Sea Water in 

Coastal Aquifers

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1613-G
Sea Water in 

Coastal Aquifers

By HILTON H. COOPER, JR., FRANCIS A. KOHOUT, HAROLD R. HENRY, and 

ROBERT E. GLOVER

RELATION OF SALT WATER TO FRESH GROUND WATER

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1613-C

A group of short papers, by the individual 

authors, on the circulation of sea water 

and the position of the salt-water front

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Goyernment Printing Office 

Washington, D.C. 20402
PREFACE

By HILTON H. COOPER, JR.

Over a period of about 35 years the sea water in the Biscayne 

aquifer of southeastern Florida advanced progressively inland, owing 

to a lowering of the fresh-water head. The drainage of the Everglades, which began shortly after the turn of the century (Parker and 

others, 1955, p. 584-591), was the principal cause of this lowering. 

Applying the Ghyben-Herzberg relation, Parker (Parker and others, 

1944, p. 18) predicted that the salt water at the base of the aquifer 

would continue to advance "* * * until ultimately it [would] come 

to rest in equilibrium with the fresh water where the average annual 

two and one-half foot contour on the water table [occurred]."

If it had advanced as predicted, the wedge of salt water would 

eventually have enveloped numerous water-supply wells, including 

those of the Miami municipal supply in the Miami Springs well field. 

Early in the 1950's, however, the advancement of the salt water appeared to cease, although the front of the wedge was still as much as 

8 miles seaward of the predicted ultimate position. Whether the 

front had stabilized or whether its advance had merely slowed was 

not only a question of economic importance but also a matter of 

scientific interest because the seemingly premature stabilization 

could not be explained by any known theory. The question therefore became the topic of several discussions among the writer and his 

colleagues, including R. R. Bennett, G. G. Parker, N. D. Hoy, and 

F. A. Kohout. From these discussions evolved the idea that whereas 

the salt water was continuing to flow inland as expected, the wedge 

front was continually being "eroded" by a seaward flow of mixed 

water in the zone of diffusion. This idea led to the concept that 

perpetual circulation of sea water in coastal aquifers was a necessary 

consequence of the dispersion of salts in the zone of diffusion (Cooper, 

1959). ' , At the time, this concept seemed to be novel, but Todd (1960) and 

Bear and Todd (1960) have found mentions of inland flow of salt 

water by several investigators. For example, Nomitsu and others 

(1927, p. 288) state:

Near the contact surface, agitated by the moving fresh water, the salt water 

will tend to be transported seawards, although it may be very very slowly. 

By way of compensation, sea water in the lower part will continue to infiltrate 

landwards.
IV PREFACE

Todd (1960), believing that Nomitsu and his associates were the 

first to postulate the concept, proposed that the dispersion-induced 

circulation be recognized as the Nomitsu circulation. However, 

Nomitsu and his associates may not have recognized dispersion as 

the motivating factor. Their statement that salt water along the 

interface is "agitated" by the moving fresh water implies that they 

visualized a circulation maintained through forces of viscosity at the 

interface. In an open body of water, such as an estuary, such a 

circulation could be appreciable, but the extent to which such forces 

could operate to maintain a macroscopic circulation through the 

pores of an aquifer is not clear. Moreover, a circulation was postulated much earlier by Drabbe and Badon Ghyben (1889) in thenreport describing what was later to become known as the GhybenHerzberg principle. This report contains the passage:

Perhaps one is therefore not far from the truth in assuming that, in the deeper 

layers also, two currents exist and have existed for many centuries. The one, by 

far the more powerful, consisting of sweet water going from the dunes to the 

Zuiderzee; under this at the Zuiderzee coast, a weak, brackish water current 

directed inland that mingles there with the lowest part of the first mentioned 

stream.

The theoretical considerations or field information that led to this 

conclusion are not clear from this report. There can be no doubt, 

however, that both Senio (1951, p. 175) and Carrier (1958, p. 484) properly related a circulation to dispersion. Senio, on the basis of an 

analysis of field data, postulated a landward flow which he explains 

as follows:

This may be due to the diffusion across boundary of the brine water, the diffusion 

being more intense in moving water than in still water, * * *. But the diffusion 

is not the only cause, as the upper region above the Herzberg's boundary is occupied with fresh water. It may be related to the fluctuation of water level and to 

the water of Lake Nakanoumi.

Carrier reasoned that the seaward discharge of salty water from the 

zone of diffusion must be balanced by an inflow of sea water.

Aside from purely scientific aspects, the circulation of sea water in a 

coastal aquifer has several practical implications. First, the landward 

part of this circulation, being accompanied by head losses, causes the 

toe of the salt-water wedge to be displaced seaward of the position of 

the interface that would exist in the absence of dispersion. Second, 

the landward flow, persisting over a long period of time, could alter 

the lithology of some aquifer materials through ion exchange. Third, 

the circulation would warrant consideration in connection with proposals to inject radioactive wastes into the salt water in coastal aquifers under the supposition that such water is essentially static.
PREFACE V

The Geological Survey, therefore, began to investigate the phenomenon in 1957. The investigation consisted of a field study of the 

patterns of flow, head, and salt concentration in the Biscayne aquifer 

near Miami, Fla., and a mathematical program to obtain solutions 

for the patterns of flow and concentration when considering the effects 

of dispersion. Also, to provide a basis for comparing the position of 

the diffused salt-water front with that of the interface that would 

occur in the absence of diffusion, solutions for the interfaces under a 

variety of conditions were derived.

The idea of a circulation due to the dispersion constitutes the first 

section of the report (p. C1-C12). The second section (p. C12-C32) 

gives the results of the field investigation of this phenomenon. The 

third (p. C32-C35) and fourth (p. C35-C70) sections present mathematical solutions for the position of the sharp interface that would 

occur in the absence of diffusion, and the fifth treats the effects of 

dispersion (p. C70-C85). The ideas and findings have been published 

in part and in preliminary form in the Journal of Geophysical Research and in Publication 52 of the International Association of 

Scientific Hydrology. (See papers by Cooper, Kohout, Henry, and 

Glover in "References.")

CONTENTS

Page

Preface, by Hilton H. Cooper, Jr ____________________________________ in

A hypothesis concerning the dynamic balance of fresh water and salt water

in a coastal aquifer, by Hilton H. Cooper, Jr________________________ Cl

Abstract___ _ ________________________________________________ 1

Introduction._ ________________________________________________ 1

Acknowledgments _____________________________________________ 6

Mechanics of the cyclic flow of sea water.________________________ 6

Dispersion.___________________________________________________ 8

Summary_ _ _________________________________________________ 11

The flow of fresh water and salt water in the Biscayne aquifer of the Miami

area, Florida, by Francis A. Kohout____________-________---__-____ 12

Abstract. _:_____.________________________________ 12

Introduction._________________________________________________ 12

Acknowledgments _____________________________________________ 14

Geologic and hydrologic characteristics of the Biscayne aquifer_ _ _ _ _ _ 14

The zone of diffusion_________________________________________ 14

Seaward flow of salty water___-____________-_________-_--__-_-_- 14

Dispersion._ __________________________________________________ 18

Hydraulic gradient in the salt-water zone.________________________ 20

Silver Bluff area______________________________ 20

Cutler area_____________________________________ 22

Potential in the fresh-water and salt-water environments______---_- 23

Patterns of flow of fresh and salt water___________________----_- 29

Summ ary_ ___________________________________________________ 32

The pattern of fresh-water flow in a coastal aquifer, by Robert E. Glover._ 32

Abstract____________________________________________________ 32

Introduction __________________________________________________ 32

Flow net._______________________________________ 33

Interfaces between salt water and fresh water in coastal aquifers, by

Harold R. Henry_--__---__-__-___________________________ 35

Abstract __ __________________________________________________ 35

Introduction._________________________________________________ 36

Acknowledgments _____________________________________________ 36

Artesian aquifers___________-___________-___-_-_--___--__-----_ 37

Notation _________________________________________________ 40

Governing equations and boundary conditions.______________ 40

Case 1, vertical outflow face__________-__-___---_-----_----- 43

Case 2, horizontal outflow face.__________________---__--_--_ 48

Computations for semi-infinite aquifers, cases 1 and 2_________ 49

Computations for finite aquifers, case 1_______________-.._-_- 52

Computations for finite aquifers, case 2______________-__---_ 53

Discussion of results.__-__-__-__-_______-____--_--_-__----- 59

Water-table aquifers receiving vertical recharge__________-_--_--_- 62

Governing equations and boundary conditions_______________ 62

Numerical solution________________________.____----------- 65

Discussion of results_---__-__-___-_---____----___---------- 69

vn
VIII CONTENTS

Page 

Effects of dispersion on salt encroachment in coastal aquifers, by Harold

R. Henry______.________________________________________________ C70

Abstract__ _ _________________________________________________ 70

Introduction..________________________________________________ 71

Acknowledgments. _________-______---__-_-_-_-_-______________ 73

Notation _____________________________________________________ 73

Field equations and boundary conditions _________________________ 74

Fourier-Galerkin solution.______________________________________ 76

Numerical solution_____________________________________________ 78

Discussion of results________--__-__..-____--__--___-_-_-________ 81

References..______________________________________________________ 82

ILLUSTRATIONS

FIGURE 1. Balance between fresh water and salt water__________^_ C2

2. Circulation of salt water___________________________ 3

3. Map of Waimano Springs, Oahu, Hawaii, showing the

chloride, content of water from orifices.______________ 4

4. Chloride content of water in a test well at Pearl Harbor. 5

5. Section through the Cutler area, near Miami, Fla., showing the zone of diffusion.__________________________ 7

6. Amplitude of tide-produced motion of water.__________ 11

7. Map of the eastern part of Bade County, Fla__________ 13

8. Section through the Silver Bluff area, near Miami, Fla.,

showing the zone of diffusion__-__________---_--____ 15

9. Section through the Cutler area, near Miami, Fla., showing the position of the line of zero horizontal gradient. 16 

10. Graph showing fluctuations of chloride content and water

level in well G 519A________._______________ 17

11-13. Hydrographs of 

11. Landward decrease in tidal fluctuation _____________ 19

12. Daily-average water level and head in well F 200

and Biscayne Bay_____-______----________-_-_ 21

13. Wells G 906, G 919, and G 936__._-_-____-_--__-- 22 

14-19. Sections through the Cutler area, near Maimi, Fla., showing 

14. Lines of equal head.____________________________ 24

15. Lines of equal fresh-water potential ________--_-_ 25

16. Lines of equal fresh-water potential--.. ---------- 26

17. Lines of equal fresh-water potential for a low head

condition ____________________________________ 27

18. Lines of equal fresh-water potential for a high head

condition ____________________________________ 30

19. Patterns of flow of fresh and salt water______---__- 31

20. Flow pattern near a beach.__________________________ 34

21, 22. Schematic sketches and complex planes______-___--__ 38

23, 24. Numerical solutions for a short sand formation________- 46

25. Numerical solution for semi-infinite artesian aquifers... 51

26. Interfaces for vertical outflow face.______________----- 54

27. Interfaces reaching to recharge reservoir._________----_ 56
CONTENTS IX

Page 

FIGUKE 28. Alternative transformations of /__-__________________. C57

29. Interfaces intersecting base of aquifer near ocean.______ 58

30. Comparison of interfaces____________________________ 60

31. Section through the Cutler area near Miami, Fla., showing isochlors---___________--____________________. 61

32. Schematic sketch and complex planes.-_________:._____ 63

33. Numerical solutions for interfaces___________________ 69

34. Flow and concentration patterns____________________ 80

35. Mean salt concentration in vertical section___________ 81

TABLES

TABLE 1. Parameters for computations plotted in figure 26____-_______- C53

2. Parameters for the two shortest interfaces in figure 27________ 55

3. Parameters for the four longest interfaces in figure 27_______ 58

4. Parameters for the interfaces in figure 29__________________ 58

719-580i

RELATION OF SALT WATER TO FRESH GROUND WATER

SEA WATER IN COASTAL AQUIFERS

A HYPOTHESIS CONCERNING THE DYNAMIC BALANCE 

OF FRESH WATER AND SALT WATER IN A COASTAL 

AQUIFER

By HILTON H. COOPEB, JB.

ABSTRACT

The dispersion of salts produced by reciprocative motion of the salt-water front 

in a coastal aquifer induces a flow of salt water from the floor of sea into the zone 

of diffusion and back to the sea. The head losses that accompany the landward 

flow tend to lessen the extent to which the salt water occupies the acquifer.

INTRODUCTION

Published explanations of the steady-state balance between salt 

water and fresh water in a coastal aquifer begin with Badon Ghyben 

(1889) and Herzberg (1901) and commonly assume that the salt water 

is static. Under this assumption the balance would be as shown in 

figure 1. The depth below sea level to a point on the interface would 

be (Hubbert, 1940, p. 872)

_ pf ,, £ ' fly

P* Pf

where p< is the density of the fresh water, ps is the density of the sea 

water, and h is the head of fresh water above sea level at the point on 

the interface. An equation for determining the shape and position of 

the interface with a known rate of discharge of fresh water under one 

set of boundary conditions has been devised by Glover (see p. C32-C35, 

this report) through an adaptation of a solution by Kozeny (1953) for 

an analogous problem of free-surface gravity flow. Equations for the 

interface under several sets of boundary conditions have been derived 

by Henry. (See p. C35-C70.)

It is the thesis of this paper that where a zone of diffusion exists, the 

salt water is not static but flows perpetually in a cycle from the floor of 

the sea into the zone of diffusion and back to the sea. This flow tends 

to lessen the extent to which the salt water occupies the aquifer.

This cycle must exist to some degree if salty water continuously discharges from the zone of diffusion into the sea (fig. 2). If such a discharge exists, the sea itself must be the source of the salts that are 

carried back to the sea, and the salts must be transported from the 

floor of the sea through the aquifer into the zone of diffusion. Any

Cl
Fresh water

FIGURE 1. Balance between fresh water and salt water in a coas
Land surface

Fresh water

 . >-. :;.;.v£-g

FIGURE 2. Circulation of salt water from the sea to th
C4 RELATION OF SALT WATER TO FRESH GROUND WATER

form of dispersion or diffusion as the principal mode of transportation 

of the salts from the sea to the zone of diffusion may be eliminated 

because these processes occur only in the presence of a concentration 

gradient, and the concentration gradient that exists across the zone of 

diffusion will not generally extend in sufficient magnitude all the way 

back to the sea floor. Therefore, the salts are transported largely by 

a flow of the salt water, as indicated in figure 2, with a consequent loss 

of head in the salt-water environment.

One example of a discharge of salty water from the zone of diffusion 

is the flow of springs around Pearl Harbor, Hawaii. Ordinarily, most 

of the seaward discharge of water from a coastal aquifer occurs at the 

floor of the sea, but around Pearl Harbor the discharge issues from 

terrestrial springs as the result of the presence of the caprock, a nearly 

impervious blanket of weathered volcanic debris that underlies and 

rims the harbor and forces some of the ground water from the basaltic 

aquifer to discharge in springs along its inland perimeter. Typical of 

these is Waimano Springs (fig. 3), consisting of several orifices that 

yield water of different salinities. The total flow from the several 

orifices of Waimano Springs ranges from 15 to 20 million gpd (gallons

162

190 160

100 200 300 FEET

160

PEARL HARBOR

FIGURE 3. Map of Waimano Springs, Oahu, Hawaii, showing the chloride content of water from orifices, 

1957; lines and figures represent the chloride content, in parts per million.
SEA WATER IN COASTAL AQUIFERS C5

100

200

300

400

500

600

700

z 800

I

Q 900 

1000 

1100 

1200 

1300 

1400 I I I I I I I I I

0 2 4 6 8 10 12 14 16 18 20 

CHLORIDE CONTENT, IN 1000 PARTS PER MILLION

FIGURE 4. Chloride content of water in a test well at Pearl Harbor, 1958.

per day). The water from these orifices increases in salinity toward 

the sea, and the chloride content ranges from less than 200 ppm (parts 

per million) at the orifices farthest inland to as much as 3,000 ppm at 

the one nearest the shore. The chloride content of sea water at most 

places is about 19,000 ppm.

The great thickness of the zone of diffusion beneath the Pearl 

Harbor area is indicated by figure 4, which shows the choride content
C6 RELATION OF SALT WATER TO FRESH GROUND WATEK

of water at various depths as determined from the drilling of a deep 

test well. The zone beings at a depth of about 200 feet and ends at 

perhaps 1,200 feet; thus it has a thickness of 1,000 feet or more.

Another place at which there is evidence of discharge from the zone 

of diffusion is the Cutler area near Miami, Fla. The zone of diffusion 

in the Biscayne aquifer of this area an aquifer that consists predominantly of cavernous limestone is represented by the isochlors in 

figure 5. The control points for the isochlors are shown by the black 

dots which represent the bottoms of fully cased wells. These wells 

were drilled to extract water samples and to measure presssure heads at 

isolated depths, as a part of an investigation by the U.S. Geological 

Survey of the various factors relating to the hypothesis described in 

this paper. The thickness of the zone of diffusion is limited by the 

thickness of the aquifer, but its horizontal breadth at the base of the 

aquifer, as reckoned from the isochlors, is probably about 2,000 feet. 

The zone extends several hundred feet beyond the shoreline. The 

distributon of salts is doubtless influenced by the presence of beds of 

low permeability in the aquifer.

As most of the seaward discharge from the Biscayne aquifer occurs at 

the floor of the sea, there is no opportunity to obtain representative 

samples to determine the salinity of the water being discharged. 

Nevertheless, that salty water is being discharged can be inferred from 

the fact that the shallow wells which end in the zone of diffusion 

beneath the floor of the sea tap salty water under sufficient head to

rise above sea level.

ACKNOWLEDGMENTS

The author is grateful to D. A. Davis and F. N. Visher of the U.S. 

Geological Survey at Honolulu for their generous permission to publish 

figures 3 and 4 prior to their own publication of their findings. He is 

indebted to F. A. Kohout of the U.S. Geological.Survey at Miami for 

the collection and compilation of the data shown in figure 5, and to 

R. E. Glover of the U.S. Geological Survey at Denver for suggesting 

the equation for the tide-produced displacement of water in a coastal 

aquifer. Richard Skalak, of Columbia University, reviewed the 

manuscript and made helpful suggestions.

MECHANICS OF THE CYCLIC FLOW OF SEA WATER

What causes the sea water to move in its cycle through the aquifer? 

Since the water has the same fluid potential when it reenters the sea 

as when it leaves, one might reason that there can be no hydraulic 

gradient and hence no flow. To understand how a flow may exist 

under these circumstances, consider the processes that operate in the 

zone of diffusion. Sea water and fresh water become intimately 

mixed in the zone of diffusion by the mechanism that creates this zone.
1500 1000

DISTANCE FROM 

FIGURE 5. Section through the Cutler area, near Miami, Fla., 
C8 RELATION OF SALT WATER TO FRESH GROUND WATER

The effect of this is the same as if some of the salt ions were extracted 

from the sea water and injected into the flowing fresh water. The 

diluted sea water, becomes less dense than native sea water and rises 

along a seaward path. The resulting circulation is analogous to the 

circulation in thermal convection, differing only in that changes in 

density are produced by changes in concentration rather than by 

changes in temperature. Meanwhile, the salts that are introduced 

into the fresh-water environment are carried back to the sea by the 

flow of the fresh-water system.

The circulation, therefore, is induced by the transfer of salts out of 

the salt-water environment. The forces that effect this transfer must 

be powerful enough to recreate the zone of diffusion continuously as it 

is dissipated by the flow of mixed water to the sea. Molecular 

diffusion is much too feeble for this.

DISPERSION

A mechanism that appears powerful enough to cause sufficient 

mixing is the reciprocative motion of the salt-water front resulting 

from ocean tides and from the rise and fall of the water table due to 

variations in recharge and other forces, including pumping. Palmer 

(1927, p. 51-52) and Wentworth (1948) theorized that this to-and-fro 

motion creates the zone of diffusion. The process by which two 

miscible liquids interfuse about their boundary when hydraulic flow 

causes the boundary to move is known as dispersion. In laminar flow 

through permeable porous media, dispersion results from the combined 

effects of convection (transfer of a fluid into the region of another due 

to variations in velocities within the interstices) and molecular diffusion (Taylor, 1953, p. 187). During a movement of the salt-water 

front in either direction, the convection component of dispersion causes 

elements of each fluid to be transferred into the opposite environment, 

wherein to a large extent they become inseparably blended with the 

other fluid by mixing and molecular diffusion. If the displaced 

elements were not blended, but if, instead, they were to remain discrete 

during and after the transfer, the potential gradients acting upon 

them in the foreign environments would drive each back to its original 

environment. Furthermore, the dispersion would be mostly negated 

at each reversal of motion if the elements were to retain their identity 

in continuous filaments and retrace their path. Thus, both convection and molecular diffusion are important parts of the dispersion 

process convection in producing large transfers, and molecular 

diffusion in completing the blending.

From the results of their experiments with Ottawa and Monterey 

sands, Rifai, Kaufman, and Todd (1956) concluded that the coefficients of longitudinal dispersion (dispersion in the direction of flow)
SEA WATER IN COASTAL AQUIFERS C9

with unidirectional flow in granular permeable media is practically 

proportional to the mean velocity of flow, so that D=Mvn, where D is 

the coefficient of dispersion, M is a constant whose magnitude depends 

on the properties of the medium, v is the mean interstitial velocity, 

and the exponent n is approximate unity. They found the values of 

Mto be 0.063 cm (centimeter) for Ottawa sand and 0.13 cm (computed 

from their table 5; the value of 0.013 cm given in their fig. 18 appears 

to be incorrect) for Monterey sand. Experiments by Orlob and 

Radhakrishna (1958) indicate that the medium dispersion constant 

M increases with the uniformity coefficient of the medium and becomes 

as high as 2.79 cm for a sand having a uniformity coefficient of 3.88. 

If the coefficient of longitudinal dispersion is proportional to the 

first power of the mean interstitial velocity, it may, in the case of 

dispersion due to ocean tides, be expressed as

jD=4MA/«o, (1)

where A is the amplitude, and t0 is the period of the displacement of 

water in the aquifer caused by ocean tides.

The amplitude of the tide-produced displacement of water may be 

related approximately to the amplitude of the tide and the distance 

from the shoreline as follows. The tide-produced change in the 

artesian head with reference to its mean in a semi-infinite artesian 

aquifer is (Jacob, 1950, p. 365; Ferris, 1951, p. 149)

h=h0 exp ( 

where

ho= amplitude of the tide, 

x= distance from the shoreline, 

to= period of the tidal cycle,

J=time referred to the beginning of a tidal cycle, 

S= coefficient of storage, and 

T= coefficient of trangmissibility.

The gradient producing the displacement will therefore be

The displacement will be zero at a time ti when dh/dx is a maximum 

or when the quantity in parentheses is ir/^+nir. It will be extreme 

at a time ^ when d^/da;=0 or when the quantity in parentheses is 

. Thus, with the substitutions
CIO RELATION OF SALT WATER TO FRESH GROUND WATER

the amplitude of the displacement will be

i i rs*/4

= (Kho/ff)T/toS/4rT exp ( xVxS//bT)- (sin M + COS u}du

Jv/i

T exp (-nfcSJUT) , (2)

where K is the permeability and 8 is the effective porosity of the 

aquifer.

The coefficients of tide-produced dispersion at given distances from 

the shoreline in a typical coastal aquifer may be estimated by using 

equations 1 and 2 and assuming that the medium dispersion constants 

obtained from experiments with unidirectional flow are applicable 

and that the displacements of the interface will be the same as the 

displacements that would occur if the water in the aquifer were all 

of one density. The latter assumption seems reasonable because, in 

most aquifers, the interface will move very slowly in response to 

changes in head and its maximum displacement will be only a fraction 

of that which would be required for it to adjust completely to the 

extremes of the tide. Equation 2 was derived for artesian aquifers 

but probably will give a fair approximation of the amplitude of the 

displacement in a nonartesian aquifer if an appropriate value for the 

coefficient of storage is used.

The amplitude of the oscillation of water at various distances from 

the shoreline in a hypothetical nonartesian aquifer of sand is illustrated 

in figure 6. If the medium dispersion constant of the sand were 

1.0 cm, the coefficient of dispersion would be 100 cm2 per day at about 

300 feet from the shoreline and 10 cm2 per day at about 900 feet from 

the shoreline. Beyond 1,400 feet it would not be significantly larger 

than 1 cm2 per day, which is approximately the coefficient of molecular 

diffusion of sodium chloride (Hodgeman, 1945, p. 1695).

It is likely, however, that in aquifers that consist of alternate beds 

of high and low permeabilities, as practically all aquifers do, there is 

a mixing mechanism that will produce rates of dispersion much larger 

than those indicated by laboratory experiments on unidirectional flow 

through homogeneous sand. Suppose, for example, that an aquifer 

is made up of alternate beds having permeabilities in the ratio of 

10 to 1 and that initially a sharp interface lies diagonally across the 

beds. If, with a rise of the tide, the interface were to move landward 

a distance of x units in a bed of high permeability, it would move only 

about O.lx units in an adjacent bed of low permeability. Consider, 

now, what would happen if the fresh water were then to begin flowing 

in a direction diagonally across the beds in an upward path toward 

the sea. Apparently, the water in the various beds would become
SEA WATER IN COASTAL AQUIFERS Oil

/0=0.5 day

T = 100,000 ft 2/per day 

/C=1000ft/perdaX 

9=0.35

100 200 300 400 

DISTANCE FROM SHORELINE, IN FEET

FIGURE 6. Amplitude of tide-produced motion of water in a coastal aquifer.

fairly well mixed, so that there would result a zone nearly x units wide 

in which the concentration of salts would be some fraction of that of 

sea water. This fraction would be dependent on the relative thicknesses and porosities of the beds. With each additional tidal cycle 

the zone would widen further until ultimately the distribution of 

salts would come to a steady state wherein the average rate at which 

the salts were carried into an element of the aquifer by dispersion 

would be in balance with the rate at which they were carried out of 

it by hydraulic flow. The rate of dispersion produced by this process 

would doubtless be greater in an aquifer of cavernous limestone or 

basalt than in one of laminated sand.

The reader will understand that the hydraulics are more complicated 

than postulated here. The example given is merely a device to illustrate the idea of dispersion by differential displacement and cross-bed 

mixing. The displacements and mixings will not, of course, occur as 

separate alternate events but will operate simultaneously much of the 

time, each playing a more prominent part in one phase of the tidal 

cycle than in another.

SUMMARY

Wherever a zone of diffusion exists in a coastal aquifer, sea water 

will flow from the floor of the sea into a zone of diffusion. The flow
C12 RELATION OF SALT WATER TO FRESH GROUND WATER

may be interrupted or reversed during low stages of the tide or high 

stages of the water table, but on the average it will persist in a landward direction. Apparently, the only question in a given case is one 

of magnitude. The magnitude of the flow evidently will be governed 

chiefly by that of the dispersing mechanism that induces it, and it 

may be large enough in some places to produce head losses in the saltwater environment that would lessen appreciably the extent to which 

the salt water occupies the aquifer.

THE FLOW OF FRESH WATER AND SALT WATER IN THE 

BISCAYNE AQUIFER OF THE MIAMI AREA, FLORIDA

By FRANCIS A. KOHOUT

ABSTRACT

Investigations in the coastal part of the Biscayne aquifer, a highly productive 

aquifer of limestone and sand in the Miami area, Florida, show that the salt-water 

front is dynamically stable as much as 8 miles seaward of the position computed 

according to the Ghyben-Herzberg principle. This discrepancy results, at least in 

part, from the fact that the salt water in the Biscayne aquifer is not static, as 

explanations of the dynamic balance commonly assume. Cross sections showing 

lines of equal fresh-water potential indicate that during periods of heavy recharge, 

the fresh-water head is high enough to cause the fresh water, the salt water, and 

the zone of diffusion between them to move seaward. When the fresh-water 

head is low, salt water in the lower part of the aquifer intrudes inland, but some 

of the diluted sea water in the zone of diffusion continues to flow seaward. Thus, 

salt water circulates inland from the floor of the sea through the lower part of the 

aquifer becoming progressively diluted with fresh water to a line along which 

there is no horizontal component of flow, after which it moves upward and returns 

to the sea.

This cyclic flow is demonstrated by a flow net which is constructed by the use 

of horizontal gradients determined from the low-head equipotential diagram. 

The flow net shows that about seven-eights of the total discharge at the shoreline 

originates as fresh water in inland parts of the aquifer. The remaining oneeighth represents a return of sea water entering the aquifer through the floor of 

the sea.

INTRODUCTION

Investigations during the past 20 years confirm that the salt-water 

front in the Biscayne aquifer is dynamically stable as much as 8 miles 

seaward of the position computed by either the Ghyben-Herzberg 

principle or the theory of dynamic equilibrium (fig. 7). The concept 

of dynamic balance between flowing fresh water and static sea water 

in a coastal aquifer is shown in figure 1.

Recent studies indicate that the lack of agreement results from the 

fact that two assumptions inherent in the Ghyben-Herzberg and 

dynamic-equilibrium concepts are not fulfilled in the Biscayne 

aquifer. These assumptions are (1) that a sharp interface exists 

between fresh water and salt water in a coastal aquifer and (2) that 

the salt water in a coastal aquifer is static. As hypothesized by Cooper
SEA WATER IN COASTAL AQUIFERS C13

Well field pumping 

| 65,000,000 gpd of fresh 

j water

I 

I 

I TAMIAMI CANAL

SILVER BLUFF AREA

jGhyben-Herzberg position

| of interface at base of

aquifer

FIGURE 7. Map of the eastern part of Dade County, Fla., showing the theoretical 

Ohyben-Herzberg position and the actual position of salt water at the base of 

the Biscayne aquifer.
C14 RELATION OF SALT WATER TO FRESH GROUND WATER

(see p. C1-C12, this report), a circulation of salt wate*r is induced 

by the dispersion of salts produced by the reciprocative motion of the 

salt-water front due to tidal action. Henry (see p. C70-C84) has 

confirmed this circulation analytically.

This section of the report describes the characteristics of flow in the 

Biscayne aquifer of the Miami area and, by a flow net constructed 

from field data, shows the pattern of fresh and salt water flow for a 

low-head condition.

ACKNOWLEDGMENTS

The writer is grateful to H. H. Cooper, Jr., N. D. Hoy, and Howard 

Klein of the U.S. Geological Survey for advice and review of the 

paper. He is indebted to Richard Skalak of Columbia University 

for suggesting the method of constructing the flow net.

GEOLOGIC AND HYDBOLOGIC CHARACTERISTICS OF THE

BISCAYNE AQUIFER

The Biscayne aquifer consists of solution-riddled limestone and 

calcareous sandstone. It is a water-table aquifer and extends from 

land surface to an average depth of 100 feet below mean sea level. 

In general, the coefficient of permeability ranges from 50,000 to 

70,000 gpd per sq ft (Parker, 1951, p. 824).

THE ZONE OF DIFFUSION

In the Biscayne aquifer the zone of diffusion is a thick zone in 

which there is a gradation of salt content from that of fresh water, 

which contains 16 ppm chloride, to that of sea water, which contains 

about 19,000 ppm chloride. Figures 8 and 9 are cross sections through 

the zone of diffusion in the Silver Bluff and Cutler areas; the location 

of these areas is indicated in figure 7.

The distance from the bay to the inland tow of the saltwater wedge 

is more than 12,000 feet in the Silver Bluff area but only about 

1,600 feet in the Cutler area. The toe of the wedge has a blunted shape 

in both areas. As pointed out by Henry (see p. C70-C84) this configuration results partly from the boundary requirement that no dispersion 

or diffusion may occur across the impermeable base of the aquifer, so 

that isochlors must approach the base of the aquifer perpendicularly.

SEAWARD FLOW OF SALTY WATER

The fluctuation of chloride content in well G 519A, which is 400 

feet from Biscayne Bay in the Silver Bluff area, is shown in figure 10 

(see fig. 8 for the position of the open-hole part of well G 519A). 

The rapid decrease in chloride content at the three sampling depths 

during October 1953 resulted from a large increase in fresh-water 

head following heavy rainfall in early October. After the rain,
12,000 10,000

DISTANCE FROM SHORELINE, IN F

8000 6000 4

 Isochlor.in parts per million 

Dashed where approximate; 

interval irregular

FIGURE 8. Cross section through the Silver Bluff area, near Miami, Fla., sho
DISTANCE FROM SHOR

1600 1400 1200 1000 800 600 

FIGURE 9. Cross section through the Cutler area, near Miami, Fla., showing the 

within the zone of diffusion, Sep
FEET IN HEAD, AVERAGE DAILY 

LEVEL BAY BELOW OR ABOVE 

ppppi-i->-

IN CONTENT, CHLORIDE 

MILLION PER PARTS 

~ I-** 5 

ZTO
CIS RELATION OF SALT WATER TO FRESH GROUND WATER

salt water was rapidly expelled from the aquifer, and the zone in which 

the concentration gradient is steep (immediately below well G 519A in 

fig. 8) was depressed downward and seaward.

A ground-water velocity test using fluorescein dye as a tracer was 

performed at the site of well G 519A on January 4, 1954. The results 

of the test indicated that water containing 1,500 to 2,000 ppm chloride 

(in the open-hole part of well G 519A, fig. 8) was flowing seaward at 

a rate of more than 70 fpd (feet per day). This is significant in that 

it demonstrates a large discharge of salt from the zone of diffusion.

A rough calculation of the quantity of salt being discharged at 

the shoreline is pertinent. If the base of the zone in which water 

flows seaward in the Silver Bluff area is assumed to be at the 5,000 

ppm isochlor (fig. 8), the thickness of this zone at well G 519A is about 

35 feet. If the average velocity through this thickness is 70 fpd 

and the effective porosity of the limestone is 0.2, then the seaward 

discharge of water in a vertical strip 1-foot wide is 490 cubic fpd.

From figure 8, the average chloride content of the water discharging 

at the shoreline is about 1,900 ppm. Ten units of water having a 

chloride content of 1,900 ppm contains about the same quantity of 

salt as one unit of sea water, which has a chloride content of about 

19,000 ppm. Therefore, in the 1-foot strip, roughly 49 cubic fpd 

of ocean water must become incorporated into the seaward flow. 

This calculation indicates that as much as 10 percent of the total 

seaward flow may be sea water.

DISPERSION

As observations show that the diffused zone remains essentially 

unchanged while large quantities of salt are flushed back to the sea, 

a mechanism much stronger than molecular diffusion is acting to 

recreate the zone of diffusion.

In recent years, extensive studies have been made of the process 

of dispersion. This process consists of two separate mechanisms: 

convection, the mechanical transfer of one fluid into the region of 

another, and molecular diffusion (Bosworth, 1949, p. 465). During 

the to-and-fro movement of the salt-water front that results from 

ocean tides, variations of fluid velocity in the pores of a permeable 

medium cause an intermingling of fluids of different concentration, 

after which the blending is completed by molecular diffusion. As 

shown in figure 11, tidal fluctuations decrease landward. (See fig. 

9 for the location of the bottoms of wells.) Consequently, the rate 

of dispersion created by the tide also decreases landward.

As shown by Cooper (see p. C1-C12), the coefficient of dispersion 

due to tides can be as much as 100 times greater than molecular 

diffusion in an aquifer of homogeneous sand and even greater in an
SEA WATER IN COASTAL AQUIFERS C19

TIME, IN HOURS 

06 08 10 12 14 16 18 20 22

FIGURE 11. Hydrographs showing the landward decrease 

in tidal fluctuation in the Cutler area, near Miami, Fla., 

September 18,1958.
C20 RELATION OF SALT WATER TO FRESH GROUND WATER

aquifer of nonuniform permeability. In the suggested mechanism, 

elements of salt water under tidal stimulus move greater horizontal 

distances in permeable beds than in adjacent less permeable beds, 

and the salt-water projections thus formed will be integrated by the 

upward cross-bed flow of fresh water.

Movement of ground water caused by the tide has both horizontal 

and vertical components. Clearly, a mechanism that permits very 

rapid transportation and dispersion of salt is available.

HYDRAULIC GRADIENT IN THE SALT-WATER ZONE

SILVER BLUFF AREA

Evidently the dispersion must occur at a rate great enough to 

maintain the zone of diffusion while a large quantity of salt water 

discharges seaward. To maintain this equilibrium, some means of 

transporting the salts from the floor of the sea through the aquifer 

and into the zone of diffusion must be available. However, in the 

regions below and seaward of the zone of diffusion (figs. 8, 9) the concentration gradient is too small to make possible the transportation 

of a large quantity of salt by dispersion. Therefore, the transportation of salts must be by a flow of wate^r and be accompanied by a loss 

of head.

In figure 12, the daily-average water level and head in well F 200 

is compared with the daily-average water level and head in Biscayne Bay. The chloride content of water in the well ranged from 

18,300 to 18,800 ppm during the period shown. The water level in 

the well, which closely represents the head of ocean water in the aquifer, is higher than the surface of the bay during periods of heavy 

rainfall, as shown by the unblackened intervals in figure 12A, and 

lower than the bay during dry periods, as shown by the blackened 

intervals. In figure 12B, the daily-average head of the water in the 

bay has been algebraically subtracted from the daily-average head 

in well F 200, so that the head in the well each day is referred to that 

in the bay for that day. When the head in figure 125 is positive, 

salt water flows seaward; and when it is negative, salt water flows 

inland. Clearly, the negative heads reflect the head losses due to 

the inland flow of sea water through the aquifer.

The head in well G 519A (fig. 10), computed in the same manner as 

that for well F 200, is never negative. This indicates that the movement of water is always seaward at that point in the aquifer. Therefore, during the intrusion part of the salt-water flow cycle, there 

must be a line somewhere between the bottoms of these two wells 

(within the zone of diffusion shown in fig. 8) where the water has 

the same head as the ocean and where the water is flowing neither 

inland nor seaward. The water along this line cannot be stagnant,
1953

AUG SEPT OCT NOV DEC JANFEB 

FIGURE 12. Hydrographs of daily-average water level and head in well F 200 and in 

B, daily-average head in well comp
C22 RELATION OF SALT WATER TO FRESH GROUND WATER

however, because continuity requirements would then be violated; 

so salt water along this line must be flowing vertically upward. 

Thus, a cyclic flow of salt water inland from the floor of the sea 

to the zone of diffusion and then seaward through the 'fresh-water 

flow section has been demonstrated.

CUTUER AREA

Wells in the Cutler area were drilled as part of the investigation 

of factors relating to the cyclic flow. These wells were fully cased to 

make possible the collection of water samples and the measurement of 

pressure heads at isolated depths. The bottoms of the wells are indicated as dots in the cross sections of the Cutler area (figs. 9, 14-19).

In figure 13, the daily-average fresh-water heads in the salt-water 

region are referred to the daily-average heads of Biscayne Bay.

FEB MAR APR MAY JUNE JULY AUG SEPT OCT NOV I DEC

1958

FIGURE 13. Hydrographs of wells O 906, O 919, and O 936 in terms of equivalent fresh-water head.
SEA WATER IN COASTAL AQUIFERS C23

Where the density varies from point to point, measurements of head do 

not indicate the direction of movement directly. For this reason 

the observed salt-water heads in the wells have been converted to 

fresh-water heads by computation. Also, the hydrographs of wells 

G 906 and G 919 have been adjusted for the density of the columns of 

salt water between the bottoms of the wells (which are 97.9 and 101.5 

ft below mean sea level, respectively) and the horizontal plane at 100 ft 

below sea level in the aquifer. Well G 936 required no adjustment 

because it terminated at the plane. The heads of the wells as related 

to the location of the wells (see fig. 9) shows how the gradient in 

terms of water of constant density along the plane 100 ft below mean 

sea level fluctuated during 1958.

A 13-inch rain in May 1958 produced a large seaward gradient. 

As the head declined after this rain, the hydrographs crossed at about 

2.4 ft; this indicates reversal to a landward gradient. The fresh-water 

head in static sea water at a depth of 100 ft below mean sea level is 2.5 

ft; thus, the gradient in the salt-water region of the aquifer reverses 

direction at a fresh-water head that compares closely with the theoretical head. The landward gradients show that salt is transported 

inland from the floor of the sea to the zone of diffusion by a flow of 

water.

Along the bottom of the aquifer, the salt disperses continuously 

landward in the direction of decreasing concentration. A seaward 

flow of water opposes the dispersion, and a landward flow supports it.

POTENTIAL IN THE FRESH-WATER AND SALT-WATER ENVIRONMENTS

Ground water of uniform density moves in the direction of decreasing 

head, but where the density varies from point to point the distribution 

of head does not indicate the direction of movement directly. This is 

illustrated in figure 14, where the figures at the bottoms of the well 

casings are daily-average water levels referred to the daily-average bay 

level. The concentration of salt water in the casing is the same as that 

in the aquifer at the bottom of the casing; the head values, as shown, 

are the original data. Obviously, all flow cannot converge upon the 

sink surrounded by the 0.2-foot contour, and diagrams constructed 

from the original data are not usable.

In figures 15 to 17 the lines of equal fresh-water potential are 

shown for high, low, and average tide on September 18, 1958. For 

wells containing salty water, the equivalent head of fresh water has 

been computed, so that all heads are the same as if the casings had 

been filled with fresh water at the time of measurement.

719-580 O 64 5
1600 1400 1200

DISTANCE FROM SHOR

1000 800 600 

_, 20

UJ

a 

§ o

20

o: 

O 40

UJ

i< 

h- 60

80

100

Base of Bisca

FIGURE 14. Cross section through the Cutler area, near Miami, Fla., showing li

September 18,195
1600 1400 1200

DISTANCE FROM SHO

1000 800 600 

FIGURE 15. Cross section through the Cutler area near Miami, Fla., showing 

September 18,1
1600 1400 1200

DISTANCE FROM SHO

1000 800 600 

20

20

O 40

UJ

8m

<

h- 60

80

'100

 2.49

f

 2.15

IV 

0

 

 1.89 

M r 

3 b «° 

1 

FIGURE 16. Cross section through the Cutler area, near Miami, Fla., showing 

September 18,1
1600 

 I 

1400 1200

DISTANCE FROM SHO

1000 800 600 

20

I 20

yj 

CD

O 40

60

80

J 100

Land surfac

_GH> BEN^HERZBERG J

xBase of Biscayn

PIGUBE 17. Cross section through the Cutler area, near Miami, Fla., showi

average for Septembe
C28 RELATION OF SALT WATER TO FRESH GROUND WATER

Conversion of observed salt-water head to fresh-water head in a 

given well is accomplished by application of the equation

p=pgl

where p is the pressure at the bottom of the casing, p is the density 

of the water in the casing, g is the acceleration due to gravity, and I 

is the measured length of water column above the casing terminus. 

Equating the right term of the above equation for fresh-water and 

salt-water columns,

Pffflf P'ffl*

where the subscripts / and s refer to fresh water and salt water, 

respectively. The density of fresh water is assumed to be 1,000 

gm per cm3 .

The following table (which uses observed daily-average data for 

well G 906, September 18, 1958, fig. 9) gives a typical computation:

Salt-water head, in feet above mean sea level ___ ______ 0. 60

Depth to bottom of casing, in feet below mean sea level_ _ 97. 90 

Length of salt-water column (lt) , in feet__ ______________ 98. 50

Chloride content of water in casing, in ppm______ _______ 18, 000

Density (p») of water in casing________________________ 1. 0240

Length of equivalent fresh-water column (I,), in feet___. __ 100. 86 

Fresh-water head, in feet above mean sea level (obtained

by subtracting depth to bottom of casing from If) ______ 2. 96

Daily-average level of Biscayne Bay, in feet above mean

sea level__________________________________________ . 90

Daily-average fresh-water head, in feet above dailyaverage level of Biscayne Bay _ ____________________ 2. 06

The equipotential lines in the upper, fresh-water part of the aquifer 

indicate the potential of fresh water in a fresh-water environment 

and hence indicate comparative potentials. As flow lines must be 

nearly perpendicular to these equipotential lines, a seaward movement of fresh water is indicated.

In the lower and seaward part of the aquifer, the equipotential 

lines indicate the potential of fresh water in a region occupied by 

salt water. The fresh-water equipotential surfaces in a region occupied by salt water will be horizontal if the salt water is static. As 

the equipotential lines in the salt water regions of figures 15 to 17 

are not horizontal but slope inland in figure 15, seaward in figure 16, 

and inland in figure 17, the salt water is not static but must be in 

motion in the direction of the slope. The daily-average equipotential 

diagram (fig. 17) indicates that the instantaneous movements occurring throughout the day average out in such a way as to produce a net 

inland movement of salt water on this date. On the other hand, the
SEA WATER IN COASTAL AQUIFERS C29

seaward slope throughout the aquifer in the daily-average equipotential diagram for May 29, 1958 (fig. 18) indicates that a high head, 

resulting from heavy recharge (see fig. 13), causes all water in the 

aquifer to move seaward.

In the low-head equipotential diagram (fig. 17), the pattern of 

fresh-water equipotential lines serves as a guide for separating the 

region of seaward-flowing water from that of the inland-flowing water. 

Such a separation is formed by a line passed through the points of 

horizontality of the individual equipotential lines. This line is shown 

in figure 17. Clearly, along this line water is flowing neither inland 

nor seaward, but as the flow must be continuous between the lower 

and upper regions, the water at all points along this line must be 

flowing upward. The position of this line of zero horizontal gradient 

in the zone of diffusion (see fig. 9) indicates that water containing 

as much as 16,000 ppm chloride may have a seaward horizontal component of flow. The Ghyben-Herzberg line in figure 9 indicates the 

position of a sharp interface according to the Ghyben-Herzberg 

principle.

PATTERNS OF FLOW OF FRESH AND SALT WATER

A flow net for a low-head condition is shown in figure 19. The 

gradients for the horizontal components of flow were determined from 

the head distribution along horizontal planes in the fresh-water equipotential diagram of figure 17. These horizontal gradients were 

plotted at vertical lines spaced at intervals of 100 feet starting from 

the shoreline. Stream tubes then were constructed by maintaining 

the product of the vertical thickness of the stream tube and the horizontal gradient constant throughout the flow net.

The permeability of the Biscayne aquifer in the Culter area is not 

uniform, as indicated by the irregularly spaced lines in the upper part 

of the equipotential diagrams (figs. 15-18). As construction of the 

flow net depends on homogeniety, adjustment for the variations in 

permeability was necessary.

The flow net was constructed to scale and was reduced horizontally 

in the preparation of figure 19. In the reduction, alternate streamlines were omitted, except for'the dashed lines at the midpoints of 

flow tubes in the landward-flow region; these were retained to show 

details of the flow pattern there.

The seaward flow of water at the shoreline is represented by 16 

flow tubes, of which 14 originate inland and 2 originate at the floor 

of the sea. Thus, about seven-eighths of the total discharge at the 

shoreline is fresh water moving seaward from inland parts of the 

aquifer, and the remaining one-eighth represents a return of sea water 

that entered the aquifer at the floor of the sea.
1600 1400 1200

DISTANCE FROM SHOR

1000 800 600 

FIGURE 18. Cross section through the Cutler area, near Miami, Fla., showin

average for May 29
1600 1400 1200

DISTANCE FROM SHOR

1000 800 600 

FIGURE 19. Cross section through the Cutler area, near Miami, Fla., showing t

September 18.1968.
C32 RELATION OF SALT WATER TO FRESH GROUND WATER

Significant features in the flow net are the zero-horizontal-gradient 

line, which separates the seaward and landward horiEontal components 

of flow, and the streamline that separates the water into two regions 

according to whether it originated as fresh water or sea water. These 

two lines intersect at the base of the aquifer.

SUMMARY

Lines of equal fresh-water potential in wells show that the salt water 

in the Biscayne aquifer of the Miami area, Florida, is not static as 

explanations of the dynamic balance commonly assume. When the 

fresh-water head is high after heavy recharge, water in, all parts of 

the aquifer moves seaward, and large volumes of salt water are 

expelled from the aquifer. As the head declines, a landward gradient 

forms in the lower part of the aquifer, and salt water flows inland 

into the zone of diffusion to a line along which there is no horizontal 

component of flow; the salt water then moves upward and returns to 

to the sea. The cyclic flow of salt water that takes place during 

intrusion acts to limit the extent to which salt water occupies the 

aquifer because part of the inland flow of salt is continuously returned 

to the sea.

THE PATTERN OF FRESH-WATER FLOW IN A COASTAL

AQUIFER

By ROBERT E. GLOVER

ABSTRACT

Formulas are developed for the flow pattern followed by the seaward-moving 

fresh ground water as it nears a beach. Under steady-flow conditions, a sharply 

defined interface is formed between the fresh and salt water. Along the interface 

the pressure of the static salt water, owing to its greater density, is counterbalanced 

by the pressures which drive the fresh water seaward. The fresh water escapes 

through a gap between this interface and the shoreline, and an increase in the 

flow of fresh water widens the gap. Tidal action causes a diffusion of salt water 

across the interface, but this salt is carried back to sea with the fresh-water flow.

INTRODUCTION

Where permeable beds underlie a land area near the sea and extend 

some distance seaward from the shoreline, the infiltration of rainfall 

causes a continuous flow of fresh water toward the sea. Under these 

conditions an extensive body of fresh water, often a valuable source of 

water supply, commonly is present beneath the land. Because the 

flow of fresh water toward the sea must balance the supply from rainfall infiltration, a seaward gradient must be present. The water 

table under the land, therefore, has the form of a mound. The 

density difference between sea water and fresh water is only about one-
SEA WATER IN COASTAL AQUIFERS C33

fortieth of the density of fresh water, and for this reason the freshwater body has a thickness below sea level of about 40 feet for each 

foot of elevation of the water table above sea level (Badon Ghyben, 

1889; Herzberg, 1901). Near the seashore, however, dynamic 

factors become significant. If static conditions alone were to prevail 

here, the fresh-water body would taper to a knife edge at the beach 

and there would be no way for the fresh water to escape. When the 

dynamic factors are considered, it is found that the fresh water 

flows through a narrow gap between a fresh water-salt water interface 

and the water-table outcrop at the beach (Hubbert, 1940). In the 

analyses that follow, however, the flow through the seepage surface 

above sea level is assumedly negligible. A short distance back from 

the beach the static conditions are closely met. Under steady-flow 

conditions the fresh water-salt water interface would be sharply 

defined, but tidal action and the rise and fall of the water table 

maintain a zone of diffusion between the fresh and salt water.

PLOW NET

A close representation of the flow conditions near a beach can be 

obtained by modifying a solution previously obtained for the flow of 

ground water under gravity forces (Kozeny, 1953). The flow net 

for the present conditions can be obtained from the relationship

(1)

*7V

where

o;=the distance measured horizontally landward from the shoreline, in feet; 

y=the distance measured vertically downward from sea level, in feet;

and

If

# = the fresh-water flow per unit length of shoreline, in square feet per second; 

X = the permeability of the strata carrying the fresh-water flow, in feet per

second; 

and

y the excess of the specific gravity of sea water over that of fresh water

(dimensionless), 

then

and

(3)

The interface between the fresh water and sea water can be plotted 

from the expression
C34 RELATION OF SALT WATER TO FRESH GROUND WATER

The width of the gap through which the fresh water escapes to the 

sea is

= _Q_

" 2yK (5)

As used here, the potential <j>, expressed in feet of fresh water, can 

be identified with the pressure head which drives the fresh water. 

The flow function \f/ also is expressed in feet of fresh water. The 

product K$ then has the same dimensions as Q. The stream function 

has the property that, for a selected value ^=^1, the product K\[/i, 

substituted in expression 3, yields a streamline. The fresh-water 

flow above this streamline is nQ.

The potential along the line ?/=0 is

 _ /o /!«./ f\1 /2 /£N 0Q= i g'Y^JX/J\.) ' . \Q)

This is the height of the water table at the distance x back from the 

shoreline. A plot of this flow net is shown in figure 20.

Some features of this solution deserve comment. If the supply of 

fresh water to the aquifer decreases so that Q decreases, then the width 

of the gap through which the fresh water can escape also decreases. 

The seaward flow of fresh water is proportional to the square of the 

potential <£0 measured at a selected distance x=x from the shoreline.

Sea water (For ordinary water 

= 2.0 y=0.025 approx)

Total flow is 0 

2 5 F'°w above any streamline is nQ

For a measured potential <t>, atj/=0 andx=x.

FIGURE 20. Flow pattern near a beach.
SEA WATER IN COASTAL AQUIFERS C35

In times of drought, the fresh-water body is conserved because the 

seaward flow is diminished. Thus, once established, the fresh water 

does not quickly waste away.

The pattern of flow described in this paper is based on the assumption that the salt water is essentially static. Cooper has hypothesized 

(see p. C1-C12) that where a zone of diffusion exists between the 

salt water and the fresh water, the salt water is not static but flows 

in a cycle from the floor of the sea to the zone of diffusion and back to 

the sea. Cooper reasoned that this flow is induced by the dispersion 

of salt produced by a reciprocative motion of the salt-water front. 

Such a circulation would alter the balance between the two fluids to 

an extent that depends on the magnitude of the circulation.

INTERFACES BETWEEN SALT WATER AND FRESH WATER 

IN COASTAL AQUIFERS

By HAKOLD R. HENKY

ABSTRACT

In a coastal aquifer, a steady flow of fresh water toward the sea limits the 

encroachment of the sea water. Under the assumption that the salt water and 

fresh water do not diffuse into one another so that a sharp interface exists, this 

action is treated for steady two-dimensional flow.

Exact solutions for the position of the interface and the boundary velocities 

in artesian aquifers are derived for several sets of boundary conditions. A 

hodograph plane is used to circumvent the boundary problem that arises because 

the position of the interface is not known in advance, and a complex potential is 

related to the hodograph by conformal mapping. Also derived is an equation 

for the interface in a water-table aquifer receiving uniform vertical recharge. 

The derivation involves physical approximations to avoid an additional boundary 

problem due to the unknown position of the water table.

Interfaces are computed for artesian aquifers of finite thickness and semiinfinite length that have vertical and horizontal seepage faces and for a range of 

fresh-water discharges in several geometries of artesian aquifers of finite thickness 

and length. Only when the salt water extends very short distances into the aquifer 

do the interfaces in finite artesian aquifers differ significantly from those in semiinfinite artesian aquifers with the same discharge. Where the salt-water wedge 

extends inland from the shore a distance many times the thickness of the aquifer, 

the position of the interface, except near the shore, can be obtained approximately 

by using a Dupuit-type assumption. Comparisons with field data indicate that 

the computed interface corresponds approximately with the position of the zone 

of diffusion.

The position of the interface in a water-table aquifer receiving uniform vertical 

recharge is found by employing the relaxation technique in a modified hodograph 

plane. Interfaces are computed for an extremely large rate of recharge and, more 

realistically, for a smaller rate. For the large rate of recharge the position of 

the interface is much deeper and farther seaward than that obtained by use of the 

Dupuit assumption, but for the smaller rate the difference between the interfaces 

obtained by the two methods is negligible.
C36 RELATION OF SALT WATER TO FRESH GROUND WATER

INTRODUCTION

The location of the interface between the fresh water and sea water 

in a coastal aquifer is obviously significant in the practical use of 

ground water. The seaward flow of fresh water in the aquifer is 

necessary to keep salt water from encroaching farther into the aquifer. 

Exact solutions for the location of the interface have been obtained for 

several boundary conditions under the assumptions that the flow is 

steady, that the salt and fresh water do not diffuse into one another, 

and that there is no interfacial tension. The effects of diffusion are 

considered on pages C70-C84.

The first analyses (Badon Ghyben, 1889; Herzberg, 1901) treated 

the salt-encroachment phenomenon as a simple buoyancy problem in 

which static fresh water floated on the heavier sea water. In this 

method, hydrostatic conditions are assumed to apply vertically below 

a point of observation of piezometric level in the fresh water. This 

determines the elevation of a point on the interface where the pressure 

is balanced by the hydrostatic conditions prevailing in the salt water. 

If the assumption that constant piezometric head exists on a vertical 

line is used together with a one-dimensional application of Darcy's 

law, the position of the interface can be estimated as a function of 

steady fresh-water discharge (Kitagawa, 1939; Todd, 1953) by the 

method used by Dupuit (1863) for gravity seepage in a two-dimensional aquifer.

In nature the interface is not sharp; instead, the salt and fresh 

water merge gradually through the process of mechanical dispersion 

aided by chemical diffusion. As discussed by Cooper on pages C8-C11, 

the width of the zone of dispersion depends on the pore characteristics 

of the aquifer and on the relative magnitude of the fluctuations due to 

recharge and tidal activity. If this zone is narrow as it is under some 

field conditions (Brown, 1925), the assumption of a sharp interface 

might be used to obtain a first approximation of the fresh water flow 

pattern and the location of the dispersion zone.

ACKNOWLEDGMENTS

The writer thanks Prof. Richard Skalak of Columbia University 

and the U.S. Geological Survey for his advice throughout the preparation of this paper. The study was suggested by H. H. Cooper, Jr., 

of the Geological Survey. Thanks are also due E. M. Laursen of 

Michigan State University for helpful discussions.

This paper is based in part on the writer's Ph. D. dissertation at 

Columbia University. 1

1 Henry, H. R., 1960, Salt intrusion into coastal aquifers: New York, Columbia Univ. Ph. D. dissertation.
SEA WATER IN COASTAL AQUIFERS C37 

ARTESIAN AQUIFERS

The type of aquifer to be considered here is one that is overlain 

in the coastal ares by an impervious bed that confines the water under 

pressure and locally prevents recharge from rain. In nature, water 

would enter such an aquifer in an inland area of recharge and would 

flow into the coastal segment approximately horizontally. The 

idealized configurations analyzed are shown in figures 21 and 22. In 

each of these it is supposed that water enters horizontally along the 

line EF and flows out along the seepage face AB. In figure 21 the 

seepage face is vertical; in figure 22 it is horizontal. In each case, 

general expressions are derived for a finite aquifer as shown. Then the 

equations are modified for aquifers that extend to infinity in a landward direction. Another limiting case which is treated is that of an 

aquifer that not only extends to infinity in the landward direction but 

also is of infinite thickness.

The general expressions are used to compute the location of the 

interfaces for the semi-infinite aquifers with horizontal and vertical 

seepage faces and for several finite values of the aquifer-aspect ratio 

£=l/d and of the discharge parameter a=Q/(kid). Here Q is the 

fresh-water discharge per foot of shoreline and ki is the product of

the permeability k of the aquifer and the buoyancy ratio >

where ps and po represent, respectively, the densities of the salt and 

fresh water.

The inflow surface EF, assumed to be vertical for analytical convenience, apparently differs widely from the nearly horizontal outcrop 

areas found in most field conditions. However, if the recharge were 

to occur a great distance inland from the toe of the salt-water wedge, 

the streamlines of fresh-water flow would be almost straight and parallel immediately upstream from the nose of the wedge. Under this 

condition the flow in the aquifer would correspond closely to that in 

the present schematization. If other factors remain constant, one 

would expect the position of the interface .to remain almost constant 

as I varies, since there would probably be little consequent change in 

the fresh-water flow pattern at D, the toe of the wedge. This expectation is borne out by the computations.

Although this analysis is for artesian aquifers, the computed interfaces may apply approximately under certain conditions to a watertable aquifer. If the water table is a streamline that is, if the quantity of recharge in the vicinity of the salt wedge is small compared with 

the total recharge and if the water table is very flat in comparison 

to the interface, the present computations should provide an estimate 

of the interface position.
C38 RELATION OF SALT WATER TO FRESH GROUND WATER

Fresh water Fresh water

A Schematic sketch of Case 1 (zplane)

D

B

B Hodograph (q plane)

e=-f

A

* 

B

b 

c *

1.0 e=-f 

 1 J ^ ,

e=f

D E f

e=o

A

FIGURE 21. Schematic sketch and complex planes for artesian aquifer with vertical outflow face.
SEA WATER EST COASTAL AQUIFERS

Fresh water

A Schematic sketch of Case 2 (2 plane)

C39

A F E D

B

U

kl -1.0

n D

~_.-L 1

i.o 1 1 

 I 1 | 

A F E D

B Hodograph (q plane) C t plane,/=

B 

A

Oi D E h^

i/,*

F * - V -

F n

_- a l 

-1.0

fl

1.0

 1 

A

V**

 1 

D /"plane, f= <t> + / ^ E wl plane, wl =sn [(2// + l)/(] 

FIGURE 22. Schematic sketch and complex planes for artesian aquifer with horizontal outflow face.
C40 RELATION OF SALT WATER TO FRESH GROUND WATER

NOTATION

The symbols used most frequently herein are defined below:

a=Q/kid, dimensionless.

A, B, C, D, E, F= points on the boundary of the aquifer.

a, b, c, d, e=coordinates of points in the complex plane.

d=depth of aquifer, ft.

/(z) = complex velocity potential=<f>+i^, ft2 per sec.

/'(z) = first derivative of / with respect to z, ft per sec.

/"(z) =second derivative of / with respect to z, sec" 1 .

/i = piezometric head, in feet of fresh water.

zW13!.

/=the elliptic modular function.

k = permeability of sand, ft2 .

k = transmission coefficient = kpQ g/n, ft per sec.

 p 8 po 

k\ transmission coefficient times the density difference ratio k 

ft per sec.

K=complete elliptic integral of first kind.

K' = complete elliptic integral of first kind with complementary modulus. 

k*=modulus of elliptic integral.

Q=net fresh-water discharge per unit length of beach, ft2 per sec. 

1=length of aquifer, ft, for artesian aquifer. 

= half width of island, ft, for water-table aquifer. 

p pressure, Ibs per ft2. 

g=complex velocity=u + iv, ft per sec. 

g=conjugate complex velocity = u iv, ft per sec.

q vector velocity, ft per sec.

t, w, \=complex variables.

u = horizontal velocity, ft per sec.

v=vertical velocity, ft per sec.

x, y=coordinates, ft (when used as subscripts,these denote differentiation).

z complex variable.

., . , . 3 C 1 p(t) ,. a=the integral, ^- I 7~r dt

j8=central angle in hodograph plane. 

p = density of solution, slugs per ft3 . 

Po=density of fresh water, slugs per ft3 . 

ps = density of sea water, slugs per ft3 . 

<t> velocity potential ==kh, ft2 per sec. 

^=stream function, ft2 per sec. 

T + i8=the auxiliary variable, ln[ /"(z)]. 

£ aspect ratio=l/d, dimensionless.

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

It is assumed that there is no dispersion of salts, so that a sharp 

interface separates the salt water from the fresh water. Darcy's law 

for a homogeneous fluid then applies to the flow of the fresh water, 

and under steady conditions the salt water in the aquifer below the 

interface will be static. In the mathematical statement of the
SEA WATER IN COASTAL AQUIFERS C41

problem, the interface is treated as a free boundary which must be 

located so that the pressure on its fresh-water side is equal to the 

static pressure on its salt-water side.

The use of Darcy's law (Darcy, 1856) for the velocity components 

u and v,

in the continuity equation

 + =0 (2)

yields Laplace's equation on 0

vV=0, (3) 

where 0 is the usual velocity potential for laminar groundwater flow:

where y is measured downward.

In terms of the fresh-water potential 0, the boundary condition to 

be satisfied on the unknown interface BD (see figs. 21, 22) and on the

seepage face AB is: p=( = -\-y ) po9=ps9y-^-B, where B is a constant, 

VZiC /

or

(4)

where ki=k - and 0o is a constant. Equation 4 may be written 

Po

in dimensionless form using 0'=^ and y'=yxd:

(4a)

The dimensionless parameter ^=a, which occurs here, is designated

Kid/

the discharge parameter. For a given discharge parameter, all flows 

with similar aquifer geometry will be similar. This can be easily 

demonstrated by dimensionless analysis also.

A stream function ^ can be defined as usual by ^= fudy-\- fvdx, 

so that \l/ and 0 satisfy the Cauchy-Rieman equations and a complex 

velocity potential /(g) exists where

/(z) = </>+*>. (5) 

The derivative of /(2) yields the conjugate of the complex velocity

-f'(z) = u-iv = q. (6)
C42 RELATION OF SALT WATER TO FRESH GROUND WATER

The method of solution used is a conformal mapping technique 

which begins with a conformal mapping of the boundaries of the 

fresh-water flow system onto a hodograph or q plane. For this procedure, the values of u and v or a relationship between u and v is 

required for every point on the boundary. In this problem there are 

boundaries of four different types namely, boundaries of constant 

head, boundaries which are fixed streamlines, the interface, and the 

seepage surface. Referring to figures 21 and 22, EF is of the first 

type, on which 0 is constant; and FA and ED are of the second type, 

on which \l/ is constant. On the interface DB and on the seepage 

surface AB the values of 0 are controlled by the hydrostatic pressure 

in the salt water as expressed in equation 4. On the interface \l/ is 

constant, but on the seepage surface the distribution of ^ is unknown.

For a straight boundary on which the head or potential is constant 

in the original plane of the flow, the velocity vector is everywhere 

perpendicular to the boundary, and the corresponding boundary in 

the hodograph plane is of the form u= (constant) v. For a straight 

boundary in the original plane of the flow which is also a streamline, 

the hodograph line is parallel to the original boundary and also of 

the form u= (constant) v.

The equation of the boundary in the hodograph plane corresponding 

to the seepage face AB can be obtained as follows. The velocity ur 

along the face is

ur=u cos a-\-v sin a = > (7) 

or

where r is the distance measured along the surface of seepage and a 

is the angle between the seepage surface and the horizontal. The

value of a is - for figure 21A and zero for figure 22 A. From equation

4 -zr = ki ^r=ki sin a. Substitution of this into equation 7 yields 

or or

the desired relation:

ki sin a+u cos a+v sin a=0. (8)

The hodograph equation for the interface is also obtained by use of 

equation 4. Differentiation of equation 4 with respect to the distance 

s along the interface BD yields:

Since the interface is a streamline, ^ is the total velocity (w2+t?2) 1/2.

o s

Multiplying equation 9 by ^ gives:

O S

0. (10)
SEA WATER IN COASTAL AQUIFERS C43

Thus, in the hodograph plane the interface is represented by a semicircle of diameter ki. The hodographs for figures 21 A and 225 are 

shown in figures 215 and 225.

CASE 1, VERTICAL, OUTFLOW FACE

The hodograph for case 1, which is shown in figure 215, may be 

transformed into the upper half of the X plane shown in figure 21(7 

by use of the auxiliary functions qi and g2 as defined below and by 

use of the elliptic modular function as defined by Nehari (1952) and 

by Jahnke and Emde (1945). Using

fc=l~» (12)

the required transformation function \=\(q1 ) is given by

/(«.), (13a)

where J(g2) is the elliptic modular function of g2 . The values of X 

corresponding to the flow boundaries can be computed by using the 

inverse of the elliptic modular function. Thus,

ft" K(\) ~ K '

where K is the complete elliptic integral of the first kind and Kf is 

the complete elliptic integral of the first kind with complementary 

modulus. Values of K' '/K for X real and between zero and unity are 

given by Hayashi (1930). Within this range, arbitrary values of X 

may be substituted into equation 13b and corresponding values of g 

computed. For other ranges, the use of certain identities which follow 

from the properties of elliptic modular functions is required to permit 

use of the tabular values. For <»<CX^O, one may use

-=/(ft) (14a) 1-x

 #Y 1 \ tiAh\ *=*-K(I=X) (14b)

and for I ^ X<C « one may use

l=/(ft-l) (15a)

(15b)

Equations 13, 14, and 15 may be considered alternate forms of the 

same transformation between and X.
C44 RELATION OF SALT WATER TO FRESH GROUND WATER

To complete the solution of the problem, it is necessary to express 

the complex potential /, or some function of /, in terms of q so that 

equation 6 or some similar equation can be integrated. Because the 

distribution- of ^ on AB in figure 21 A is unknown, it is not possible 

to relate the boundaries of the flow in the / plane directly to those in 

the X plane. An indirect method developed by Hamel (1934) is 

utilized instead. Use is made of the function r-{-i6 defined by

r+iO=-ln(-f"(z)]. (16)

It follows that 8 is the argument of f'(z}. -The second derivative 

 i"(z) may be written

From equation 17 and an inspection of the hodograph, it is found that 

04B=-|; 8BD= -j; 9DE=-->r; «w= |; ««=o. (18)

The angle ft in the second of equations 18 is the angle indicated in the 

hodograph in figure 215.

Figure 2lC shows that the values of 0 are known at all points of the 

real axis of the X plane. Values of 0 can then be obtained for all 

points of the upper half plane by a Fourier integral. Subsequently, 

values of T are obtained by integrating the Cauchy-Bieman equations 

for T and 6. Muskat (1937) gave details of this procedure, which 

yields

-

where TO is an arbitrary constant and the integral is taken along the 

real axis.

Evaluating the integral in 19 and substituting in 16 yields

-VX(c-X)(6-X 3 =j=jr e5

where b and c are the coordinates of points F and E, respectively, in 

the X.plane. Substituting the derivative of equation 6-into equation 

20 and solving for z yields

£= Cer+»dq + constant, (21)

Tff J

where f\, a constant of integration, serves as a reference length that 

may vary with changes in b and.c, and where

r _ 1-X ea (22) 

e ~Vx(c-X)(6-X)
SEA WATER IN COASTAL AQUIFERS C45

and

Hamel and Gunther (1935) and Muskat (1935) gave tabulated and 

graphical values of ea as a function of X. Thus, the velocities at the 

boundaries of the flow and the location of the interface can be computed by substituting arbitrary real values of X into the proper one 

of equations 13, 14, and 15 and, subsequently, graphically or numerically integrating equation 21. Assigning numerical values to b and c 

in equation 22 corresponds to assigning values to the parameters

l/d=j-, the aspect ratio of the aquifer, and -r^-j o,, the discharge Kid

parameter.

If the relative positions of the levels of the fresh water and the salt 

water in figure 21 A are such that point D coincides with E, that is, if 

the interface reaches to the upstream reservoir, then <t>E=<i>D and

(24)

This case is mathematically identical with that of seepage through a 

rectangular dam with no tailwater, in which the permeability k has 

been replaced by k\. Muskat (1935) gave numerical solutions for 

the discharge Q and the height of seepage surfaces AB for four such 

situations. Muskat plotted computed boundary velocities for two 

cases and also drew in the free surface without calculation by following 

the general features computed by Hamel and Gunther (1935) for a 

dam with a tailwater. Muskat's curves have been inverted and 

relabeled to fit the present application in figures 23 and 24.

When the aquifer extends an infinite distance to the left in figure 

21 A, the point E in the hodograph coincides with point F, 

and eT becomes

~ /-. (25)

(c-X)Vx

Moreover, if the aquifer also extends an infinite distance downward, 

the hodograph points D, E, and F coincide, and eT becomes

er = -L g«. (26)

Vx 

The numerical solution for this case is given in a later section.
C46 RELATION OF SALT WATER TO FRESH GROUND WATER

O

PUB '
0.4 0.6 

0.8 0.6 0.4 0.2 0

FIGURE 24. Numerical solution for a sbort sand for
C48 RELATION OF SALT WATER TO FRESH GROUND WATER 

CASE 2, HORIZONTAL OUTFLOW FACE

Figure 22A represents a flow in which the seepage face AB is 

horizontal. The hodograph or g plane for this case is shown in 

figure 225. The hodograph can be transformed to the upper half 

of the T plane of figure 22(7 by

(27a) 

9

or, inversely, by

q= -* *'. (27b) 

cosh"1 <

In the complex-potential plane /=<£-H^, the boundaries of the 

flow form the rectangle shown in figure 22D, where the potential along 

the seepage face AB is assumed to be zero. The rectangle in the / 

plane is now mapped into the upper half of the t plane. This is 

accomplished by first mapping / into the upper half of the Wi plane 

of figure 22E using the Schwarz-Christoffel theorem

(28a)

or, inversely,

/ Q-2i(~H _1 /sn-%i 

 V' \

where

(28c)

and

K=sn-*(l, **), (28d)

where k* is the modulus of the elliptic integral.

The Wi plane can be mapped into the t plane of figure 22 C by the 

linear transformation

(29)

d\ t

where di is the coordinate of point D in the Wi plane. The relation 

between the coordinates a and 6 of the t plane and the values of di and 

k* of the Wi plane are found by substituting the respective values of 

Wi and t into equation 29:

J I 7.*

(30)

(31)
,SEA WATER IN COASTAL AQUIFERS C49

Thus the problem is in principle solved, since q is expressed parametrically in terms of/ through equations 27b, 28b, and 29, and, 

in principle, equation 6 can be integrated to give

-7^= U=7^+constant. (32)

The numerical solution of a particular problem requires given values 

of either a and b or di and k*. Assuming either pair of values corresponds in effect to assigning values to the two dimensionless quantities 

Q/(kid) and |.

For the horizontal strip of aquifer semi-infinite in length in figure 

22 A, the points E and F are at an infinite distance to the left, the two 

corresponding points in the hodograph are coincident, and, in the t 

plane, a=b. Also, in figure 22D, E and F would be at infinity to the 

right. In the Wi plane this requires &*=0 and the points E and F 

at infinity. Equation 28b, transforming the Wi plane into the/ plane, 

degenerates into

/=^ cosh-i Wl. (33)

The relation between the Wi and t plane is still expressed by equation 

29, and equations 30 and 31 degenerate into a=b=di.

Finally, if in figure 22 A, in addition to / becoming infinite, the 

vertical thickness d becomes infinite also, the points D, E, and F will 

coincide in the hodograph. Then the mapping of the q plane into 

the/ plane is obtained by

In this instance, equation 32 can be integrated to yield

This is mathematically identical to a seepage problem solved by 

Kozeny (1953) for a free-surface flow in a semi-infinite aquifer draining 

downward through a horizontal outflow face when ki is substituted 

for k. This correspondence between the Kozeny solution and the 

salt-intrusion problem for the semi-infinite aquifer was noted and 

utilized by R. E. Glover on pages C32-C35.

COMPUTATIONS FOR SEMI-INFINITE AQUIFERS, CASES 1 AND 2

For case 1, equations 18 and 26 are substituted into equation 21 

and values of q are computed from the appropriate one of equations 

13, 14, 15. Then equation 21 is integrated numerically. Along
C50 RELATION OF SALT WATER TO FRESH GROUND WATER

AB, X varies from «> to 0 and the corresponding values of q are 

computed from equation 14b, which in this case is

u K'

Values of -^ can be found in Hayashi's tables (1930). Values of ea

are taken from Hamel and Gunther (1935). The integration indicated by equation 21 was performed by the trapezoidal rule. The 

computed depth ys of the seepage surface AB expressed in dimensionless form is ysk1 /Q=Q.74:l, as shown in figure 25. 

For the interface BD, equation 10 yields

|.=:zi (e*_ 1)} (37) 

and use of equations 18 and 26 yields

J

pT+ii Cpa-ift/2 

V^-P^O'- <38)

Solving for the x and y coordinates from equation 21 gives, in dimensionless form,

a 

t -, e° cos ss

X-XB 1 f ____ 2 . Q , 

 == I p dp (39a) 

 n 2 J, \

(39b)

Values of # corresponding to given values of X were computed using 

equation 13b, which in view of equations 11 and 37 may be rewritten

tan|=^(X). (40)

The discharge is obtained by integrating u dy over the seepage surface 

AB or, in terms of the present parameters,

Equations 39a and 39b are divided by equation 41 to obtain the

dimensionless coordinates in the form ^ and ^p as shown in figure

v V

25. The integrals of equations 39a, 39b, and 41 were computed by 

the trapezoidal rule to give the coordinates of the interface.
SEA WATER IN COASTAL AQUIFERS C51

5.0 

4.0 

3.0 

2.0 

1.0 

0

I \ \ I I I I I

Curves for vertical outflow face

Curves for horizontal outflow face

2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4

~Q 

FIOUBE 25. Numerical solution for semi-infinite artesian aquifers with vertical and horizontal outflow faces.

The piezometric head hx on the boundary AF is indicated by the

line labeled -, x * in figure 25. This line indicates what the posi- (Ps Po)Q

tion of the interface would be if hydrostatic conditions prevailed along 

vertical sections in the fresh-water body. This computation requires 

first the computation of u versus x as shown by the solid curve in the 

upper graph. This is determined by a graphical integration of equa-
C52 RELATION OF SALT WATER TO FRESH GROUND WATER

tion 21 by using equation 15 for the relation between X and <fr. The 

rhead is subsequently computed by

Po

f' = 

Jo k

(42a)

or, in dimensionless form,

t^-|t« d (^)=P^«^, (42b) 

Q Jo *i V Q / Jo ki ki

in which hx represents the head on AF at any abscissa and hA represents the head at A, in feet of fresh water. Equation 42b was integrated graphically to derive the curve shown in figure 25.

Computations for the semi-infinite aquifer of case 2, in which the 

outflow face is horizontal, were performed as follows. The width of 

the seepage surface AB was first obtained by substituting y=0, 

x=xs, ^=Q, 0=0 into equation 35. The result, in dimensionless

x k\ 1 form, is ~7r=~o' as shown in figure 25. The interface was com-

(£ 6

puted by substituting \I/=Q and <t>=kiy into equation 35, giving

-'-" (43>

which is plotted as the dashed curve labeled "interface" in figure 25. 

The pie/ometric head on AF in the form yokJQ is obtained by 

substituting ^=0, y=0 into equation 35, which yields

(44)

The velocity along AF is given by differentiating -<j> as obtained from 

equation 35:

"=-±^=-^0/2^. (45) 

ki ki da;

COMPUTATIONS FOR FINITE AQUIFERS, CASE 1

The range of X, values of 0, the relation between g_ (or u, v) and 

X, and the specific form of equation 21 for each part of the boundary 

are given below in the most useful form for computations of finite 

aquifers with vertical outflow faces. For the outflow face AB (see 

fig. 21A),

' (46)

' 2' ki K \1 

For the interface BD;

x XB

(47)
SEA WATER IN COASTAL AQUIFERS C53

For the boundary segment DE;

K \X

 

f

I u eT -=- dw

Jo fci (48)

Equations 46, 47, and 48 are sufficient to define the relative magnitudes of the length arid depth and, therefore, the aspect ratio £ 

and also the location of the interface. Similar equations on EF and 

FA could be applied as a check on the overall dimensions. Computation of the discharge parameter a=Q/kid requires the use of 

equation 41, which applies equally to finite and semi-infinite aquifers. 

The specific values of the constants 6 and c (see equation 22 and 

fig. 21C) used in the computations and the corresponding values of 

a and £ are given in table 1. The integrations indicated were performed by the trapezoidal rule using a choice of increments and scales 

intended to maintain three-place accuracy. The results shown graphically in figure 26 are referred to the same discharge by dividing the 

dimensionless coordinates given by equations 46, 47, and 48 by the 

dimensionless discharge given by equation 41.

TABLE 1. Parameters for computations plotted in figure 26

b

10

5

2

1.2

1. 00235

1. 00200

c

1

1

1

1

1

1. 00100

a=0/kid

1 021

896

709

663

309

340

t=l/d

0.49

.56

.70

.93

1. 64

2. 14

The first four series of parameters in table 1, which correspond to 

the dashed curves in figure 26, are values from Muskat's (1935) cases 

3, 4, 5, and 6 for gravity seepage of water through a rectangular dam 

having an impermeable base and no tailwater. His cases 3, 4, 5, and 6 

are mathematically identical to the present problem when the interface 

reaches to the recharge reservoir, but he computed only the dimensions 

of the aquifer, the width of the outflow face, and the discharge; so the 

shape of the interfaces for these four cases are estimated.

The interface for the semi-infinite aquifer is shown also in figure 26. 

Only for a very short aquifer or a very short encroachment distance 

does the interface for the limited aquifer differ significantly from the 

corresponding part of the interface for the semi-infinite aquifer.

COMPUTATIONS FOR FINITE AQUIFERS, CASE 2

For the horizontal outflow face (fig. 22), equations 27 through 32 

are applicable. The specific forms of the equations for the various
C54 RELATION OF SALT WATER TO FRESH GROUND WATER
SEA WATER IN COASTAL AQUIFERS C55

segments of the boundary and of the interface are as follows. The 

reference by Byrd and Friedman (1954) was helpful in determining the 

most convenient forms for the elliptic integrals. For the outflow 

face AB:

(49)

For the interface BD:

(50)

where

fc'=Vl k*2 and A 

For the upper boundary AF:

,_,. (51)

where up and <& are values of velocity and potential for which Wi=\jk*. 

The segment DE of length I is given by

j ~* ~*i/»* r»~^ -^ * !_* &! liki r+akid<f> 4> sn~1 (A,kf) -d^Wl ^l/k*; 0±u±uB; <=-cosh ; -g-J^ ~f ; §- 3* '

(52)

where it^ and <f>s are values of velocity and potential that yield 

Wi=l/k*. The two shortest interfaces in figure 27 were computed by 

use of the foregoing theory and the values of the parameters in 

table 2.

TABLE 2. Parameters for the two shortest interfaces in figure 27

di

2

I/sin 89°

ft*

1/2

sin 89°

a

1.56

.578

f

0. 13

.89

The integrals were evaluated by using the trapezoidal rule. The 

difficulty presented by the infinite value of u/v at the nose of the 

interface was circumvented by locating point D by computing li 

along ED after obtaining I along AF.

For £ approximately equal to or greater than unity, formulas 49 

through 52 lead to elliptic integrals with moduli between sin 89° and 

sin 90°. To use the tabulated values of the elliptic integrals more
C56 RELATION OF SALT WATER TO FRESH GROUND WATEK

I I I I

0
SEA WATER IN COASTAL AQUIFERS C57

efficiently, the following alternative parametric relations between 

/ and t were used

Q~K' ___ 

K

(53a)

. i=

where d2 and d3 are the coordinates of the point D in the w2 and 

planes, respectively, as shown in figure 28, and dz= TTJ"'

'~ K U9

-\/k* -1.0

 1 1 

F E

d2

 I 

D

1.0 +!/**

H 1 

B A

,4 w2 Plane, f/Q=KlK\\ -!Hl^_] +1 

K

-I/** -1.0 1.0

FIGURE 28. Alternative transformations of /for horizontal outflow face.

Equation 53a is suited for computation of the outflow face AB, the 

interface BD, and the segment DE. The special form of/ in equation 

53a used for computation along AB is:

(54)

The form for the interface:
C58 RELATION OF SALT WATER TO FRESH GROUND WATER

The form for the segment DE:

K

K, (56)

Equation 53b is useful for computation of the total length of the 

aquifer along face AF for which

_1<^<_i. 1=1*= - i I >Q~K' +

, **)

K' (57)

The assumed values of d2 and k* and resulting computed values of 

the aspect ratio £ and discharge parameter a, using these alternative 

transformations, are given in table 3 for d^= l.

TABLE 3. Parameters for the four longest interfaces in figure 21

dt

-1

_ i

-1

-1

**

sin 89°

sin 88°

sin 86°

sin 80°

a

0. 145

. 164

. 194

. 251

*

3.43

2.99

2. 55

1.96

The corresponding interfaces reach to the recharge reservoir and are 

the four longest shown in figure 27.

Four instances in which d^> 1 and the interface do not extend 

to the recharge reservoir were computed and are shown in figure 29; 

the parameters are given in table 4.

xk i

0 

22 20 18 16 14 12 10 8 6

1 1 1 1 1 1 1 1 1

( ytttttHHMftft <««,««y«

I = 5.07 4.43 3.70 2.76 

2 0315 \ a =0.357 &

= n OQ7 a \J.-3LO bi>rr»rr»»n»t»

420

1 1 1

=0.435 ^/

 , ,,,r,f^, ,,,,,, ,

-Semi-infinite aquifer

FIGURE 29. Interfaces intersecting base of aquifer near ocean, horizontal outflow face. 

TABLE 4. Parameters for the interfaces in figure 29

*L

-sin 11°

-sin 14°

-sin 21°

-sin 27°

**

sin 89°

sin 88°

sin 86°

sin 80°

a

0.297

.315

.357

.435

£

5.07

443

3.70

2. 76
SEA WATER IN COASTAL AQUIFERS C59

In both figures 27 and 29 the results are presented in dimensionless form, which may be interpreted as holding the discharge constant 

in all instances. The interface for the semi-infinite aquifer is shown 

for comparison.

DISCUSSION OF RESULTS

Two significant conclusions are evident from these results. The 

first is that when the length of the salt-water wedge is greater than 

the aquifer depth, the interface in a finite aquifer is very nearly the 

same as that in a semi-infinite aquifer except at the lower end, and 

the difference between them may be neglected for most engineering 

purposes. The second conclusion is that changing the value of the 

aspect ratio of the aquifer has no appreciable effect upon the position 

of the interface, which is primarily determined by the dimensionless 

discharge parameter a=Qjkid.

The results for semi-infinite aquifers may also be compared to the 

interface derived under assumptions analogous to those used by 

Dupuit (1863) for gravity seepage in a two-dimensional aquifer. 

The Ghyben-Herzberg assumption that the pressure is hydrostatic 

in a vertical section is adopted, and one-dimensional flow according 

to Darcy's law is assumed in the horizontal direction. After integration, the coordinates of the interface are found to be

3=4^1+constant. (58) / y

The constant of integration cannot be determined for this solution 

without a field measurement of the coordinates of at least one point 

on the interface. If the constant is chosen equal to Qj2k, equation 

58 would be identical with the equation for the interface for the semiinfinite aquifer having a horizontal outflow face. However, a common, though usually tacit, assumption is that the constant is zero 

(Kitagawa, 1939; Todd, 1953). Equation 58, with the constant set 

equal to zero, is plotted in figure 30 along with the curves for the semiinfinite aquifers of cases 1 and 2. Except near the ocean, this result 

is likely to be accurate enough for practical purposes of estimating 

the position and shape of the interface. The uncertainties in estimating the discharge and the aquifer permeability may be greater 

than the differences shown in figure 30.

To illustrate the application of equation 58 to a particular field condition, use is made of data obtained by Kohout for the Biscayne aquifer in the Cutler area near Miami, Fla. (See p. C12-C32, this report.) 

Figure 31 shows Kohout's isochlors, on which the interfaces for high, 

mean, and low tides are superposed. The interfaces were computed 

as if the flow were completely steady at each of the three instantaneous 

stages of the tide. The data necessary to plot the coordinates of the
24 22 20 18 16

0 

14 12 10

Dupuit assumption (cons

Vertical outflow face 

Horizontal outflow face

FIGURE 30. Comparison of interfaces in semi-infinite 
SEA WATER IN COASTAL AQUIFERS C61

DISTANCE FROM SHORELINE, IN FEET 

1400 1200 1000 800 600 400 200 200

BISCAYNE BAY

20

m 40

OC

o

S! 

8 60

80

$100

UJ

d

jg^g S. t 1

 ~~Base of Biscayne aquifer

Isochlor, in parts per million 

Interval is irregular

FIGUEK 31. Section through the Cutler area near Miami, Fla., showing isochlors on September 18,1958, 

and interfaces computed as if conditions were steady at high, mean, and low tides.

interface axe values of Qjki for the three conditions and are estimated 

from the piezometric slopes shown by Kohout. For an average permeability of 8,000 feet per day as indicated by Kohout, the three 

interfaces represent steady .conditions for Q=76, 440, and 840 foot2 

per day, respectively.

The interface corresponding to the mean bay level indicates the 

approximate location of the center of the dispersion zone except very 

near the ocean and at the inland extremity. With the same exceptions, the interfaces for high tide and low tide effectively circumscribe 

the dispersion zone. The solution for a sharp interface thus seems to 

give a first approximation of the location of the zone of diffusion. It 

is necessary, of course, to include the effect of dispersion to reproduce 

the gradual change from fresh to salt water. The inclusion of the 

dispersion also results in the change from the sharp toe of the saltwater wedge to the snub nose shape of the isochlors that intersect the 

bottom of the aquifer in figure 31. The condition including dispersion is discussed on pages C70-C84.
C62 RELATION OF SALT WATER TO FRESH GROUND WATER 

WATER-TABLE AQUIFERS RECEIVING VERTICAL RECHARGE

Where vertical recharge is absent, as assumed in the treatment of 

artesian aquifers, the vertical components of the velocity are small; 

so conditions are favorable for use of the Dupuit assumption. However, where vertical recharge exists, the influence of vertical components of the velocities may be appreciable. This is illustrated 

schematically in figure 32 A, which shows the approximate position 

of the streamlines for fresh water seepage into a long oceanic island 

or peninsula. Here, x is the horizontal distance from the shore, I is 

the half width of the island, y is the depth below sea level to the interface, and V0 is the rate of uniform vertical recharge per unit area.

The Dupuit equation for Q, the discharge per unit length of shore 

line, is

Q=--%g, (59)

where k is the transmission coefficient, the piezometric head h is 

assumed to be constant in a vertical section, and y is assumed to include the entire depth of flow. The effect of a uniform recharge is to 

cause Q to increase with distance toward the shore according to the

relation

Q=V0 (l-x). (60)

Since the Dupuit equation makes no provision for vertical velocities, 

its application here requires the assumption that the recharge V0 

enters the aquifer through a continuous distribution of recharge 

wells from which flow occurs uniformly over the depth y. Using this

assumption and the equality h=~ y, the substitution of equation

Po 

60 into equation 59 and subsequent integration yields the following

approximate relation between the coordinates of the interface:

TPs Po where ki=k > ps is the density of salt water, and po is the density

Po 

of fresh water.

The object of the present paper is to find the position of the interface more precisely by use of the methods for potential flow in two 

dimensions.

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

In figure 32A, Laplace's equation applies to the region ABDF'A, 

where AB is the outflow seepage face on the ocean floor, BD is the
SEA WATER IN COASTAL AQUIFERS C63

Interface

A Schematic sketch of the flow pattern beneath an oceanic island 

receiving uniform vertical recharge (not to scale)

A

. iv

e=- 3*

A 9 = 0-

B hodograph (q plane) for region/4 BDF

9 = 0

v '

9=-^

B A

V2 9=0

F D

9=0

(6,0)

~~T

(o,r2 )

D Region of applicability of analytic 

C Transformed hodograph (w plane) approximation for T

FIGURE 32. Schematic sketch and complex planes for a water-table aquifer.

interface separating the static salt water from the flowing fresh water, 

DF' is a line of symmetry, and F'A is the water table at which the 

uniform recharge V0 enters the zone of saturation. The seepage face 

that occurs above sea level at A is small in relation to AB and is 

neglected in the analysis.
C64 RELATION OF SALT WATER TO FRESH GROUND WATER

In satisfying the requirements imposed upon <£ by the physical 

conditions at the boundaries of the flow system, a difficulty arises 

from the fact that the positions of both the water table F'A and the 

interface BD are unknown. The hodograph is used to circumvent 

the difficulty pertaining to the position of the interface. The condition $=kiy on the interface leads to a semicircular hodographic 

representation of BD with diameter k\ as shown in figure 32.5 and 

explained previously. The difficulty due to the unknown position 

of the water table is avoided by taking the x axis as the upper boundary 

of the flow region and by assuming that the vertical component of 

the velocity everywhere along the positive x axis is V0 - This is equivalent to ignoring the slight inclination of the streamlines in the 

saturated region AFF'A above sea level. This approximation probably will not affect the computed position of the interface appreciably 

because the distance between the water table and the x axis is small 

in comparison to the total depth of flow. The approximation is 

similar to an assumption made by Kirkham (1958) in the solution of 

a drainage problem. Under these assumptions, the horizontal line 

FA in the hodograph represents the upper boundary of the flow region. 

The other lines delineating the boundaries of the hodograph are 

obtained by inspection of the flow boundaries in figure 32 A. To yield 

the vertical line AB on the hodograph, the ocean bed AB is assumed 

to be horizontal. Thus, each segment of the boundary of the flow 

pattern has a known location in the hodograph plane.

The velocity potential 0 which satisfies Laplace's equation in the 

z plane (the x, y plane in fig. 32-4) also satisfies Laplace's equation in 

the g plane (fig. 325). Values of <f> are known only on a part of the 

boundary namely, AB, the seepage face on the horizontal ocean 

floor. However, a method of solution is available by use of the 

auxiliary variable r-\-i6 introduced by Hamel (1934) and defined by

»r' L a(2/»?) J

where 77 is a constant of integration which serves as a reference length.

mi t .L- IT i ^ f dQ du idv JLne function 6 is seen to be the complex argument of j j \ j '

values of which can be obtained for each segment of the boundary. 

The boundary values of 6 are indicated in figure 325.

They are obtained by inspection except on the interface. The

O ft

details of obtaining the value 6 £ on the interface are similar

z

to those given by Muskat (1935) in the solution for the free surface 

of seepage through a dam and to those of obtaining equations 18. If
,SEA WATER EST COASTAL AQUIFERS C65

values of T can also be obtained along the boundaries, then the coordinates of the interface can be obtained by integrating equation 62. 

Thus,

«/,= fe-(r+w |?+ constant. (63)

To obtain values of T along the boundaries, values of 0 must first be 

obtained in the interior of the region, after which the Cauchy-Riemann 

equations can be integrated for T along the boundaries.

NUMERICAL SOLUTION

To facilitate the numerical determination of 6 and the subsequent 

integration for T, the hodograph is transformed into the rectangle in 

the w=<r-\-i£ plane shown in figure 32(7. The mapping function is

w=ln '-. ' (64) Vo iq ha

This function maps the outflow face AB onto the a axis in the interval 

(0, 0) to (b, 0) ; the upper boundary of the region FA onto the £ axis 

in the interval (0, 0) to (0, T) ; the centerline of the island DF on the 

line f= T in the interval (0, T) to (b, T); and the interface BD 

on the line o-=^in the interval (b, 0) to (b, T). The value of a in

equation 64 is given by a=- sinh b where ft^cosh" 1 (-7 °+l)- 2, - \ KI /

Since r-\-id is an analytic function of g and thus of w, the variation 

of 8 throughout the rectangle in the w plane is described by Laplace's

&2d c^29 equation, >~3+^2=0. The boundary values of 6 in the w plane are

the same as those at the corresponding points of the g plane. Before 

a solution for 6 throughout the rectangle can be obtained the values 

of 6 must be obtained for corresponding values of f on BD. These 

are found by substituting numerical values of #= 26/3 into the 

following equations, obtained by applying the interface conditions to 

equation 64:

f=tan"

The solution for 6 throughout the rectangle can then be obtained by 

any suitable numerical procedure. The procedure used herein consists of improving an initial guess by the relaxation technique applied 

to the finite-difference form of Laplace's equation in an appropriate 

grid system. After 0 is determined, its normal derivatives at the
C66 RELATION OF SALT WATER TO FRESH GROUND WATER

boundaries are obtained graphically and values of T are computed by 

numerical integration of the Cauchy-Reimann equations along the 

boundaries. The integrals for the four boundary segments are:

on BD:

* (66a)

onDF:

r* o0 T TD= I -z~d<r (66b) 

J& of

on AB:

T TB= I J^.d(r (66c) 

*/ b ^s

on FA:

^r <# (66d)

The variation of T is from the minimum value TD at D to infinity 

at A. In the application of equation 63, the reference length rj 

absorbs the constant e~ . The infinite value of T at A necessitates 

a modification in the computations in the immediate vicinity of A. 

An analytical approximation for T is obtained from the solution of 

the boundary value problem for the infinite quarter plane indicated 

in figure 32Z>, in which cl and f2 represent the limits of validity of the 

approximation. Along AB in the region 0< <r< cri,

T _ TlSS _3in fL (67a) 

where TI is the value of T at <TL Similarly along FA, in the region

T -T2 «-3Zn (67b)

S2

The values av and f2 are chosen within the range of overlap of 

applicability of equations 66c and 67a, and 66d and 67b.

After the boundary values of T are determined, equation 63 may 

be integiated to yield the coordinates of the interface. Thus:

cos j8/2dis (68)

(69)
SEA WATER IN COASTAL AQUIFERS C67

The integration indicated in these equations is performed numerically 

after T has been determined by a numerical integration of equation 

66a. Equations 68 and 69 apparently determine the shape of the 

interface but not its coordinates because v is unknown. The value of 

yB is zero because the ocean floor is assumed to coincide with the x 

axis. The value of XB, the distance from A to B, is given by the sum 

of the following two equations:

=_ r"'Vd<-./*D (70) 

Jl

Equations 70 and 71 are developed, respectively, from the two 

equations 66c and 67a used to determine r on AB. Equation 70 is 

integrated numerically for values of the variables corresponding to 

o->o-] (see fig. 32D). For values of the variables corresponding to 

o-<Co-i, equation 71 is used. The values Vi and xi correspond to the 

value <T=(TI. The values of v/ki are determined as a function of o--by 

applying the outflow face conditions to equation 64a which yields

- vfki = a coth (a/2) - V0/ki. (72)

The infinite integral 71 is evaluated by first obtaining an analytic 

approximation to d( v/ki) near o-=0 by a series expansion:

*

,, . adff 2ada ,- ,. f"--- (73)

By letting the range of application of the approximations 73 and 

67a be identical, the integral in 71 reduces to

<r\

The dimensionless half width of the island lj.fi is obtained by subtracting the length of the outflow face AB from the total horizontal

length - of the interface. A division of the dimensionless "Q

coordinates x/rj and y\i\ by l/rj produces the more conventional dimensionless coordinates x/l and y/l. Thus, for any given V0/ki, the 

dimensionless coordinates can be obtained. Where desirable the 

absolute coordinates can then be obtained by multiplying these by 

the half width of the island.
C68 RELATION OF SALT WATER TO FRESH GROUND WATER

The total depth of the fresh-water lens can be computed independently, except for the evaluation of r, by another method. Applying to equation 63 the conditions on DF yields

yy yD_\ - -__T j/../ 7. N (75\

where T is determined by equation 66b. The values of vfki are 

determined by applying the conditions on DF to..equation 64, which 

yields

»/fc,= F0/fci-atanh~ (76)

£

Equation 75 is integrated numerically. The half width I of the 

island can also be computed independently, except for the evaluation 

of T, by summing the following two equations:

" Vd(- W/fcO (77)

£z3i=g-T « f° (L\*d(-ulki). (78) 

 n J-V*1 ^2 '

Equations 77 and 78 correspond, respectively, to the two .equations 

66d and 67b used to determine T on DF. Equation 77 is integrated 

numerically for values of the variables corresponding to f<f2 (see 

fig. 32/>). For values of the variables corresponding to f>f2 equation 

78 is used. The values u^ and x2 correspond to the value f=f2. 

The values of ufki are determined as a function of f by applying the 

conditions on FA to equation 64. This yields

u/ki=a cot (r/2). (79)

The infinite integral 78 is evaluated by first obtaining an analytic 

approximation to d(u/ki) near f=0 by a series expansion:

(80)

If one lets the range of application of the approximations 80 and 67b 

be identical, the integral in 78 reduces to

(81)

The depth of the fresh-water lens at D as computed from equation 75 

should be the same as the maximunr ordinate of the interface as 

computed from equation 69. Similarly, the half width of the island
SEA WATER IN COASTAL AQUIFERS C69

as computed by summing equations 77 and 78 should be the same as 

the maximum abscissa of the interface as given by equation 68 less 

the distance from A to B as given by the sum of equations 70 and 71.

DISCUSSION OF RESULTS

The procedure outlined above was applied to two values of the 

recharge parameter V0/k1} namely, 0.755 and 0.0062. The degree 

of accuracy of the computations is indicated by the fact that the 

values of I and yD computed by the two alternative potential-theory 

methods agreed withir 3 percent.

For an aquifer having a transmission coefficient of k=4Q feet per

day and for a density difference ratio of =1/40, the larger

Po

value of V0/ki would correspond to a rate of recharge of 9 inches per 

day. Such a rate is of course unrealistically high, except possibly for 

artificial recharge. The purpose of this computation was to investigate how the position and shape of the interface are affected by a 

high rate of recharge. The effects are shown in figure 33&. As one

Interface obtained by use of 

Dupuit assumption

Interface obtained by 

potential theory

"Interface obtained by potential theory 

= 0.078 and by use of Dupuit assumption

B Interface for VQ /*, =0.0062

FIGURE 33 Numerical solutions for interfaces in water-table aquifers.
C70 RELATION OF SALT WATER TO FRESH GROUND WATER

would expect, the maximum depth of the interface DF and the width 

of the outflow face AB are both large in relation to the half width of 

the island. However, the significant fact demonstrated is that with 

very large recharge the interface computed from potential theory is 

much deeper and more seaward than that computed by use of the 

Dupuit assumption. 

The rate of recharge corresponding to the smaller value of V0/ki,

with the same values of k and as used above is 27 inches per

Po

year. Although this conforms more nearly to that which could occur 

naturally, it is still extremely large. Rarely in nature would the 

recharge parameter V0/ki be as large as 0.0062. Even so, the position 

of the interface computed by potential theory and by use of the Dupuit 

assumption are so nearly the same that they are indistinguishable in 

the plot shown in figure 336. Therefore, the use of the Dupuit assumption appears to give results that are sufficiently accurate for many 

purposes. Only where / is large and the subject of interest is the 

position of the interface near the shore would the use of the Dupuit 

assumption be likely to cause an appreciable error. Other significant 

features of figure 336 are the maximum depth of the interface DF, 

which is 0.078J, and the width of the interface AB, which is so small as 

to be imperceptible on the scale of the figure.

EFFECTS OF DISPERSION ON SALT ENCROACHMENT 

IN COASTAL. AQUIFERS

By HAROLD R. HENRY

ABSTRACT

The problem of salt encroachment, including the effects of dispersion, 

is treated for an artesian aquifer in which there is a steady seaward 

flow of fresh water. Darcy's equation with variable density and the 

continuity equations for water and for salt are the governing equations. 

In terms of the dimensionless stream function $' and the salt concentration c', these equations reduce to

and

where ki is the transmission coefficient times the density difference 

ratio (PS PO)/PQ, d is the depth of the aquifer, Q is the net fresh-water 

discharge, and D is the dispersion coefficient. The effect of tidally 

induced motion in producing a large effective D is discussed.
SEA WATER IN COASTAL AQUIFERS C71

The nonlinear-boundary-value problem is reduced to an infinite set 

of nonlinear algebraic equations by the assumption of Fourier series 

representations of $' and c' and a subsequent application of the 

Galerkin procedure to yield equations for the Fourier coefficients. 

A truncated set of these algebraic equations is solved by an iterative 

scheme, and numerical values of c' and if/f are presented for a specific 

problem. The pattern of the concentration c' is compared to the 

sharp interface that would exist in the limiting case of no diffusion. 

The pattern of c' for the other extreme, diffusion with no density difference, also is presented for comparison.

The computations show that dispersion induces a landward flow of 

salt water in the lower part of the aquifer and reduces the extent of

salt encroachment.

INTRODUCTION

The extent of encroachment of sea water and the location of the zone 

of brackish water are significant factors in determining the use of 

ground water from coastal areas. Using the description of conditions 

at the interface in terms of the velocity potential of the fresh water, 

Hubbert (1940) gave the boundary conditions at an interface. These 

conditions are analogous to those prevailing at the free surface of 

water seeping by gravity through an aquifer. Glover (see p. C32- 

C35, this report) utilized an exact solution of this free-surface problem 

by Kozeny (1953) to represent the solution of the analogous saltencroachment problem for an artesian aquifer of infinite depth. The 

writer solved the potential problem for finite aquifers with horizontal 

and vertical outflow faces. (See p. C35-C70.)

All these analyses treat the interface between the fresh water and 

salt water as if it were sharp and well defined, whereas in reality the 

salt water merges with the fresh water in a zone of diffusion. If this 

zone is a narrow band, as it is in some coastal aquifers (Brown, 1925), 

the solution for a sharp interface may be used to obtain a first approximation to the fresh water flow pattern. If, on the other hand, the 

zone of diffusion is extensive, as in the Biscayne aquifer of southeastern 

Florida (see p. C12-C32, this report), an analysis which takes into 

account the dispersion process and the entrainment of salt water by 

the moving fresh water is necessary.

The dispersion of salt water and fresh water in porous media is the 

result of two phenomena. The first is the mechanical mixing which 

results because, under a given potential gradient, the fluid in the larger 

pores will move faster than that in the smaller ones, and also because 

the fluid near the center of a pore will move faster than that at the 

periphery. Thus, where a potential gradient exists, filaments of fresh 

water will extend into the salt water, and vice versa. The second
C72 RELATION OF SALT WATER TO FRESH GROUND WATER

phenomena is molecular diffusion, which completes the process by 

diffusing the filaments on a microscopic scale. Although the dispersion process is dominated by mechanical mixing, it may nevertheless 

be described by a Fickian-type law similar to that used for molecular 

diffusion (Barrer, 1951) that is, the dispersive mass transfer is of the 

form DVc. Whereas the coefficient of molecular diffusion is related 

to a velocity and a length, both characteristic of the molecular activity, 

the coefficient of dispersion D depends upon the macroscopic fluid 

velocity and a length which is characteristic of the aquifer pores.

Several recent studies have increased understanding of the physics 

of the dispersion in granular media. Wentworth (1948) analyzed the 

zone of diffusion under the hypothesis of a rinsing action due to tides. 

Scheidegger (1954, 1958) applied statistical procedures to derive a 

modification of Darcy's law which includes effects of dispersion. Day 

(1956) used this approach also in analyzing experimental data of onedimensional dispersion. Other experiments (Rifai and others, 1956; 

Ogata; 2 Simpson 3) indicated that the dispersion coefficient in the 

direction of the velocity is approximately proportional to the velocity that is, D Mu and that the dispersion coefficient in the lateral 

direction is much smaller. Here, M is a proportionality constant 

which depends upon the pore characteristics of the aquifer, and u is 

the velocity that appears in Darcy's law.

The magnitude of the dispersion is determined principally by the 

movements of water in an aquifer produced by tides and seasonal 

fluctuations in recharge. However, if inland from the range of tidal 

effect the fresh water flows seaward at a steady rate and no further 

recharge near the beach occurs, then one can define "steady-flow" 

quantities as being the temporal averages of instantaneous quantities 

averaged over a period of many tidal cycles. In this sense the flow 

may be considered steady, the sea level constant, and the tides only 

as a mechanism producing dispersion, as discussed by Cooper (see 

p. C1-C12). Cooper theorized that the dispersion is sufficient to cause 

a large amount of salt water to become entrained in the otherwise 

fresh water flowing seaward and that this induces a motion of salt 

water from the ocean into the zone of diffusion. This circulating 

motion of the salt water has been confirmed by the field measurements 

of Kohout. (See p. C12-C32, this paper.)

The present paper takes into account the effects of dispersion at 

the salt-water front and derives the flow and concentration patterns. 

The macroscopic point of view is assumed, and Cooper's ideas concerning the effect of the tides are utilized to determine the approxi2 Ogata, Akio, 1968, Dispersion in porous media: Evanston, HI., Northwestern Univ. Ph. D. dissertation. 

* Simpson, Eugene, 1960, Transverse dispersion in liquid flow through porous media: New York, Columbia Univ. Ph. D. dissertation.
SEA WATER IN COASTAL AQUIFERS C73

mate magnitude of the dispersion coefficient under certain field 

conditions. Experiments indicate that the dispersion is directional 

and that its magnitude perpendicular to the velocity is much smaller 

than that in the direction of the velocity. It is assumed here that 

the flow pattern is approximated by using for D an effective value 

that is scalar and does not vary with direction or with the magnitude 

of the velocity but is constant throughout the aquifer. Although 

these assumptions are broad, they enable analytical demonstrations 

of certain significant consequences of dispersion which to date have 

been inferred only on a qualitative basis. These consequences are 

the circulation of salt water mentioned above and a reduction of the 

inland extent of salt water.

ACKNOWLEDGMENTS

This paper is based upon the writer's doctoral dissertation at Columbia University.4 The problem was suggested by H. H. Cooper, 

Jr., and the research was supported by the U.S. Geological Survey. 

The writer is grateful to Prof. Richard Skalak for. his criticism throughout the preparation of this paper. Mrs. Judy Tasman programmed 

the computations for the IBM 650 computer.

NOTATION

The symbols used most frequently herein are defined below.

a=Q/kid 

b=D/Q

c=concentration of salt, in mass per unit volume of solution, slugs per ft3 

c, = concentration of salt in sea water, slugs per ft3 

c'=c/c, 

C=(ef-xlQ

d=thickness of aquifer, ft 

D=dispersion coefficient, ft2 per sec

E= dimensionless coefficient relating the concentration to the density of 

the solution

g = vector acceleration of gravity, ft per sec2 

k=permeability of sand, ft2

k=transmission coefficient of aquifer= > ft per sec

fcj=transmission coefficient times the density-difference ratio=k >

po

ft per sec

1=length of aquifer, ft 

p=pressure, Ibs per ft2 

Q = net fresh-water discharge per unit length of beach, ft2 per sec

q= vector fluid velocity, ft per sec

< See footnote 1, p. C36.
C74 RELATION OF SALT WATER TO FRESH GROUND WATEK

u = horizontal velocity, ft per sec

v = vertical velocity, ft per sec

x, y= coordinates, ft (when used as subscripts, a; and y denote differentiation)

x'=x/d

y' = yfd

p = density of solution, slugs per ft3

PO = density of fresh water, slugs per ft3

ps = density of sea water, slugs per ft3

Pu, = mass of water contained in a unit volume of solution, slugs per ft3

p /_  

P PO 

Ps PO

M= viscosity of water, Ib-sec per ft2 

^' = dimensionless stream function

% = l/d= aspect ratio

FIELD EQUATIONS AND BOUNDARY CONDITIONS

When dispersion is taken into account, the equation of motion, 

which is Darcy's equation, must be written with density as a variable 

since the density varies with the salt concentration. The general 

vector form of Darcy's equation is (Hubbert, 1953; Offeringa and 

Van der Poel, 1954):

?=--(vp- P1). (1)

M

In addition to equation 1, equations of continuity must be satisfied 

for the water and the salt. The conservation of mass of water in 

steady flow is expressed by

-» 

V- P«,g=0, (2)

where pw is the mass of pure water per unit volume of the solution. 

The conservation of the mass of salt in solution in steady flow is

expressed by a similar equation, V-cge =0, where c is the mass per

unit volume of the salt and ge is the effective vector velocity of the 

motion of the salt. This velocity is the sum of the average fluid 

velocity q and that due to dispersion, Z>Vc. Thus, the equation of 

continuity for the salt becomes

-» 

V-cq V-DVc=0. (3)

It is assumed here that the unsteady effects of the tides and recharge 

can be averaged and superposed on the steady seaward flow by using 

a suitable dispersion coefficient D.

In the treatment of equations 1,2, and 3 it is convenient to introduce the dimensionless quantities

. ud , vd . x . y . c , , P PO (A\ u'=-7r, »'=-7r> x' = -v y'=y c' ~> and p' (4) Q Q d a d c. P. PO
SEA WATER IN COASTAL AQUIFERS C75

Equation 2 can be simplified by using the empirical relation between 

the density p of the mixture and pw of the pure water (Baxter and 

Wallace, 1916) as indicated by

P=P«,+c=po+(l-#)c, (5)

where E is a dimensionless constant which has the approximate value 

0.3 for concentrations to as much as that of sea water. Inserting 

equation 5 into equation 2 yields

v-?- (v-c'?)=0. (6)

Po

The quantity c' has an order of magnitude of unity, so that V-c'q is

EC of the same order of magnitude as v-q. For sea water, - is about

Po

0.008, and hence the order of magnitude of the second term in equation 6 is smaller than that of the first. When the second term is 

dropped, v-<?=0, which implies the existence of a stream function 

that will be defined as a dimensionless quantity by

Pressure can be eliminated from equation 1 by writing the scalar 

forms of equation 1 and differentiating u with respect to y, v with 

respect to x, and subtracting. A substitution of the derivations of 

equation 7 then eliminates u and v in favor of ^', and the use of p'=c' 

obtained from equation 5 yields

**" $£ (8)

In equation 3 the dispersion coefficient D is a function of the velocity 

(Rifai, Kaufman, and Todd, 1956; Simpson 5 ) and, therefore, cannot 

in principle be determined before the problem is solved. Also, in 

applying equations 1, 2, and 3 to the salt intrusion problem, which 

is unsteady due to tidal activity, the burden of absorbing all the 

unsteady terms, which would appear in a more detailed analysis, is 

put upon D. However, it is assumed for purposes of the analysis 

that a representative average value of D is given and that its use 

as a constant scalar throughout the field of flow will not destroy the 

essential features of the problem. Assuming D constant and sub-

' See footnote 3, p. 072.
C76 RELATION OF SALT WATER TO FRESH GROUND WATER

stituting for u and v in terms of ^ in equation 3 yields the following 

dimensionless form:

^W-^.'^-^-'-^-' (9) 

Q V dy' da;' da;' £>*/'' w

Equations 8 and 9 reveal the two significant parameters of the problem, 

kd/Q and D/Q.

The model treated in this study is a rectangular confined aquifer 

in contact with a fresh-water reservoir on one end and the ocean on 

the other, as shown in figure 34. Along the impermeable boundaries 

at the top and bottom of the aquifer the normal velocity must be 

zero, so that ^ is constant. Moreover, along these boundaries the 

noimal diffusive movement of salt must also be zero, so that the

be'

derivative r dy': vanishes. The latter condition requires that the isochlors approach the base of the aquifer vertically. This explains at 

least in part the prominent blunted shape of the toe of the salt-water 

wedge observed by Kohout. (See fig. 31.)

Along the vertical boundaries the pressure distributions are hydrod^' 

static, so that ;=0. Also, along the fresh-water side c'=0, and

O3/

along the ocean side c'=l.

The boundary conditions are summarized below. (See fig. 34.)

At y' = 0:*' = 0, |^ = 0 At x' = 0:^ = 0, c' = 0

' (10)

FOURIER -GALERKIN SOLUTION

A method of solution similar to that used by Foots (1958) in a 

problem of heat transfer is utilized to solve the nonlinear boundary 

value problem defined by equations 8, 9, and 10. First, new variables 

are introduced to simplify the boundary conditions: let

(11)

With the primes omitted; with the substitutions £=l/d, a=Qfkid, 

b=D/Q; and with the use of subscripts to denote differentiation, the 

problem in terms of ^ and C becomes

C,+| (12)

' +!* +£*+1 (13)

At j/=0, 1:^=0, Cy=0 Atz=0, ^:*=0, C=0. (14)
SEA WATER IN COASTAL AQUIFERS C77

The functions ^ and C can be represented by double Fourier series 

which satisfy the boundary conditions, equation 14, identically:

* = S JC 4».» sin 0»»V) cos (n*xl$ (15)

'.« cos ( y) sin

r=0 *=1

If the coefficients Am>n and Br, g are chosen in such a manner that the 

differential equations 12 and 13 are satisfied, then equations 15 and 

16 will constitute the solution. This is accomplished by substituting 

the Fourier representations for ^ and C into equations 12 and 13 

and applying Galerkin's method (Galerkin, 1915; Duncan, 1938). 

Galerkin's method in this instance consists of multiplying equation 

12 by 4 sin (gin/) cos (hirxl£) and equation 13 by 4 cos (gin/) sin (hirxl£) 

after substituting from equations 15 and 16 for C and >P, and then 

integrating each equation over the rectangular domain. This gives 

an infinite set of algebraic equations for the Fourier coefficients Ag,n 

and B0fh, as follows:

, h [V+|n £=ib Br.k'h'N(g, r) +- W(g, h) (17) 

L ? J r=o *"

7»=1 «=0 r=0

) + - W(h,g). (18)

4=1 1r

The notation in equations 17 and 18 is defined below where 8 is 

the Kronecker delta:

'-! (-!)*-"-*-! (-1) »+"+«- 1 (-1)

fe+n s ft n s fe+n+s ft n s

(-1) >+"-«-! (-1) >-»-«- 1 (_

fe+n s ft n s k+n+s k n 

Oif

{

2i 

i.

2 if fe= 

if
C78 RELATION OF SALT WATER TO FRESH GROUND WATER

A truncated set of equations 17 and 18 was solved numerically. 

This gave an approximation to the true solution in a least squares 

sense for the truncated set of coefficients retained.

NTJMBRICAL SOLUTION

The selection of the parameters Z=-j> a=Q/kid, and b=D/Q for the

numerical solutions was governed by two considerations. On one 

hand it is desirable that the parameters correspond as nearly as possible with field data. For the data presented by Kohout (see p. 

C12-C32, this report) the field values were approximately £= 16, a=0.04, 

0.01 <b< 0.001. On the other hand, parameters must be chosen that 

will make the numerical procedures converge. The parameters which 

will lead to a significantly accurate solution within a manageable 

truncation of equations 17 and 18 cannot be determined in advance. 

However, two general observations can be made before making any 

trials. The first is that values of £ greater than unity will probably 

require the values of the index h to be greater than those of g, the 

difference increasing with £. The second is that small values of b 

should yield a solution approaching that for an interface, which would 

require terms with large values of g and h to describe accurately the 

sharp increase in concentration. With these considerations in mind 

and after some preliminary computations, the values £=2, a=0.450, 

6=0.100 were chosen for initial computations to be performed with 

the use of a slide rule and a desk calculator. The value 6=0.100 is 

about 10 times the average value caused by the tides in the Biscayne 

aquifer. The value a=0.450 yields a corresponding interface which 

strikes the aquifer bottom approximately at the midpoint of its length 

for £=2. An additional case with £=2, a=0.263, 6=0.100 was chosen 

to show the effect of varying fresh water discharge. The value of 

a=0.263 yields a corresponding interface which reaches to the upstream reservoir when £=2 as shown in figure 345.

The computations with slide rule and desk calculator were performed for the following truncation: Ait0 through A.ioJ A2,i through A2i7 ; 

BQi 1 through B0> u', Bit i through Bit n ; B2. i through B2, 7. An iterative procedure was used wherein the nonlinear terms in equation 18 were 

first assumed to be zero. The resulting linear equations yielded 

approximations for Ag, h and Bg>h which were then used to compute 

numerical values of the quadratic terms. These were substituted in 

equation 17, and the resulting set of linear equations were solved for 

new approximations of A0ih and Bg>h. Three iterative cycles were 

carried out. The linear systems were rearranged to give a strong 

main diagonal for the matrix of the coefficients and were then solved 

by relaxation.
SEA WATER IN COASTAL AQUIFERS C79

Convergence of the numerical procedure was indicated by the computations with slide rule and desk calculator, and the streamlines and 

isochlors obtained from three iterative cycles on the Fourier coefficients 

conformed generally to the pattern measured in the field by Kohout. 

(See fig. 9.) However, the values of $' and c f determined by these 

computations were not sufficiently accurate at all places in the aquifer, 

and it appeared that the labor involved in obtaining sufficient accuracy with slide rule and desk calculator would be prohibitive. It 

was desired that \f/' and c' be determined within two significant figures, 

and to accomplish this the problem was programmed for the IBM 650 

digital computer. The machines at the Watson Computing Laboratories and the Electronic Research Laboratories at Columbia University were used.

Two sets of parameters, £=2, a=0.450, 6=0.100 and £=2, a=0.263, 

6=0.100 were used hi the enlarged truncation: Aitlo through AiiU ; 

A2 ,i through A2 ,w; A3>0 through A3i5 ; Aitl through Aii3; A& t0 through 

A5 ,2 ; -B0il through B0i20 ; Biti through BliW; B2<i through 52 ,5 ; -B3 ,i 

through -B3i3 ; and 54il through B± t2 . In the initial machine computations the quadratic terms in equation 18 were assumed to be zero, 

the resultant linear system consisting of equations 17 and 18 were 

solved for the Agih and the Bgih, and these were used to compute 

numerical values for the quadratic terms. Using these terms as 

constants, the linear system was then solved for new values of the 

coefficients. After three iterative cycles using the first set of parameters and five using the second set had been run, it was apparent that 

the numerical procedure was not converging. This was indicated by 

the fact that the successive computed values for any particular 

B0i)t or Ag ,h fluctuated, apparently without tending toward any limit. 

In addition, the computed values of $' and c' after successive iterations became progressively more different from what one would expect 

by comparison with the field data of Kohout. Furthermore, after 

the fifth iteration for the second set of parameters, the computed 

values of c' were as high as 1.9, whereas they should not exceed 1.00. 

The computation of the quadratic terms for each iteration took 

approximately 14 hours of machine time.

As a result of this experience, a modified procedure employing the 

digital computer was applied to the set of parameters £=2.0, a=0.263, 

6 = 0.100. In this procedure, the quadratic terms were first computed 

from values of the A0th and B0ih obtained from an assumed distribution 

of $' and c'. An iterative solution of equation 18 was used in which 

the B0th were held constant and new values of the Bg>h (gC>0) were 

computed by several subiterations and vice versa. The subiterative 

cycle consisted of recomputing the values of the quadratic terms using 

the revised values of Bg<h (#>0) while continuing to hold the B0th
C80 RELATION OF SALT WATER TO FRESH GROUND WATER

v/v/vx5 ////////////////////////////////////////

0 d 1.0 2.0 

A Streamlines

1.0 

5 Isochlors

FIGURE 34. Flow and salt concentration patterns In an idealized mathematical model.

constant and then using these values to recompute the Bg<h 

During an iteration for the B0ih, the roles of B0 , h and B0ih (</>0) were 

reversed. Two to four subiterative cycles were used each time the 

new values of the B0th or Bg ,h (gC>0) were computed.

The procedure was carried out for several iterative cycles. Convergence was obtained in the subiterations as indicated by the fact 

that the sum of the changes in absolute magnitude of the coefficients 

Bg ih decreased for each subcycle. This same sum also decreased for 

each overall iterative cycle. The validity of the numerical procedure 

was also indicated by the fact that the computed values of the mean
SEA WATER IN COASTAL AQUIFERS C81

concentration on a vertical section (fig. 35) remained zero within 

± 0.005 for the last three iterations in the range 0 >x/d> 0.8 and began 

to show an appreciable rise starting at x/d=Q.9. The value x/d=Q.9 

corresponds to the inland limit of the landward flow along the base 

of the aquifer (fig. 34). Furthermore, the computed positions of the 

streamline ^'=0 from the last three iterations fell within a band which 

departed from a mean position by only ±5 percent of the length of 

the salt-water wedge.

The computations appear to have bracketed the solution sufficiently 

accurately so that several significant conclusions can be drawn as 

given below.

DISCUSSION OF RESULTS

Figures 34^4. and 345 show the streamlines and isochlors for the set 

of parameters £=2.0, o=0.263, 6 0.100. These are interpolated 

from the computed values of $' and c' and are drawn to eliminate local 

variations that appear due to truncation of the Fourier series. Superposed on the streamline and isochlor patterns in figure 34 B is the 

corresponding sharp interface for £=2.0, a=0.263, obtained from the 

solution given on pages C35-C70 of this report. Figure 35 shows for 

the problem under discussion three curves representing the mean salt 

concentration in vertical sections. Curve a is for dispersion with no 

density difference as computed from the diffusion equation, the solution of which is c' = (exf/l> l)/(ef/6 1). Curve b is for a density 

difference with no dispersion and is identical to the interface shown 

hi figures 34,4 and 345. Curve c is for density difference with dispersion. The ordinate of curve c at any x/d is the average" of c' taken

a Dispersion with no density difference 

b Density difference with no dispersion 

c Density difference with dispersion

"3 

FIGTHIE 36. Mean salt concentration in vertical sections.
C82 RELATION OF SALT WATER TO FRESH GROUND WATER

from figure 345 at that x/d. Most of the curve c falls between curves 

a and b, as expected.

Several features of the numerical solutions are significant. As 

observed in figure 34,4, the position of the equivalent sharp interface 

approximately separates the streamlines of the through flow that 

originate at the fresh-water end of the aquifer from those of the 

circulating flow that originate at the ocean end. The isochlors in 

figure 345 approach the base of the aquifer vertically in such a way 

as to give the toe of the salt-water wedge a blunted shape similar 

to that observed by Kohout in the Biscayne aquifer. (See p. 14, 

this report, under the heading "The zone of diffusion.") This shape 

results from the boundary condition that the concentration gradient 

normal to the base of the aquifer be zero. Perhaps the two most 

significant consequences of dispersion that are demonstrated by the 

solutions are the landward flow of salt water in the lower part of the 

aquifer and the concomitant reduction of the landward extent of salt 

water. These two features were predicted by Cooper and were 

observed in the field by Kohout. (See p. C11-C12.)

REFERENCES

Barrer, R. M., 1951, Diffusion in and through solids: Cambridge Univ. Press,

464 p. 

Baxter, G. P., and Wallace, C. C., 1916, Changes in volume upon solution in

water of the halogen salts of the alkali metals, IX: Am. Chem. Soc. Jour., v.

38, p. 70-104. 

Bear, Jacob, and Todd, David K., 1960, The transition zone between fresh and

salt water in coastal aquifers: Berkeley, California Univ., Water Resources

Center Contrib. 29, 156 p.

Bosworth, R. C. L., 1949, The mechanisms of diffusional processes: Royal Australian Chem. Inst. Jour, and Proc., p. 460-482. 

Brown, J. S., 1925, A study of coastal ground water, with special reference to

Connecticut: U.S. Geol. Survey Water-Supply Paper 537, 101 p. 

Byrd, P. F., and Friedman, M. D., 1954, Handbook of elliptic integrals for

engineers and physicists: Berlin, Springer Verlag, 355 p. 

Carrier, G. F., 1958, The mixing of ground water and sea water in permeable

subsoils: Jour. Fluid Mechanics, v. 4, p. 479-488. 

Cooper, H. H., Jr., 1959, A hypothesis concerning the dynamic balance of fresh

water and salt water in a coastal aquifer: Jour. Geophys. Research, v. 64,

no. 4, p. 461-467. 

Darcy, Henri, 1856, Les Fontaines Publique de la Ville de Dijon [The public

fountains of the city of Dijon]: Paris, Victor Dalmont. 

Day, P. R., 1956, Dispersion of a moving salt-water boundary advancing through

saturated sand: Am. Geophys. Union Trans., v. 37, no. 5, p. 595-601. 

Drabbe, J., and Badon Ghyben, W., 1889, Nota in verband met de voorgenomen

putboring nabij Amsterdam [Notes on the probable results of the proposed

well drilling near Amsterdam]: The Hague, Koninkl. Inst. Ing. Tijdschr.,

1888-89, p. 8-22. 

Duncan, W. J., 1938, The principles of Galerkin's method: Aeronaut. Research

Comm. Rept. and Mem. 1848.
,SEA WATER IN COASTAL AQUIFERS G83

Dupuit, Jules, 1863, Etudes the"oriques et pratiques sur le mouvement des eaux

dans les canaux de couvert et d travers les terrains perme"ables. [Theoretical

and practical studies on the movement of water in closed channels and

through permeable ground]: 2d ed., Paris, Dunod, 304 p. 

Eriksson, Erik, 1958, A note on the dispersion of a salt-water boundary moving

through saturated sand: Am. Geophys. Union Trans., v. 39, no. 5, p. 937-938. 

Ferris, J. G., 1951, Cyclic fluctuations of water level as a basis for determining

aquifer transmissibility: Union ge"od. ge"ophys. Internat., Assoc. internat.

d'hydrologie Sci. Assembled ge"n. Bruxelles, v. 2, p. 148-155 [Internat. Union

Geodesy and Geophysics, Internat. Assoc. Sci. Hydrology, Brussels, v. 2, p.

148-155], 

Galerkin, B. G., 1915, Ssries solutions of some problems of elastic equilibrium of

rods and plates: Moscow, Vestnik Inzhenerov, v. 1, p. 879-908 [in Russian]. 

Glover, R. E., 1959, The pattern of fresh-water flow in a coastal aquifer: Jour.

Geophys. Research, v. 64, no. 4, p. 457-459. 

Hamel, G., 1934, Uber Grundwasserstromung [On groundwater flow]: Zeitschr.

angew. Math. u. Mech., v. 14, p. 130-157. 

Hamel, G., and Giinther, E., 1935, Numerische Durchrechnung zu der Abhandlung iiber Grundwasserstromung [Numerical computations in connection

with the paper on groundwater flow]: Zeitschr. angew. Math. u. Mech., v.

15, p. 255-265. 

Hayashi, K., 1930, Tafeln der Besselschen, Theta, Kugel, und anderer Funktionen

[Tables of Bessel, Theta, Spherical, and other functions]: Berlin, Springer

Verlag, 150 p. 

Henry, H. R., 1959, Salt intrusion into fresh-water aquifers: Jour. Geophys.

Research, v. 64, no. 11, p. 1911-1919.

 1960, Salt intrusion into coastal aquifers: Internat. Assoc. Sci. Hydrol., 

Comm. Subter. Waters Pub. 52, p. 478-487.

Herzberg, Alexander, 1901, Die Wasserversorgung einiger Nordseebader [The 

water supply on parts of the North Sea coast]: Munich, Jour. Gasbeleucht. 

u. Wasserversorg., v. 44, p. 815-819, 842-844.

Hodgeman, C. D., 1945, Handbook of chemistry and physics: Cleveland, Chemical Rubber Publishing Co., 2640 p.

Hubbert, M. K., 1940, The theory of ground-water motion: Jour. Geology v. 48, 

no. 8, pt. 1, 785-944.

 1953, Entrapment of petroleum under hydrodynamic conditions: Am. 

Assoc. Petroleum Geologists Bull., v. 37, no. 8, p. 1954-2026.

Jacob, C. E., 1950, Flow of ground water, chap. 5 in Rouse, Hunter (ed.), Engineering hydraulics: New York, John Wiley and Sons, p. 321-385.

Jahnke, Eugene, and Emde, Fritz, 1945, Tables of functions: New York, Dover 

Pub., 304 p.

Kirkham, Don, 1958, Seepage of steady rainfall through soil into drains: Am. 

Geophys. Union Trans., v. 39, p. 892-908.

Kitagawa, K., 1939, Un aspect du developement des Etudes des eaux souterraines 

au Japon [An aspect of the development of the studies on underground 

waters in Japan]: Japanese Jour. Astronomy and Geophysics, v. 17, p. 

141-155.

Kohout, F. A., 1960a, Cyclic flow of salt water in the Biscayne aquifer of southeastern Florida: Jour. Geophys. Research, v. 65, no. 7, p. 2133-2141.

 1960b, Flow pattern of fresh water and salt water in the Bi&cayne aquifer 

of the Miami area, Florida: Internat. Assoc. Sci. Hydrol., Comm. Subter. 

Waters Pub. 52, p. 440-448.
C84 RELATION OF SALT WATER TO FRESH GROUND WATEB

Kozeny, Josef, 1953, Hydraulik: Vienna, Springer Verlag, 588 p. 

Muskat, Morris, 1935, The seepage of water through dams with vertical faces: 

Physics, v. 6, p. 402-415.

 1937, The flow of homogeneous fluids through porous media: New York,

McGraw-Hill Book Co., 763 p.

Nehari, Zeev, 1952, Conformal mapping: New York, McGraw-Hill, 396 p. 

Nomitsu, T., Toyohara, Y., and Kamimoto, R., 1927, On the contact surface of

fresh and salt water near a sandy sea shore: Kyoto Imp. Univ., Mem. Coll.

Sci., ser. A., v. 10, no. 7, p. 279-302. 

Offeringa, Jan, and Vander Poel, G., 1954, Displacement of oil from porous media

by miscible liquids: Am. Inst. Mining Engineers Trans., v. 201, p. 310-316. 

Orlob, G. T., and Radhakrishna, G. N., 1958, The effects of entrapped gases on

the hydraulic characteristics of porous media: Am. Geophys. Union Trans.,

v. 39, no. 4, p. 648-659. 

Palmer, H. S., 1927, The geology of the Honolulu artesian system: Honolulu

Sewer and Water Comm. Rept. 

Parker, G. G., 1951, Geologic and hydrologic factors in the perennial yield of the

Biscayne aquifer: Am. Water Works Assoc. Jour., v. 43, no. 10, p. 817-835. 

Parker, G. G., Ferguson, G. E., and Love, S. K., 1944, Investigations of water

resources of southeastern Florida: Fla. Geol. Survey Rept. of Inv. 4, 39 p.

 1955, Water resources of southeastern Florida: U.S. Geol. Survey WaterSupply Paper 1255, 965 p. 

Foots, G., 1958, Heat transfer by laminar free convection in enclosed plane gas

layers: Quart. Jour. Mech. and Appl. Math., v. 11, pt. 3, p. 257-273. 

Rifai, M. N. E., Kaufman, W. J., and Todd, D. K., 1956, Dispersion phenomena

in laminar flow through porous media: Berkeley, California Univ., Canal

Seepage Research Prog. Rept. 2, 157 p. 

Scheidegger, A. E., 1958, The random-walk model with auto-correlation of flow

through porous media: Canadian Jour. Physics, v. 36, p. 649-653. 

Senio, K., 1951, On the ground water near the seashore: Union g6od. ge"ophys.

Internal, Assoc. Internat. d'Hydrologie Sci. Assembled ge"n. Bruxelles, v. 2,

p. 175-177. [Internat. Union Geodesy and Geophysics, Internat. Assoc.

Sci. Hydrology, Brussels, v. 2, p. 175-177.] 

Taylor, G. I., 1953, Dispersion of soluble matter in solvent flowing slowing through

a tube: Royal Soc. London Proc., Ser. A, v. 219, p. 186-203. 

Todd, D. K., 1953, Sea water intrusion in coastal aquifers: Am. Geophys. Union

Trans., v. 34, no. 5, p. 749-754.

 1960, Salt intrusion of coastal aquifers in the United States: Internat.

Assoc. Sci. Hydrol., Subter. Waters Comm. Pub. 52, p. 452-461. 

Wentworth, C. K., 1948, Growth of the Ghyben-Herzberg transition zone under

a rinsing hypothesis: Am. Geophys. Union Trans., v. 29, no. 1, p. 97-98.

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