United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada, OK 74820
Research and Development
EPA/600/SR-94/210
EPA Project Summary
March 1995
Demonstration of the Analytic
Element Method for Wellhead
Protection
Hendrik M. Haitjema, Otto D.L. Strack, and Stephen R. Kraemer
A new computer program has been
developed to determine time-of-travel
capture zones In relatively simple geo-
hydrological settings. The WhAEM
package contains an analytic element
model that uses superposition of
(many) closed form analytical solutions
to generate a ground-water flow solu-
tion. W/iAEM consists of two
executables: the preprocessor GAEP,
and the flow model CZAEM. W/iAEM
differs from existing analytical models
in that it can handle fairly realistic
boundary conditions such as streams,
lakes, and aquifer recharge due to pre-
cipitation.
The preprocessor GAEP is designed
to simplify the procedures for getting
data into a ground-water model; spe-
cifically it facilitates the interactive pro-
cess of ground-water flow modeling
that precedes capture zone delineation.
The flow model CZAEM is equipped
with a novel algorithm to accurately
define capture zone boundaries by first
determining all stagnation points and
dividing streamlines in the flow domain.
No models currently in use for well-
head protection contain such an algo-
rithm.
This Project Summary was developed
by USEPA's Robert S. Kerr Environ-
mental Research Laboratory, Ada, OK,
to announce key findings of the Re-
search Project that Is fully documented
In separate reports and supporting soft-
ware (See Project Report ordering In-
formation at the back).
Introduction
The delineation of capture zones re-
quires precise determination of stream-
lines. In most numerical methods, such
accurate determination is difficult because
the velocities are computed on the basis
of values of piezometric heads that are
known only at the nodes of a mesh. This
deficiency stimulated the development of
a number of computer models which imple-
ment elementary analytic solutions for
ground-water flow problems. These ana-
lytic models are capable of producing more
or less approximate shapes of the bound-
aries of capture zones for any given time;
e.g., the USEPA's original wellhead pro-
tection model WHPA. However, even when
the velocity components are known pre-
cisely, accurate determination of the
boundaries of capture zones still requires
that both stagnation points and dividing
streamlines are known.
The delineation of capture zones in com-
plex settings is currently done either by
the use of discrete numerical models or
analytic element models. The discrete nu-
merical models, such as MODFLOW/
MODPATH of the US Geological Survey,
and analytic element models, such as
Strack's MLAEM, require detailed knowl-
edge of the setting and advanced model-
ing expertise to run; and they do not cur-
rently have advanced algorithms for delin-
eation of capture zones.
The Wellhead Analytic Element Model
(W/7AEM) falls between the two aforemen-
tioned technologies, ft does contain an
advanced algorithm for determining cap-
ture zones for any well at any time based
on precise knowledge of the locations of
all stagnation points and dividing stream-
lines, it has features that make the inclu-
sion of open or closed, head-specified
boundaries possible (for example to model
streams), but lacks the power of advanced
Printed on Recycled Paper
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discrete numerical or analytic element
models. The authors believe that the newly
developed model will serve ground-water
professionals who wish to determine cap-
ture zones in relatively simple
geohydrological settings.
W/?AEM consists of two executables:
the preprocessor GAEP and the flow
model CZAEM (see Figure 1). In order to
facilitate data entry in the computer pro-
gram CZAEM (Capture Zone Analytic Ele-
ment Model), a separate computer pro-
gram was developed called GAEP, (Geo-
graphic Analytic Element Preprocessor).
This program makes it possible for most
users not familiar with the input structure
of analytic element models to concentrate
on modeling aspects, rather than on the
intricacies of preparing input data files.
W/jAEM was developed under a coop-
erative agreement between EPA and Indi-
ana University and the University of Min-
nesota.
The Analytic Element Method
The analytic element method was de-
veloped at the end of the seventies by
Otto Strack at the University of Minne-
sota. For a detailed description of the
method, the reader is referred to Ground-
water Mechanics, O.D.L. Strack, 1989,
Prentice Hall, while a brief review follows.
This new method avoids the discretiza-
tion of a ground-water flow domain by
grids or element networks. Instead, only
the boundaries of the surface water and
aquifer features in the domain are
discretized and entered into the model.
Each of these boundary segments is rep-
resented by closed form analytic solu-
tions—the analytic elements. The com-
prehensive solution to a complex, regional
ground-water flow problem is obtained by
superposition of all analytic elements in
the model, from a few hundred to thou-
sands.
Traditionally, modeling ground-water
flow by use of analytic functions was con-
sidered to be limited to homogeneous aqui-
fers of constant transmissivity. However,
by formulating the ground-water flow prob-
lem in terms of appropriately chosen dis-
charge potentials rather than piezometric
heads, the analytic element method be-
comes applicable to both confined and
unconfined flow conditions as well as to
heterogeneous aquifers.
The analytic elements are chosen to
best represent certain hydrologic features.
For instance, stream sections are repre-
sented by line-sinks; lakes or wetlands
may be represented by areal sink distribu-
tions. Streams and lakes that are not fully
connected to the aquifer are modeled by
WhAEM
Figure 1. The modeling process using WhAEM.
area sinks with resistance to flow between
surface water and the aquifer. Disconti-
nuities in aquifer thickness or hydraulic
conductivity are modeled by use of line
doublets (double layers). Other analytic
elements may be used for special fea-
tures, such as drains, fractures, and slurry
walls.
The analytic element method differs fun-
damentally from most classical numerical
models, for instance:
1. The solution is analytical; no interpo-
lation is required to compute heads
or velocities at any point in the do-
main. This makes the method insen-
sitive to scale, allowing contour plots
and streamlines to be produced in
areas varying in size from several
square feet to hundreds of square
miles.
2. Since the velocity field is calculated
analytically, inaccuracies in capture
zone boundaries and isochrones of
travel times are due solely to approxi-
mations made in the conceptual
model and its implementation in the
program; they are not the conse-
quence of a model grid resolution
and the associated numerical (ap-
proximate) differentiation process.
3. The aquifer is unbounded in the hori-
zontal plane; there are no artificial
model boundaries that may influence
the solution.
W/7AEM
Time-of-travel capture zone delineation
is conducted by ground-water flow model-
ing. Modeling in this context is an interac-
tive process of data acquisition, data analy-
sis, and running a computer model. Initial
modeling results prompt changes in the
conceptual model, which in turn will lead
to new modeling results. In the absence
of definite data, hypothesis testing and
sensitivity analysis will be important as-
pects of the modeling process. As a re-
sult, the modeler will usually make fre-
quent changes to the input data file (an
ASCII file of program instructions) during
the modeling process.
WhAEM was designed to facilitate this
process. GAEP greatly improves the pro-
cess of data entry, especially when used
in combination with USGS topographic
maps and a digitizer (optional). The GAEP
generated electronic background map be-
comes the template for "on-screen" de-
sign of the ground-water flow model. GAEP
communicates with the flow solver CZAEM
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through an ASCII file that contains the
command script for delineating capture
zones.
Program CZAEM
CZAEM is a single layer model for simu-
lating steady flow in homogeneous aqui-
fers. The mathematical framework under-
lying the model is based on the Dupuit-
Forchheimer assumption, where the verti-
cal resistance to flow is negligible, such
as for shallow aquifer flow. The imple-
mentation of the analytic element method
in CZAEM is elementary, supporting only
a few basic analytic elements. These ele-
ments can be used to simulate river bound-
aries, streams, lakes, wells, uniform flow,
and uniform infiltration over a circular area.
Line-sinks are used to model river
boundaries, streams, and lakes. Line-sinks
are mathematical functions that simulate
a constant rate of extraction along a line.
The sink densities (strengths) of the line-
sinks in the model are determined such
that the heads at the center of the line-
sinks are equal to specified values (usu-
ally chosen to equal the water levels in
the streams or lakes). The accuracy with
which the ground-water inflow (or outflow)
along a stream can be modeled improves
with a finer subdivision of the stream in
line-sink segments.
The well function (Thiem equation) is
used to model wells with given discharge
(pumping rate). Unlike numerical models,
the piezometric head distribution and the
velocity field near a well remain accurate,
since there is no discretization of the aqui-
fer by a grid or element network.
A special function, the "pond" function,
is used to model areal recharge due to
precipitation. Since CZAEM models steady
state flow, this recharge rate is a yearly
average. The "pond" function is a circular
element with an areal "source" density
equal to the recharge rate. The circular
pond overlays the domain of interest, the
well field and surrounding surface waters,
to simulate the desired aquifer recharge.
The uniform flow function may be used
to replace the combined effects of areal
recharge and surface water boundaries,
similar to the WHPA program. Since
W/iAEM allows the explicit representation
of these boundary conditions, the uniform
flow approximation is less often used in
applications to field problems.
The computer program CZAEM is an
elementary analytic element model with
the capability to generate capture zones
of wells. The program has the following
modules:
1. AQUIFER, for the input of aquifer
data.
4.
5.
6.
2. GIVEN, for the input of uniform fbw
and areal recharge.
3. REFERENCE, for the input of the
head at one point in the aquifer.
WELL, for the input and implemen-
tation of wells.
LINE-SINK, for the input and imple-
mentation of line-sinks.
GRID, for the generation of a grid of
piezometric heads to be contoured.
7. PLOT, for piezometric contour plot-
ting.
8. TRACE, for streamline tracing.
9. CAPZONE, for the generation of
time-of-travel capture zones.
10. CURSOR, for cursor controlled in-
teraction with elements.
11. CHECK, for monitoring the values of
parameters.
12. IO, for binary write and save of solu-
tions.
13. PSET, for sending graphics to out-
put devices.
Capzone Module
The capzone module has been designed
to define the capture zone boundaries as
well as the travel time isochrones inside
these capture zone boundaries, called
timezones for arbitrary arrangements of
wells and line-sinks (stream boundaries).
Rather than merely tracing a number of
streamlines from the well, the capzone
module logic first determines all stagna-
tion points in the flow domain, determines
whether they are connected to the wells
by streamlines, and uses them as the
basis for defining the capture zone bound-
aries. Under multiple well scenarios, one
well may have several different capture
zones, termed subzones, which have their
own travel time isochrones.
Figure 2 illustrates capture zones for
five different wells surrounded by two
stream branches and a tributary (solid
lines). The dashed lines are background
map features for orientation purposes. The
capture zones are pear shaped and of
finite extent: the wells receive all their
water from areal recharge.
In Figure 3, travel time isochrones are
presented for two wells in a uniform flow
field. Notice that the isochrones wrap all
the way around the well; between the well
and its stagnation point, all travel times
between zero and infinity occur. Also no-
tice that the capture zone of the right-
hand well wraps around the capture zone
of the other well, causing a discontinuity
Figure 2. Capture zone envelopes for five wells
in a regional setting defined by streams and areal
recharge
Figure 3. Travel time isochrones for two wells in
a uniform flow field.
in the residence times across the latter
capture zone envelope.
Figure 4 shows capture subzones for a
well near a river. The well is receiving
80% of its discharge from the far-field and
20% from the river.
The module CAPZONE has been writ-
ten specifically for this project. The source
code (FORTRAN) of this module is avail-
able from USEPA and contains documen-
tation to facilitate its inclusion in other
ground-water models.
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Figure 4. Subzones for a well near a river. The
dotted lines represent hydraulic head contours.
Preprocessor GAEP
The absence of a grid or element net-
work in analytic element models makes it
unnecessary to translate hydrography data
(stream locations and associated stream
levels) into cell specific data, which is the
main function of preprocessors for numeri-
cal models. In contrast, the stream levels
and geometry are entered into the ana-
lytic element model, through simple com-
mands which are in an ASCII file. These
commands can be input directly from the
keyboard (command line mode) or read in
from an ASCII file (batch mode) (see Fig-
ure 5). The line-sinks representing streams
and lakes are represented in the file by
their end coordinates and a specified head
at the center (stream level). The tradi-
tional procedure for creating such an in-
put data file is to sketch the line-sink lay-
out on a map, write the stream elevations
near these line-sinks, and use a digitizer
to produce the coordinate pairs listed in
Figure 5. The syntax of the various com-
mands in Figure 5 must be consistent
with the requirements of CZAEM, or one
or more errors occur when the file is read
by CZAEM. Any change in the input data,
as part of the interactive modeling proce-
dure requires editing of the input file,
whereby it is often necessary to revisit the
topographical map and digitize new line-
sinks.
To facilitate this process, the prepro-
cessor GAEP separates the digitizing ac-
tivities from the creation of analytic ele-
ments (e.g. line-sinks) to be included in
the input data file for CZAEM. GAEP,
therefore, has two functions:
1. Creation of a digital map of all
streams, lakes, wetlands, well lo-
cations, and background map in-
formation (roads, city boundaries,
outlines of surface geological fea-
tures, etc.). The surface water fea-
tures have associated with them
the water levels as reported on
the topographical map (intersec-
tions of elevation contours with the
stream beds).
2. Creation of an input data file for
CZAEM with all aquifer data and all
analytic elements (line-sinks and
wells) needed for the model.
The first activity, creation of the digital
map, is a routine procedure that does not
require any modeling expertise or hydro-
logical knowledge. The digital map is saved
on disk for future use by the modeler. A
digital map, as displayed on screen by
GAEP, is reproduced in Figure 6. By point-
ing at a feature with the mouse, its name
is displayed for easy identification.
The modeler will use GAEP and a digi-
tal map previously saved on disk to create
a CZAEM input data file. To represent a
stream by line-sinks, the modeler merely
points at the stream with the mouse and
selects it by "clicking" the mouse button.
By moving the pointer over the stream
and clicking on intended line-sink end
points, a string of line-sinks is created
with heads computed at their centers us-
ing the stream elevations stored with the
digital map. In Figure 7, a string of line-
sinks is illustrated. The numbers printed
near the line-sinks represent the average
stream elevations (heads at the center of
the line-sinks). GAEP will also prompt for
AQUIFER
BOTTOM 330
THICKNESS 100
PERMEABILITY 350
POROSITY 0.20
RETURN
AQUIFER
REFERENCE 0 656160 410
RETURN
Rain Element
5INKDISK
DISCHARGE
21983 -6193 43881 -24755 -0.00411
RETURN
TabletltemlD: wabash east
.INESINK
HEAD
-13409 -18389 -12778 -10574 395.2 [we1
-12778 -10574 -8046 -5825 396.1 [we2
-8046 -5825 -5927 -1257 396.8 [we3;
-5927 -1257 -2322 367 397.2 [we4
aquifer data; and when instructed to cre-
ate the CZAEM input data file, write an
ASCII file to disk that can be read directly
by CZAEM. In fact, the GAEP generated
input data file will introduce the data, solve
the problem, and create a grid with piezo-
metric heads. The modeler then enters
the TRACE and CAPZONE modules in
CZAEM to generate capture zone bound-
aries.
Conclusions and
Recommendations
The deliverable of this project consists
of an analytic element modeling package
for simulating steady flow in homogeneous
aquifers, with the primary objective to de-
lineate capture zones in settings with
streams, rivers, lakes, infiltration and wells.
New algorithms have been developed for
the accurate delineation of capture zone
boundaries. These algorithms are imple-
mented in the computer code CZAEM.
The algorithms make accurate delineation
of capture zone boundaries possible. A
preprocessor program, GAEP, has been
developed to facilitate the entry of field
data into CZAEM. GAEP simplifies the
process of modeling considerably. Spe-
cifically, GAEP separates the time con-
suming (but routine) task of digitizing hy-
drography data from the creation of con-
ceptual models and subsequent analytic
element input data files. With GAEP, the
modeler is free to concentrate on interpre-
tation of modeling results rather than the
details of data modification and entry into
CZAEM.
The W/iAEM package is documented in
various ways. The primary documentation
is contained in a program manual, which
includes installation instructions, program
descriptions and a tutorial for the inte-
grated use of GAEP and CZAEM. Refer-
ence manuals for both GAEP and CZAEM
are provided in the WftAEM manual. A
tutorial for stand-alone use of the program
CZAEM is available as a separate docu-
ment. Finally, both GAEP and CZAEM
codes support on-line help.
The W/iAEM package should only be
used by ground-water hydrologists quali-
fied to address wellhead protection delin-
eation problems. W/iAEM is designed to
assist hydrologists in learning as much as
possible about the geohydrological prob-
lems they face. Model predictions must
always be interpreted critically, with the
simplifying assumptions in mind. For com-
plex geohydrological settings, it may be
necessary to apply a more powerful model
than WhAEM, requiring more experience
from the ground-water flow modeler.
Figure 5. Part of a CZAEM input data file
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Mouse button 1: add point Keyboard F3: Done; Esc: CANCEL
A
360
470
Hardware Requirements
• 386 or 486 PC
• 2.5 MB RAM
• MS compatible mouse
• digitizer (optional)
• printer (optional)
Software Requirements
• MS DOS version 5.0 or higher
• Windows 3.1 (optional)
Figure 6. Kelso Creek with elevation marks where contours cross the stream
Mouse button 1: add point Keyboard F3: Done; ESC: CANCEL
interpolated
line sink head
Figure 7. Line-sinks created along Kelso Creek.
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Hendrik M. Haitjema is with the School of Public and Environmental Affairs, Indiana
University, Bloomington, IN 47405.
Otto D.L Strack is with the University of Minnesota, Civil and Mineral Engineering,
Minneapolis, MN 55455.
Stephen R. Kraemer (also the EPA Project Officer see below), is with the U.S.
Environmental Protection Agency, Ada, OK 74820
The complete report consists of two volumes, entitled "CZAEM User's Guide:
Modeling Capture Zones of Ground Water Wells Using the Analytic Element
Method," (Order No. PB95-194189; Cost: $19.50, subject to change) and
"WhAEM: Program Documentation for the Wellhead Analytic Element Model,"
(Order No. PB95-167375; Cost $27.00, subject to change) will be available only
from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Robert S. Ken Environmental Research Laboratory
U.S. Environmental Protection Agency
Ada, OK 74821
United States
Environmental Protection Agency
Center for Environmental Research Information
Cincinnati, OH 45268
Official Business
Penalty for Private Use $300
BULK RATE
POSTAGE & FEES PAID
EPA
PERMIT No. G-35
EPA/600/SR-94/210
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