared and natural neighbor approaches overestimate streamflow depletion compared to the MODFLOW 

results, and the inverse distance to valley segment weighting 

tends to underestimate the depletion near the stream compared to the MODFLOW results. The transmissivity-weighted 

approaches (not shown) are not significantly better than the 

inverse distance to valley segment weightings and require 

slightly more computational effort and more book-keeping to 

gather and use the transmissivity of each neighboring valleysegment catchment. The buffer weighting and distance to 

center weighting (also not shown) did not produce interpolated 

maps that were reasonable matches to the MODFLOW results, 

and these methods were rejected. Statistical and ranked comparisons of the weighting methods based on the differences 

between each method and the MODFLOW results showed that 

each of the methods performed the best for some of the hypothetical well points and were outperformed at others. None of 

the methods produced a statistically significant difference in 

error characteristics compared with the MODFLOW results to 

help guide the decision. As a result of this testing, the inverse 

distance to the valley segment method was selected for the 

screening tool because it produces a reasonable overall pattern 

of streamflow depletion compared with the MODFLOW tests, 

it is the most straightforward to implement in the Web application, and it has some theoretical basis in Wilson’s analysis 

(1993).
Appendix 2 35

A B

C D

EXPLANATION

Streamflow depletion, in percent

4.6 - 10

10.1 - 20

20.1 - 30

30.1 - 40

40.1 - 50

50.1 - 60

60.1 - 70

70.1 - 80

80.1 - 90

90.1 - 100

0 1 MILES

0 0.8 KILOMETERS

Michigan

Michigan valley

segment catchment 3544

Kalamazoo, Mich.

-

Base watershed from Brenden and others, 2006

Figure 2–5. Estimated streamflow depletion from valley segment 3544 as a percentage of pumping rate after 5 years of 

pumping. A, results from a sequential application MODFLOW. Results from sequential application of the analytical model 

with B, the inverse distance to the stream, C, inverse distance to the stream squared, and D, natural neighbor distribution 

methods. 
36 Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool

References Cited in Appendix 2

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Wang, Lizhu, Aichele, S.S., Bissell, E.G., and Stewart, 

J.S., 2006, A GIS framework for collecting, managing, 

and analyzing multi-scale landscape variables across large 

regions for river conservation and management, in Hughes, 

R.M., Wang, Lizhu, and Seelbach, P.W., eds., Landscape 

influences on stream habitats and biological assemblages, 

Proceedings, American Fisheries Society Symposium, 

Madison, Wis., August 25–26, 2004: Bethesda, Md., American Fisheries Society Symposium 48, p. 49–74.

Harbaugh, A.W., Banta, E.R., Hill, M.C., and McDonald, 

M.G., 2000, MODFLOW–2000, the U.S. Geological Survey modular ground‐water model—User guide to modularization concepts and the ground‐water flow process: 

U.S. Geological Survey Open‐File Report 00–92, 121 p.

Hill, M.C., Banta, E.R., Harbaugh, A.W., and Anderman, 

E.R., 2000, MODFLOW-2000, the U.S. Geological Survey 

modular ground-water model—User guide to the observation, sensitivity, and parameter-estimation processes and 

three post-processing programs: U.S. Geological Survey 

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map potential capture of surface water by ground-water 

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p. 204–208.

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Scientific Investigations Report 2004–5054, 65 p.

Sibson, Robin, 1980, A vector identity for the Dirichlet tessellation: Mathematical Proceedings of the Cambridge 

Philosophical Society, v. 87, pt. 1, p. 151–155.

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interpolation, in Barnett, Vic, ed., Interpreting multivariate 

data: Chichester, U.K., John Wiley & Sons, p. 21–36.

Tucker, Corey, 2005, ArcGIS 9—Writing geoprocessing 

scripts with ArcGIS: Redlands, Calif., ESRI, 91 p.

van Rossum, Guido, and Drake, F.L., Jr., eds., 2006, Python 

reference manual, release 2.5: Python Software Foundation, 

accessed November 21, 2007, at http://docs/python.org/

index.html

Wilson, J.L., 1993, Induced infiltration in aquifers with ambient flow: Water Resources Research, v. 29, no. 10, p. 3503–

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Reeves and others—Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool—Scientific Investigations Report 2009–5003