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Sea Water in Coastal Aquifers By HILTON H. COOPER, JR., FRANCIS A. KOHOUT, HAROLD R. HENRY, and ROBERT E. GLOVER RELATIO
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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 ____________________________________ A hypothesis concerning the dynamic balance of fresh water and salt water in a coastal aquifer, by Hilton H. Cooper, Jr________________________ Abstract___ _ ________________________________________________ Introduction._ ________________________________________________ Acknowledgments _____________________________________________ Mechanics of the cyclic flow of sea water.________________________ Dispersion.___________________________________________________ Summary_ _ _________________________________________________ The flow of fresh water and salt water in the Biscayne aquifer of the Miami area, Florida, by Francis A. Kohout____________-________---__-____ Abstract. _:_____.________________________________ Introduction._________________________________________________ Acknowledgments _____________________________________________ Geologic and hydrologic characteristics of the Biscayne aquifer_ _ _ _ _ _ The zone of diffusion_________________________________________ Seaward flow of salty water___-____________-_________-_--__-_-_Dispersion._ __________________________________________________ Hydraulic gradient in the salt-water zone.________________________ Silver Bluff area______________________________ Cutler area_____________________________________ Potential in the fresh-water and salt-water environments______---_Patterns of flow of fresh and salt water___________________----_Summ ary_ ___________________________________________________ The pattern of fresh-water flow in a coastal aquifer, by Robert E. Glover._ Abstract____________________________________________________ Introduction __________________________________________________ Flow net._______________________________________ Interfaces between salt water and fresh water in coastal aquifers, by Harold R. Henry_--__---__-__-___________________________ Abstract __ __________________________________________________ Introduction._________________________________________________ Acknowledgments _____________________________________________ Artesian aquifers___________-___________-___-_-_--___--__-----_ Notation _________________________________________________ Governing equations and boundary conditions.______________ Case 1, vertical outflow face__________-__-___---_-----_----Case 2, horizontal outflow face.__________________---__--_--_ Computations for semi-infinite aquifers, cases 1 and 2_________ Computations for finite aquifers, case 1_______________-.._-_Computations for finite aquifers, case 2______________-__---_ Discussion of results.__-__-__-__-_______-____--_--_-__----Water-table aquifers receiving vertical recharge__________-_--_--_Governing equations and boundary conditions_______________ Numerical solution________________________.____----------Discussion of results_---__-__-___-_---____----___----------
vn
in Cl 1 1 6 6 8 11 12 12 12 14 14 14 14 18 20 20 22 23 29 32 32 32 32 33 35 35 36 36 37 40 40 43 48 49 52 53 59 62 62 65 69
VIII
CONTENTS
Page Effects of dispersion on salt encroachment in coastal aquifers, by Harold R. Henry______.________________________________________________ Abstract__ _ _________________________________________________ Introduction..________________________________________________ Acknowledgments. _________-______---__-_-_-_-_-______________ Notation _____________________________________________________ Field equations and boundary conditions _________________________ Fourier-Galerkin solution.______________________________________ Numerical solution_____________________________________________ Discussion of results________--__-__..-____--__--___-_-_-________ References..______________________________________________________
C70 70 71 73 73 74 76 78 81 82
ILLUSTRATIONS FIGURE
1. Balance between fresh water and salt water__________^_ 2. Circulation of salt water___________________________ 3. Map of Waimano Springs, Oahu, Hawaii, showing the chloride, content of water from orifices.______________ 4. Chloride content of water in a test well at Pearl Harbor. 5. Section through the Cutler area, near Miami, Fla., showing the zone of diffusion.__________________________ 6. Amplitude of tide-produced motion of water.__________ 7. Map of the eastern part of Bade County, Fla__________ 8. Section through the Silver Bluff area, near Miami, Fla., showing the zone of diffusion__-__________---_--____ 9. Section through the Cutler area, near Miami, Fla., showing the position of the line of zero horizontal gradient. 10. Graph showing fluctuations of chloride content and water level in well G 519A________._______________ 11-13. Hydrographs of 11. Landward decrease in tidal fluctuation _____________ 12. Daily-average water level and head in well F 200 and Biscayne Bay_____-______----________-_-_ 13. Wells G 906, G 919, and G 936__._-_-____-_--__-14-19. Sections through the Cutler area, near Maimi, Fla., showing 14. Lines of equal head.____________________________ 15. Lines of equal fresh-water potential ________--_-_ 16. Lines of equal fresh-water potential--.. ---------17. Lines of equal fresh-water potential for a low head condition ____________________________________ 18. Lines of equal fresh-water potential for a high head condition ____________________________________ 19. Patterns of flow of fresh and salt water______---__20. Flow pattern near a beach.__________________________ 21, 22. Schematic sketches and complex planes______-___--__ 23, 24. Numerical solutions for a short sand formation________25. Numerical solution for semi-infinite artesian aquifers... 26. Interfaces for vertical outflow face.______________----27. Interfaces reaching to recharge reservoir._________----_
C2 3 4 5 7 11 13 15 16 17 19 21 22 24 25 26 27 30 31 34 38 46 51 54 56
CONTENTS
FIGUKE
28. 29. 30. 31. 32. 33. 34. 35.
Alternative transformations of /__-__________________. Interfaces intersecting base of aquifer near ocean.______ Comparison of interfaces____________________________ Section through the Cutler area near Miami, Fla., showing isochlors---___________--____________________. Schematic sketch and complex planes.-_________:._____ Numerical solutions for interfaces___________________ Flow and concentration patterns____________________ Mean salt concentration in vertical section___________
IX
Page C57 58 60 61 63 69 80 81
TABLES TABLE 1. 2. 3. 4.
Parameters Parameters Parameters Parameters
719-580i
for for for for
computations plotted in figure 26____-_______the two shortest interfaces in figure 27________ the four longest interfaces in figure 27_______ the interfaces in figure 29__________________
C53 55 58 58
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 ,,
£
'
P*
fly
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
FIGURE 1. Balance between fresh water and salt water in a coastal aquifer in which the salt water is static.
Fresh water
. >-. :;.;.v£-g
.Salt watervA :.: :: ! £ ; :;V>
FIGURE 2. Circulation of salt water from the sea to the zone of diffusion and back to the sea.
Fresh water
Land surface
Q
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 160
190
162
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.
C5
SEA WATER IN COASTAL AQUIFERS
100 200
300
400
500
600
700
z 800 I Q 900
1000
1100
1200
1300 1400 0
2
I I I I I I 4 6 8 10 12 14 16 CHLORIDE CONTENT, IN 1000 PARTS PER MILLION
I
I 18
I 20
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.
1000
DISTANCE FROM SHORELINE, IN FEET 500 500
BISCAYNE BAY
Isochlor, in parts per million Dashed where approximate; interval irregular
FIGURE 5. Section through the Cutler area, near Miami, Fla., showing the zone of diffusion, September 8,1958.
1500
o
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
= (Kho/ff)T/toS/4rT exp ( xVxS//bT)T exp (-nfcSJUT) ,
rs*/4 Jv/i
(sin M + COS u}du (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 DISTANCE FROM SHORELINE, IN FEET
400
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
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.
C13
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 BISCAYNE
CHARACTERISTICS AQUIFER
OF
THE
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,
10,000
Isochlor.in parts per million Dashed where approximate; interval irregular
DISTANCE FROM SHORELINE, IN FEET 8000 6000 4000 2000
FIGURE 8. Cross section through the Silver Bluff area, near Miami, Fla., showing the zone of diffusion, November 2,1954.
12,000
O
1400
1200
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 400
Dashed where approximate; interval irregular
Isochlor, in parts per million
200
FIGURE 9. Cross section through the Cutler area, near Miami, Fla., showing the position of the line of zero horizontal gradient (traced from fig. 17) within the zone of diffusion, September 18,1958.
1600
h-»
o Oi
DAILY AVERAGE HEAD, IN FEET ABOVE OR BELOW BAY LEVEL
CHLORIDE CONTENT, IN PARTS PER MILLION
ppppi-i->-
5
I-**
~
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
06
08
10
TIME, IN HOURS 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.
C19
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,
SEPT
OCT
1953 NOV
DEC
JAN
FEB D
MAR
JUNE
JULY
AUG
Water level in well lower than Biscayne Bay
Water level in well higher than Biscayne Bay
1954 APR MAY
FIGURE 12. Hydrographs of daily-average water level and head in well F 200 and in Biscayne Bay. A, Daily-average water level in well and in bay; B, daily-average head in well compared to bay level.
AUG
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
_, 20
1400
1200
Base of Biscayne aquifer/
0.16
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 200
400
interval 0. 1 foot
Dashed where approximate;
Bottom of fully cased well, with average observed head, in feet Equipotential line
FIGURE 14. Cross section through the Cutler area, near Miami, Fla., showing lines of equal head in terms of environmental water; average for September 18,1958.
100
80
< h- 60
i
UJ
o: O 40
20
a § o
UJ
1600
1400
1200
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 400
Bottom of fully cased well, with fresh-water head at high tide, in feet
200
FIGURE 15. Cross section through the Cutler area near Miami, Fla., showing lines of equal fresh-water potential at 1335 E3T (bay high tide), September 18,195$.
1600
to
O
2.49
1400
f0
1200
3
IV
2.15
M b
/jygf
/ « 1 CO
r «° 1
200
400
I 0-82, 0.83 0.82J I 0.95
0.67
Interval 0.1 foot
Equipotential line
BISCAYNE BAY
Bottom of fully cased well, with fresh-water head at low tide, in feet
7~r7TTir~r~ / 0-91 V®(lf 1 /'III LI 79 1 II / 1.76
1.89
.1.54
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200
FIGURE 16. Cross section through the Cutler area, near Miami, Fla., showing lines of equal fresh-water potential at 0750 EST (bay low tide), September 18,1958.
'100
80
< h- 60
8m
UJ
O 40
20
20
1600
1200
Land surfac
1400
xBase of Biscayne aquifer
_GH> BEN^HERZBERG JJiN-
0.2-
1.42
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 400 I
. . Q '42 >
0.32'-^^
2.2
2.0
1.8
1.6
/.*
1.2
1.0
0.8
0.6
0.4
« "- *-^0.2 030^26 0.19
0.50.^_^^
.0.44-1
0.2$K. 0.15
_BISCAYNE BAY
Equipotential line Interval is 0. 7 foof
Bottom of fully cased well, with average fresh-water head, in feet
200 I
PIGUBE 17. Cross section through the Cutler area, near Miami, Fla., showing lines of equal fresh-water potential for a low-head condition; average for September 18,1958.
J 100
80
60
O 40
yj CD
I 20
20
1600 I
to
O
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.
1400
1200 .1.84
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 400
Equipotential line Interval is 0.1 foot
Bottom of fully cased well, with average fresh-water head, in feet
200
FIGURE 18. Cross section through the Cutler area, near Miami, Fla., showing lines of equal fresh-water potential for a high-head condition average for May 29,1968.
1600
o CO o
1400
1200
DISTANCE FROM SHORELINE, IN FEET 1000 800 600 400 200 400
Streamline
Bottom of fully cased well
200
FIGURE 19. Cross section through the Cutler area, near Miami, Fla., showing the patterns of flow of fresh and salt water for a low-head condition, September 18.1968.
1600
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 *7V
(1)
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_ (5) "
2yK
As used here, the potential , 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 0Q= i g'Y^JX/J\.) ' .
/£N
\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 .
(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 u r =u cos a-\-v sin a
=> or
(7)
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. or or the desired relation:
Substitution of this into equation 7 yields
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. os Multiplying equation 9 by ^ gives: OS
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
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 - as obtained from equation 35: "=-±^=-^0/2^. ki ki da;
(45)
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), '
'
2' ki
(46)
K \1
For the interface BD;
x XB
(47)
SEA WATER IN COASTAL AQUIFERS
C53
For the boundary segment DE; f dw I u eT -=Jo fci
K \X
(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
c
a=0/kid
t=l/d
10 5 2 1.2 1. 00235 1. 00200
1 1 1 1 1 1. 00100
1 021 896 709 663 309 340
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 sn~13* (A,kf) ' -d^ ^l/k*; 0±u±uB; 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 »'=-7r> , vd x'. = -v x y'=y . y c'. ~> c and, p' , P PO u'=-7r, Q Q d a d c. P. PO
(A\ (4)
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-