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

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