A STUDY AND EVALUATION OF SALTWATER INTRUSION
IN THE FLORIDAN AQUIFER
BY MEANS OF A HELESHAW MODEL
By
ANDREW JOSEPH EVANS, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
ACKNOWLEDGEMENTS
To my supervisory committee, I wish to extend my
sincere appreciation for their efforts on my behalf. A
special thanks goes to Dr. B. A. Christensen and Dr. J. H.
Schaub for their continuing support and advice. It's not
an easy task to pull an oar in Hagar's longboat.
The efforts of Mr. C. L. White, Mr. William
Whitehead, and the staff of the Mechanical Engineering
machine shop in the construction of the model are grate
fully acknowledged, as well as the assistance of Mr. Richard
Sweet and Mr. Tom Costello'in the operation of the model, and
the mathematical advice of Dr. Jonathan Lee. The participa
tion of Mr. Floyd L. Combs in all aspects of the project was
especially valuable.
Without the financial support of the Office of Water
Resources Research, United States Department of the Interior,
and the administrative assistance of Dr. W. H. Morgan and
the staff of the Florida Water Resources Research Center, this
project would not have been launched.
Finally, to my mother and my many dear friends who
have watched the "ins and outs" of my academic and profes
sional career, bless you for your interest and encouragement.
I want to leave the reader with this closing thought,
which I believe is paraphrased from Benjamin Franklin:
"Waste not want not, you never miss the water 'till the
well runs dry."
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ...................................... ii
LIST OF TABLES ........................................ vi
LIST OF FIGURES .... ........................... ........ vii
KEY TO SYMBOLS OR ABBREVIATIONS ........................ x
ABSTRACT .............................................. xiii
Chapter
I. INTRODUCTION AND STATEMENT OF PROBLEM ......... 1
Topography ................................. 2
Western Highlands ....................... 3
Marianna Lowlands ........................ 6
Tallahassee Hills ....................... 7
Central Highlands ....................... 7
Coastal Lowlands ........................ 7
Climate .................................... 8
Geology .................................... 12
II. MODELS, NUMERICAL AND PHYSICAL ................ 15
A: NUMERICAL METHODS ............................. 15
Method of Finite Differences ................ 15
Method of Finite Elements ................... 17
Relaxation Methods ......................... 18
B: PHYSICAL MODELS ............................... 19
Sandbox Model .............................. 21
HeleShaw Analog ............... ............. 22
Electric Analog .............................. 23
Continuous Electric Analog .............. 23
Discrete Electric Analog ................ 24
Ion Motion Analog ....................... 24
Membrane Analog .............................. 25
Summary .................................... 26
TABLE OF CONTENTSContinued
Chapter Page
III. THE HELESHAW MODEL ........................... 28
Viscous Flow Analog ........................ 29
Scaling .................................... 37
Time .................................... 39
Anisotropy .............................. 40
Leakage ................................. 50
Storativity ............................. 54
Discharge ............................... 55
Accretion ............................... 59
Volume .................................. 59
IV. SITE SELECTION AND PROTOTYPE GEOLOGY
AND HYDROLOGY .............................. 61
Site Selection ............................. 61
Prototype Geology .......................... 67
Prototype Hydrology ......................... 69
V. DESIGN, CONSTRUCTION, AND OPERATION
OF MODEL ................................... 71
Design ..................................... 71
Prototype ............................... 71
Model ................................... 71
Construction ............................... 86
Frame ................................... 86
Plexiglas Plates and Manifolds .......... 105
Saltwater System ........................ 125
Freshwater System, General .............. 125
Freshwater System, Accretion ............ 132
Freshwater System, Wells ................ 132
Freshwater System, Flow Meters .......... 132
Operation ........ .......................... 156
VI. RESULTS, CONCLUSIONS, AND RECOMMENDATIONS ..... 158
Results .............. ....................... 158
Conclusions and Recommendations ............ 191
Appendix
A. UNIFORM FLOW THROUGH A CONDUIT OF
RECTANGULAR CROSS SECTION .................. 193
B. FLOW METER CALIBRATION ........................ 195
BIBLIOGRAPHY .......................................... 210
BIOGRAPHICAL SKETCH ................................... 213
LIST OF TABLES
Table Page
2.1 APPLICABILITY OF MODELS AND ANALOGS .......... 27
5.1 PROTOTYPE PARAMETERS ......................... 72
5.2 MODEL PARAMETERS ............................. 81
5.3 SIMILARITY RATIOS ............................ 82
6.1 DEPTH TO SALTWATER ........................... 186
LIST OF FIGURES
Figure Page
1.1 Topographic divisions of Florida ................ 5
1.2 Mean annual precipitation ...................... 11
3.1 Free body flow diagram for HeleShaw model ..... 32
3.2 Section of anisotropic grooved zone in
HeleShaw model ............................. 44
3.3 Head loss in grooved anisotropic zone ........... 46
3.4 Section of leaky zone in HeleShaw model ....... 53
3.5 Storage manifold for HeleShaw model ........... 57
4.1 Regional area of prototype ..................... 64
5.1 GhybenHerzberg interface model ................ 75
5.2 Cradle and cradle dolly ........................ 88
5.3 Cradle rotation ................................ 91
5.4 Frame and model setup .......................... 93
5.5 Air hose and valve arrangement ................. 96
5.6 Stub shaft and pillow block arrangement ........ 98
5.7 Back up air supply .............................. 100
5.8 Internal support and sealing system ............ 102
5.9 Model mounting system .......................... 104
5.10 Model back and front plates .................... 107
5.11 Detail of the model front and back plate ....... 109
5.12 Detail of the model back plate ................. 111
vii
LIST OF FIGURESContinued
Figure Page
5.13 Front plate with accretion manifolds ........... 114
5.14 Accretion manifolds ............................. 116
5.15 Detail of accretion manifolds .................. 118
5.16 Connections between model and fluid
supply system ............................... 120
5.17 Saltwater constant head tank ................... 122
5.18 Back and front plate clamp up .................. 124
5.19 Fluid supply network schematic ................. 127
5.20 Saltwater reservoir and pump ................... 129
5.21 Freshwater supply system ....................... 131
5.22 Freshwater reservoir and accretion pump ........ 134
5.23 Well supply manifold and pump .................. 136
5.24 Well supply manifold and accretion
supply manifold ............................. 138
5.25 Opposite view of Figure 5.24 ................... 141
5.26 Flow meter bank ................................ 143
5.27 Flow meter detail .............................. 145
5.28 Flow meter to model connections ................ 148
5.29 Flow meter pressure sensing lines .............. 150,
5.30 Flow meter switching device and
pressure transducers ........................ 152
5.31 Carrier demodulator and strip chart
recorder .................................... 155
6.1 Interface location, tm = 0 min. ................. 160
6.2 Interface location, tm = 32 min. ............... 162
6.3 Interface location, tm = 48 min. ................ 164
viii
LIST OF FIGURESContinued
Figure
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
location,
location,
location,
location,
location,
location,
location,
location,
location,
location,
tm = 62 min.
t = 79 min.
t = 92 min.
m
tm = 102 min
tm = 107 min
t = 117 min
t = 132 min
m
tm = 147 mir
tm = 162 mir
tm = 172 mir
L.
L.
t.
L.
6.14 Wedge location at time zero .....
Page
............... 166
............... 168
............... 170
.............. 172
........ ..... 174
..... ......... 176
.............. 178
.............. 180
.............. 182
.............. 184
............... 188
6.15 Time overlay of wedge location ..
Interface
Interface
Interface
Interface
Interface
Interface
Interface
Interface
Interface
Interface
190
KEY TO SYMBOLS OR ABBREVIATIONS
Symbols
A = area, Fourier coefficient
b = width of model and prototype
B = body force, Fourier coefficient
f = subscript denoting freshwater, Floridan aquifer
g = acceleration due to gravity
h = height of water
j = summation limit
k = intrinsic permeability
.K = hydraulic conductivity
1 = distance between storativity tubes, subscript
denoting leaky layer
L = distance center to center of groove in anisotropic
zone, flow meter tube length
m = subscript denoting model
n = effective porosity
p = pressure, subscript denoting prototype
q = specific discharge
Q = total flow
r = subscript denoting ratio
R = accretion
R = Reynolds number
KEY TO SYMBOLS OR ABBREVIATIONSContinued
Symbols
S = specific storage
t = time
T = transmissivity
U = volume
V = velocity
x = horizontal direction parallel to test section
y = horizontal direction perpendicular to test section
z = vertical direction
1,2 = subscripts denoting zone 1 and zone 2
a = width adjustment factor
y = unit weight
A = length adjustment factor
p = absolute viscosity
E = geometric parameter in anisotropic zone
p = mass density
v = kinematic viscosity
= potentiometric head
= velocity potential, potential
Abbreviations
D. substantial derivative
Dt
V2 = Laplace operator
A = difference operator
C = Centigrade
KEY TO SYMBOLS OR ABBREVIATIONSContinued
Abbreviations
cfs = cubic feet per second
*F = Fahrenheit
fps = feet per second
g/cm3 = grams per cubic centimeter
gpd = gallons per day
Hg = Mercury
I.D. = inner diameter
MGD = million gallons per day
msl = mean sea level
O.D. = outer diameter
pcf = pounds per cubic feet
psid = pounds per square inch differential
psig = pounds per square inch gage
RPM = revolutions per minute
xii
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
A STUDY AND EVALUATION OF SALTWATER INTRUSION
IN THE FLORIDAN AQUIFER
BY MEANS OF A HELESHAW MODEL
By
Andrew Joseph Evans, Jr.
August, 1975
Chairman: B. A. Christensen
Major Department: Civil Engineering
Continuing development of the coastline zone in the
middle Gulf area of Florida is increasing the demand for
groundwater supplies and, in turn, increasing the probability
of saltwater intrusion. Methods must be developed to make
longrange predictions on the effects of increased demands
on the Floridan aquifer.
A HeleShaw model is a physical model which fits the
requirements for longrange planning. It is wellsuited for
handling anisotropic aquifers, difficult boundary conditions
and can simulate years of field conditions in minutes of
model time.
xiii
The section selected for study lies in a line from
the Gulf coast near Tarpon Springs to a point near Dade City
and passes through the EldridgeWilde well field. The
EldridgeWilde well field is the major water producer for
Pinellas County. This region has experienced several years
of dry weather, and pumping has lowered the water levels in
the aquifer by a significant amount. This loss of fresh
water head is certain to induce saltwater intrusion.
A HeleShaw model has been built for this area, and
all pertinent geological and hydrological features of the
area are included. Steady state characteristics of the
aquifer system have been considered. In particular, the
longterm effects due to pumping and artificial recharge
were examined.
xiv
CHAPTER I
INTRODUCTION AND STATEMENT OF PROBLEM
Florida, with the possible exception of California,
is the fastest growing state of the United States. The
rapid influx of people since World War II has greatly
increased the demands for land and water. In the past,
there has been an almost total lack of wide range planning
for the uses of these resources. Even fewer investigations
have been made into the consequences of their rapid and
unordered development. Recently, water supplies have had
to be rationed in South Florida. Overall, land and wet
lands required for fish and wildlife have so diminished
that, in some instances, there has been a marked decrease
in their numbers. It seems reasonable to conclude that in
some areas of the state, land and water resources cannot
support much larger populations with current locally avail
able supplies without almost irrecoverable damage to the
groundwater system in the form of saltwater encroachment.
With the growing affluence of the American people,
and the availability of economically priced air conditioning
units, it can be expected that even more people will leave
the colder northern climates for the southern and western
states. Florida can expect to receive more than its share
of the migration. Frequently now, environmental protection
groups are making forecasts of impending doom. At worst,
their predictions may come true and people are beginning to
look at all growth with a jaundiced eye.
It is doubtful, however, that growth and development
can be stopped. The history of man indicates a continual
effort to better his lifestyle, his private environment.
There is little doubt that this has sometimes caused a
degradation of other portions of his world. Unless the
cessation of all growth and development is acceptable, new
ways must be found of forecasting, or predicting, the results
of all growth so as to combat possible undesirable results.
Consequences of all growth must be known, even of those
resulting when pragmatic shortterm solutions are used.
Hopefully, the remainder of this study will present
a modeling method which will be useful in forecasting the
results of pumpage and use of groundwater in our coastal
zones so that we may better plan their usage. But first,
a little background on Florida.
Topography
Florida lies between latitudes 2440' and 3100'
north, and longitudes 8000' and 87*40' west, and is the
most southerly unit of the continental United States. In
its southernmost extension it is less than 10 of latitude
north of the Tropic of Cancer.
Florida is bounded on the east by the Atlantic Ocean;
on the south by the Straits of Florida and the Gulf of Mexico;
on the west by the Gulf of Mexico and the state of Alabama;
and on the north by Alabama and Georgia. The shape of the
state in relation to the remainder of the United States
suggests two distinctive parts: the Floridan panhandle and
the peninsula of Florida. The panhandle is a strip roughly
225 miles long that stretches in an eastwest direction.
The peninsula is a southsoutheast extension at approximate
bearing S 17 east. From the northern boundary of the state
to the tip, not including the chain of keys, the peninsula
is approximately 415 miles long and includes twothirds of
the land mass of the entire state. Its coastline, some
1350+ miles long, is the longest with the exception of
Alaska. No place in the interior of Florida is more than
60 miles from either the Gulf or the Atlantic coast.
Cooke (1939) divided the terrain of Florida into
five sections, Figure 1; the Western Highlands, the Marianna
Lowlands, the Tallahassee Hills, and a narrow band of Coastal
Lowlands, which comprise the panhandle, and the Central High
lands and Coastal Lowlands, which comprise the peninsula.
The topography of each is described briefly.
Western Highlands
Extending eastward from the Perdido River (the
western boundary of Florida) to the Apalachicola River, the
northern part of this region near the Alabama state line
FIGURE 1.1
Topographic divisions of Florida.
Western Highlands
Marianna Lowlands
Tallahassee Hills
Central Highlands
Coastal Lowlands
86
0 50 100 150
Scale of Miles
After: Cooke (1939)
is not much higher than 300 feet. It is considered to be
hilly when compared to the broad gently rolling southern
parts of this region which drop to 100 feet elevation as
one approaches the coastal lowlands. The highest elevation
in the state, 345 feet, is found in this region in the north
west corner of Walton County. The western highlands are
underlain with the sand of the Pliocene Citronelle Formation.
The steepness of the bankslopes at the headwaters of the
many streams is the most unique physiographic characteristic
of this section.
Marianna Lowlands
This roughly triangularshaped region of Holmes,
Jackson, and Washington counties, with somewhat smaller
contributions from Bay and Calhoun counties, lies between
the Tallahassee Hills and the Western Highlands. It is
difficult to recognize this area of gently rolling hills
as lowlands. Cooke (1945) attributes the lower elevations
to the solubility and consequent degradation of the under
lying limestone. This area is one of the two in the state
where the Ocala Formation is exposed to the surface and the
only area of the state where the Marianna limestone, the
soft white limestone of the Oligocene Age, is found exposed.
The region is dotted with sinks, sinkhole lakes, and springs.
Tallahassee Hills
From the Apalachicola River east to the Withlacoochee
River, the Tallahassee Hills extends along the Georgia
Florida border and is only 25 miles in width. The western
section is a nearly level plateau some 300 feet above mean
sea level. The remainder consists of rolling hills carved
out of the Citronelle Formation. In addition to this, a
red clayey sand and Fuller's earth of the Hawthorn Formation
are found in this area. This is a fertile farming region.
Central Highlands
The Central Highlands forms the backbone of the
Floridan peninsula and extends from the Georgia line between
the Withlacoochee and St. Mary's rivers southsoutheastward
some 250 miles into Glades County west of Lake Okeechobee.
This region is highly diversified. It includes high swampy
plains, hills, and innumerous lakes. Soils are sandy. Many
of them were derived from Pleistocene (Ice Age) marine ter
races. However, a distinguishable amount comes from the
Miocene Hawthorn and Pliocene Citronelle Formations. The
lakes and sinks which dot the entire area indicate the
presence of limestone below the surface. Elevations of
this region average just slightly more than 150 feet; however,
they vary from less than 100 feet to approximately 300 feet.
Coastal Lowlands
The Coastal Lowlands, or Coastal Plains as it is
sometimes called, borders the entire 1350 mile Florida
coastline. Flanking on both sides of the Central Highlands,
the Coastal Lowlands is widest just south of Lake Placid and
narrowest between the western border and the Choctawhatchee
Bay just south of the Western Highlands. The elevations
everywhere within this region are less than 100 feet. The
soil for the most part is sandy except in the Everglades and
Big Cypress Swamp locales where Pliocene limestone, muck,
and peat prevail near the surface. The keys, which extend
some 100 miles into the Straits of Florida, are mostly sandy
oolitic limestone like that of the mainland; however, some
limestone with coral heads is found. The islands seldom
reach 15 feet elevation. The entire region is generally
flat, typical of recently deposited material with little or
no erosion.
Climate
The sea surface temperatures east and west of Florida
average, respectively, 780 and 77 Fahrenheit. Water tem
peratures range from 740 to 83 Fahrenheit in the east and
700 to 84* Fahrenheit in the west. The coldest month in
both cases is February; the warmest month is likewise
August. The relatively homogeneous distribution of sea
temperature, the lack of high relief and the peninsula shape
of Florida contribute to its climate. The temperature is
everywhere subtropical. Mean annual average temperature in
the north is 68 Fahrenheit, and in the southern tip 750
Fahrenheit.
The tradewinds, which shift from northern Florida to
southern Florida and back semiannually, bring a mildly mon
soon effect to Florida. In November, the tradewinds are at
their southernmost extension and Florida's climate is con
trolled by frontal, or cyclonic, activity moving in from the
continental United States. Rainfall during this period is
of low intensity and long duration.
Beginning in early May, the tradewinds move north,
again bringing with them the moist warm air of the Atlantic.
The cyclonic activity is greatly reduced over the state and
convectional instability begins to become established. June
through September is known as the rainy season in Florida.
The thunderstorms of this period are intense and very
specially varied. They usually occur during the hottest
part of the day and only on rare occasions last longer than
two hours. About 60 percent of the total average annual
rainfall occurs during this period, Figure 2. The mean
average rainfall of Florida is in the neighborhood of 53
inches. It varies from 38 to 40 inches in the lower keys
to over 65 inches in the southeast corner of the peninsula
and the western portion of the panhandle. Most of the
interior, that is the central highlands, receives approxi
mately the mean annual average.
FIGURE 1.2
Mean annual precipitation (in inches).
I Over 66
S62 66
S58 62
I 54 58
50 54
S46 50
LZ Less than 46
0 50 100 150
Scale of Miles
^ < 
Note: Precipitation normals
compiled from records
published in Climatological
Data: Florida Section,
December, 1971.
_I~_ _
~
Geology
The Floridan peninsula and the offshore submerged
lands above 50 fathoms, which Vaughan (1910) called the
Floridian Plateau, have existed for several million years.
The region has not been subject to violent earth movement
and, consequently, there has been a gentle doing resulting
in the formation of an oval arch above the basement rock.
The rock of the core underlying the plateau is hypothesized
to be preCambrian; however, no drill has penetrated the
core. The oldest rocks penetrated, to date, are quartzites
found at about 4500 feet below the surface in Marion County.
The borehole encountered another 1680 feet of quartzite
before drilling was suspended. This metamorphized rock,
believed to be a continuation of the Piedmont region of
Georgia, was assigned by Cooke (1945) to the Pennsylvanian
period. The arch above the metamorphized basement, composed
of almost pure porous limestone, is known as the Lake City,
Avon Park, and Ocala Formations. Dated in the Eocene period,
the Ocala Formation has an estimated maximum thickness of
360 feet. It is found at, or above, mean sea level through
out northeast and north central Florida and is this section's
principal aquifer. In southern Florida, in the vicinity of
the Everglades, the Ocala is found at depths approaching
1300 feet. The Lake City and Avon Park limestones found
below the Ocala are the principal aquifer used by
agricultural interests in central and south central Florida
and are known locally as the Floridan aquifer.
Above the Eocene series are the formations of the
Oligocene epoch. These are represented by the Marianna
limestone and the Byran limestones found and mined in the
Marianna Lowlands of the northern part of the state, and
the Suwanneelimestone found over the Ocala Formation as far
south as Hillsborough County.
The next higher formations are those of the Miocene
epoch. These are well represented by the Tampa limestones
of the early Miocene which are found above the Suwannee and
Ocala limestone in south Florida, the Chipola and Shoal
River Formations of the Alum Bluff group found in northwest
and north central Florida, the Hawthorn Formation, and the
Duplin marls. The latter three formations, Alum Bluffs,
Hawthorn, and Duplin, chiefly are sands, clays, and marls
that form a confining layer over the Eocene and Oligocene
limestones.
The Hawthorn, with the possible exception of the
Ocala, is the most extensive formation within the state.
It occurs at, or near, the surface in most of north Florida.
It overlies the Tampa limestone formation in Hillsborough
County, and is, itself, overlain by the Duplin marls and
younger deposits in the south central and southern parts
of the state.
14
The surface material of most of the coastal lowlands
are of the Pliocene, Pleistocene, and Recent periods. The
most widely distributed are the sands formed along the old
shorelines of previous ocean levels. Cooke (1945) defines
seven of these marine terraces. Some small deposits of
coquina, oolite, coral reef limestone, and freshwater marls
are found among these deposits.
CHAPTER II
MODELS, NUMERICAL AND PHYSICAL
The purpose of this chapter is to enumerate some
of the more widely used modeling techniques in groundwater
flow, along with a brief description of each. The reader
is referred to Bear (1972) for additional information and
references.
A: NUMERICAL METHODS
Numerical methods are used in many cases where the
partial differential equations governing flow through porous
media cannot be solved exactly. Various techniques have
been developed for obtaining numerical solutions.
Method of Finite Differences
The method of finite differences is one such
technique. The first step is to replace the differential
equations by algebraic finite difference equations. These
difference equations are relationships among values of the
dependent variable at neighboring points of the applicable
coordinate space.
The resulting series of simultaneous equations
is solved numerically and gives values of the dependent
variables at a predetermined number of discrete or "grid"
points throughout the region of investigation.
If the exact solution of the difference equations
is called D, the exact solution of the differential equation
is called S, and the numerical solution of the difference
equation is called N, two quantities of interest may be
defined. They are the truncation error, IS DI, and the
roundoff error, ID NI. In order for the solution to
converge, it is necessary that IS DI 0 everywhere in
the solution domain. The stability requirement is such
that ID NJ > 0 everywhere in the solution domain. The
general problem is to find N so that IS NJ is smaller
than some predetermined error. Noting that (S N)
= (S D) + (D N), it is seen that the total error is
composed of the truncation error and the roundoff error.
The arbitrary form selected for the finite difference
equation leads to the truncation error. This error is
frequently the major part of the total error.
The actual computation proceeds by one of two
schemes. They are the explicit, or forwardintime,
scheme and the implicit, or backintime, scheme. The
explicit scheme is simpler but more time consuming than
the implicit scheme due to the stability constraint. The
implicit scheme is more efficient but requires a more
complicated program as compared to the explicit scheme.
Method of Finite Elements
The finite element technique employes a functional
associated with the partial differential equation, as
opposed to the finite difference method which is based on
a finite difference analog of the partial differential
equation. A correspondence which assigns a real number
to each function or curve belonging to some class is termed a
functional.
The calculus of variations is employed to minimize
the partial differential equation under consideration. This
is done by satisfying a set of associated equations called
the Euler equations. Thus, one seeks the functional for
which the governing equations are the Euler equations and
proceeds to solve the minimization problem directly, rather
than solving the differential equation.
The procedure is continued by partitioning the flow
field into elements, formulating the variational functional
within each element and taking derivatives with respect to
the dependent variables at all nodes of the elements. The
equations of all the elements are then collected. The
boundary condition is expressed in terms of nodal values
and incorporated into the equations. The equations are
then solved.
Relaxation Methods
This method may be applied to steady state problems
which are adequately described by the Laplace or Poisson
equations. The process involves obtaining steadily improved
approximations of the solution of simultaneous algebraic
difference equations.
The first step of the procedure is to replace the
continuous flow domain under investigation by a square or
rectangular grid system. The governing differential
equation is also replaced by corresponding difference
equations. Next, a residual, say R is defined corre
sponding to the point o on the grid. Ro represents the
amount by which the equation is in error at that point.
If all values of the equation are correct, R0 will be zero
everywhere. In the initial step, values are assigned at
all grid points and, in general, the initial residuals will
not be zero everywhere. The process now consists in
adjusting values at each point so that eventually all
residuals approach zero, or to at least some required
accuracy.
The reduction of residuals is achieved by a
"relaxation pattern" which is repeated at different grid
points so as to gradually spread the residuals and reduce
their value.
B: PHYSICAL MODELS
As implied in section A, direct analytical solutions
are frequently inadequate or impractical for engineering
application. In many cases, the analytical solutions which
are found are difficult to interpret in a physical context.
In an attempt to circumvent some of the shortcomings of a
purely mathematical approach, model and analog methods are
frequently employed. The analog may be considered as a
single purpose computer which has been designed and built
for a given problem.
Modeling, then, is the technique of reproducing the
behavior of a phenomenon on a different and more convenient
scale. In modeling, two systems are considered: the proto
type, or system under investigation, and the analog system.
These systems are analogus if the characteristic equations
describing their dynamic and kinematic behavior are similar
in form. This occurs only if there is a onetoone corre
spondence between elements of the two systems. A direct
analogy is a relationship between two systems in which
corresponding elements are related to each other in a
similar manner.
A model is an analog which has the same dimensions
as the prototype, and in which every prototype element is
reproduced, differing only in size. An analog is based
on the analogy between systems belonging to entirely
different physical categories. Similarity is recognized
in an analog by two characteristics: (1) for each dependent
variable and its derivatives in the equations describing
one system, there corresponds a variable with corresponding
derivatives in the second system's equations, and (2) inde
pendent variables and associated derivatives are related to
each other in the same manner in the two sets of equations.
The analogy stems from the fact that the characteristic
equations in both systems represent the same principles of
conservation and transport that govern physical phenomena.
It is possible to develop analogs without referring to the
mathematical formulation; an approach which is particularly
advantageous when the mathematical expressions are exces
sively complicated or are unknown.
Analogs may be classed as either discrete or
continuous with respect to space variables. In both cases,
time remains a continuous independent variable.
The need for complete information concerning the
flow field of a prototype system is obvious, and no method
of solution can bypass this requirement. However, in many
practical cases involving complicated geology and boundary
conditions, it is usually sufficient to base the initial
construction of the analog on available data and on rough
estimates of missing data. The analog is then calibrated by
reproducing in it the known past history of the prototype.
This is done by adjusting various analog components until a
satisfactory fit is obtained between the analog's response
and the response actually observed in the prototype. Once
the analog reproduces past history reliably, and within a
required range of accuracy, it may be used to predict the
prototype's response to planned future operations.
Sandbox Model
A reduced scale representation of a natural porous
medium domain is known as a sandbox model, or a seepage tank
model. Inasmuch as both prototype and model involve flow
through porous media, it is a true model.
A sandbox model is composed of a rigid, watertight
container, a porous matrix filler (sand, glass beads, or
crushed glass), one or several fluids, a fluid supply system
and measuring devices. The box geometry corresponds to that
of the investigated flow domain, the most common shapes being
rectangular, radial, and columnar. For onedimensional flow
problems, the sand column is the most common experimental
tool. Transparent material is preferred for the box construc
tion, especially when more than one liquid may be present and
a dye tracer is to be used. Porosity and permeability
variations in the prototype may be simulated by varying the
corresponding properties of the material used as a porous
matrix in the model according to the appropriate scaling
rules. The porous matrix may be anisotropic. In order to
measure piezometric heads and underpressures, piezometers
and tensiometers may be inserted into the flow domain of
the model.
Wall effects are often eliminated by gluing sand
grains to the walls of the box. This effect can also be
reduced by making the porous matrix sufficiently large in
the direction normal to the wall. Inlets and outlets in the
walls connected to fixed level reservoirs or to pumps are
used to simulate the proper boundary and initial conditions
of the prototype.
Water is usually used in models which simulate ground
water aquifers, although liquids of a higher viscosity may be
used to achieve a more suitable time scale.
The sandbox model is used extensively because of its
special features which permit studies of phenomena related
to the microscopic structure of the medium such as hydro
dynamic dispersion, unsaturated flow, miscible and immiscible
displacement, simultaneous flow of two or more liquids at
different relative saturations, fingering, wettability, and
capillary pressure. The capillary fringe in a sandbox model
is disproportionately larger than the corresponding capillary
rise in the prototype and, for this reason, the sandbox model
is usually used to simulate flow under confined rather than
phreatic conditions.
HeleShaw Analog
The HeleShaw or viscous flow analog is based on the
similarity between the differential equations governing two
dimensional, saturated flow in a porous medium and those
describing the flow of a viscous liquid in a narrow space
between two parallel planes. In practice, the planes are
transparent plates, and the plates are usually mounted in
a vertical or horizontal orientation.
The vertical HeleShaw analog was selected for this
study because it is more appropriate for the prototype system
under investigation. Also, it is not possible to model a
free goundwater table or percolation in a horizontal model.
A detailed description of this analog is presented
in Chapter III of this study.
Electric Analog
Three types of electric analogs are powerful tools
in the study of flow through porous media. They are the
continuous electric analog, the discrete electric analog
and the ion motion analog.
Continuous Electric Analog
This analog takes two forms: the electrolytic tank
and the conducting paper analogs. The analogy rests on the
similarity between the differential equations that govern
the flow of a homogeneous fluid through a porous medium, and
those governing the flow of electricity through conducting
materials.
In particular, Darcy's law for flow in a porous
medium and Ohm's law for the flow of an electric current
in a conductor may be compared. Also, the continuity
equation for an incompressible fluid flowing through a
rigid porous medium may be compared with the equation for
the steady flow of electricity in a conductor. One concludes
from this comparison that any problem of steady flow of an
incompressible fluid having a potential may be simulated by
the flow of electric current in an analog.
Discrete Electric Analog
This analog also takes two forms: the resistance
network analog for steady flow, and the resistancecapacitance
network for unsteady flow.
In this analog, electric circuit elements are
concentrated in the network's node points to simulate the
properties of portions of the continuous prototype field
around them. The unknown potentials are the solution of
the problem, and they can only be obtained for those points
which correspond to the nodes of the analog network. The
discrete electric analog is based on the finitedifference
approximation of the equations to be solved; therefore, the
errors involved in the discrete representation are the same
as those occurring in this approximation.
The electric resistor corresponds to the resistance
of soil to flow through it, and capacitors are used at the
nodes to simulate storage capacity of the prototype.
Ion Motion Analog
This analogy uses the fact that the velocity of ions
in an electrolytic solution under the action of a DC voltage
gradient is analogous to the average velocity of fluid
particles under imposed potential gradients in a porous
medium. In this case, both electric and elastic storativi
ties are neglected. The primary advantage of the ion motion
analogy is that, in addition to the usual potential distri
bution, it permits a direct visual observation of the movement
of an interface separating two immiscible fluids. In ground
water interface problems where gravity is involved, this
analog cannot be used. Scaling for the analog is based on
the similarity between Darcy's law and Ohm's law governing
the ion motion in an electrolytic solution.
Physically the analog consists of an electrolytic
tank having the same geometry as the investigated flow
domain. Inflow and outflow boundaries are simulated by
positive and negative electrodes, and two and three
dimensional flow domains may be investigated.
Membrane Analog
The membrane analog consists of a thin rubber sheet,
stretched uniformly in all directions and clamped to a flat
plane frame. The achievement of.equilibrium of various
forces and stresses in the membrane (caused by distorting
the frame or transversal loads) leads to the Laplace equation
and to the Poisson equation. The analogy is based on the
similarity between these two equations and the corresponding
equations that describe the flow in the prototype.
This method is applicable mainly to cases of steady
twodimensional flow involving complicated boundary geometry
and point sources and sinks within the flow field.
Summary
Following Bear (1972), Table 2.1 is presented as a
summary of the models and analogs discussed in section B of
this chapter. In section A, the numerical methods discussed
are most likely to be carried out on a digital computer. It
is important for the investigator to examine both the cost
and the applicability of these various numerical and physical
methods to his particular case. An analog is usually pre
ferred to a digital solution when the accuracy and/or amount
of field data is small. In many simple cases, the analog is
likely to be less expensive than a digital computer; whereas,
for large regions or unsteady threedimensional problems, the
computer may be less expensive.
The HeleShaw model also has definite advantages when
demonstration of the saltwater intrusion phenomenon to a
public body, or other laymen involved in political decision
making, is considered. This type of model allows for direct
observation of the phenomenon without the numerical interpre
tations used in the computer models.
TABLE 2.1
APPLICABILITY OF MODELS AND ANALOGS
Characteristic
Dimensions of field
Steady or unsteady flow
Sandbox Model HeleShaw Analog
Vertical Horizontal
two
both
Simulation of phreatic surface yes'
Simulation of capillary yes
fringe and capillary pressure
Simulation of elastic yes,
storage di
Simulation of anisotropic yes
media'
Simulation of medium yes
inhomogeneity
Simulation of leaky formations yes
Simulation of accretion yes
Flow of two liquids with appr
an abrupt interface
Simultaneous flow of two yes
immiscible fluids
Hydrodynamic dispersion yes
Observation of streamlines yes,
dime
tran
for
sion
or three
two two
both both
yes* no
yes no
Electric Analogs
Electrolytic RC Network Ion Motion
two or
steady
yes1
no
three two or three
both
no'
no
for two yes yes yes, for two yes
mensions dimensions
yes yes yes yes
kx5kz kx ky
yes5 yes5 yes yes
oximately
yes yes yes' yes
yes yes yes, for two yes
dimensions
yes yes no6 no6
no no
r
for two I
nsions, near
parent walls
three dimen
s
no no
yes no
two(horizontal)
steady
no
no
no
yes
kxk y
yes
no
no
yes
no
no
no
I Subject to restrictions because of the presence of a capillary fringe.
z By trial and error for steady flow.
I By trial and error for steady flow, or, as an approximation, for relatively small phreatic surface fluctuations.
By scale distortion in all cases, except for the RC network and sometimes the HeleShaw analog where the hydraulic
conductivity of the analog can be made anisotropic.
s With certain constraints.
6 For a stationary interface by trial and error.
Membrane
Analog
two(horizontal)
steady
no
no
no
yes
kx y
yes
no
yes
no
no
no
no
CHAPTER III
THE HELESHAW MODEL
The viscous flow analog, more commonly referred to
as a parallelplate or HeleShaw model, was first used by
H. S. HeleShaw (1897, 1898, 1899) to demonstrate two
dimensional potential flow of fluid around a ship's hull
and other variously shaped objects. The analog is based on
the similarity of the differential equations which describe
twodimensional laminar flow, or potential flow for that
matter, of a viscous fluid between two closely spaced parallel
plates; and those equations which describe the field of flow
below the phreatic surface of groundwater, namely Darcy's
law:
qx = K (3.la)
x x ax
qz = K (3.1b)
where
qx, qz = Darcy velocity of specific discharge in the
xdirection and zdirection, respectively
K K = hydraulic conductivity in the xdirection
Kx z and zdirection, respectively
x = horizontal direction (major flow direction)
z = vertical direction
= potentiometric head
and, by use of the conservation of mass principle, the
Laplace equation
2 + 822 = 0 (3.2)
ax2 az2
Viscous Flow Analog
To demonstrate the analogy of model and prototype,
the equations of motion and continuity for laminar flow of
a viscous fluid between two closely spaced parallel plates
will be developed and then compared to equations 3.1a,
3.1b, and 3.2.
Consider a viscous incompressible fluid flowing
ever so slowly between two parallel plates which are spaced
such that the Reynolds' number, R based on the interspace
width is less than 500 (Aravin and Numerov, 1965). In
Cartesian coordinates, the general NavierStokes equations,
i.e., the equations of motion, are
DV B + 1 I + V2Vx (3.3a)
DVy = B + + V2V (3.3b)
Dt y+ y ]
Vz = B + 1 3P + v2V (3 .3c)
Dt z p DZ
where
D = substantial derivative = + V a
Dt at x ax
+ V a + V z 
V2 = Laplace operator = 2 + y2 +
Vx, V Vz = velocity in the x, y, and zdirections,
respectively
B B Bz = body forces in the x, y, and
xz zdirections, respectively
p = pressure
p = density of the fluid
p = absolute viscosity of the fluid
y = horizontal direction (minor flow
direction)
t = time
Referring to the freebody diagram of the idealized
flow regime shown in Figure 3.1, if no slip conditions
(adherence to the walls) of the fluid particles are assumed
in the molecules closest to the walls of the parallel plates,
it is easily seen that the velocity gradient in the
ydirection is much larger than the velocity gradient in
either the x or zdirections. Thus., the first and second
order partial derivatives taken with respect to both x and
z may be neglected when compared to those taken in the
ydirection. Secondly, because of the very low velocities
("creeping" motion) the intertia terms, that is the terms
FIGURE 3.1
Free body flow diagram for HeleShaw model.
b
m
_~I
_ _II~
_I
__
on the left side of equations 3.3, are very small when
compared to the viscous terms, those on the right side of
equations 3.3, and may be neglected. Thirdly, because of
the restriction to two dimensions, the velocity in the
ydirection is taken to be zero; consequently, all rates
of change of velocity in the ydirection.must be zero.
Finally, the only noncancelable body force acting on the
fluid is gravity which acts only in the vertical. Mathe
matically, Bx x (gz) = 0; By = (gz) = 0; and
Dz ~ a (gz) = g = 32.17 ft./sec.2. Incorporating all
of the above arguments and values into equations 3.3, the
equation of motion becomes:
32V
3+ 0 (3.4a)
2= 0 (3.4b)
82V
pg a = 0 (3.4c)
Defining the potentiometric head, or potential, i = z + p/y,
where y equals the unit weight of the fluid, and taking the
partial derivative with respect to x, y, and z, the following
results are obtained after multiplying through by the unit
weight of the fluid:
Y M =_ ap (3.5a)
Y ax ax
Y  = DP (3.5b)
Tay ay
Y (y y 'I
Introducing these relationships into equations 3.4 and
dividing through by the unit weight yields:
32V
3t = P2 x (3.6a)
3x y ay2
D 0 (3.6b)
ay
32V
S_ 1 2 z (3.6c)
2z y ay2
It is evident from equation 3.6b that the potentiometric head
is constant in the ydirection. It is possible then to inte
grate the first and third equations of equation 3.6 with
respect to y. After separating variables and integrating
3V
once, using the boundary condition y = 0, x = 0, and
3V ay
z = 0, the following equations are obtained:
ay
y a 1 x (3.7a)
a x y 8y
y = z (3.7b)
Yz y ay
Integrating, once again, using the second boundary condition
y = b/2, Vx = 0, Vz = 0 (no slip) the above becomes, after
solving for the respective velocities:
Vx y2 ) (3.8a)
Vz 2 (2 ] (3.8b)
Note, where b is the spacing between the plates, that if a
potential = y b4 is defined, equations 3.8 can
be written:
V (3.9a)
x ax
V (3.9b)
P, the velocity potential, is dependent only on y. Inte
grating the velocity profiles established by equations 3.8
between the limits of b/2, and dividing by b, the
directional specific discharges are obtained:
q +b/ 2 y= b/2
qx b _/2 x y b = x N b/2
= b2 Y (3.10a)
f+b/2
qz b jb/2 vz dy 2i z (3 jb/2
b2 Y (3.10b)
TZ T az
It is obvious that for a model of constant spacing b, the
quantity b2 does not vary in either the x or zdirection.
Defining the model hydraulic conductivity as K = K
= b* ~ equations 3.10a and 3.10b become:
qx = Kxm x (3.11a)
qz = Km (3.11b)
which, of course, is analogous to equations 3.1.
Consider, now, the twodimensional continuity
equation for flow between parallel plates:
aV 3V
_x + z 0 (3.12)
ax az
The specific discharge, or Darcy velocity, is
related to the velocity by the vector equation n q = V,
where ne is the effective porosity of the flow media. In
the model, ne equals 1. From the analogy Vx = q x; Vz = qz'
and substituting the relationship obtained from equation
3.11 into equation 3.12:
x m ax x Kzm = z (3.13)
or dividing by Km and recalling that for a model Km = Kxm
= Kzm
9 + 2i (3.14)
ax2 az2 (3.14)
which is clearly analogous to equation 3.2.
The similarity of equations 3.1 and 3.11 and
equations 3.2 and 3.14 establish the analogy.
Scaling
The twodimensional equation along the free surface,
or water table, of an anisotropic porous media given by Bear
(1972) is:
K p .2 + K 2 = n (3.15)
xp xp zp z az ep at
Sj p p
where the subscript p denotes the prototype. For a Hele
Shaw model, using the subscript m, the equation can be
written as:
2 Mjt 2
K Im + K = n mm (3.16)
xm jxm zm azm a m em atm
Introducing the similitude ratios, denoted by the
subscript r, of the corresponding parameter of model and
prototype:
K
Kxr = K (3.17a)
xp
K
Kzr = (3.17b)
zp
x
Xr x (3.17c)
P
z
zr = z (3.17d)
P
r I (3.17e)
P
n
ner = n (3.17f)
ep
t
tr = t (3.17g)
P
and substituting these relationships into equation 3.15, the
following is obtained:
Kxm [( m ) 2 Kzm [*a(m/r) 2 (m r)
T Tv K+ _zz
Kxr m/x Kzr Zm/Zr) (Zm/Z
nem < tm/tr) (3.18)
The ratios of model to prototype quantities are constant and
can be removed from behind the differential; therefore,
equation 3.18 can be rearranged in the following way:
Kxr r2 xm 2 Z r m 2
K r 2 xm ax m zm K ^ 2 3z
Zr m'I tr m
Kzr r zmJ nerzr nem tm (3.19)
Comparing the equations 3.16 and 3.19, it is evident that, if
the equations are identical, the following must be true:
x 2 z 2 z t
1 K r r r r (3.20)
xr r zr r zr'r er r
Solving the third equality for zr, the following important
relationship is found:
zr =r (3.21)
The second equality, after crossmultiplying, yields:
S xr K (3.22)
zr Kzr
Recalling the definitions of Kxr and Kzr (equations 3.17a
and 3.17b) and remembering that K = K in an isotropic
xm zm
model, the above equation can be rewritten:
x r 2 K/Kxp K
r xm xp zp (3.23)
r K zm zp Kxp
K
The ratio of zK is called the ratio or degree of anisotropy
xp
of the prototype.
Time
Using the fourth equality of equation 3.20, the time
ratio of the model and prototype is established:
n z
t er r (3.24a)
r K
zr
n x 2
t er r (3.24b)
r K rr
Substituting the vertical ratio, zr, for the potentiometric
head established by equation 3.20 and the similitude ratios
of time, hydraulic conductivity and porosity into equation
3.24b results in
tm nem X r (3.25)
tp nep Kxm z
The effective porosity of an isotropic model, nem, is unity.
The hydraulic conductivity of the model was defined pre
viously as p?, thus the time scale for the model is finally
written as:
K x 2
t 12 vK xx r t (3.26)
m g ep zr P
Anisotropy
The HeleShaw model is normally isotropic. This is
because of the nonvariance of the. spacing of the parallel
plates. There are, however, two methods for simulating
anisotropy in a model. Equation 3.23 gives a clue as to
the first possibility of simulating anisotropy:
r xm x_ zp (3.23)
r zm /Kzp xp
Since Km = Kzm, the x or z ratio can be adjusted so that the
model's hydraulic conductivities are kept equal. This is
usually done by choosing a suitable horizontal ratio. Know
ing the prototype parameters, a vertical scale for the model
is computed so that the aforementioned conductivities are
kept equal, demonstrating:
zr zm X m (3.27)
p zp p
Solving for zm:
K x
zzm p xm z (3.28)
zp p
Unfortunately, the geometric distortion method is adequate
for modeling only one ratio of anisotropy. If there is a
second aquifer, within the prototype which has a different
vertical or horizontal hydraulic conductivity, the second
aquifer cannot be correctly simulated unless, of course,
the second aquifer's ratio of anisotropy is the same as the
ratio of the first. This restriction would severely limit
the use of the HeleShaw analog in modeling of regional
groundwater problems unless another method were available
to correct the ratio of anisotropy.
PolubarinovaKochina (1962) suggests using a grooved
plate within the model to correct the ratio of anisotropy of
the second flow zone. The plate may be grooved in any one
of several methods. It matters little whether a grooved
plate is sandwiched between the parallel plates, or if
rectangular bars are attached to the front or back plate.
The degree of anisotropy of the second aquifer and the amount
of geometric distortion used to model the first flow zone
determines the directions in which the grooves, or bars,
are placed; however, the grooves are normally placed
horizontally or vertically. Collins and Gelhar (1970) have
developed the conductivity equations for the flow zone in
which PolubarinovaKochina's grooved plate is used. The
analysis assumes onedimensional flow and can be used
equally well with either vertical or horizontal orientation
of the grooves.
Following Collins and Gelhar (1970), consider flow
in a grooved portion of a model. For simplicity, assume
Figure 3.2 is a section of the grooved zone. Assuming such,
the horizontal direction then corresponds to the xdirection
and the grooves, which are vertical, lie in the zdirection.
Area 1 is associated with the wider spacing of length ab.
Area 2 is associated with the narrower spacing of length b.
Since flow area 1 is the much larger of the two areas, most
of the frictional head loss occurring through the total
length L is developed in flow area 2 which has length
(1 M)L. Lambda, X, is a length correction factor..
Referring to Figure 3.3, the potentiometric gradient
ax across area 2 is:
FIGURE 3.2
Section of anisotropic grooved zone in HeleShaw model.
Back Plate
Rectangular
Bar
AL Front Plate
L (1 A)L
Plan View
Perspective View
FIGURE 3.3
Head loss in grooved anisotropic zone.
A 1)
x or z
XL
1
 
__
ax (1 A)L (3.29)
For high values of a:
___ A = 2C(3 .30)
ax L L
but, from equation 3.29, 2 = (1 A) 3 so that:
a = (1 ) ~2 (3.31)
ax ax
Applying Darcy's law to area 2:
q= K (3.32)
qx x2 ax
and, substituting the previous expression for :2
ax
K
S x2 1 (3.33)
x (1 A) ax
The effective hydraulic conductivity in the xdirection
then is:
K Kx b2g (3.34)
xm (1 X) 12v 1 ) (334)
Consider vertical flow through the grooved zone
illustrated in Figure 3.2. In particular, consider flow
downward through areas 1 and 2. The total discharge
through these areas can be written as the sum of the dis
charge through each, that is, Qzm = Q1 + Q2. Applying
Darcy's law for the total discharge, Q:
48
Q, = K (abAL) (3.35)
Q2 = K ( ) Lb (3.36)
z2 az
Adding Qi and Qz:
Qm= K (aX) + K (1 )]bL (3.37)
zm z1 z2z z
For flow area 2 it is not unreasonable to assume that the
frictional forces in the fluid boundary to either side of
area 2 are negligible. Therefore, the vertical conduc
tivity in this area is the same as defined by the earlier
analysis, that is:
K b2 = b2 (3.38)
Z2 = l 12
where v is the kinematic viscosity. Furthermore, if b/ab
<< 1, the flow in area 1 can be assumed roughly equivalent
to flow through a rectangular hole. According to Rouse
(1959), the equation of motion through a rectangular
cross section of length XL and width ab is given by:
00
v x(x XL) + X sin JTx
z 2i 3z jl AL
x A. cosh 1J + B. sinh (3.39)
S1AL j AL
49
where
A. 2y(AL) 2 (cos j I) (3.40)
and,
cosh jab 1(
B. = A. (3.41)
sinh jab3
XL
By integrating Vz over the area and dividing by the total
area ALab, the mean velocity is given by:
V= (a b)2 (3.42)
z az
where
S= 192 b j (cos (j ) 1)2 _
= 2 L j( 4j5 2cab
(3.43)
Since the terms of the infinite series decrease as j5s, only
the first term of the series need be considered and retained,
so that:
1 19 tanh fXL (3.44)
Additional information may be found in Appendix A. From
equation 3.42, the equivalent hydraulic conductivity in
area 1 is given by:
K Y (ab)2 (3.45)
Kz P 12
Finally, introducing the values found for K and K into
zE Z2
equation 3.37:
Qzm =] (b) (aX) + b (1 U) bL (3.46)
or,
Qzm = (3 + 1 A) bL (3.47)
from which it is seen that the effective vertical hydraulic
conductivity is given by:
Kzm = (a 3X1 + 1 X) (3.48)
after defining Qzm/bL = Vz, where Vz = qz is the effective
vertical specific discharge. Equations 3.34 and 3.48 give
the second method available to correct the hydraulic con
ductivity of a model so that it can simulate the true
ratios of anisotropy found in the prototype.
Leakage
An aquiclude can be defined as a soil stratification
in which the hydraulic conductivities are zero. In certain
geohydrologic problems, it is convenient to assume such
conditions. However, in reality few soil masses are truly
impervious. The degree of perviousness in a stratum is
referred to as leakance and it is generally assumed that the
direction of flow is only vertical. There is no horizontal
flow, that is,
Kxp = 0 (3.49)
Bear et al. (1968) suggest the use of vertical slots
to model such a semipervious layer. To accomplish this, the
spacing between the parallel plates of the HeleShaw analog
is filled with a slotted middle plate. See Figure 3.4.
The analysis to determine the effective vertical
hydraulic conductivity of a model's leaky layer closely
parallels that for an anisotropic grooved zone. Again,
following Collins (1970), Darcy's law for flow through a
vertical slot is:
QZ = ALab (3.50)
z = z 3z
The effective specific discharge through the slot found by
integrating the Rouse equation (equation 3.39) is the same
as equation 3.42 from which is found the hydraulic conduc
tivity:
Kz (b)2 (3.51)
and, introducing the above into equation 3.50:
Qz = (b)2 Lab (3.52)
z U 2 aL z
FIGURE 3.4
Section of leaky zone in HeleShaw model.
y
Back Plate
Flowspace
Solid ab
x l b
FI v
PL (1X)L t
Front Plate L
Plan View
Perspective View
Again, the effective specific discharge, or mean velocity,
is equal to:
z 3 b (3.53)
Vz bL 12 az 353)
so that the effective hydraulic conductivity of a leaky layer
in the model is:
Kzm a b2 (3.54)
Storativity
While the problem of storage has not been completely
solved, it has, in general, been neglected by most researchers.
Bear (1960) suggests that discrete tubes attached to either
the front or back plate and connected to the aquifer be used
to model the specific storage of a confined aquifer. For a
nonisotropic aquifer, the right hand side of equation 3.13
is not zero, but, in fact, equals the specific storage, S ,
times the rate of change of the potentiometric head. Rewrit
ing the twodimensional equation 3.13 for both model and
prototype to include the above gives the following:
K 2 + K  = S  (3.55)
xp ax 2 zp aZ 2 op at
p p p
K 2m + K m m (3.56)
xm Dxm2 zm 3Z%2 om atm
m m m
Defining a ratio of storativity:
S
Sor = (3.57)
op
it follows from inspection that,
z K z
K Sr  (3.58)
xr x2 zr or t
or that,
K t S
zr r om (359)
or z S (3.59)
r op
Referring to Figure 3.5, the storage represented by
the model in the discrete length 1m is equal to:
A
S om_ m (3.60)
mo m b 1 z
mmm
where A is the crosssectional area of the storativity tube.
Introducing the above into equation 3.59 and solving for Am
t
A bl z Sop K r (3.61)
m mmm op zr z 2
r
Discharge
The discharge scales are.obtained from Darcy's law.
Written for both prototype and model with the usual sub
scripting, these are in the xdirection:
Q K  b z (3.62)
xp = xp ax pp
FIGURE 3.5
Storage manifold for HeleShaw model.
Area, A
bm m
Front
Plate
Back
Plate
End Section
I I
Elevation
and,
(3.63)
xm xm 3x mm
m
Dividing equation 3.63 by equation 3.62 and recalling the
definitions for the various parameters' ratios, it follows
that:
z z 2
S K r b r=K b r
xr xr r r x xr rx
Similarly, in the zdirection,
x
Q = K r b r K b x
zr zr r r zr zr r r
r
(3.64)
(3.65)
Solving equation 3.22 for the hydraulic conductivity in the
xdirection:
Kx K
xr zr
S2
Xr
r
(3.66)
and, substituting
that:
Qxr = Kzr
this result into equation 3.64, it follows
r b r K b x
r r
or,
Qxr = Qzr = Q
*xr zr r
(3.67)
(3.68)
Accretion
Accretion, R, is the rate at which a net quantity
(precipitation and surface inflow minus evapotranspiration,
runoff, etc.) of liquid is taken into the flow system at
the phreatic surface. It is measured as a volume per unit
horizontal area per unit time, that is:
Q
Rr (3.69)
rr
From equations 3.64 or 3.65, it follows that:
K z 2
R x Kzr (3.70)
r
Volume
On occasion, volume, U, is of some importance. The
volume scale follows directly from continuity, that is:
Ur = Q tr (3.71)
Substituting the values found from equations 3.65 and 3.24a
for Qr and tr, respectively, the above equation becomes:
z
r zr r r er K r er r r (3.72)
zr
As inferred earlier in this section's opening sentence, the
volume scale is usually neglected; however, in the case of
free surface water bodies, lakes, rivers, etc., if the volume
exchange of liquid is of interest and has to be modeled, the
60
volume scale requires an additional restriction. In the
following analysis, the bar above the width dimension
indicates the free water surface of a river, lake, ocean,
or such.
In the portion of the model simulating the body of
water, the spacing of the model is increased to maintain
hydrostatic pressure distributions within the model. The
narrower spacing of the model is, of course, a measure of
the hydraulic conductivity of the aquifer. In the proto
type, however, the width of the open water and the aquifer
are equal and this leads to the following (Bear, 1960) for
the model and prototype, respectively:
U = n b x z (3.73a)
r er r r r
U = n b x z (3.73b)
r er r r r
The same volume ratio must be applicable to both the narrow
and the enlarged interspace; therefore, Ur Ur. It follows
that:
n er b = ner b (3.74)
but,
n
n em 1 (3.75)
er
n
ep
so,
br = ner br (3.76)
Note that for an anisotropic media, nem does not necessarily
equal one.
CHAPTER IV
SITE SELECTION AND PROTOTYPE GEOLOGY
AND HYDROLOGY
Site Selection
The site selected for this study is the middle Gulf
area of Florida. This region has a rapidly expanding popu
lation with a corresponding growth in water demand. The
increased pumping to satisfy this demand also increases
the likelihood of saltwater intrusion, and, in fact, a
number of municipal supply wells in the coastal zone have
been shut down in recent and past years due to chloride
contamination.
Black et al. (1953) list eight factors responsible
for saltwater intrusion. They are:
1. Increased water demands by municipalities.
2. Increased water demands by agriculture.
3. Increased water demands by industry.
4. Excessive drainage.
5. Lack of protective works against tidewater in
bayous, canals, and rivers.
6. Improper location of wells.
7. Highly variable annual rainfall with insufficient
surface storage during droughts.
8. Uncapped wells and leakage.
Of these eight factors, numbers 1, 2, 3, 6, and 7 would apply
to this area. The city of St. Petersburg is an outstanding
example of these problems. Their original water supply was
from local artesian wells, but increased demands caused salt
water intrusion and forced the closing of these wells. In
1929 the present CosmeOdessa field was located farther
inland to escape this problem.
One of the major water supply systems in this region
is the Pinellas County Water System, and this study is
centered around the EldridgeWilde well field of this system.
The location of EldridgeWilde in relation to several of the
population centers of this region is shown in Figure 4.1.
It is about 8 miles east of the Gulf of Mexico and encom
passes an area in the northeast corner of Pinellas County,
at the intersection of the boundaries of Pinellas, Hills
borough, and Pasco counties.
This system was instituted in 1937 to supply the
towns along the Gulf coast from Belleair Beach to Passa
Grille. Its original form was that of a raw water reservoir,
and the first wells were not drilled until 1946 in the
McKay Creek area. These wells were soon contaminated with
salt water, and investigations were begun in 1951 to locate
the well field at its present site. This well field has
grown over the years, and in 1970 (Black, Crow and Eidsness,
Inc., 1970), the waterworks facilities at EldridgeWilde
included: sixtyone water wells, over 11 miles of raw water
FIGURE 4.1
Regional area of prototype.
il MEXIUI0GQ;:1;:.: ,
iiii  / ko
V ,^ / 10'
. .
. ..
Dade City
60' /
40'
, \
70'
0
S80'
^ ./ I
K
N " 50'
S 30'
EldridgeWilde Well Field
Lake Tarpon
0O Tampa
:: :1 :i:: Peter .sbug : ::? :::: ::::::l
::.. : : ... .. : :::::: ::::: :
...................i 9
collection piping, water treatment facilities consisting of
aeration and chemical treatment, including chlorination and
fluoridation, and high service pumping units.
All wells are open hole and penetrate the Floridan
aquifer at depths from 140 to 809 feet below ground surface,
averaging 354 feet. The design capacity, of the field at the
present time is 69 million gallons per day, although the
maximum allowable pumpage has been set by the Southwest
Florida Water Management.District at 28 million gallons per
day on the average with a maximum day of 44 million gallons
per day.
In selecting the prototype location within the site
area, two characteristics of the vertical HeleShaw analog
must be considered. The first characteristic is that there
can be no general flow normal to the parallel walls of the
model. This means that the flow from one end of the model
to the other is streamline flow. The second characteristic
is that the ends of the model are finite. Therefore, the
prototype must be along a streamline in the flow domain and
have boundary conditions which are "infinite" reservoirs or
water divides.
The prototype selected meets the above requirements
and includes the point of interest, i.e., EldridgeWilde
well field. The center line of the prototype is shown in
Figure 4.1 as the unbroken line passing through Eldridge
Wilde in a southwest to northeast direction. The dotted
contours in the figure define the potentiometric surface of
the Floridan aquifer in feet above mean sea level as of May,
1971. They were obtained from a map publication entitled
"Potentiometric Surface of Floridan Aquifer Southwest Florida
Water Management District, May, 1971" prepared by the U. S.
Geological Survey in cooperation with the Southwest Florida
Water Management District and the Bureau of Geology, Florida
Department of Natural Resources. Now, in a flow field,
streamlines are perpendicular to potentiometric lines. As
can be seen from the figure, the prototype orientation
reasonably satisfies the streamline requirement. The proto
type is terminated on the southwestern end at the 15 feet
depth contour in the Gulf of Mexico, and it is assumed that
this satisfies the infinite reservoir boundary condition.
The northeastern terminus is located in the center of the
80 feet contour, southwest of Dade City. This location
satisfies the water divide boundary condition. The area in
the vicinity of the 80 feet contour is known as the Pasco
High. The overall length of the prototype is 36 statute
miles. The width of the prototype is taken to be 3.5
statute miles. This dimension is sufficient to include the
cone of depression caused by pumping in EldridgeWilde well
field, and is based on the results of a study by Mr. Evans
(employing a numerical model) for Black, Crow and Eidsness,
Inc. The land surface contours of the prototype were
obtained from U. S. Geological Survey topographic maps.
The bottom boundary of the prototype is taken to be the base
of the Lake City Formation, with depths being determined
from available well logs of wells in the prototype vicinity.
The maximum depth from highest land surface to deepest point
is 1340 feet.
Prototype Geology
Stewart (1968) identifies eight formationsas being
of interest in terms of water production in the prototype
area. They are in descending order, the Undifferentiated
Deposits, Tampa Limestone, Suwannee Limestone, Crystal
River Formation, Williston Formation, Inglis Formation,
Avon Park Limestone, and Lake City Limestone. Underlying
the Lake City Limestone is the Oldsmar Limestone which is
not used as a source of water at present.
The Undifferentiated Deposits are interbedded sand,
silt, and clay of PostMiocene age and range in thickness
from zero near the Pasco High to 60 feet in the Eldridge
Wilde well field. The thickest deposits are in northeast
Pinellas County around the north end of Lake Tarpon where
sand dunes, as much as 40 feet high, overlie alternating
layers of clay, thin limestone beds, and sand greater
than 70 feet. thick.
The Tampa Limestone is a hard, dense, sandy, white
to light tan, or yellowishtan fossiliferous limestone of
Miocene age. This limestone is near the surface in the
area of the Pasco High and about 80 feet below land surface
at the EldridgeWilde well field. At EldridgeWilde, the
thickness varies erractically from about 20 to 240 feet.
The Tampa Limestone is a poor to fair producer of water.
The Suwannee Limestone is a soft to hard, nodular
or grandular, fossiliferous white to tan limestone of
Oligocene age and is about 200 feet thick. The Suwannee
and Tampa Limestones are the major water producers for
wells in the area.
The Crystal River, Williston, and Inglis Formations
comprise the Ocala Group of late Eocene age. The Crystal
River and Williston Formation are lithologically similar
units of white to cream, porous, soft, coquinoid limestone
and are generally poor producers of water. The Inglis
Formation is a hard, cream to brown to gray fossiliferous
limestone and is generally a good producer of water.
The Avon Park and Lake City Limestones are litho
logically similar units of soft to hard, cream to brown,
fossiliferous limestone with beds of dolomitic limestone
and some gypsum. Both formations are good producers of
poor quality water.
The Oldsmar Limestone is a fragmental dolomitic
limestone with lenses of chert, thin shale beds, and
some gypsum.
In this study, two formations are considered,
the Undifferentiated Deposits and the Floridan aquifer.
The Floridan aquifer is considered to contain all formations
from the Tampa to, and including, the Lake City Limestone.
The transmissivity of the Floridan aquifer ranges
from about 165,000 to 550,000 gallons per day per foot,
and the coefficient of storage ranges from about 0.0005
to 0.0015. The coefficient of leakage is approximately
0.0015 gallons per day per cubic foot.
Based on groundwater discharge and water levels,
the estimated recharge (leakance) to the Floridan aquifer
was computed (Stewart, 1968) to be about 103 million
gallons per day. Based on aquifer test data, the estimated
recharge for a 250 square mile area was 90 million gallons
per day.
The Undifferentiated Deposits act as a confining
layer, and the Floridan aquifer is thus under artesian
conditions.
Prototype Hydrology
The surface waters of the area consist of many
lakes and few streams. Because of the flat topography,
little water runs off into streams, and swampy wetlands
are numerous. Most rainfall evaporates or is transpired
by plants.
The Floridan aquifer is recharged through the
Undifferentiated Deposits by surface and groundwater
derived from local rainfall. Many millions of gallons
of water are also admitted to the aquifer by numerous sink
holes in the region. Water levels in the Floridan aquifer
respond to rainfall since this is the recharge source.
This response is not immediate, but usually fluctuates with
the wet and dry seasons. Water levels in wells which are
not directly affected by local pumping show yearly lows in
the dry season, April and May, and yearly highs during the
wet season, late summer or early fall (Black, Crow and
Eidsness, Inc., 1970).
The aquifer recharge has been estimated (Black,
Crow and Eidsness, Inc., 1970) from available data and the
use of the following formula:
Aquifer Recharge = P + SWI + GWI ET R GWO
The basin area is 575 square miles, changes in
storage are assumed zero, and evapotranspiration is assumed
to be 75 percent of the precipitation. The applicable
values are listed below in million gallons per day:
P = Precipitation
SWI = Surface Water Inflow
GWI = Groundwater Inflow
ET = Evapotranspiration
R = Runoff
GWO = Groundwater Outflow
Aquifer Recharge
This value is in reasonable
reported values (Stewart, 1968).
= +1492
= + 0
= + 0
= 1119
= 218
= 37
= + 118
agreement with previous
CHAPTER V
DESIGN, CONSTRUCTION, AND OPERATION OF MODEL
Design
Prototype
The selection of the prototype area was discussed
in Chapter IV. Table 5.1 is a summary of the prototype
characteristics. The leaky layer is synonymous with the
undifferentiated deposits. The top and bottom of the
Floridan aquifer were determined by straightline extrapo
lation from available well logs.
The hydraulic data are within the reported range
of values and are the result of a trial and error process
to stay within the range and stillproduce a reasonable
model.
Model
The purpose of the HeleShaw analog in this study
is to model saltwater intrusion." Before discussing the
model design, it seems appropriate at this point to pro
vide some background about saltwater intrusion. Water,
in general, whether it be surface water or groundwater,
is continually migrating towards the sea, where an
equilibrium, or moving freshwater/saltwater interface,
TABLE 5.1
PROTOTYPE PARAMETERS
Floridan
Parameters Aquifer Leaky Layer
Geometric
x (ft.)
z (ft.)
yp = bp (ft.)
Hydraulic
Txp (gpd/ft.)
Tzp (gpd/ft.)
Kxp /Kzp
Leakance (gpd/ft.3)
Kzp (gpd/ft.2)
Kxp (gpd/ft.2)
S
v @ 77* F (ft.2/sec.)
gp = 32.2 ft./sec.2
190,080
1,340
18,480
225,000
184,426
1.218
161.4
196.7
0.00158
0.965 (105)
190,080
55
18,480
0
0.0015
0.09
0
0.965 (105)
is established. The two fluids are miscible, but because
of the difference in densities and the very low velocities,
the interface is formed. Across the interface, the salinity
varies from that of the fresh groundwater to that of the
ocean. The transition zone, as it is called, is due to
hydrodynamic dispersion and, although it is anything but
abrupt, it is usually assumed to be. The interface then
is generally selected to occur at some measured electric
conductivity or salt (chloride) concentration.
The earliest investigations of saltwater encroch
ment were made by BadonGhyben (1888) in Holland and
Herzberg (1901) in Germany. Working independently, both
investigated the equilibrium relationships between the
shape and position of the freshwater/saltwater interface.
Figure 5.1 shows a coastal phreatic aquifer and the
GhybenHerzberg interface model. BadonGhyben and Herz
berg assumed static equilibrium and a hydrostatic pressure
distribution in the fresh groundwater and stationary saline
groundwater near the interface.
Considering a point P on the interface, and choosing
mean sea level as the datum, the pressure at point P is:
pp = h 5 (5.1)
where
h = vertical distance from mean sea level to
s point P
s = unit weight of sea water
FIGURE 5.1
GhybenHerzberg interface model.
75
Water Table
Vertical
h h
pdcA Interface
Y^ \pdA Y
This pressure may, also, be expressed by:
p = {hf + hs] f (5.2)
in which hf equals the vertical distance from mean sea level
to the phreatic line at the location of p and yf equals the
unit weight of freshwater. Equating the, preceding two
equations:
hSys = hfyf + hsf (5.3)
and, rearranging,
hsYs hf = hff (5.4)
Solving for hs :
yf
hs Y= Yf hf (5.5)
Introducing y = pg, where p is the density, factoring and
canceling out the gravity term, the GhybenHerzberg
relation is found:
f
h = f hf (5.6)
S PS f
for a saltwater density of 1.981 lb sec.2/ft.4 and a fresh
water density of 1.933 lb sec.2/ft.4, @ 250 C, the quantity
pf/ps pf) = 40. The implications of equation 5.6 are
rather dramatic. For instance for every foot of freshwater
above the datum, there is 40 feet of freshwater below
the datum. More importantly however, consider the effects
of lowering the phreatic surface. For every one foot drop
of the water table, the interface raises 40 feet. It must
be remembered that the above analysis assumes static
conditions. This, in fact, is not always the case. The
position of the interface is a function of dynamic conditions
rather than static. Even so, in cases where flow is quasi
horizontal, i.e., the equipotential lines are nearly
vertical, equation 5.6 is valid.
Many investigators have incorporated dynamic forces
into the analysis of the stationary interface. Hubbert
(1940) was able to ascertain a more accurate determination
of the shape of the interface near the coast line. He
assumed that at the interface the tangential velocity was
zero in the saltwater, but increases with horizontal dis
tance in the freshwater as the coast line is approached.
This then is the cause for the interface to tilt upwards
as the sea is approached and the greater depths found than
those estimated by the GhybenHerzberg relationship.
Hubbert showed that the GhybenHerzberg equation holds
between points on the water tableand the interface along
an equipotential line, rather than along a vertical plane.
R. E. Glover (1959) modeled an infinitely deep
coastal aquifer by assuming no flow in the saltwater
region, a horizontal water table and a horizontal seepage
face located seaward of the coast line. He found an exact
solution for the shape of the wedge, giving the following
relationship:
z2 2qx + Pf 2 (5.7)
K s f K2 S f
Pf Pf
where x and z are the horizontal and vertical directions,
respectively, q is the seepage rate per unit width and K
is the hydraulic conductivity. De Wiest (1962) using
complex variables and a velocity potential of D = Kx .
p Pf)/pf derived the same equation.
Bear and Dagan (1964b), using the Dupuit assumptions
and the GhybenHerzberg equation, developed the approximate
shape of the interface for a shallow aquifer of constant
depth.
All of the above investigated the equilibrium
position of the saltwater/freshwater interface.
If there is a change in the freshwater flow regime,
a transition period is caused during which the interface
moves to a new point of equilibrium. The nonlinear
boundary conditions along the interface make the solution
for the shape and position of the transient interface all
but impossible except for the simplest geometries. Bear
and Dagan (1964a), as well as other investigators, have
used the Dupuit assumptions to approximate the rate of
movement of an interface in a confined aquifer. Following
PolubarinovaKochina's (1962) suggestion, they assumed
quasisteady flow and were able to approximate the inter
face shape and position for both a receding motion and
landward motion of the interface.
Characteristically, the solutions obtained by
investigators to date have all had simple,geometries and
involved simplifying assumptions, some of which have had
little resemblance to actual conditions. Therein lies the
advantages of a HeleShaw analog, complex geometries and
boundary conditions can be modeled with relative ease.
In order to satisfy additional similitude require
ments for the flow of two liquids with an abrupt interface,
as in this study, and to provide a suitable time ratio,
two liquid silicone fluids were chosen to be used in the
model. Dow Corning Corporation Series 200 silicone fluid
was used to model fresh water. Series 510 silicone fluid
from the same company was used to simulate salt water.
The 200 Series fluid and the 510 Series fluid have densi
ties of 0.977 gm/cm3 and 1.00 gm/cm3, respectively, at
250 C. Both fluids have kinematic viscosities of 500
centistokes at 250 C. Dow Corning 200 fluid is a clear
dimethyl siloxane which is characterized by oxidation
resistance, a relatively flat viscositytemperature slope
and low vapor pressure. Dow Corning 510 fluid is a clear
phenylmethyl polysiloxane which also has a relatively flat
viscositytemperature slope. In order to locate and follow
the interface movement, the denser fluid was dyed blue and
the lighter fluid dyed orange.
The dimensions of the model were selected so that a
unit of reasonable size would be produced. These dimensions
are 11.75 feet long, 0.5 inch wide inside, and 2 feet deep.
The parameters xr, Zr, and br are therefore set, and the
result is a distorted model. This distortion requires the
use of a slatted inner zone as discussed in Chapter III.
Tables 5.2 and 5.3 list the applicable parameters.
The following analysis is shown for the design of
the model Floridan aquifer, leaky layer and storage
coefficient:
A. Slatted Anisotropic Zone for the Floridan Aquifer.
1. Compute k xm/kzm from equation 3.22
x 2 Kxr
r _= xr (3.22)
Sr Kzr
Noting that k = K v (5.8)
g
Then 3.22 becomes,
x '2 k k
zr XM Z kz (5.9)
r ixp zm
and finally,
p [ 6.182 1002
xMm = (1.22) 0 [1 = 0.00209
zm Kzp zr1.493 x 1
TABLE 5.2
MODEL PARAMETERS
Floridan
Parameters Aquifer Leaky Layer
Geometric
x (ft.)
z (ft.)
ym (ft.)
a
A
b (in.)
L (in.)
ab (in.)
XL (in.)
(1 A)L (in.)
Hydraulic
kxm (in.2)
kzm (in.2)
xm /kzm
S (ft.1)
vm @ 77* F (ft.2/sec.)
gm = 32.2 ft./sec.2
11.75
2.0
0.0417
16.50
0.8097
0.030
1.235
0.6905
0.495
1.00
0.235
3.94. x 104
.1884
.00209
0.0369
5.382 x 103
11.75
0.082
0.0417
0.25
0.3846
0.500
1.235
0.8349
0.125
0.475
0.760
0
1.05 x 104
0
5.382 x 103
82
TABLE 5.3
SIMILARITY RATIOS
This is the value which the slatted zone must
produce.
2. Select dimensions for the slatted zone as shown
in Figure 3.2, as follows:
hold ab = 0.5 inch
select b
find a
select (1 X)L
select XL
find L
find x
3. Compute kxm b2 ( 1 (3.34)
4. Compute kzm = ((3AE + 1 A) (3.48)
5. Compute kxm/kzm and compare to the results of
step (1).
6. Repeat the process until the result of step (5)
equals the result of step (1).
B. Slotted Leaky Layer. Refer to Figure 3.4 for this
section.
kxlm = 0 (5.10)
b2 (3.54)
kzlm = iZ a (3.54)
84
Also, continuity of flow between the leaky layer and
the Floridan aquifer requires that:
(5.11)
Qlr = Qfr
Also,
Q = Qzr = Qxr = K bzrb x
"r zr "xr zr rr
Therefore,
K b x =K b x
zlr rr zfr rr
so,
Kzlr = Kzfr
or,
kzlr = kzfr
(3.67) & (3.68)
(5.12)
(5.13)
(5.14)
Therefore,
kzlm = kzlp kzfr
Now rewrite equation 3.54 as,
a3 k 12
zlm U_
1. Set b = 0.5 inch.
2. a3A = constant, since b is set and kzlm can be
computed from previous information.
(5.15)
3. Select ab and find a.
4. Select XL and L and find X.
5. Compute E.
6. Compute a3A.
7. Compare the result of step (6) to the result
of step (2).
8. Repeat the process until the result of step (6)
equals the result of step (2).
9. Make sure the physical size of this layer is
compatible to the slatted layer, especially
in terms of slot spacing, i.e., blockage.
C. Storage Coefficient Manifolds. Refer to Figure 3.5
for this section (bm = 0.5").
1. Take average storage coefficient and average
depth in prototype to compute S
2. Select convenient time ratios, in this case
1 minute = 1 year, and compute Sor from
equation 3.59.
3. Compute S om
4. Equation 3.60 is now used to compute A for
m
various 1 's with z averaged over 1 .
m m m
In this case the model was apportioned into five
zones with one manifold per zone.
Construction
Because the HeleShaw analog is capable of modeling
complex geometries and boundary conditions, it is desirable
that it be as adaptable to as many different prototype
geometries and hydraulic parameters as possible. This
would facilitate model construction and investigations of
many different areas in the state of Florida where saltwater
intrusion is, or in the future might be, a problem. A
reduction in cost of investigation would also be achieved
if many of the parts were reusable.
A list of general specifications would then be as
follows:
1. The HeleShaw model should be housed in a frame
in which it can be easily installed and removed.
2. The front and back plates with interior model
parts should not be permanently sealed together.
3. The front and back plates should be as adaptable
as possible to different situations.
4. The model should have as few opaque parts as
possible.
5. The model should be mobile.
Frame
As shown in Figure 5.2, the frame is composed of
two assemblies; the cradle and the cradle dolly. The
function of the cradle is to support and orient the
Plexiglas plates. It also contains the inflatable neoprene
hose which seals the plates. It is fabricated of 21/2"
x 2" x 3/8" steel angles which are welded into a channel
