••• 

		 								

(200) 

WRi 

no. 2400? -ti- c . 

Estimating Hydraulic Properties 

of the Floridan Aquifer System by 

Analysis of Earth-Tide, Ocean-Tide, 

and Barometric Effects, Collier and 

Hendry Counties, Florida 

U.S. GEOLOGICAL SURVEY 

Water-Resources Investigations Report 03-4267 

1111111 I 111111 I I I I 1 1 I 11111 

EET ABOVE SEA 

LEVEL 

NOVEMBER 1999 

reared in cooperation with the 

South Florida Water Man ent Di , 

DEPOSITORY 

science for a ch 

Estimating Hydraulic Properties of the 

Floridan Aquifer System by Analysis of 

Earth-Tide, Ocean-Tide, and Barometric 

Effects, Collier and Hendry Counties, 

Florida 

By Michael L. Merritt 

U.S. GEOLOGICAL SURVEY 

Water-Resources Investigations Report 03-4267 

Prepared in cooperation with the 

South Florida Water Management District 

U.S. GEOLOGICAL SURVEY 

RESTON. VA 

APR 2 1 2004 

SR 

L I B Ft A H Y 

USGS 

science for a changing world 

Tallahassee, Florida 

2004 
U.S. DEPARTMENT OF THE INTERIOR 

GALE A. NORTON, Secretary 

U.S. GEOLOGICAL SURVEY 

Charles G. Groat, Director 

Use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Geological Survey. 

For additional information Copies of this report can be purchased 

write to: from: 

U.S. Geological Survey U.S. Geological Survey 

2010 Levy Avenue Branch of Information Services 

Tallahassee, FL 32301 Box 25286 

Denver, CO 80225-0286 

888-ASK-USGS 

Additional information about water resources in Florida is available on the World Wide Web at 

http://fl.water.usgs.gov 
CONTENTS 

Abstract 

Introduction 

Purpose and Scope 

Acknowledgments 

Description of the Study Area 3 

Hydrogeology of the Study Area 4 

Naturally Occurring Stresses on Aquifers 7 

Solar and Lunar Stresses 7 

Atmospheric Stresses 10 

Estimates of loading efficiency and relation to tidal and barometric efficiencies 12 

Determination of barometric efficiency by Clark's method 14 

Determination of loading efficiency by Clark's method 22 

Harmonic Structure of Tidal and Atmospheric Stresses 23 

Analysis for amplitude and phase of harmonic components 29 

Analytical Methods for Computing Aquifer Parameters from Naturally Forced Periodic Data 35 

Using Heads Influenced by Ocean Tides for Estimates of the Hydraulic Diffusivity of an Aquifer 35 

Inland extent of ocean-tide influence 40 

Choice of method for estimating hydraulic diffusivity 42 

Estimates of hydraulic diffusivity 44 

Analysis with numerical model 50 

Using Heads Influenced by Earth Tides for Estimates of Specific Storage. Porosity, and Bulk Modulus 

of the Aquifer 54 

Recent research into methods of estimating specific storage, porosity, and bulk modulus 56 

Using Heads in Open Wells Penetrating Confined Aquifers for Estimates of the Transmissivity and Storage 

Coefficient of an Aquifer 58 

Using Heads or Water Levels to Estimate Vertical Hydraulic Conductivity 63 

Conclusions 63 

Summary 66 

References Cited 68 

FIGURES 

. Map showing location of study area and selected sites, southwestern Florida 3 

2. Schematic showing generalized geology and hydrogeology of southwestern Florida 5 

3-27. Plots showing: 

3. The ocean tide at the National Oceanic Atmospheric Administration site, Naples. Florida. 

November 1999 8 

4. The ocean tide at Naples. Florida, and the measured head in well C-1111, November 20-26, 1999 9 

5. The measured head and total head in well HE-1087, and the atmospheric pressure at station FPWX, 

November 20-26, 1999 9

6. The measured head and total head in well C-1112, and the atmospheric pressure at station FPWX, 

November and December 1999 11 

7. The relation between matrix compressibility and loading efficiency for three values of matrix 

porosity. based on Jacob's formula for loading efficiency 14 

8. Computation of barometric efficiency based on measured-head data from MZ-2, well C-1111 15 

9. Computation of barometric efficiency based on measured-head data from MZ-3, well C-1112 16 

Contents III 
	 	

10. Computation of barometric efficiency based on measured-head data from MZ-4, well C-1112 16 

11. Uncorrected measured-head data from MZ-2, well C-1112, data corrected for the effect of 

long-term, aperiodic atmospheric-pressure fluctuations, and the trend line based on the data, 

November and December 1999 17 

12. Computation of barometric efficiency based on measured-head data from MZ-1, well C-1111 19 

13. Computation of barometric efficiency based on measured-head data from MZ-2, well C-1111 19 

14. Computation of barometric efficiency on measured-head data from MZ-3, well C-1111 20 

15. Head increment sums in data from the monitor zone in well HE-1087 as a function of 

barometric-pressure increment sums 20 

16. Application of trial-and-error method for correcting measured-head data from the monitor zone 

in well HE-1087 for the influence of long-term, aperiodic atmospheric-pressure variations 21 

17. Periodogram of harmonic frequencies present in ocean-tide data collected by the National Oceanic 

Atmospheric Administration at Naples, Florida 24 

18. Detail of Naples ocean-tide periodogram showing minor harmonics having frequencies nearly 

equal to one cycle per day 25 

19. Detail of Naples ocean-tide periodogram showing minor harmonics having frequencies nearly 

equal to two cycles per day 25 

20. Periodogram of harmonic frequencies present in atmospheric-pressure data collected at 

South Florida Water Management District station FPWX 26 

21. Periodogram of harmonic frequencies present in head data from MZ-3. well C-1111 27 

22. Periodogram of harmonic frequencies present in head data from MZ-2, well C-11 P 27 

23. Periodogram of harmonic frequencies present in head data from the monitor zone in well HE-1087 28 

24. The measured and total heads in MZ-2. well C-1112. and atmospheric pressure at station FPWX for 

July and August 1999 29 

25. Detrended tidally influenced and barometrically influenced total-head data from MZ-2, 

well C-1112, for November and December 1999 31 

26. Detrended total-head data from MZ-2, well C-1112, and head oscillations estimated from 

regression analysis for November 21-27, 1999 34 

27. Individual harmonic components determined by regression analysis of total-head data from 

MZ-2, well C-1112 for November 21-27, 1999 34 

28. Schematic showing sea-aquifer models for which solutions for head (h(x)) as a function of distance 

inland (x) have been derived 36 

29. Plot showing distance from the coastline at which 90 percent and 99 percent reductions of the amplitude 

of the 01 component of the ocean tide would be observed in wells in a confined aquifer, based on the 

equation of Van der Kamp (1972) 41 

30. Plot showing distance from the coastline at which 90 percent and 99 percent reductions of the amplitude 

of the component of the ocean tide would be observed in wells in a confined aquifer, based on the 

equation of Van der Kamp (1972) 42 

31. Map showing peninsular Florida and contour lines showing the extent and slope of the Florida Platform 43 

32-37. Plots showing: 

32. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage 

coefficient of 1 x 10-5 and zero leakance, by the method of Van der Kamp (1972) 45 

33. Estimates of transmissivity based on total-head data from MZ-2, well C-1111, assuming a storage 

coefficient of 1 x 10-5 and zero leakance, by the method of Van der Kamp (1972) 46 

34. Estimates of transmissivity based on total-head data from MZ-3, well C-111 I, assuming a storage 

coefficient of 1 x 10-5 and zero leakance, by the method of Van der Kamp (1972) 46 

35. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage 

coefficient of 1 x 1(Y5 and a leakance of 1 x 10-6 by the method of Li and Jiao (2001a) 48 

36. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage 

coefficient of 1 x 10-5 and a leakance of 2 x 10-5 d-i , by the method of Li and Jiao (2001a) 48 

37. Estimates of transmissivity based on total-head data from MZ-2, well C-1111, assuming a storage 

coefficient of 1 x 1(15 and a leakance of 2 x 10-5 d-I , by the method of Li and Jiao (2001a) 49 

38. Schematic showing model grid design used for testing concepts related to the inland influence of ocean 

tides on pressure heads in wells 51 

IV Contents 
39. Plot showing simulated ocean-tide induced head oscillations in MZ-3 at the approximate inland distance 

of well C-1111 (5 miles), assuming various values of vertical hydraulic conductivity for overlying and 

underlying confining layers 53 

40. Plot showing simulated ocean-tide induced head oscillations in MZ-3 at the approximate inland distance 

of well C-1111 (5 miles), assuming various values of transmissivity for the aquifer 53 

41. Schematic showing idealized representation of an open well in which water-level oscillations are 

caused by oscillation of artesian pressure 59 

42. Plot showing the ratio of the amplitudes of the well water-level and aquifer pressure-head oscillations 

as a function of dimensionless transmissivity, for various values of redefined stoarage coefficient 61 

43. Plot showing the phase lag between the well water-level and aquifer pressure-head oscillations 

as a function of dimensionless transmissivity, for various values of redefined storage coefficient 61 

TABLES 

I. Locations and identification numbers of monitor wells providing data for this study 6 

2. Description of monitor zones providing tidally influenced head data for the study 7 

3. Barometric-efficiency and loading-efficiency determinations 18 

4. Harmonic components of tides 23 

5. Amplitudes and phases of the five major harmonic components in Naples ocean-tide data and in 

total-head data from monitor zones in wells C-1111. C-1112. and HE-1087 32 

6. Summary of transmissivities computed for the three monitor zones in well C-1111 using the phase 

lag calculation methods of Van der Kamp (1972) and Li and Jiao (2001a), and the values of loading 

efficiency at which the amplitude ratio calculation would yield the same estimate s(1 

7. Some parameters for diurnal and semidiurnal equilibrium tides 55 

8. Summary of calculations for specific storage (Ss) and porosity based on head data from monitor zones in 

wells C-1112 and HE-I087 56 

9. Summary of methods evaluated, parameters determined by each method, and applicability in 

southern Florida 64 

Contents V 
	 	

	 	

CONVERSION FACTORS AND VERTICAL DATUM 

Multiply By To obtain 

Length 

inch (in.) 2.54 centimeter 

foot (ft) 0.3048 meter 

mile (mi) 1.609 kilometer . 

Area 

acre 0.4047 hectare 

square mile (mil) 2.590 square kilometer 

Mass 

pound (1b)1 0.4536 Ikilogram 

Flow Rate - cubic foot per second (tt-1 /s) 0.02832 cubic meter per second 

million gallons per day (Mgal/d) 0.04381 cubic meter per second 

inch per year (in/yr) 25.4 millimeter per year 

Pressure 

1 millimeter of Hg (mercury) 0.044602 feet of water at 4 degrees Celsius 

1 millimeter of Hg (mercury) 1.33322 millibars 

1 millimeter of Hg (mercury) 0.019337 pounds per square inch 

Hydraulic Conductivity 

foot per day (ft/d) 0.3048 meter per day 

*Transmissivity 

foot squared per city (ft'-/d) 0.09290 meter squared per day 

Leakance 

foot per day per foot [(ft/d)/ft] 1.0 meter per day per meter 

*The standard unit for transmissivity is cubic foot per day per square foot times foot of aquifer thickness 1(f13/d)/112 111 In this 

report, the mathematically reduced form, foot squared per day (ft2/d), is used for convenience. 

Temperature in degrees Fahrenheit (° F) may be converted to degrees Celsius (° C) as follows: °C.(' F-32)/1.8. 

Vertical coordinate information is referenced to the National Geodetic Vertical Datum of 1929 (NGVD 29). 

Horizontal coordinate information (latitude-longitude) is referenced to the North American Datum of 1927 (NAD27). 

Acronyms: 

bls below land surface 

CFSW Clewiston Field Station weather site 

NOAA National Oceanic and Atmospheric Administration 

SFWMD South Florida Water Management District 

USGS U.S. Geological Survey 

VI Contents 
Estimating Hydraulic Properties of the Floridan 

Aquifer System by Analysis of Earth-Tide, 

Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

By Michael L. Merritt 

ABSTRACT 

Aquifers are subjected to mechanical 

stresses from natural, non-anthropogenic, processes such as pressure loading or mechanical 

forcing of the aquifer by ocean tides, earth tides, 

and pressure fluctuations in the atmosphere. The 

resulting head fluctuations are evident even in 

deep confined aquifers. The present study was 

conducted for the purpose of reviewing the 

research that has been done on the use of these 

phenomena for estimating the values of aquifer 

properties, and determining which of the analytical techniques might be useful for estimating 

hydraulic properties in the dissolved-carbonate 

hydrologic environment of southern Florida. 

Fifteen techniques are discussed in this report, of 

which four were applied. 

An analytical solution for head oscillations 

in a well near enough to the ocean to be influenced by ocean tides was applied to data from 

monitor zones in a well near Naples, Florida. The 

solution assumes a completely non-leaky confining unit of infinite extent. Resulting values of 

transmissivity are in general agreement with the 

results of aquifer performance tests performed by 

the South Florida Water Management District. 

There seems to be an inconsistency between 

results of the amplitude ratio analysis and independent estimates of loading efficiency. A more 

general analytical solution that takes leakage 

through the confining layer into account yielded 

estimates that were lower than those obtained 

using the non-leaky method, and closer to the 

South Florida Water Management District estimates. A numerical model with a cross-sectional 

grid design was applied to explore additional 

aspects of the problem. 

A relation between specific storage and the 

head oscillation observed in a well provided estimates of specific storage that were considered 

reasonable. Porosity estimates based on the specific storage estimates were consistent with values 

obtained from measurements on core samples. 

Methods are described for determining aquifer 

diffusivity by comparing the time-varying drawdown in an open well with periodic pressure-head 

oscillations in the aquifer, but the applicability of 

such methods might be limited in studies of the 

Floridan aquifer system. 

INTRODUCTION 

Since its inception, an important part of the 

science of ground-water hydrology has involved the 

estimation of values for aquifer properties, volumeaveraged theoretical variables that quantify in a general way the water-transmitting and water-storage 

properties of subsurface formations. Estimates are 

usually based on measured data describing the change 

in the pressure field in a body of subsurface water 

caused by the application of mechanical stresses that 

alter the rate of water movement and release of water 

from storage within the rock matrix. One such stress is 

the forcing of water into and out the formation by 

Introduction 1 
pumping from a well, or by injection of water into a 

well at a known rate. Doing this in conjunction with a 

planned program to collect pressure data (an aquifer 

test) is the most common method used for determining 

aquifer property values. Hydrologists have devised a 

variety of analytical methods to analyze the relation 

between the pumping or injection rate and changes in 

the pressure field. 

Aquifers are also subjected to mechanical 

stresses from natural, non-anthropogenic, processes. 

These include pressure loading or mechanical forcing 

of the aquifer by ocean tides, earth tides, and pressure 

fluctuations in the atmosphere. Where an aquifer is 

overlain by an ocean, the tidal oscillations in the ocean 

are associated with a pressure loading or unloading of 

the formations underlying the ocean. One effect is that 

heads (pressure heads or water levels) in wells near the 

ocean will rise and fall. The same motion of the sun 

and moon that causes ocean tides also causes dilatation of subsurface formations (expansion and contraction of the rock matrix). These effects are referred to 

as earth tides. As the rock matrix expands and contracts, the pressure in water contained within the rock 

pores falls and rises, which affects the heads measured 

in wells. Atmospheric pressure fluctuations also cause 

a pressure loading or unloading of subsurface formations that is manifest in fluctuations of heads in wells. 

The effects are evident even in deep confined aquifers 

that maintain a substantial head difference from surface formations across tight confining layers. 

Hydrologists early in the 20th century realized 

that ocean tides and atmospheric-pressure fluctuations 

caused corresponding fluctuations in heads in wells 

and began to pursue the goal of using these natural 

processes to deduce information about aquifer properties. Papers by Jacob (1940) and Ferris (1951) were 

among the first to describe the use of tidally influenced 

heads to estimate aquifer properties. The research has 

continued to the present day (2003), and recent papers 

represent new contributions to existing technology. 

The present study: (1) reviews the research on 

the use of tidally influenced and other naturally forced 

head fluctuations for estimating aquifer properties, 

(2) determines which of these methods are useful in 

the hydrologic environment of southern Florida, and 

(3) applies selected algorithms to a data set of heads 

from wells in Collier and Hendry Counties. The study 

was conducted by the U.S. Geological Survey (USGS) 

in cooperation with the South Florida Water Management District, who also supplied the data set. 

Using water-level fluctuations caused by natural 

processes for the cited purposes could represent a significant cost savings over the substantial costs associated with well drilling, instrumentation, and 

manpower required by more conventional aquifer testing. Even where aquifer testing is done, use of waterlevel fluctuations caused by natural processes can help 

to validate the results of the aquifer testing. If the data 

are collected during a period of monitoring prior to 

aquifer testing, they can be used to aid in the design of 

the tests. The methods described in this report appear 

to have potential usefulness for hydrologists in southern Florida and elsewhere. 

Purpose and Scope 

The purpose of this report is to document results 

of the review of research into the use of tidally influenced and other naturally forced head data to estimate 

values of aquifer properties. The report serves as a 

digest of the results of research previously documented in the technical literature, and presents results 

of the application of selected methods to head data 

obtained from wells located in Collier and Hendry 

Counties. 

The review of research groups the various methods into four main categories according to the specific 

objectives of the research. The application of selected 

methods is preceded in the report by a presentation and 

explanation of the field data used in the analysis, and 

includes a description of the data analysis and other 

calculations that are made to yield numerical estimates 

based on the field data. Where research methods are 

not applied, a brief description is included, with appropriate references, as a guide to readers who may wish 

to investigate the alternative or more advanced 

methods. 

Acknowledgments 

The author expresses appreciation to Emily 

Richardson, South Florida Water Management District, for supporting the study, and to Michael J. Bennett, South Florida Water Management District, for 

providing data. The author is grateful to Paul Hsieh, 

National Research Program, USGS, Menlo Park, California, for providing analytical software, review, and 

consultative assistance, and to Edwin P. Weeks, 

National Research Program, USGS, Lakewood, Colorado, for report review and helpful suggestions. 

Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

2 
		

		

DESCRIPTION OF THE STUDY AREA ally between 81 and 82 degrees of longitude and 

between 26 and 27 degrees of latitude and includes all 

The study area is best characterized as that part or most of Lee, Hendry, and Collier Counties. The 

of southwestern Florida that includes the location of topography is flat, and wetlands are present throughout 

test wells providing data for the study and the loca- much of Hendry and Collier Counties. The Big 

tions of additional sites where atmospheric and ocean Cypress National Preserve extends into the southern 

tide data were acquired (fig. 1). This region lies gener- part of the study area (fig. 1). Two Native American 

82°00' 81'00' 

27°00' 

HIGHLAN S -

COUNT 

I WTI 

CO 

5ke 

CHARLOTTE 1

5tkeechobee 

COU Y 

---r JI • 

CFSW-4 

1 • 

HENDRY CO JNTY • PA M BEACH 

HE-1087 UNTY 

12 

-— -— • 

BROWARD 26°00' COLLIER COUNTY COUNTY 

BIG CYPRESS 

NATIONAL 

PRESERVE — 

DADE 

COUNTY 

COUNTIY 

Base modified from U.S. Geological Survey digital data, 1:2,000,000, 1990 

Universal Transverse Mercator projection, zone 17 

0 20 40 MILES 

1 i 

I 

I 

I 

1 I 

I 

0 20 40 KILOMETERS 

EXPLANATION 

• FLORIDAN AQUIFER 

MONITORING WELL 

A NOAA ATMOSPHERIC AND 

TIDAL DATA STATION 

SFWMD ATMOSPHERIC 

DATA STATION 

NOAA (National Oceanic Atmospheric Administration) 

SFWMD (South Florida Water Management District 

Figure 1. Location of study area and selected sites, southwestern Florida. 

Description of the Study Area 3 
Reservations are located in southern Hendry and western Broward Counties. Most of Hendry County and 

eastern Lee County is rural, but the central and western part of Lee County is heavily urbanized (cities of 

Fort Myers and Cape Coral). 

Like most of the State, the region is subtropical 

with hot, wet summers and cool, dry winters. Only 

rarely does the air temperature go below freezing. 

Winters in the region are marked by the frequent passage of air masses of high and low pressure (warm and 

cold fronts). Winter air pressure extremes of 997 to 

1,034 millibars (mb) were measured by the National 

Oceanic and Atmospheric Administration (NOAA) at 

Naples (fig. 1) during the 1998-2000 calendar years. 

Summers have less atmospheric-pressure variation 

(non-cyclonic summer extremes of 1,006 to 1,021 mb 

were measured at Naples in 1998-2000), but the occasional passage of tropical disturbances in late summer 

and fall can bring about markedly low atmospheric 

pressure. A pressure of 991.7 mb was measured at 

Naples during the passage of Tropical Storm Mitch on 

November 5, 1998, and a pressure of 998.5 mb was 

measured during the passage of Hurricane Irene on 

October 15, 1999. 

Hydrogeology of the Study Area 

Southern Florida is underlain by rocks of 

Cenozoic age to a depth of about 5,000 feet (ft) 

(Meyer, 1989, p. 5). The rocks are principally carbonates (limestone and dolostone), with minor amounts of 

evaporites (gypsum and anhydrite) in the lower part 

and elastics (sand and clay) in the upper part. The 

movement of ground water from the inland areas to the 

ocean and the reverse (the inland movement of seawater in the saline-water zone postulated by Meyer) 

occurs primarily through the carbonate rocks (Meyer, 

1989, p. 5). The brief description of lithology and 

stratigraphy that follows is a summary of the more 

extensive discussion included in Reese (2000). 

The Floridan aquifer system in southwestern 

Florida includes (from oldest to youngest) the upper 

part of the Cedar Keys formation of Paleocene age, the 

Oldsmar Formation of early Eocene age, the Avon 

Park Formation of middle Eocene age, the Ocala 

Limestone of late Eocene age, and the Suwannee 

Limestone of early Oligocene age (fig.2). Overlying 

the Suwannee Limestone is the Hawthorn Group of 

Miocene and late Oligocene age, which (as defined 

by Scott (1988)) is divided into the Peace River 

Formation in the upper part and the Arcadia Formation 

in the lower part. The Hawthorn Group is largely composed of elastics and acts as an effective confining bed 

for aquifers in underlying formations. Reese (2000) 

prepared 10 hydrogeologic sections through southwestern Florida that show the stratigraphy in detail at 

many control points, including the test/monitor wells 

that provided tidally influenced head data for this 

study. The predominant rock types found in each formation are described (fig. 2) by Reese (2000). 

The Floridan aquifer system is one of three principal aquifer systems in southwestern Florida. The 

other two are the surficial and intermediate aquifer 

systems. In southwestern Florida, the surficial aquifer 

system includes the water-table aquifer and the lower 

Tamiami aquifer (fig. 2). The intermediate aquifer system lies within the Hawthorn Group and includes, in 

descending order, the sandstone aquifer and the midHawthorn aquifer. The sandstone aquifer is present in 

Lee County and in western Hendry and Collier Counties (Reese, 2000). The mid-Hawthorn aquifer is 

absent in most of Hendry County. 

The Upper Floridan aquifer includes the lower 

part of the Hawthorn Group, the Suwannee Limestone, 

the Ocala Limestone, and the upper part of the Avon 

Park Formation. Reese (2000) refers to production 

zones, where present, in the lower part of the Hawthorn Group as the lower Hawthorn producing zone. 

Generally, the Upper Floridan aquifer in southwestern 

Florida consists of several thin water-bearing zones of 

relatively high permeability interlayered with thick 

zones of much lower permeability. The Upper Floridan 

water-bearing zones usually are composed of limestone or limestone with interbedded dolomite, and are 

characterized by secondary porosity, and are not differentiated into named aquifers by Reese (2000). 

The Lower Floridan aquifer includes the dolomite-evaporite unit described by Reese (2000) that 

may be within the lower part of the Avon Park Formation or the upper part of the Oldsmar Formation, and 

extends down into the Cedar Keys Formation. Reese 

(2000) regards the Avon Park and Oldsmar Formations 

as being quite difficult to differentiate. The Lower 

Floridan aquifer also is characterized by thin waterbearing zones of high permeability interlayered with 

thick zones of low permeability. The zones of high 

permeability commonly are composed of sucrostic 

dolomite that collapses when drilled into. Some of the 

collapsed dolomitic zones used for waste injection 

have remarkably high permeability. 

Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

4 
Approximate Approximate Geo l ogic Series thickness Lithology Hydrogeologic unit thickness Unit (feet) (feet) 

Quartz sand, silt, clay, and Undifferentiated 0 - 70 shell Water-table 20 - 100 Intermediate aquifer Surficial aquifer FLORIDAN AQ

UIFERSYSTEM system syste

m

aquifer 

Confining beds 0 - 60 

Lower Tamiami 

aquifer 25 - 160 

Confining unit 20 - 100 

Sandstone aquifer 0 - 100 

Confining unit 10 - 250 

Mid-Hawthorn 0 - 130 aquifer 

Confining unit 100 - 400 

Holocene 

to Silt, sandy clay, micritic Tamiami limestone, sandy, shelly 

limestone, calcereous sandPliocene 

Formation 0 - 175 

stone, and quartz sand 

Interbedded sand, silt, Peace gravel, clay, carbonate, River 50 - 400 

Hawthorn Group

and phosphatic sand Formation 

Miocene 

Sandy limestone, shell beds, Arcadia dolomite, phosphatic sand 400 - 550 Formation and carbonate, sand, silt, 

and clay Lower Hawthorn 0 - 300 producing zone 

Early Suwannee 0 - 600 Fossiliferous, calcarenitic Oligocene Limestone Upper Floridan 700 - 1,200 

aquifer 

500 - 800 

Middle confining 

unit 

1,400 - 1,800 

limestone 

w Ocala Chalky to fossiliferous as 0 - 400 ..., J Limestone calcarenitic limestone 

a) Avon Park Eocene 900 - 1,200 Formation Fine-grained, micritic to

ears,Aprils aqlp uowiposaa

...- fossiliferous limestone, ? dolomitic limestone, dense

>, Oldsmar dolomite, and gypsum 800 - 1,400 Lower 

Floridan Boulder 075 Li.i Formation 400 zone aquifer 

Paleocene Cedar Keys 

Formation 

500 - 700 

1,200? 

Dolomite and dolomitic 

limestone 

Massive anhydrite beds Sub-Floridan 

confining unit 1,200? 

Figure 2. Generalized geology and hydrogeology of southwestern Florida (from Reese, 2000). CJ1 
The conceptualization of the Floridan aquifer 

system as upper and lower aquifers separated by a middle confining unit is a somewhat inadequate generalization. Thin, water-bearing zones can be found 

anywhere within the carbonate sequence extending 

downward from the lower part of the Hawthorn Group 

to the Cedar Keys Formation. The concept of a middle 

confining unit attempts to describe an interval between 

the one to several limestone water-bearing zones of 

secondary porosity in the upper part of the sequence 

and the collapsed dolomitic water-bearing zones in the 

lower part of the sequence, an interval in which the 

occurrence of water-bearing zones is less in a vertically 

spatially averaged sense than in the upper and lower 

parts of the sequence. It is difficult to select upper and 

lower elevations for such a confining unit at individual 

well sites, and also difficult to generalize confiningunit boundaries on a regionally extensive scale. 

Water in the upper Floridan producing zones is 

under artesian pressure and flows from wells open at 

land surface. The equivalent freshwater head decreases 

from a high near the center line of the peninsula to sea 

level at the coasts (Meyer, 1989). The head measured 

in a well is an inverse function of the density of the 

water column in the well. Assuming that the watercolumn density is representative of that of the screened 

zone, or open hole, of the well, the water density will 

increase with depth of the open-hole zone because the 

dissolved solids concentration of the formation water 

increases with depth. In the Floridan aquifer system in 

the study area, brackish water (less than 10,000 milligrams per liter (mg/L) of total dissolved solids) overlies and is separated from water of seawater salinity 

(more than 35,000 mg/L of dissolved solids) by a salinity transition zone that is 100 ft or less in thickness at 

many sites. The salinity transition zone is deeper than 

2,000 ft in the central part of the Florida peninsula and 

as shallow as 1,000 ft on the mainland near the southwestern coast of Florida. Hydraulic gradients in variable-density ground water are discussed by Hickey 

(1989). The relation of freshwater and environmental 

heads is discussed by Lusczynski (1961). Concepts of 

variable-density flow are presented by Mason and Kipp 

(1998). 

Tidally influenced head data for this study were 

acquired from monitor zones in three wells extending 

into the dolomite-evaporite unit of the Lower Floridan 

aquifer. The locations and various identification numbers assigned to the wells are provided in table 1. 

These wells were drilled by the South Florida Water 

Management District (SFWMD) as part of a program 

to investigate the Floridan aquifer system in southwestern Florida as a possible future source of water for public supply. The depth intervals of the zones and 

characteristics of the formations from which the data 

are acquired are given in table 2. Depth intervals 

extend from 690 ft in the basal Hawthorn unit to 

2,350 ft in the dolomite-evaporite unit. The lower 

heads in the deepest monitor zones in wells C-1111 

and C-1112 probably do not imply a hydraulic potential across an intervening confining unit, but probably 

are consistent with the hydraulic influence of the 

saline-water density of the lower zones, as explained 

above. 

Table 1. Locations and identification numbers of monitor wells providing data 

for this study 

(Longitude of the ocean coast west of C-1111 is 81"48' 49"; USGS, U.S. Geological Survey] 

Florida Latitude Longitude Bureau of USGS local (degrees, (degrees, Location Geology identification minutes, minutes, identification number seconds) seconds) number 

1-75 Canal near Naples, 

Florida 

W-17405 C-1111 26° 10' 12" 81" 43' 51" 

Immokalee Waste Water 

Treatment Plant, 

Immokalee, Florida 

W-17391 C-1112 26° 24' 48" 81" 25' 24" 

L-2 Canal near 

Clewiston, Florida 

W-17093 HE-1087 26° 36' 30" 80° 56' 58" 

Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

6 
Table 2. Description of monitor zones providing tidally influenced head data for the study 

IC-int is 4.9 miles from the west coast of Florida (LSD = 9.87 feet); C-1112 is 28 miles from the west coast of Florida (LSI) = 31.45 feet); 

and HE-1087 is 57 miles from both east and west coasts of Florida (LSD = 17.84 feet). LSD is land surface datum in feet above NGVD 291 

Depth Monitor Head Water-quality interval Formation zone (feet) zone (feet) 

C-1111 (1-75 Canal site) 

MZ- I 690-760 34-35 Basal Hawthorn and upper Suwannee Limestone brackish-water zone 

MZ-2 905-1,050 34-35 Suwannee Limestone base of brackish-water zone 

MZ-3 2,300-2,350 7-10 Dolomite-evaporite unit saline-water zone 

C-1112 (Immokalee site) 

MZ-2 1,060-1,140 53-55 Top of Ocala Limestone brackish-water zone 

MZ-3 1,752-1,880 53-56 Mid to lower Avon Park Formation brackish-water zone 

MZ-4 2,134-2,354 30-37 Dolomite-evaporite unit saline-water zone 

HE-1087 (L-2 Canal site) 

MZ 1,400-1,810 57-59 Avon Park Formation brackish-water zone 

NATURALLY OCCURRING STRESSES 

ON AQUIFERS 

Many of the mechanical stresses applied to the 

materials (water and rock) that comprise the subsurface have naturally occurring causes rather than the 

anthropogenic ones familiar to hydrologists (aquifer 

tests, pumping for water supply, or injection). The subsurface is stressed by forces that change during the 

movement of the sun and moon relative to the earth. 

and by changes in loading arising from changes of the 

weight of the overlying atmosphere (atmospheric pressure). Many of these forces are periodic in nature 

( tidal), such as those exerted by lunar and solar 

motions and by periodic variations in atmospheric 

pressure, hut some are aperiodic, such as the change in 

loading accompanying episodic changes in atmospheric pressure occurring over random periods of 

days or weeks that are caused by the movement of air 

masses of higher and lower pressure across the surface 

of the earth. 

The effects of pressure loading from ocean tides 

and atmospheric-pressure changes are evident in deep 

confined aquifers that maintain a substantial head difference from surface formations across tight confining 

layers. Pressure loading effects also are evident from 

changes in the altitude of the water table related to 

recharge and draining or drying of surficial aquifers. 

The downward transmission of pressure usually is considered to be instantaneous (Spane. 2002), transmitted 

by grain-to-grain contact through the rock matrix 

(Rojstaczer, 1988), and is not associated with the vertical movement of water, except when related to the 

movement of the water table or in other special cases 

that are described near the end of this report. (When 

the water-table altitude changes, there may also he a 

concomitant change in the rate of downward leakage 

through confining layers.) The effects of pressure 

loading from atmospheric-pressure changes usually 

are nearly uniform over relatively large regions. so that 

resulting lateral flows are slight. 

Solar and Lunar Stresses 

The gravitational influences of the sun and 

moon cause sea level to rise slightly at some locations 

on the surface of the earth and to fall at other locations, 

depending on the geometric relation of each location 

with the astronomic bodies and the mass of the earth. 

These sea-level oscillations vary periodically with the 

changing positions of the sun and moon at a given 

location and are referred to as ocean tides. The range 

of sea level tidal oscillation varies across the earth as a 

function of many factors, such as the configuration of 

coastlines and local hathymetry. In the study area, the 

normal range of oscillation generally is from 2 to 4 ft 

(fig. 3). The sea level tidal oscillation causes an oscillation in the pressure loading on materials of the 

underlying subsurface, which causes oscillations of 

pressure or head in wells. The inland part of the 

Naturally Occurring Stresses on Aquifers 7 
3 —r 

C) 

> 2 

z 

w 

0 

co 

H 

w 

Li.1 0 

Uz V 'V 

w 

0 -1 

I t 

Cf) 

_J 

0 

H 

1 2 3 4 5 6 7 8 9 10' 11' 12' 13 ' 14 ' 15' 16' 17 18' 19 20 21 22 231 24 25 261 27 28 29 30 

NOVEMBER 1999 

Figure 3. The ocean tide at the National Oceanic Atmospheric Administration site, Naples, Florida, November 1999. 

formation, at a distance from the surface loading, 

responds hydraulically as if an oscillatory pumping/injection sequence were being performed in the 

subsurface beneath the ocean at the coastline. 

Such a case is illustrated in figure 4. The head in 

MZ-3 in C-1111, east of Naples (the 1-75 Canal site 

test well) and 4.9 miles from the ocean, is a subdued 

image of the ocean tide measured by NOAA at Naples, 

although a slight phase shift is apparent in the well 

data. The approximate range of the tidally induced 

head oscillations is 0.5 ft. The monitor zone is a 

saline-water zone at 2,300 ft below land surface (bls) 

(table 2). Higher brackish-water monitor zones at 690 

and 905 ft his have average heads of 34-35 ft above 

NGVD 29 (land surface is about 10 ft above NGVD 

29), demonstrating that all of the zones are well-confined from the surface where the ocean tides occur. 

When pressure loading from the surface occurs, 

part of the stress is absorbed by the rock matrix, so that 

only part of the stress is transmitted to the water contained within the formation. Van der Kamp and Gale 

(1983) define the ratio of the change in pore pressure 

(Opt) to an areally uniform change in pressure loading 

at land surface (Apt) as the "loading efficiency" (y). 

Earlier writers (Jacob, 1940; Ferris,1951), considering 

the case of ocean-tide loading at the surface, referred 

to the change in pressure head within the formation 

(Ah1) to the tidal stage change at the surface (Alit) as 

"tidal efficiency" (Te). As will be shown later, tidal 

efficiency and loading efficiency are not the same. 

Inland wells respond hydraulically to pressure 

oscillations at the coastline that have been reduced in 

amplitude by the factor Te from the ocean tide oscillations at the surface of the ocean at the coast. If the subsea outcrop of the formation occurs near the coastline, 

the pressure oscillations in the aquifer at the coastline 

will just be the ocean-tide oscillations. 

Solar and lunar motions also cause stretching 

and contracting of the subsurface rock matrix, decreasing and increasing the pressure of the water contained 

within the rock pores. This is evidenced in oscillations 

in the pressure or head of wells, even in mid-continental locations far from the influence of ocean tides. 

Such phenomena are termed earth tides. Heads 

recorded in the monitor zone in well HE-1087, located 

virtually equidistant (57 miles) from the east and west 

coasts of Florida, show tidal oscillations having a 

range of about 0.1 ft (fig. 5), that represent the 

combined influences of earth tides and periodic atmospheric-pressure fluctuations. 

Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

8 
			

						

11 

TIDAL STAGE, INFEET ABOVE NGVD 29 

Ocean tide at 

rn Naples 

10 H 

DJ 

z 

w m

> 1 m O 

CO 

z I— m 

m

H 

co W 0 z 1 

cf) z 

MZ-3, C-1111, m < -

a measured head

1-= co 

20 21 22 23 24 25 26 

NOVEMBER 1999 

Figure 4. The ocean tide at Naples, Florida, and the measured head in well C-1111, November 20-26, 1999. 

59.1 34.1 

Atmospheric pressure, 

FPWX site 59.0 34.0 

z 

58.9 33.9 71 0 cn rT

m m 

H Measured water level, 

HE-1087, L-2 Canal site M

58.8 33.8 co 

o

< 

m D:1 

58.7 m 33.7 > 

H 

me 

DJ jj 

m 

58.6 Total pore pressure head, 33.6 

HE-1087, L-2 Canal site 

(less 34 feet) 

58.5 33.5 

20 21 22 23 24 25 26 

NOVEMBER 1999 

Figure 5. The measured head and total head in well HE-1087, and the atmospheric pressure at station FPWX, 

November 20-26, 1999. 

Naturally Occurring Stresses on Aquifers 9 
Gregg (1966) theorized that earth tides are 

exactly out of phase with the ocean tides. This concept 

also is stated by Meyer (1974. p. 16). The reasoning 

was that, when the moon passes over a location on the 

earth's surface where a confining layer overlies an 

aquifer beneath the ocean, the bulge in the earth's crust 

would cause a lateral stretching of the earth's material 

and an increase in pore space, reducing hydraulic pressure. The same lunar passage causes a rise in the ocean 

stage, increasing the pressure loading from the surface 

on the aquifer. Because of the complexity and interrelation of lunar and solar movements and the differences in their effects, this hypothesis probably needs 

further investigation. No other publications known to 

the author subsequent to Meyer (1974) have included a 

study of this subject. 

Direct measurements of earth tides are not possible as in the case of ocean tides. However, the theoretical tidal potential, W,(04,t), where 0 is the latitude. 

(I) is the longitude, and t is time, can be used to derive 

the range of the tidal oscillation at points on the earth's 

surface. W,(0,0,t) will be defined in a later section. 

Atmospheric Stresses 

The other major class of natural stresses acting 

on subsurface formations are the pressure loading and 

unloading caused by changes in the weight of the 

atmosphere. These changes are linked to periodic 

(diurnal and semidiurnal) changes in atmospheric 

pressure and to aperiodic, longer-term movements of 

masses of air of higher and lower pressure across the 

surface of the earth. The periodic oscillations of head 

(fig. 5) shown for well HE-1087 in November 1999 

are caused by both periodic atmospheric-pressure 

oscillations and by earth tides. 

Also shown in figure 5 is the atmospheric pressure measured at station FPWX (fig. 1). The inverse 

correlation between the periodic variations of the 

atmospheric pressure and head measured in the well is 

evident for this 7-day period, typical of many parts of 

the period of record. Slight differences in the phases of 

the corresponding highs and lows are related to the 

earth tide influence. Where the periodic atmospheric 

pressure and earth tide oscillations are out of phase, 

the relation of periodic atmospheric-pressure oscillations to the measured heads is more complex. 

A description of the exact causes of the periodic 

atmospheric-pressure variations is beyond the scope of 

this report, but the variations probably are related to 

the heating and cooling, expansion and contraction, 

and rising and falling of the air mass near land surface. 

Ritzi and others (1991) discounted the possibility that 

solar and lunar effects on the atmosphere explain diurnal variation. Chapman and Lindzen (1970) explain 

the phenomenon of barometric tide in terms of the 

cyclic heating of atmospheric water vapor and ozone. 

Most investigators note that daily lows tend to occur in 

the early morning and early afternoon and that daily 

highs occur in the late morning and late afternoon. 

Clark (1967) described atmospheric pressure as "being 

high at about 10 am and pm and low at about 4 am and 

pm.-An investigation of data for January 1998 and 

July 1999 made as part of this study indicates that the 

timing of peaks and lows is quite variable within a 

2-3 hour period from day to day, and is often hard to 

determine precisely because of fluctuations about the 

peak or low within a 2-hour period containing the peak 

or low. 

The movement of air masses of varying pressure 

causes fluctuations in heads in wells that are aperiodic 

and of the same duration (days or weeks) as the presence of an air mass of a given characteristic pressure in 

the area. There is an inverse correlation between the 

average atmospheric pressure typical of these time 

periods and heads measured in wells for the same periods (fig. 6). In the study area, fluctuations in atmospheric pressure caused by movement of air masses. 

and the consequent inverse fluctuations of heads measured in wells, are more frequent in winter, when 

episodic "cold fronts" (higher pressure) alternate with 

"warm fronts" (lower pressure). As previously noted, 

pronounced low pressure events may occur in late 

summer or early autumn in conjunction with tropical 

storms. 

Barometric pressure measurements in the study 

area from four stations operated by SFWMD and from 

one station operated by NOAA (fig. 1) were evaluated. 

Data from the various stations were plotted against 

each other for selected time periods and were found 

to be in close agreement with, at most, only a slight 

(1 millimeter of Hg (mercury)) bias. 

As previously noted, when pressure loading 

from the surface occurs, part of the stress is absorbed 

by the rock matrix so that only part of the stress is 

transmitted to the water contained within the formation. On the other hand, the surface of the water column in an open well is subjected to the full extent of 

the atmospheric-pressure change, so that the water 

level in the well is determined by both the pore 

10 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
			 		

56 T i I 1 i 11111-T 36 

Measured water level, 

MZ-2, C -1112 2 

55 Total pore pressure head, ' 35 m 0

MZ-2, C-1112 (less 55 feet) mm 

H 

• M 

CO 3:3 

O 0 

< -0 

M 

M 

54 34 H (f) 

M c 

m 

Atmospheric pressure. 

station FPWX 

53 1111111 111_111111 33

5 10 15 20 25 5 10 15 20 25 

NOVEMBER 1999 DECEMBER 1999 

Figure 6. The measured head and total head in well C-1 112. and the atmospheric pressure at station FPWX. 

November and December 1999. WELL WATER, IN FEET ABOVE NGVD 29 

pressure (absolute pressure) change in the aquifer and 

the atmospheric-pressure change at the water surface. 

In this report, the ratio of the change in the water level 

in an open well (AIL,) to the atmospheric-pressure 

change at the surface, converted to feet of water (Aha ), 

is termed barometric efficiency (Br ). Open wells at 

inland locations, therefore, show measured head fluctuations that have been reduced in amplitude by the 

factor Be from the atmospheric-pressure fluctuations at 

land surface. 

The artesian monitor zones of this study were 

shut in, and pressure was measured by means of pressure transducers located in the wellhead. If the pressure transducers had also been shut in and not vented 

to the atmosphere, the measured pressures would have 

been pore (absolute) pressures. Many pressure sensors 

are designed to measure pore pressure. However, pressure-head data supplied by the SFWMD for this study 

were measured using In-Situ Troll 4000 pressure 

recorders containing pressure transducers that were 

vented to the atmosphere (M. Bennett, South Florida 

Water Management District, written commun., 

December 2000) in a way that prevented the discharge 

of water under artesian pressure. Because the transducers were vented, the pressure-head measurements are 

relative to atmospheric-pressure ("gage pressure"), and 

have the same interpretation as water levels that would 

be obtained from wells open to the atmosphere. 

The instrument readout showed heads in feet of 

freshwater computed from the measured pressures in 

pounds per square inch (psi) using a conversion factor 

of 2.31 ft of water/psi that assumes the water was of 

freshwater density. Use of a single standard value 

(2.31) for the head conversion regardless of the salinities of water in the various monitor zones was not considered a source of error, because calculations (to be 

described later) used small tidal head variations of the 

order of 0.1-1.0 ft. 

The inverse correlation between measured head 

and atmospheric pressure (figs. 5-6) is a result of the 

head data being referenced to atmospheric pressure, 

and is also characteristic of open-well water levels. 

Pore pressure heads (heads computed from absolute 

pressure in the aquifer) should increase as atmospheric-pressure loading increases. The increase in the 

aquifer, however, is damped by an amount represented 

by the loading efficiency. Considering the case of an 

open well, because the pressure at the water surface in 

the well changes by an amount equal to the atmospheric-pressure increase, the net pressure change at 

the well water surface is an increase in downward 

pressure, so that the water surface is lowered (lower 

Naturally Occurring Stresses on Aquifers 11 
		

head in the well) as the atmospheric and pore 

pressures increase. In the case of a vented transducer, 

the damped pore pressure is opposed by the atmospheric-pressure change, resulting in the same inverse 

correlation. 

That this scenario is the case for head data from 

the SFWMD wells (fig. 1) is demonstrated (figs. 5-6) 

by plotting the pore pressure head, computed by adding atmospheric pressure, converted to feet of water, to 

the head measured in the well. The positive correlation 

between aperiodic changes in atmospheric head and 

pore pressure head is evident (fig. 6). The periodic 

changes (fig. 5) have a more complex interpretation, 

being negatively correlated in the first 2 days shown in 

the figure and positively correlated in the last 3 days. 

This is likely the result of the influence of earth tides. 

Because heads measured in the wells are referenced to atmospheric pressure, barometric efficiencies 

computed based on these data have a different significance from loading efficiencies based on true formation pressures. These concepts are clarified in the 

following section. 

Estimates of loading efficiency and relation to tidal 

and barometric efficiencies 

An "amplitude factor" is presented by Jacob 

(1950) and Ferris (1951) to describe the change of formation pressure caused by an areally distributed 

change of pressure at land surface. "Amplitude factor" 

was a new term that replaced the term "tidal efficiency" used by Jacob (1940) in presenting the same 

factor. The amplitude factor of Jacob and Ferris was 

defined as: 

(1) + 

where 

a is the vertical compressibility of the skeletal 

aquifer (psi-I ) (inverse pounds per square 

inch), 

13 is the compressibility of water (psi-1 ), and 

r is porosity (unitless). 

A more rigorous expression for the formation 

pressure change caused by surface loading that takes 

the possible compressibility of the solid fraction of the 

aquifer into account was presented by Van der Kamp 

and Gale (1983) and referred to as "loading efficiency" 

(Y): 

B(1 +v) (2) Y = 3(1 —v)-2AB(1 —2v)' 

where 

A = 1— -- , and (3) K 

B (I 1 .. IF,1 )+ 11( " MI (4) iLvc K \K KslJ 

where 

K is the bulk modulus of elasticity of the formation (the inverse of compressibility), 

11 is the interconnected porosity of the formation, 

and 

Kf and K,are the bulk moduli of elasticity of water and . of "the solid fraction including the effect of 

unconnected pores" (Van der Kamp and 

Gale, 1983). 

The Poisson ratio (v), a constant for most materials, is 

the ratio of the amount of transverse contractual strain 

(compression) that occurs when a material is subjected 

to a longitudinal extensional strain (stretching) to the 

amount of the longitudinal strain. 

Van der Kamp and Gale (1983) state that these 

equations "constitute a rigorous expression for the 

undrained response of pore pressure to a change 

of surface load if horizontal displacements are 

negligible." These authors showed that if the solids 

are incompressible (1/K, = 0) and the confined 

modulus of elasticity is written as: 

K, = 3K(1 —v) (5) 1 +v 

then equation 2 reduces to Jacob's amplitude factor, 

(ofJ(a+Tif3). Because values for v for most rocks do not 

deviate appreciably from v = 0.25 (P. A. Hsieh, 

U.S. Geological Survey, written commun., 2003), then 

1.8 K. Van der Kamp and Gale (1983) note that B 

(equation 4) reduces to a form that also appears analogous to Jacob's amplitude factor when 1/K, = 0. They 

note, however, that "the distinction between B and y 

resides in the fact that B represents the ratio of change 

of pore pressure to change of octahedral normal stress 

while y represents the ratio of pore pressure change to 

change of vertical stress when horizontal deformations 

12 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
are negligible." Van der Kamp and Gale (1983) also 

state that "in field applications, the loading efficiency 

y is a more useful parameter, since generally only the 

load (or change of vertical stress) is known." Only values of the confined modulus K' should be used in 

equation 1 (E. P. Weeks, U.S. Geological Survey, written commun., 2003), since Jacob (1940) assumed a to 

represent vertical compressibility. Values of bulk modulus (K) should be converted to confined modulus by 

use of equation 5 before use in equation 1. 

In the terminology and notation adopted for use 

in this report, Van der Kamp's "loading efficiency" (y ) 

is understood to be, as defined, the true loading efficiency that takes into account compressibility of the 

solid fraction of the aquifer. Jacob's "amplitude factor" 

is understood to be an approximation for y that 

assumes the solids to be incompressible (1/K, = 0). 

Both Jacob's factor and a data-based estimate of the 

pore pressure response to surface loading are termed 

"loading efficiency" and denoted Le. If the solid fraction is, in fact, incompressible, Le= y. 

The term "tidal efficiency" (Te) has a more specialized application, representing the pore pressure 

change at the coast caused by ocean tides. As will be 

shown, tidal efficiency has a numerical value lower 

than loading efficiency. Some authors (Gregg, 1966; 

Erskine, 1991) use the terms "tidal efficiency" or "tidal 

efficiency factor" in a spatially varying sense, including the exponential decrease with distance from tidewater in the mathematical expression and in estimates 

based on well data. 

The loading efficiency relation of Jacob and Ferris (equation 1) is used to estimate a range of probable 

values of Le based on values cited for rock compressibility by various authors. The standard value for the 

compressibility of water is 3.34 x 10-6 psi-1 (inverse 

pound-force per square inch) at normal atmospheric 

pressure and 20 degrees Celcius. In the carbonate 

Upper Floridan aquifer, porosities of the order of 

35 percent commonly are measured on cores or estimated from geophysical logging (solid dolomite sections commonly have a lower porosity, possibly no 

more than 15 percent). Domenico and Schwartz (1997) 

suggest 1.6 x 10-8 to 3.3 x 10-7 psi- ' as a typical range of 

values for the compressibility of fissured or jointed 

rock. Using this range of values, and assuming a 

porosity of 35 percent, the loading efficiency estimate 

would range from 1.4 to 22 percent. 

Robinson and Bell (1971) cite Birch (1966) in 

stating a range of I x 10" to 3 x 10" dynes per square 

centimeter (dynes/cm2) for the bulk modulus of aquifer 

matrix material "representative of common sedimentary rocks" and said to be based on analysis of rock 

samples (Rhoads and Robinson, 1979). Assuming a 

Poisson ratio (v) value of 0.25, and using equation 5 to 

convert to confined modulus, this is equivalent to a 

range of vertical compressibilities of 0.13 x 10-6 to 

0.38 x 10-6 psi'. The corresponding range of the loading efficiency estimate (4) would be 10 to 25 percent. 

However, results of a field study by Rhoads and Robinson (1979) led to computed estimates of bulk modulus (4.05 x 10 to 13.87 x 1010 dynes/cm-, equivalent 

to a vertical compressibility range of 0.28 x 10' to 

0.94 x 10-6 psi- '). The corresponding range of Le would 

be 19 to 45 percent. Rhoads and Robinson (1979) suggest that their lower estimates of bulk modulus could 

be the result of larger fractures and solution cavities in 

the aquifer and not present in rock samples. 

The relation between aquifer compressibility (a) 

and loading efficiency, based on the value for water 

compressibility (p) cited above, is shown for an aquifer compressibility range of 1 x 10-7 to I x 10' psi-1 

and for aquifer porosities of 15, 30, and 45 percent 

(fig. 7). The graph shows that, in highly compressible 

formations, pressure changes are nearly equal to the 

imposed pressure loading, but that relatively incompressible formations absorb most of the pressure 

loading. 

Jacob (1940) also derived a relation for the 

open-well barometric efficiency: 

B = 813 = 1 — (6)

u + 013 

and showed, as a consequence, that open-well barometric efficiency (Be) and loading efficiency (Le) 

summed to unity. Citing this work, Van der Kamp and 

Gale (1983) state that "in analyzing the barometric 

response of the water level in a well open to the atmosphere, the influence of barometric pressure acting 

directly on the water surface in the well should be born 

in mind." Rhoads and Robinson (1979) note that "an 

increase in atmospheric pressure . . . causes a change 

in pore pressure in the aquifer, which is equal to the 

change in atmospheric pressure plus the change in 

water level in the well." 

Naturally Occurring Stresses on Aquifers 13 
		

				 				

				

1.0 

LOADING EFFICIENCY, INPERCENT 

Water compressibility is 6 

_ ...I 

assumed to be 3.34 x 10 psi 

0.8 

0.6 

POROSITY = 15 PERCENT 

0.4 

POROSITY = 30 PERCENT 

POROSITY = 45 PERCENT 

0.2 

1._ 1 1_ 1 1 LL_ 0.0 

2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 -7 -5 -4 

10 10 -6 10 10 

ROCK COMPRESSIBILITY, IN SQUARE INCHES PER POUND 

Figure 7. The relation between matrix compressibility and loading efficiency for three values of matrix porosity, based on 

Jacob's formula for loading efficiency. 

Determination of barometric efficiency by Clark's 

method 

It is often useful to know the open-well barometric efficiency of a given set of barometrically 

affected head data; such knowledge is helpful in 

obtaining estimates of loading efficiency needed for 

application of methods used in this study. Clark (1967) 

devised such a method based on aperiodic, long-term 

pressure variations arising from the movement of air 

masses and corresponding measured head changes in 

the well. This method was investigated as part of this 

study, and was found to work well with some data sets 

and to provide inaccurate or meaningless results in 

others. 

In order to summarize the steps of Clark's 

method, time series of barometric-pressure and head 

measurements are assumed to have: (1) the same 

beginning and end times; (2) the same, equally spaced 

time increment for independent measurements (1 hour 

in the example below); and (3) no missing values. 

Barometric-pressure measurements are converted to 

feet of water. (A factor of 0.044602 converts millimeters of mercury to feet of water, and a factor of 

1.33322 converts millimeters of mercury to millibars.) 

Proceeding incrementally from one time step (t,) to the 

next, concurrent sums of barometric-pressure (h,) and 

head (k) changes are generated according to the following criteria: 

Ab; = bi — , 

Ahi 1 

13 ; = Ah. Ahi , 

- 1 , = Si, + A b, 

if 13 >0 , 

1 Ah, if < 0 , and 

14 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
		

in figure 6. The Clark method was applied to the entire = Sti I if Abi = 0 . (13) 

period of record of heads used in this study (April 

In Clark's description of his method, the sign 

convention was to consider AI?, to be positive when 

head is rising and to consider Oh; to be positive when 

barometric pressure is falling. Equations 7-13 implicitly include this convention. The procedure was 

devised to separate incremental changes in head 

caused by atmospheric-pressure fluctuations from 

changes caused by other influences. The time series of 

S,,' is plotted as a function of the corresponding time 

series Sbi, and a regression line is drawn through the 

data. (The linear fit function of the S-Plus software 

was used for this study.) The slope of the regression 

line is the calculated barometric efficiency Be. 

The Clark method was used with good results 

for head data measured in two of the three monitor 

zones in well C-1112, 28 miles from the coast of the 

Gulf of Mexico north of Naples, and with limited success in the third (figs. 8-10). Head data from MZ-2 

(1,060-1,140 ft bls) measured in November and 

December 1999 and corresponding atmosphericpressure measurements from station FPWX are shown 

1999-November 2000). The linear fit to the plot of the 

two computed sums (fig. 8) for head data from MZ-2 

indicates a barometric efficiency of 63 percent. 

After barometric efficiency is estimated, the 

value is partly verified by using it to remove the atmospheric-pressure effects from the well water-level data. 

Software developed for this study used barometricpressure changes and the estimated barometric efficiency (Be) to correct the head data for the barometric 

influence. Initial corrections to the data often created a 

slight bias between the averages of the original and 

corrected data sets. This bias was found to be inconvenient in making graphical comparisons of the original 

and corrected head data sets. Therefore, the software 

was executed a second time with the bias removed by 

adding a constant correction value (COR). The correction procedure is as follows: 

814,i = —(B Abi) , (14) 

SU

MOFWATER-LEVELCHANGES, IN FEET 

125 

Analysis for barometric efficiency by Clark's method 

(Clark, 1967). Slope of linear fit is the estimated 

barometric efficiency (Be ). 

Linear-fit equation is -5.26 + 0.6296'x 

100 Be = 63 percent 

75 

50 

25 

0 

0 50 100 150 200 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

Figure 8. Computation of barometric efficiency based on measured-head data from MZ-2. well C-1112. 

Naturally Occurring Stresses on Aquifers 15 
	 	

150 

I— Analysis for barometric efficiency by Clark's method LLI (Clark, 1967). Slope of linear fit is the estimated 

LL- 125 barometric efficiency (Be). 

Linear-fit equation is -5.429 + 0.7076*x (.6 

Be = 71 percent 

100 

w 75 

_J 

1— 50 

IL 

25 

CI) 

0 

0 50 100 150 200 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

Figure 9. Computation of barometric efficiency based on measured-head data from MZ-3, well C-1112. 

35 

Analysis for barometric efficiency by Clark's method 

(Clark, 1967). Slope of linear fit is the estimated 30 barometric efficiency (Be). 

Linear-fit equation is 0.5614 + 0.4813*x 

25 Be = 48 percent 

20 

15 

10 

5 

0 

0 25 50 75 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

Figure 10. Computation of barometric efficiency based on measured-head data from MZ-4, well C-1112. 

16 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida SUM OF WATER-LEVEL CHANGES, IN FEET 
		 			

j=i-I (diurnal and semidiurnal) head oscillations also are = 

Dwi = 6wi 8w, , and (15) 

j=1 = 

= 

= h + + C 0 R , (16) 

j=1 

where 

6w1 is the estimated correction to the measured 

head value h1 , and 

hi` is the corrected head value. 

The corrected data (shown for November and December 1999 in fig. 11) are without the effects of longterm, aperiodic atmospheric-pressure fluctuations during that time period, although the head data show a 

decreasing trend caused by other environmental influences during that time period. The sudden drop in head 

readings (possibly erroneous data) of November 8-9 

are not affected by the procedure. The periodic 

56 

55 

54 

Trend line 

(shifted down 2 feet)

WATER LEVEL, IN FEET ABOVE NGVD 29 

52 11111111111111 

5 10 15 20 25 

NOVEMBER 1999 

altered, but some periodic oscillations remain because 

of the influence of earth tides. Because of the latter 

influence, it is difficult to determine the extent to 

which the periodic head oscillations caused by 

periodic atmospheric-pressure oscillations have been 

corrected. 

The method was applied to data from MZ-3 

(1,700-1,880 ft bls), with the same period of record 

as MZ-2, and a barometric efficiency estimate of 

71 percent was obtained (table 3). The corrected plot 

was smoother than the original, but the correction process did not appear as effective for MZ-3 as for MZ-2. 

A barometric efficiency of 48 percent was obtained 

from analysis of data from MZ-4 (2,200-2,300 ft bls), 

which were from a more limited period of record 

(April-September 1999). Atmospheric pressureinduced head fluctuations in this zone appeared to he 

subdued. The correction process appeared to smooth 

the raw data from the early part of the period of the 

data, but caused distortions in the raw data from the 

later part of the record period, when the raw data 

appeared to show almost no response to long-term, 

aperiodic atmospheric-pressure fluctuations. 

Uncorrected 

measured data 

Data corrected for 

atmospheric pressure variation 

(shifted down 1 foot) 

iiiiiiiiiiiiiii 

5 10 15 20 25 

DECEMBER 1999 

Figure 11. Uncorrected measured-head data from MZ-2, well C-1112, data corrected for the effect of long-term, 

aperiodic atmospheric-pressure fluctuations, and the trend line based on the data, November and December 1999. 

Naturally Occurring Stresses on Aquifers 17 

53 
Table 3. Barometric-efficiency and loading-efficiency determinations 

1--, not estimated] 

Monitor Well zone 

Barometric 

efficiency 

calculated by 

the Clark' 

method 

(in percent) 

Barometric 

efficiency 

calculated by 

the trial-anderror method 

(in percent) 

Loading 

efficiency 

calculated by 

the modified 

Clark method 

(in percent) 

Loading 

efficiency 

calculated by 

the trial-anderror method 

(in percent) 

Rock compressibility 

computed from 

loading efficiency 

(inverse pounds per 

square inch x 10 6) 

MZ-1 C-1111 (1-75 Canal site) 83 

MZ-2 C-1111 (1-75 Canal site) 74 to 86 

MZ-3 C-1 I 11 (1-75 Canal site) 31 85 

MZ-2 C-1112 (Immokalee site) 63 

MZ-3 C-1112 (Immokalee site) 71 

MZ-4 C-1 112 (Immokalee site) 48 

MZ HE-1087 (L-2 Canal site) 75 

'Clark, 1967. 

Clark's method was applied to head data (period 

of record November 1997-December 2000) from the 

three monitor zones in well C-1111 near Naples 

(figs. 12-14). Atmospheric data from the Naples 

NOAA station, beginning January 1, 1998, were used 

for the analysis. The linear fit computation of barometric efficiency worked well for MZ-1 (690-780 ft bls) 

and provided an estimate of 83 percent. Most of the 

variation in the head data caused by aperiodic, longterm atmospheric-pressure variations was removed in 

the correction process. The linear-fit plot was somewhat variable for MZ-2 (905-1,050 ft bls). An overall 

linear-fit value of 74 percent was computed for barometric efficiency, but two sections of the plot having a 

slope of 86 percent were separated by a section of 

nearly zero slope. Both values produced good results 

when used in the correction process, and results from 

use of the two values could not be easily distinguished 

from one another. 

Data between December 1999 and June 2000 

were absent from the record of heads from MZ-3 

(2,300-2,350 ft bls) and only data from the period 

between January 1998 and December 1999 were used 

in the analysis. The slope of the plot of the head-increment sum as a function of the barometric-pressure 

increment sum (fig. 14) was quite variable, and ranged 

from less than zero to about 90 percent in various 

sections. The overall computed slope (estimate of 

17 0.24 

26 .41 

69 IS .21 

37 .69 

29 .48 

52 80 

95 25 to 45 

barometric efficiency) was only 28 percent. Correcting 

the raw data using a barometric efficiency estimate of 

28 percent was ineffective. A value of 85 percent was 

found to be appreciably more effective in the correction process. 

Unsatisfactory results occurred when the Clark 

method was applied to head data from the monitor 

zone in HE-1087 (1,400-1,810 ft bls). The data were 

measured between September 1998 and December 

1999. The atmospheric-pressure data used for the analysis were measured at the Clewiston Field Station 

weather site (CFSW) operated by the SFWMD. The 

plot of the sum of head increments as a function of the 

barometric-pressure increment sum (fig. 15, no time 

lag) has a slope of about 43 percent until mid-April 

1999, at which time the slope becomes negative. A 

detailed examination of the data reveals that it was 

somewhat noisy, which could serve to make the relation between incremental changes in atmospheric pressure and heads less well defined than if the data were 

free of noise. There was also a strong downward, and 

then upward, trend in the atmospheric-pressure data 

between March and June 1999, but this did not appear 

correlated with the trend of the sums in the Clark 

method analysis. 

It was found that the slope could be reversed if 

several hours were added to or subtracted from the 

time coordinate of the head data. The software was 

18 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
	 			

					

300 

Analysis for barometric efficiency by Clark's method 

(Clark, 1967). Slope of linear fit is the estimated 

u_ 250 barometric efficiency (Be). 

U.1 

Linear-fit equation is -0.05783 + 0.8306*x 

LLI 

Be = 83 percent 

Z 200 

0 

_1 

U-I 150 

U.1 

L1.1 

I— 100 

0 

2 50 

Cn 

0 50 100 150 200 250 300 350 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

Figure 12. Computation of barometric efficiency based on measured-head data from MZ-1, well C-1111. 

250 

Analysis for barometric efficiency by Clark's method 

(Clark, 1967). Slope of linear fit is the estimated 

barometric efficiency (Be). 

200 

Linear-fit equation is -3.845 + 0.7427*x 

Be = 74 percent 

150 

100 

50 

0 

0 50 100 150 200 250 300 350 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

SUM OF WATERL

EVELCHANGES, IN FEET 

Figure 13. Computation of barometric efficiency based on measured-head data from MZ-2, well C-1111. 

Naturally Occurring Stresses on Aquifers 19 
	 	 	 	

125 

Analysis for barometric efficiency by Clark's method 11.1 

(Clark, 1967). Slope of linear fit is the estimated 

U_ barometric efficiency (Be). 

Z 100 

Linear-fit equation is 14.86 + 0.309*x C') 

Be = 31 percent 

I• 75 

11.1 

f± 50 

LL 

O 25 

(1) 

0 

50 100 150 200 250 300 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET OF WATER 

Figure 14. Computation of barometric efficiency based on measured-head data from MZ-3, well C-1111. 

40 

I4-HR TIME LAG 

• 30 

NO TIME LAG 

O 20 16-HR TIME LA 

I 

10 

6-HR TIME LAG 

I 

-10 

LL 

2-20 

10-HR TIME LAG 

-30 -L -L 

0 50 100 150 

SUM OF BAROMETRIC-PRESSURE CHANGES, IN FEET 

Figure 15. Head increment sums in data from the monitor zone in well HE-1087 as a function of barometric-pressure 

increment sums. 

20 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
modified to provide for a bias of the time of the head statistical alternative for obtaining estimates. Barodata, and various lengths of time were used in trial metric efficiency values of 43, 75, and 95 percent were 

computations. The plot of the sum of head increments assumed in attempts to correct the head data (fig. 16). 

as a function of the barometric pressure sum became a From viewing the figure, the first of these values 

straight line for a 4-hour time lag (fig. 15), but the appears to be too low, and the third appears to be too 

slope of the graph was only about 27 percent. For a high, resulting in an overcorrection for the atmotime lag of 6 hours, the positive and negative slopes of spheric pressure effect. The value of 75 percent 

the original graph became reversed. The plot was appears to be nearly correct, and the trend of the corstraight with a negative slope for a 10-hour time lag, rected head data appears to be largely free of the influbut became straight with a positive slope again for a ence of long-term, aperiodic atmospheric-pressure 

time lag of 16 hours, where the slope of the plot was variations.The trial-and-error method also was applied 

24.5 percent. Correcting the head data assuming a successfully to data from MZ-3, well C-1111, but not 

barometric efficiency of 24-27 percent, however, was to data from MZ-4. 

not effective in removing the atmospheric-pressure It is concluded that the Clark method can be an 

influence. effective tool for determining barometric efficiency 

Because of the failure of the Clark method to when the head data are of high quality. But the method 

provide a realistic estimate of barometric efficiency for does not appear to be entirely robust, and can provide 

the monitor zone in well HE-1087, an alternative trial- values that are too low in some data sets that might be 

and-error method was implemented. Trial-and-error affected by a high degree of noise, strong trends, or 

was mentioned by Clark (1967) as one method of timing inaccuracy. The method provided values that 

determining barometric efficiency, and his computa- were too low in data sets that did not have obvious 

tional method provided a more efficient and precise problems of these kinds. It seems prudent to apply a 

59 35 

Raw water level 

Barometric 

pressure 

JJ 

0 

5 34 m 

I 

0 

I 

Corrected water level, m 

cf) Be = 95 percent, shifted down 1 foot CD ok,..."-WANAAAAI, ki," "" mINAVArew., 57 "1".'''\\'''SVOYAYAMPAWAst"AvVVIOWv ,o 1 h.._ 33 

Corrected water level. W4vi\i,OA 

m 

Be = 75 percent, shifted rit-.,y) 1 5 fop! "A,A,,.. I 

m N m

H 

m 

m Corrected water level, 56 32 

Be = 43 percent, shifted down 2 feet 

m 

I 

EXPLANATION 

Be is barometric efficiency 

55 15 

1 

14 

1 15 31 

JANUARY FEBRUARY MARCH 

1999 

Figure 16. Application of trial-and-error method for correcting measured-head data from the monitor zone in 

well HE-1087 for the influence of long-term, aperiodic atmospheric-pressure variations. 

Naturally Occurring Stresses on Aquifers 21 
		

process of actually correcting head data sets to determine whether barometric efficiency values determined 

by the Clark method are correct. If they are not correct, 

or if the method yields ambiguous results, the trialand-error method should be applied, in which a process of arbitrarily selecting a value and using it for 

correction is repeated until results are satisfactory. 

Determination of loading efficiency by Clark's 

method 

One of the physical properties most often used 

in analytic methods for basing aquifer parameter estimates on measured tidally influenced data is the loading efficiency (Le). From equation 6, it is straightforward to estimate loading efficiency once an estimate of the barometric efficiency (Be) is obtained. It 

was desirable, however, to verify such estimates by 

applying a modified version of Clark's method to the 

time series of total pore pressure-head values, obtained 

by summing values of water level from the vented 

pressure transducers and the measured barometric 

pressure, converted to feet of water, at each time for 

which both were available. Because the total pore 

pressure and barometric pressure were positively correlated, the sign of the conditional relation was 

changed in equations 11-12, that is, the absolute value 

of Ahi was added to the sum if 13, > 0 and subtracted 

from the sum if 0, < 0. This is equivalent to reversing 

the sign of ft, or Abi (or reversing Clark's sign convention). If hi is the measured pressure head used in equation 8 and hit is the corresponding total pressure head, 

loading efficiency will be a regression line drawn 

through the plot of: 

/ = 1 = 

1AI y( A17 — IA/7; 

j = 0 j = (1 0 ., = 1 . (17) 

i = i i = i 1 = 1 

101); 1(—Ahi ) lAhi 

j = 0 j -= 0 i = 0 

This result suggests that the modified Clark method 

should produce the result predicted by equation 6. The 

values obtained by applying the modified Clark 

method to the time series of measured data (table 3) 

confirm equation 6. The plots of Si, as a function of Sb 

for the monitor zone in HE-1087 showed no sections 

of negative slope, even when the entire record was 

used. 

Estimates were checked by using them to correct fluctuations in the time series of total pressure 

head induced by long-term aperiodic atmosphericpressure fluctuations. Such fluctuations were not 

clearly defined in the data from the three monitor 

zones in well C-1111 in Naples. It was difficult to 

evaluate the effects of the small corrections (17 and 

26 percent) applied to data in MZ-1 and MZ-2. The 

69-percent correction applied to MZ-3 data clearly 

was too large; a correction of 15 percent seemed more 

appropriate, but again, results were difficult to 

evaluate. 

The estimated 37- and 29-percent corrections 

were successfully applied to total head data from 

MZ-2 and MZ-3 in well C-1112 at Immokalee, but a 

52-percent correction appeared inadequate for data 

from MZ-4. Because the latter part of the 6-month 

record of measured heads showed no response to 

aperiodic atmospheric-pressure fluctuations, adding 

the atmospheric-pressure values to generate the total 

head data caused strong aperiodic atmosphericpressure fluctuations in the latter. An 80-percent correction applied to the data successfully removed the 

fluctuations caused by atmospheric pressure, but 

appeared to amplify periodic oscillations. The raw 

data and correction results from MZ-4 appeared 

anomalous and are not considered highly reliable. A 

95-percent correction applied to the data from the 

monitor zone in HE-1087 (in the center of the peninsula) clearly was too great. Corrections ranging from 

25 to 45 percent were more effective for various parts 

of the data record. 

The estimated loading efficiencies that appeared 

realistic or that were used successfully to correct for 

long-term, aperiodic atmospheric-pressure effects 

were used for estimates of rock compressibility (a) for 

comparison with literature values cited previously. 

These values are shown in table 3. All computed compressibility values are based on loading efficiencies 

computed by the modified Clark method, except for 

MZ-3, well C-1111, where the trial-and-error value 

(0.15) was used. The values computed generally are 

within the ranges cited by Domenico and Schwartz 

(1997), Birch (1966), and Rhoads and Robinson 

(1979). 

22 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
Harmonic Structure of Tidal and 

Atmospheric Stresses 

Ocean and earth tides, being of solar and lunar 

origin, are periodic, as are the motions of those astronomical bodies relative to the earth. Because these 

motions are complex and have many components of 

different frequencies and level of influence, the tides 

also are a combination of periodic components of different frequencies and amplitudes. These components 

are expressed as distinct "harmonics," sinusoidal functions of given amplitude and frequency and phase relation to some common time reference. Many of the 

harmonics that are part of daily tidal oscillations are 

listed in table 4 together with the symbols usually used 

to refer to them and their associated frequencies. These 

frequencies are known with great precision based on 

decades of astronomical observations (Hanson and 

Owen, 1982, table 1). The frequencies are common to 

all ocean and earth tide data, but the amplitudes and 

phase relations for each component are a characteristic 

of each individual set of tide data. Further information 

about tides can be obtained from Melchior (1983) and 

Godin (1972). 

Bredehoeft (1967) cites Melchior (1964) in stating that, of all the harmonic components, only five 

(M„, S2, N2, K1 , and 01 ) are of importance geophysically, together accounting for approximately 95 percent of the tidal potential. Several other minor 

components have frequencies so close to those of one 

of the other harmonics that they often cannot be 

resolved from the other harmonic at a high probability 

level by harmonic analysis (S,-K,, K,-P1 , N2-2N2, 01 -

Q, Q1 -2Q1 , and K,-J1 ). Such resolution becomes possible at a lower probability level only when a long time 

series of relatively error-free data is available. PI and 

K2 are major components, but usually are combined 

with K1 and 5,, respectively, in analyses. Hanson and 

Owen (1982, fig. 1) illustrate the relative position and 

amplitude of minor tidal components Q1 , 2Q1 , 2N2, J1 , 

and K2. Nowroozi and others (1969) show Q1 , 2Q1 , 

001 , 2N,, and 112, but all are stated to be resolved at 

less than a 95-percent confidence level. 

Analysis of time-series data to determine the 

presence of harmonics (spectral analysis) is usually 

accomplished using the discrete Fourier transform 

technique (Shumway, 1988). A range of frequencies is 

subdivided into increments, and for each frequency 

Table 4. Harmonic components of tides 

Symbol 

Angular 

frequency 

(radians/hour) 

Frequency 

(cycles/day) 

Period 

(hours) Explanation 

0, .24335189 .92953574 25.819341 Main lunar diurnal 

K, .26251618 1.00273794 23.934469 Lunar-solar diurnal 

M, .50586804 1.93227356 12.420602 Main lunar semidiurnal 

S2 .52359878 2.00000000 12.000000 Main solar semidiurnal 

.49636693 1.89598199 12.658348 Lunar elliptic (lunar semi-diurnal 

N, component caused by monthly 

variation in moon's distance) 

P, .26108260 .99726206 24.065891 Main solar diurnal I 

K2 .52503234 2.00547579 11.967235 *IRA 

v, 

.23385075 

.49763845 

.89324406 

1.90083884 

26.868357 

12.626005 

egimaill11111 

M, .25301504 .96644625 24.833249 

J, .27201730 1.03902954 23.098477 

P2 .48813732 1.86454720 12.871758 

L2 .51536918 1.96856524 12.191620 

T2 .52288202 1.99726218 12.016449 

2N, .48686579 1.85969031 12.905375 

00, .28168047 1.07594013 22.306074 

2Q, .22434962 0.85695241 28.006223 

Naturally Occurring Stresses on Aquifers 23 
	 	 	

(increased incrementally), the sine and cosine 

functions are integrated with the data over the period 

of record used (Fourier transform). Where the data 

have a periodic component corresponding to the frequency, a Fourier transform value results that is substantially different from zero. A graph of the results of 

these computations with frequency as the abcissa is 

called a periodogram and shows the frequencies of the 

harmonics present in the data. This method of analysis 

was applied to data collected for this study to verify 

and demonstrate the presence of the usual harmonics. 

This technique was applied to hourly tidal stage 

data measured by NOAA at Naples, Florida, for nearly 

4 years (January 1997-November 2000). The software 

used was the PFFI' program included in Microsoft 

Fortran Power Station, Professional Edition. An application program was written to input the data set and to 

use PFFT and various other supporting utility programs for the analysis and storage of the results. A 

graph showing the resulting periodogram (fig. 17) 

clearly indicates the large Fourier transform values 

obtained at frequencies nearly equal to those of the 

major harmonic components of ocean and earth tides 

1,000 

EXPLANATION 

Spectral analysis for harmonic components 

in time series of data points. Labels refer to 

800 individual harmonic components and are 

defined in table 4 

600 

400 

K1 

200 

01 

0 

0.0 0.5 1 0 

FOURIER SERIESVALUE 

(the 0,, K,-P,, M„, and S,-K, components). For convenience, the K r-P, and S,-K2 components are labeled as 

the larger of the two components in each set (K, and 

S.2 ). The N, component is small, but is clearly shown at 

a slightly lower frequency than the M2 component. 

In this analysis, the availability of a lengthy data 

set of reasonably good quality has provided a good 

signal-to-noise ratio in the results of the analysis that 

enhances their graphical interpretation. At a larger 

scale (figs. 18-19), the Q, component is clearly indicated, and the P, and K, components can be separated 

from the K, and S2 components, respectively. The M,, 

J,, 2N2,1.12, and v2 components are also visible, with 

Fourier series values ranging from 0.3 to 3.0. A slight 

spike in the graph at the correct frequency might be the 

L, component. Only the 2Q, and 00, components 

cannot be found, and the T, component cannot be 

distinguished from the S, component. 

The Fourier transform technique also is applied 

to atmospheric-pressure data from station FPWX collected at quarter-hour intervals between January 1998 

and November 2000 (fig. 20). The periodogram indicates that the harmonic components are strictly diurnal 

M2 

S2 

N2 1 

1.5 20 

FREQUENCY, IN CYCLES PER DAY 

Figure 17. Periodogram of harmonic frequencies present in ocean-tide data collected by the National Oceanic 

Atmospheric Administration at Naples, Florida. 

24 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

25 
	 	 		

		

5 Il l T II I ► 11 1 1 1 1 11 T 1 1 1 1 I T 

FOURIER SERIES VALUE 

K1 EXPLANATION 01 P1 Spectral analysis for harmonic (30) 

components in time series of 

4 — data points. Labels refer to — 

individual harmonic components 

and are defined in table 4 

3 

2 _ 

1 — 

Q1 

(10) 

M1 

J1 _ 

0 AA .,) AA.A.AtiviimAiI 

0.85 0.90 

T i MAIrAtii41... TIN 4A A.kit,-

0.95 1.00 

FREQUENCY, IN CYCLES PER DAY 

_. L • A -,...i. -A.A.-MA-A.. 

1.05 

1 .- A. 

1.10 

Figure 18. Detail of Naples ocean-tide periodogram showing minor harmonics having frequencies nearly equal to 

one cycle per day. 

10 

EXPLANATION 

9 Spectral analysis for harmonic _ 

M2 components in time series of 

8 data points. Labels refer to _ 

FOURIER SERIES VALUE 7 

individual harmonic components 

and are defined in table 4 

6 N2 K2 — 

5 

4 

3 _... 

2 

1 —2N

0 

112 

,,,, ,L„ .„L i 

L2(?) ,L _ 

1.85 1.90 1.95 2.00 

FREQUENCY, IN CYCLES PER DAY 

Figure 19. Detail of Naples ocean-tide periodogram showing minor harmonics having frequencies nearly equal to 

two cycles per day. 

Naturally Occurring Stresses on Aquifers 25 

2.05 
or semidiurnal at frequencies of one and two cycles 

per day. There seems to be a slight signal at a frequency of three cycles per day. It has been customary 

for some authors (Hsieh and others, 1987) to label the 

two major components S, and S2. The S, component 

has nearly the same frequency as Ki of the Naples 

ocean-tide analysis, and the S2 component of the 

atmospheric-pressure analysis has the same frequency 

(2.00) as the S2 component from the ocean-tide 

analysis. Large Fourier series values between frequencies of 0 and 0.5 are caused by aperiodic atmosphericpressure fluctuations lasting for days or weeks that are 

treated by the Fourier transform analysis as harmonics 

having a long period (low frequency). The fact that the 

periodogram does not show harmonic components 

having lunar tidal frequencies confirms that the diurnal 

and semidiurnal atmospheric-pressure oscillations are 

not caused by the gravitational effects of lunar 

movements. Given this fact, it also is unlikely that the 

lesser gravitational effects of solar movements affect 

the data. The tidal oscillations in the atmosphericpressure data are more likely the result of solar heating 

and other factors in the earth's environment. 

Application of the Fourier series technique to 

head data from wells C-1111, C-1 112, and HE-1087 

(figs. 21-23) reveals a full suite of the major solar and 

lunar harmonic components that are found in the 

Naples ocean-tide data. The Fourier series values for 

the harmonics in the head data from monitor zone 

MZ-3 in well C-1111 (table 2), only 4.9 miles from the 

ocean near Naples, are high compared to those from 

monitor zones in the other two wells because of the 

strong ocean-tide influence. This influence is small or 

negligible at the inland wells C-1112 and HE-1087, as 

will be discussed later. 

FOURIER SERIES VALUE 

900 

S2 

800 EXPLANATION 

Spectral analysis for harmonic 

components in time series of 

700 data points. Labels refer to 

individual harmonic components 

and are defined in table 4 

600 

500 

400 

300 

200 

100 

1_ 

2 3 4 5 

FREQUENCY, IN CYCLES PER DAY 

Figure 20. Periodogram of harmonic frequencies present in atmospheric-pressure data collected at South Florida 

Water Management District station FPWX. 

26 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

0 
	 						 	 				 					

									 		

EXPLANATION 

Spectral analysis for harmonic components M2 

in time series of data points. Labels refer to 

individual harmonic components and are 

defined in table 4 

FOURIERSERIESVALUE 

3 

K1 

2 

01 

S2 

N2 

0 L Li1 

0.0 0.5 1.0 1.5 2.0 2.5 

FREQUENCY, IN CYCLES PER DAY 

Figure 21. Periodogram of harmonic frequencies present in head data from MZ-3, well C-1111. 

„ 

-. 

0.4 1 1 1 I T I 1 1 1 1 1 1 T 1 1 1 T 1 1 T r 

EXPLANATION M2 S2

Spectral analysis for harmonic -

_ components in time series of 

data points. Labels refer to 0.3 

FOURIERSERIES VALU

E

individual harmonic components 

and are defined in table 4 

0.2 

0.1 

O. 

Ki 

0 0 

0.0 III ill II1l I I I i 

0.5 1.0 1.5 

FREQUENCY, IN CYCLES PER DAY 

1116111 ILIIII III 11111 , 1 II. I . . i , , . 

2.0 

2 

1 11 1. , . 

2.5 

_ 

Figure 22. Periodogram of harmonic frequencies present in head data from MZ-2, well C-1112. 

Naturally Occurring Stresses on Aquifers 27 
	 	

F 

EXPLANATION 

Spectral analysis for harmonic 

components in time series of 

w data points. Labels refer to D 0.2 individual harmonic components _1 

and are defined in table 4 

tz 

CE 

D 0.1 

0 

01 

0.0 

M2 

S2 

0 0 0.5 1.0 1.5 .02 2 5 

FREQUENCY, IN CYCLES PER DAY 

Figure 23. Periodogram of harmonic frequencies present in head data from the monitor zone in well HE-1087. 

At inland locations, the amplitude of tidally 

influenced head oscillations in wells is largely influenced by the local hydraulic characteristics of the formations open to the wells, so variations in Fourier 

series values between C-1112 and HE-1087 do not 

have a global interpretation. The presence of the lunar 

harmonics, 0, and M„ in head data from the inland 

wells shows the influence of earth tides, as the influence of ocean tides should be negligible and the periodic oscillations in atmospheric pressure do not have 

lunar harmonics (fig. 20). The lunar harmonics at this 

inland site can be regarded as almost entirely caused 

by earth tides. On the other hand, the solar harmonics, 

K, and S2 , contain the combined influence of earth 

tides and diurnal and semidiurnal atmospheric-pressure oscillations. This fact is important for the application of analytic procedures for determining aquifer 

properties. 

The solar and lunar harmonics in head data can 

alternately augment and interfere with one another. 

For instance, the lunar and solar periods for the principal diurnal components are about 25.82 and 23.93 

hours, respectively. If the two are in phase, complementing each other, at time to, they will be exactly out 

of phase when the daily difference in period, about 

1.885 hour, sums to 12.91 hours, half the longer 

period. This will occur in about 6.85 of the longer 

periods, or at time to + 7.39 days. The principal lunar 

and solar semidiurnals periods are 12.42 and 12.00 

hours, respectively. Again, if in phase at time too, they 

will be exactly out of phase when the difference in 

period (0.42 hour) sums to half the longer period 

(6.21 hours). This will occur in 14.79 of the 12.42-

hour periods, or at time t,, + 7.65 days. Thus, both the 

principal solar and lunar components of the tide will 

go in and out of phase in 7-8 days, although the two 

interactions (solar and lunar) will not always be in 

phase with one another. This might be observable as 

increases and decreases of tidal amplitude in graphical 

representations of the tide. 

An example is obtained from the data used for 

this study (fig. 24). The measured and total heads in 

July and August 1999, are from MZ-2 in well C-1112. 

An analysis, described in the following section, which 

makes possible the reconstruction of an estimated tide 

in MZ-2, was used to determine when the solar and 

28 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
		 				

56.0 f T T 

c\i 

0 

Z 55.5 

Total heads Measured heads minus 33.5 feet 

w 

OD 

out of phase w 

z 55.0 7 in phase 

Out of phase in phase 

out of phase w out of phase in phase 

w in phase 

cc 

I— 54.5 

Atmospheric pressure 

plus 20.5 feet 

54.0 111111111 1111111.11.11

10 15 20 25 30 5 10 15 20 25 30 

JULY 1999 AUGUST 1999 

Figure 24. The measured and total heads in MZ-2, well C-1112. and atmospheric pressure at station FPWX for July 

and August 1999. 

lunar diurnal tidal components were in and out of 

phase, and when the solar and lunar semidiurnal components were in and out of phase. The arrows in figure 

24 show when the lunar (0,) and solar (K,/S,) diurnal 

components were in and out of phase. Generally, the 

tidal amplitude diminishes when these components are 

out of phase and increases when they are in phase. 

There did not seem to be an evident correlation of tidal 

amplitude with the relative phases of the lunar (M2) 

and solar (S2) semidiurnal components in this data set. 

The best estimates of the barometric and loading 

efficiencies in MZ-2, well C-1112, computed by the 

Clark method, were 63 and 37 percent, respectively 

(table 3), and 2 months of total head and atmospheric 

pressure data, converted to feet of water, are shown in 

figure 24. Because the barometric efficiency is relatively high and the loading efficiency is relatively low, 

the effect of atmospheric-pressure fluctuations should 

be more apparent in the measured-head data than in 

the total-head data. That this hypothesis is actually the 

case is shown in figure 24. 

Analysis for amplitude and phase of harmonic 

components 

The analysis of specific tidally influenced data 

sets for estimates of aquifer properties depends largely 

on being able to resolve the amplitude and phase of the 

various harmonic components. In particular, the amplitude and phase of the five harmonic components (O,, 

K,, N,, M2, and S,) that comprise approximately 

95 percent of the tidal potential must be determined. 

A means for doing this is by regression analysis, as 

described by Nowroozi and others (1966). This 

method was encoded in Fortran by Hsieh and others 

(1987), and the program was used in the present study. 

The program can estimate the amplitude and phase of 

up to 16 components (table 4), but in this study was 

used generally to estimate amplitudes and frequencies 

of only the five principal components. The frequencies 

(ca) of the components are entered as input. As previ- l 

ously noted, these frequencies are known precisely 

from many years of astronomical observations, and it 

was not necessary to determine them by spectral 

analysis. 

Naturally Occurring Stresses on Aquifers 29 
The least squares method minimizes the sum of 

squares of a set of residuals (S): 

P 

s 1 

x. "0 

2 E(acoso). t.+b. sino). .1 

1.) 

.1 1 _I , (18) 

= 1 i =1 

where: 

n is the number of data points xi(ti) (measured or 

total well pressure heads), 

ti is the time of the data point, 

co is the frequency of the fh tidal component, 

P is the number of tidal components considered 

in the analysis, and 

a a. and b are the 2P+1 unknown coefficients. 0, ,j 

Equation 18 is differentiated with respect to the 

2P+1 unknown coefficients, and the 2P+1 resulting 

equations are each set equal to zero to minimize the 

sum of the squares (S). The resulting system of linear 

equations is solved by matrix inversion to obtain the 

values of the 2P+1 unknown coefficients. The amplitude (%) and phase ((IV of each harmonic component is 

determined by: 

2 2 1/2 = (a. +b. ) , and (19) J 

(1),/ = atan . (20) 

Theft' tidal component (hit)) is estimated by: 

hi(t) = cticos(cojt + (1)i) . (21) 

The tidal height (h(t)) is estimated by: 

P 

(1„ h(t) = an + = —+ Ia•COS0.0-J t . (22) 

/ =1 

This analysis was applied to the ocean tide at Naples 

and to the tidally influenced head data in each monitor 

zone in wells C-1111, C-1112, and HE-1087. The 

analysis requires specification of an arbitrary initial time 

as a reference for the phase computation. To facilitate 

the comparison of various tidal data sets as part of 

analyses that are described in the following pages, the 

initial time for all analyses for amplitude and phase was 

midnite, the beginning of January 1, 1998. As a modification to the original software, all time coordinates in 

tidal or tidally influenced data sets were referenced to 

this time. Following the original software configuration, 

time values were converted to hours. 

Before tidal or tidally influenced data could he 

analyzed by the regression analysis, it was necessary 

to detrend the data so that fluctuations in the data 

would be strictly tidal in origin. Besides long-term, 

aperiodic atmospheric-pressure effects, possible 

causes of trends in head data in wells include temporal 

variations in aquifer recharge, regional changes in 

potentiometric level, and effects of pumping. The 

detrending procedure generally was done in two steps: 

(1) estimate a trend line by use of a low-pass filter; and 

(2) compute the detrended data set by subtracting the 

trend line from the raw data. 

The low-pass filter (that removes highfrequency harmonics from the time series) was applied 

by use of a program prepared as part of this study. This 

program computes a revised time series as the midpoints of successive 25 data-point averages of the original hourly time series. This computation was 

performed twice, first on the raw (total head) data set, 

then again on the first revised data set. In the notation 

1 2 of Godin (1972), this filter is —A 5 . An example of a 

25' 

trend line computed by this procedure is shown for 

head data from MZ-2 in well C-11I2 (fig. 11). The 

detrended data for this 2-month time interval (November-December 1999) is shown in figure 25. This twostep filter was used instead of the somewhat more 

1 2 complex filter 2 4A 25 used by Hsieh and others 

24 4j 

(1987) because a study of the results of a third filter 

pass using the filter programmed for this study (the 

combined three passes being 13A25) showed the 

25 

third smoothing to have a negligible effect on the 

result. 

Observations made by Serfes (1991) suggest a 

possible source of error in the filtering procedure 

described above. As previously noted, Hsieh and 

1 2 others (1987) used the filter 1 A that 24--A 25 

24 1--) 

involved smoothing three times, two 24-point averages 

followed by a 25-point average. Serfes (1991) cites 

Godin (1972) as explaining that the 25-point average 

30 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
			

		

0.1 

WATER LEVEL, IN FEET 

ABOVE NGVD29 

0.0 

__L._ 1 l 1 1 I l 1111111(111111l111 -0.1 10 15 20 25 10 15 20 25 

NOVEMBER 1999 DECEMBER 1999 

Figure 25. Detrended tidally influenced and barometrically influenced total-head data from MZ-2, well C-1112, for 

November and December 1999. 

is the best filter for the lunar components of the tidal 

oscillation but does not filter all of the solar variation, 

and that the 24-point average is the best filter for the 

solar components but does not filter all of the lunar 

variation. To test for the possibility of error in the procedure used in this study, two measured-head data sets, 

from MZ-1 in well C-1111 and from MZ-2 in C-1112, 

were filtered using the method of Hsieh and others 

(1987) and analyzed for amplitude and phase of the 

five major components by the least-squares analysis. 

Differences in amplitude and phase values were negligible in MZ-1, C-1111, and negligible to slight 

(0.5 percent) in MZ-2, C-1112. 

Data collected from the monitor zones in 

C-1111, C-1112, and HE-1087 contained time gaps of 

several weeks to several months, and several abrupt 

positive and negative discontinuities of more than a 

foot were present. The latter appeared to be related 

either to the servicing of instrumentation at the site 

(because they coincided with an reinitialization of the 

elapsed time channel), or to unexplained abrupt 

changes in the values from the temperature and/or 

pressure channels of the sensor. The consequences for 

the data analysis was that the computation of a trend 

line would have misleading results in the neighborhood of the these time gaps or data discontinuities. 

The resulting detrended data would show unusual 

fluctuations in these time periods, leading to unrealistically high variance values in the solution for amplitude 

and phase of tidal harmonics. 

To avoid this undesirable result, each data set 

was subdivided into smoothly varying subsets, continuous in time, for computation of individual trend lines. 

The resulting individual trend lines were then successively appended to produce a single trend line before 

subtraction from the raw data. The procedure caused 

time gaps to occur about the previous data discontinuities because smoothed data values were not computed 

for the first and last 12 points of each subset in each 

stage of the filtering, but this did not appreciably affect 

the results of the analysis for amplitude and phase. 

At first, the detrending procedure and the analysis for amplitudes and phases were applied to data sets 

of measured heads that were corrected for long-term, 

aperiodic atmospheric-pressure fluctuations. Semidiurnal solar (S2) component amplitudes computed from 

Naturally Occurring Stresses on Aquifers 31 
these detrended time series were substantially less (by 

40-80 percent) than the S2 amplitudes computed from 

data sets of measured heads not corrected for atmospheric-pressure fluctuations. The estimated phases 

for S2 also differed substantially. The cited differences 

were greatest for the upper Floridan monitor zones. 

Possibly, the atmospheric-pressure correction had the 

effect of removing a part of the S, component caused 

by diurnal and semidiurnal atmospheric-pressure fluctuations. Amplitudes and phases computed for other 

harmonic components were nearly the same for data 

sets corrected and not corrected for atmospheric-pressure fluctuations. 

Next, the detrending steps and least-squares 

analysis were applied to the data sets of total pore 

pressure heads (sums of contemporaneous values of 

barometric pressure and measured head) not corrected 

for atmospheric-pressure fluctuations. Except for the 

S2 component, resulting amplitudes and phases were 

virtually the same as for the raw measured head data 

(corrected and uncorrected for atmospheric-pressure 

fluctuations). The new S., amplitudes in the total head 

data were substantially smaller than S„ amplitudes of 

the measured-head data uncorrected for atmosphericpressure fluctuations, and phases also differed substantially. It is believed that the reduced S, amplitudes 

in the total head data reflected the relatively low 

loading efficiencies of the formations. The measuredhead data (high barometric efficiencies) contained a 

higher dependence on the atmospheric-pressure fluctuations and would be expected to have a solar semidiurnal (S,) signal of greater amplitude than the total head 

data. Amplitude and phase values from the analysis of 

the total head data (table 5) were the ones used in the 

application of algorithms described in later sections of 

this report. Another result of the detrending procedure 

was that the tidal oscillations were oriented about the 

value of zero, so estimates of at, (equations 18 and 22) 

were expected to be (and were) nearly equal to zero. 

Of the Naples ocean-tide data, only data after 

December 31, 1997, were used for estimating amplitudes and phases of harmonic components. The time 

series contained a gap of most of August and September 2000. Tidal harmonic components in the Naples 

ocean-tide data (table 5) are substantially higher in 

amplitude than those in head data from any of the well 

monitor zones. Among the latter, the amplitudes are 

significantly greater in the data from well C-1111 than 

in data from C-1112 and HE-1087 because of the 

strong ocean-tide influence at C-1111. 

Table 5. Amplitudes and phases of the five major harmonic components in Naples ocean-tide data and in total-head 

data from monitor zones in wells C-1111, C-1112, and HE-1087 

I Amplitude in feet (phase in degrees). M, mean square of fit in square feet; V, variance of data in square feet. Negative degrees (-d) are equivalent to 

negative degrees + 360 (-d+360)1 

Source Time period 

Naples ocean tide January 1998 - November 2000 

Well C-1111, MZ- I January 1998 - December 2000 

Well C-1111, MZ-2 January 1998 - December 2000 

Well C-1111. MZ-3 January 1998 - December 2000 

Well C-1112, MZ-2 April 1999 - December 2000 

Well C-1 112, MZ-3 April 1999 - December 2000 

Well C-1112, MZ-4 April 1999 - September 1999 

Well 11E-1087, MZ September 1998 - June 2000 

0, 

component 

K, 

component 

M2 

component 

S2 

component 

N2 

component MN 

0.38326 0.45690 0.89004 0.30864 0.18478 93.3 

(0.87) (-69.51) (65.16) (-0.85) (5.48) 

0.03981 0.04861 0.06259 0.00695 0.01328 84.9 

(42.13) (-26.15) (146.70) (83.52) (79.78) 

0.03754 0.04733 0.05445 0.00187 0.01126 77.9 

(49.58) (-15.60) (158.68) (158.95) (90.66) 

0.05586 0.06740 0.09859 0.02228 0.02198 83.0 

(22.30) (-47.08) (106.87) (2.14) (43.86) 

0.01074 0.01124 0.02476 0.01081 0.00416 50.9 

(-112.90) (168.42) (-105.79) (-91.40) (-140.11) 

0.01278 0.01500 0.03022 0.00883 0.00477 64.3 

(-113.96) (166.78) (-105.73) (-110.68) (-141.94) 

0.00978 0.01283 0.02679 0.00659 0.00396 46.1 

(-111.43) (155.64) (-121.79) (-103.14) (-155.94) 

0.00961 0.00691 0.01991 0.03232 0.00327 50.4 

(-85.35) (-167.14) (-85.26) (-91.16) (-130.53) 

32 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
Amplitudes and phases of tidal components for 

MZ-1 and MZ-2 in well C-1111 were generally similar, but amplitudes for MZ-3 were greater and the 

phase differences with the ocean-tide components 

were less than those of MZ-1 and MZ-2. As shown 

later, this indicates that the transmissivity of MZ-3 is 

higher than that of the other zones. For each ( th) harmonic component, the percent of variance (Pvj) was 

computed, according to: 

2 

a. 

Pvj = where V is the variance of the data: (23) 2 V , 

1 2 = - -ao/ 2) . (24)

= 1 

The values of Pvj can be used to indicate the relative significance of theft' harmonic component in the 

tidal or tidally influenced data set. The sum of the Pvj 

values is a ratio (M/V), the mean square of fit (M) 

divided by the variance of the data (V): 

P P 

M i 1 — = EPv i • = — la . (25) V 21: / 

i=1 i = I 

MN can be used as an indicator of the degree of 

success with which the tidal components identified by 

the regression analysis explain the variance in the data 

set. In this study, MN was multiplied by 100 and 

expressed as a percent (table 5). The MN value for the 

Naples ocean-tide data was 93 percent. MN values for 

the monitor zones in C-1111 range from 77.9 to 84.9 

percent. In decreasing order of magnitude of the Pv3 

values are the components M„ K1 , 01 , N,, and S,. 

When the analysis was run on measured head 

values, 52 values were only slightly less than 0, amplitudes on MZ-1 and MZ-2. Using measured head data, 

the analysis for MZ-1 was rerun with components PI , 

K,, Q,, and T, added to the solution. The MN value 

increased about 4 percent, but only the P, and Q, components added appreciably to the MN value. When 16 

of the 17 harmonic components listed in table 4 were 

included in the analysis, the additional 7 components 

added negligibly to the MN value. When the original 

five components were solved for and the data set 

length was changed by deleting the first year of data, 

the change in estimated amplitudes varied from negligible to 5 percent, the change in estimated phases varied from negligible to 8 degrees of arc, and the MN 

value increased slightly. 

In analyzing data from well C-1112, amplitudes 

computed for MZ-3 based on analysis of total heads 

are generally slightly greater than those for MZ-2 and 

MZ-4. The computed phases for MZ-2 and MZ-3 are 

similar. The amplitude of M, is appreciably higher 

than the others. MN values range from 46.1 to 

64.3 percent. The amplitudes of the S, components 

were only slightly less than that of M, when measured 

head data were used for the analysis. 

An MN value of 50.4 was obtained when analyzing total-head data from HE-1087. When the analysis was restricted to the time period (April-September 

1999) before a suspected time bias (one time series 

shifted in time relative to the other) may have affected 

the data, the MN value improved to 70.6, and the 

phases for Kl and S2 changed significantly. The largest amplitude was for the M, components. 

To partially verify the results of the component 

analysis and to check for errors, a program was written 

to construct a time series by summing the values of the 

five principal tidal components computed at successive 

times from the amplitudes and phases estimated by the 

least-squares analysis. This "simulated (detrended) 

tide" was graphically superimposed on the detrendedtide time series based on the field data so that comparisons could be made in various time periods of the data 

for error checking. An example plot (fig. 26), for MZ2 in well C-1112, shows estimated and measured tides 

for 7 days in November 1999. The match seems to be 

slightly better when tidal oscillations are greatest, as 

during November 24-26, than when tidal oscillations 

are small and affected by noise not removed by the 

detrending procedure, as in the first 2 days of this plot. 

Another view of the harmonic structure of the 

tide is obtained by computing time series of the individual harmonics and graphically superimposing them. 

The example plot shown in figure 27, for the same data 

and time period as the previous plot in figure 26, 

shows the complex interplay between the harmonics 

over a 7-day period. At the beginning, the lunar diurnal 

(01 ) and the lunar-solar diurnal (K1 ) are partly out of 

phase, but, between the 25th and 26th of the month, 

they are exactly in phase. Similarly, the lunar and solar 

semidiurnals (M2 and S2) begin the 7-day time period 

partly out of phase, but are almost completely in phase 

on the 25th. 

Naturally Occurring Stresses on Aquifers 33 
		 		

0.1 

Lu 

LuLL. 

z 

Lu 

cc 

< 0.0 

0 Lu 

0 

z

Lu 

cc 

F- Lu 

0 

Estimated water level, 

reconstructed from harmonics 

-0.1 

21 22 23 24 25 26 27 

NOVEMBER 1999 

Figure 26. Detrended total-head data from MZ-2, well C-1112, and head oscillations estimated from regression 

analysis for November 21-27, 1999. 

0.03 

0.02 

F1.1.1 

LL 

z- 0.01 

L.L.1 

L1.1 

CC 

0.00 

a 

a 

Lij -0.01 

CC 

FLU 

0 

-0.02 

EXPLANATION 

— 01 — K1 — M2 -- S2 — N2 

-0.03 

21 22 23 24 25 26 

NOVEMBER 1999 

Figure 27. Individual harmonic components determined by regression analysis of total-head data from MZ-2, 

well C-1112, for November 21-27, 1999. 

34 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

27 
ANALYTICAL METHODS FOR 

COMPUTING AQUIFER PROPERTIES 

FROM NATURALLY FORCED PERIODIC 

DATA 

Research into the analytical use of tidally 

influenced and other naturally forced head fluctuations 

to estimate values of aquifer properties tends to segregate into four principal categories: (1) use of heads 

influenced by ocean tides to estimate aquifer diffusivity (transmissivity divided by storage coefficient); 

(2) use of heads influenced by earth tides or atmospheric-pressure oscillations for estimating specific 

storage and porosity; (3) use of water levels from 

open wells penetrating a confined aquifer and influenced by earth tides or atmospheric-pressure oscillations to estimate aquifer transmissivity and storage 

coefficient; and (4) use of water levels or pressure 

heads in a vertical diffusion equation to determine 

vertical hydraulic conductivity. The order of the four 

categories corresponds generally to the sequence in 

time in which these categories were the principal 

avenue of research. 

The second research category primarily involves 

aquifer pressure changes caused by surface loading or 

dilatation of the aquifer material. Since the head 

changes are not associated with water movement, 

except possibly on a microscopic scale (Van der Kamp 

and Gale (1983) and other authors refer to these as 

"undrained processes"), they cannot serve as the basis 

for the estimation of properties such as transmissivity 

that describe water movement in the aquifer. For this 

reason, most of the research in category three involves 

a consideration of the well-aquifer system problem, in 

which pressure oscillations cause macroscopic water 

movement (a "drained process"), in this case, movement of water into and out of a well. 

The first three research categories involve a 

comparison of pressure heads or water levels with 

other data: category 1 - with ocean-tide measurements; 

category 2 - with the theoretical tide potential; and category 3 - with background pressure data, dilatometer 

data, or the theoretical tidal potential. As will be 

shown, application of many of the derived equations 

reduces to a comparison of theoretical and observed 

amplitude ratios and phase shifts. 

Using Heads Influenced by Ocean Tides 

for Estimates of the Hydraulic Diffusivity 

of an Aquifer 

Ferris (1951) presented one of the first analytical solutions to the problem of calculating the theoretical head oscillations in a well near enough to the ocean 

to be influenced by ocean tides. The solution was 

developed mainly for the case in which the aquifer 

crops out at or near the coastline (fig. 28a). In this 

case, the oscillation of sea level corresponds to an 

equal oscillation of head at the subsea outcrop, and 

water is forced into and out of the aquifer, causing 

heads in the aquifer (h(x,t)) to fluctuate with an amplitude that diminishes with distance (x) away from the 

subsea outcrop as the time lag of a given maximum 

increases. The solution assumes that the aquifer is 

homogeneous and of uniform thickness and of infinite 

extent shoreward. The subsea outcrop is straight and 

the X-coordinate direction is normal to it. The effects 

of atmospheric-pressure fluctuations and earth-tide 

oscillations are ignored in the theoretical development. 

Citing analogous problems of determining the transmission of thermal energy in solids, and assuming that 

the tidal oscillation is a simple harmonic, Ferris (1951) 

represented the problem with the following differential 

equation, previously presented by Jacob (1950): 

a2 h(x, t) Sah(x, t) (26) T at axe 

subject to the boundary conditions: 

h(x, t) = ho sin w t at x=0, and (27) 

h(0., t) = 0 , (28) 

\N'here: 

t is time (T), 

11 is the amplitude or half range of the tidal harmonic oscillation (L), 

w is the frequency of the harmonic oscillation (T-'), 

S is the aquifer storage coefficient (unitless), 

T is the aquifer transmissivity (L2/T), 

L is a length unit, and 

T is the time unit. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 35 
11 0 Sill (01 

LAND SURFACE NGVD 29 

V 

SEA 

CONFINING 

ht sin wt AQUIFER Th(x,t) LAYERS

-

-

(a) Model of Jacob (1950) and Ferris (1951) - aquifer and overlying confining layer 

crop out at or near coastline. 

11 0 sin (01 

LAND SURFACE NGVD 29 

V 

SEA 

CONFINING 

Te ho sin (01 AQUIFER Ikx,t) LAYERS 

4 

- +

0 

(b) Model of Van der Camp (1972) - a completely confined aquifer extending 

under the sea. 

h ( Sill (Ut 

LAND SURFACE NGVD 29 

V UNCONFINED 

LEAKAGE AQUIFER 

SEMI PERMEABLE LAYER 

SEA LEAKY Ls Ls CONFINED Th(x,t) 

AQUIFER 

IMPERMEABLE CONFINING LAYER 

-L 0 

(c) Model of Li and Jiao (2001a) - a leaky contained aquifer system extending 

under the sea for a distance 1.. 

Figure 28. Sea-aquifer models for which solutions for head (//(x)) as a function of distance inland (x) 

have been derived. 

36 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
The solution to the equation and boundary 

conditions is: 

-x T COS h(x. 1) = hoe 41 2 sin(0)1 -x —) = T

TES 

. s hoe s m (cot - (29) T T 

where 

is the period of the harmonic oscillation (T). 

This solution is a harmonic function with a phase shift 

that increases with distance inland (x) and with an 

exponential damping term that reduces the amplitude 

of the harmonic function with distance inland. 

The use of a simple harmonic representation of 

the ocean tide in the analysis is not limiting, as the 

ocean tide was previously shown to consist of a number of harmonics superimposed on one another, only 

five of which are dominant. Equation 26 is linear, so 

that solutions for simple harmonics can be added 

together, or tides can be analyzed for component 

amplitudes and phases, and equation 29 can be applied 

to individual components. 

Although figure 28a portrays an aquifer confined to or near the subsea outcrop, Ferris (1951) did 

not restrict the solution to that case, stating that the 

solution will apply to aquifers under water-table conditions where appreciable vertical flows do not occur 

and where the saturated thickness of the aquifer is sufficiently large. The storage coefficient (S) may be four 

or more orders of magnitude greater for unconfined 

aquifers than confined ones, so the exponential damping term will quickly reduce the amplitude of tidally 

forced head oscillations with distance from the ocean 

in surficial aquifers (Jiao and Tang, 1999). For this reason, some authors discount the use of analytical methods depending upon tidally forced water levels in 

surficial aquifers, but these methods have been applied 

successfully in certain cases (Erskine, 1991). Even so, 

Erskine (1991) reported that tidal oscillations in an 

unconfined aquifer were unmeasurable farther inland 

than 400 meters from the shore. Referring again to 

equation 29, however, if the transmissivity (T) is 

exceptionally high, as in the surficial Biscayne aquifer 

of south Florida, then measurable tidal effects may be 

observable inland even farther than 400 meters. 

Neither did Ferris (1951) ignore the case 

(fig. 28b) in which the aquifer confinement extends 

under the sea, stating that "if . . . the aquifer is confined by an extensive aquiclude, the rise and fall of 

the surface-water stage changes the total weight upon 

the aquifer. Resultant variations in compressive stress 

are borne in part by the skeletal aquifer and in part by 

its confined water." Ferris (1951) stated that "when 

the aquifer response is due to loading rather than head 

change at the suboutcrop, the amplitude factor 

should be reduced (Jacob, 1950, p. 365) by the factor 

cd(a+r0), where a is the vertical compressibility of the 

skeletal aquifer, 13 is the compressibility of the water, 

and ri is the porosity of the sand." This is the factor 

presented as the loading efficiency (Le) (equation 1). 

(However, it was shown by Van der Kamp, 1972, that 

the tidal efficiency at the coast was only half the factor 

proposed by Jacob and Ferris.) 

The assumption inherent in this case and in 

equation 29 is that head oscillations in a confined aquifer beneath the ocean at the coast are reduced in amplitude by the "tidal efficiency" (Te) from the amplitude 

of the ocean tide without any change in phase or frequency; from this location, the head oscillations 

reduced in amplitude by the factor Te are transmitted 

into the aquifer inland from the ocean with an exponential decrease in amplitude with distance, with a 

change in phase (time lag) that increases with distance, 

but with no change in frequency. 

The sea-aquifer model shown in figure 28b was 

considered in detail by Van der Kamp (1972) in order 

to obtain a solution for subaquaeous formations separated from the overlying water body by a confining 

layer. The particular objective of Van der Kamp (1972) 

was to derive a solution for narrow tidal channels, but 

the solution has wider applicability. The previous 

assumptions regarding aquifer homogeneity, extent, 

and uniformity remain in effect. As in the previous 

analysis of Ferris (1951), "it is assumed that changes 

in hydraulic potential and of the stresses on the solid 

skeleton can be transmitted horizontally only by water 

movement and not directly through the solid skeleton 

independent of water movement," an assumption that 

simplifies "treatment of the complex interactions at the 

land-sea boundary due to the abrupt difference of compressive load" (Van der Kamp, 1972). Thus, Van der 

Kamp's generalization of the physical process was 

similar to that of Jacob and Ferris in assuming that, at 

the coast, the amplitude of the oscillations of head in 

the confined aquifer are reduced from the amplitude of 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 37 
the ocean-tide oscillations by the tidal efficiency factor, and the resulting lesser oscillations are transmitted 

inland within the aquifer only hydraulically, as 

described by Darcy's Law. 

Van der Kamp (1972) stated separate governing 

equations and boundary conditions for the inland 

(x> 0) and subsea (x < 0) parts of the confined aquifer. 

The inland equation (equation 26) remained the same, 

but the subsea equation had an additional term: 

ah(x, T h(x, = Leh wcosom + (30) 

at s 

The boundary condition for the subsea region was: 

h(x, t) = Leho sinwt at x = (31) 

Solutions were required to match at the coastal boundary between the two regions, resulting in the following 

equations for tidally influenced heads in the inland and 

subsea regions: 

T-Ls. 2T caS h(x, t) = 1L2 eh0e sin ( 0)1- A- 2 l—) for x > 0, A T 

and (32) 

h(x, t) = Leho sinwt + 

1 , . 0.)S + ) L, 27 S 11 (0)1 + 7r for x < O. (33) 2 e T 

Equation 32 represents a modification of the 

result of Jacob (1950) and Ferris (1951) for the inland 

region. Van der Kamp (1972) interprets this modification as the result of considering both the subsea and 

inland parts of the aquifer and the fact that a damped 

wave of water moving in and out of the inland part is 

matched by a damped wave of opposite phase moving 

in the opposite direction into the subsea part of the 

aquifer. Equation 32 reduces the tidal efficiency at 

the coast (Te) to half the loading efficiency 

1 (Le): Te, = 2Le . The first term on the right-hand side 

of equation 33 for the subsea region is the ocean tide 

reduced in amplitude by the loading efficiency value 

(Le). 

Application of the theory is most efficiently 

done by comparing the amplitudes and phases of harmonic components in the well head data with those of 

the ocean tides. The theoretical harmonic component 

amplitude (A(x)) and phase lag (tag) in a well at a distance x inland from the coast are: 

els iTs r 2T 1 -xtst: A(x) = Lehoe = Lehoe , and (34) 2 

Wt (35) lag 

The amplitude ratio of the corresponding harmonic components of the head in a well at a distance 

inland x0 and the ocean tide is expressed by dividing 

A(x0) by the ocean-tide amplitude h0. Setting the result 

equal to the observed amplitude ratio r and rearranging 

the equation, an equation is obtained that relates transmissivity (T) to storage coefficient (S) based on data 

from the well: 

2 

T - S . (36) 

[11.1G-1--)1 2 

e 

Recognizing that the exponential term in equation 34 

cannot have values greater than one, equation 36 is 

only valid for values of Le greater than 2r. 

If the difference between the phases of a particular harmonic component of the head in the well and 

the ocean tide is denoted Coand converted to units of 

radians (23t radians = 360 degrees of arc) and set equal 

to the right-hand side of the equation for cotkig (equation 35), then the expression for the time lag, or phase 

shift, leads to a second, independent equation relating 

T to S: 

2 

T = lExo 

2 . (37) 

38 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
The actual difference in phase could be tplus a 

multiple of 2Tc. Whether or not this is true requires 

hydrologic intuition on the part of the analyst and data 

from additional wells, if available. Equation 37 has an 

advantage relative to equation 36 in that it does not 

contain a dependence on the loading efficiency (Le), 

for which estimates might not be available. Equations 

similar to equations 36-37 were presented by Meyer 

(1974), which is the only publication known to the 

author that applies the theory of Van der Kamp (1972) 

to a set of field data. 

An intermediate case in which a non-leaky confining layer extends a finite distance (L) under the sea 

was considered by Li and Chen (1991). However, their 

solution for damped tidal oscillations in a fully confined aquifer caused by an assumed simple harmonic 

ocean-tidal oscillation was corrected by Li and Jiao 

(2001a). Jiao and Tang (1999) considered the case of a 

leaky, confined aquifer separated from an overlying 

unconfined aquifer by a semipermeable layer. The 

assumption of a non-varying head at mean sea level in 

the unconfined aquifer was considered an adequate 

approximation of cases normally observed in realworld aquifer systems. The semiconfining layer and 

semiconfined aquifer were assumed to terminate at the 

shoreline with no extension under the sea. Flow was 

assumed to be horizontal in the aquifer, and storage 

within the semiconfining layer was neglected as vertical leakage occurred. 

Li and Jiao (2001a) combined these cases into a 

more general solution. The coastal aquifer system 

(fig. 28c) consists of a leaky semiconfining layer separating an overlying unconfined aquifer from an underlying semiconfined aquifer. The aquifer and 

semiconfining layer (referred to by Li and Jiao (2001a) 

as the "roof of the aquifer") extend for a finite distance 

(L) under the sea. As above, the water level in the 

unconfined aquifer is assumed constant at mean sea 

level, vertical leakage (at a rate denoted L,) occurs 

through the semiconfining layer, and storage effects in 

the latter are ignored. Flow in the semiconfined aquiferis horizontal. The governing equations, boundary conditions, and solution equations are as follows. 

Governing equations: 

Inland (x > 0): 

sah(x, t) T a2h(x, 1)— L sh(x, t) . (38) at 2 al 

Offshore aquifer (-1. < x < 0): 

sah(x, t) a2 h(x, t) + 

at ar t 

ahs(t) ST, + Ls[hs(t)- h(x, I)] . (39) at 

Boundary conditions: 

h(-L, t) = hs(t) = ho cos cot and h(oo, t) = ,(40) 

where 

hs(t) is a harmonic component of the ocean tide 

having amplitude h„ and angular frequency co. 

Also, the inland and offshore functions for h(x,t) and 

their .v-derivatives must match at the boundaries 

between the two regions. The solutions are expressed 

in terms of u and a, where u is a dimensionless leakage 

parameter and a, referred to by Li and Jiao (2001a) as 

the "tidal propagation factor," was used before in equations 29 and 32-35: 

coS' 17cS 

a = -= — and (41 ) 2T 4 T T' 

Ls 

(42) coS 

Only the solution for the inland region is presented 

here: 

h(x, t) = hoCee-pax cos (cot — qax - (p) , where (43) 

0 < = ,j(R1 +X/ 2)2 + ( j_t/ 2 )2 , (44) 

ti 

2 + I, (1 - Le )u 

x = ' and IA = , (45) ,

_ 2 u +1 u + 1 

RI = e_paL[( — X)cos(gaL) - ptsin(qaL)] 

-2paL 

+ -e [X cos(2yaL) + μsin(2qaL)] , (46) 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 39 
		

-paL 

e [(1 - X)S111(gaL) + licos(cmL)1 

-2paL 

+ [Xsin(2q a L) - μcos(2qa L)] , (47) 

p = + 112 + U and q = JAI] +u2 -u, and (48) 

2/ 1 - cp = atan (49) 2R1 + 

Li and Jiao (2001a) use the notation T; instead of L,, in 

equation 45, and refer to the parameter as "tidal 

efficiency" without additional explanation. The correct 

parameter, however, is assumed to be the loading efficiency (Le) defined as "tidal efficiency" by Jacob 

(1940). 

Li and Jiao (2001a) consider special cases 

represented by extreme values of solution parameters. 

For instance, the special case in which leakage is zero 

(Ls = 0) and the extent of the aquifer and "roof" under 

the sea are infinite (L = is the case considered by 

Van der Kamp (1972). The solution of Li and Jiao 

(2001a) reduces to that of Van der Kamp (equation 32) 

when these parameter values are used (Ce, reduces 

to L ,12,p = q =1, and cp = 0). When leakage is zero 

(L, = 0) and the aquifer and confining zone terminate 

at the shoreline (L = 0), the solution reduces to that 

of Ferris (1951) presented in equation 29, as Ce = 1, 

p = q =1, and cp= 0. 

As before, application of the theory is most 

effectively done by comparing measured harmoniccomponent amplitudes and phases of heads in wells 

with those of the ocean tides. The expressions for the 

amplitude ratio and phase lag at an inland well (for a 

given harmonic component) are: 

A(x) = hoce—pax , and (50) 

, 1 

t = -(qax + (p) . (51) tag - co 

Using the logic for comparing theoretical and 

measured amplitude ratio (r) and phase shift (0) that 

led to equations 36 and 37, the following two independent relations between T and S at inland location x0 are 

obtained: 

22 

rcp xo 

T - S , and (52) 2 

'T 1110 

[ 

.v 0 

T = S. S (53)

T (fi-1- (p)-

Substituting the values of Ce, p, q, and y that correspond to the Van der Kamp case, these relations reduce 

to equations 36 and 37. 

Li and Jiao (2001a) applied their theory to theoretical cases only, and no publication known to the 

author describes an application to field data. In a second paper, Li and Jiao (2001b) treat the aquifer system, as before, as a semiconfined aquifer separated 

from an overlying unconfined aquifer by a leaky semiconfining layer, but there is no extension under the sea 

(L = 0). Other assumptions are as before, with the 

exception that storage within the leaky semiconfining 

layer is accounted for in the governing equation. 

Inland extent of ocean-tide influence 

The equation of Van der Kamp (1972) was used 

to show approximately how far inland from the coast 

near Naples that ocean-tide oscillations propagate 

before the exponential damping term reduces the tidal 

amplitude to 10 percent and 1 percent of the oceantide amplitude measured at the coast. Van der Kamp 

(1972) assumed that a homogeneous, fully confined 

aquifer of uniform thickness extended both inland and, 

with confinement, beneath the ocean. The justification 

for adopting this assumption for analysis of Naples 

observation-well data is discussed in the next section. 

The relation for tidal amplitude at inland locations within the confined aquifer (equation 34) was 

adapted for this application. The amplitude of tidal 

oscillations at inland distance x, A(x), is divided by the 

ocean-tide amplitude Ito to obtain an expression for the 

amplitude ratio r, and then solved for the distance x at 

which this amplitude ratio will, theoretically, be 

observed: 

T T 2r 

x = I n — (54) TES Le ' 

where all symbols are as previously defined. 

40 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
				

This equation is applied to lunar harmonic components 

O, and M,. Results (figs. 29-30) are presented as functions of the hydraulic diffusivity (transmissivity (7) 

divided by the storage coefficient (S)). If a 10-ft-thick 

artesian zone having hydraulic conductivity of 10,000 

feet squared per day (ft-/d) is considered, the estimated 

storage coefficient should be of the order of 1 x 

based on the rule of thumb offered by Lohman (1979, 

p. 8) for confined aquifers (1 x 10' per foot of thickness). The hydraulic diffusivity would be 1 x ft'-/d. 

This aquifer scenario generally is representative of 

thin, discrete flow zones found within the Upper Floridan aquifer. If the aquifer were considered to be 100 ft 

thick and to have a transmissivity of 100,000 ft2/d, as is 

sometimes characterized in models, a value of 1 x 10 

might be representative of the storage coefficient 

according to Lohman (1979, p. 8). The resulting value 

of diffusivity is again, 1 x 10+9 ft'-/d. 

For the 0, component (fig. 29) and the cited 

value of hydraulic diffusivity (1 x 10+9 ft2/d), a 90-percent reduction in tidal amplitude occurs at 0.8 miles 

inland when loading efficiency (Le) is 25 percent, at 

3.2 miles inland when is Le= 50 percent, and at 

4.6 miles inland when Le = 75 percent. A 99-percent 

reduction in tidal amplitude occurs at 8.9 miles inland 

50 

EXPLANATION 

01 component (r = 1.0758 days) 

r is the period of the 01component 

40 Le is loading efficiency in percent 

Ratio is well water-level amplitude 

at the given distance inland divided 

by the ocean-tide amplitude 

at the coast 30 

20 

ratio = 0.01 

10 

0 

2 3 4 5 6 2 3 4 5 6 8 107 10

DISTANCEFROMCOASTLINE, IN MILES 

when Le= 25 percent, at 11.3 miles inland whcn Lt

.=

50 percent, and at 12.7 miles inland when Le = 75 percent. Because the period of the K, component of the 

tide is similar to that of the 0, component, the required 

distances for K, would be similar. 

Results are somewhat different for the M, component because the period (t) is about half that of the 

0, and K, components. For the M2 component (fig. 30) 

and the cited value of hydraulic diffusivity, a 90-percent reduction in tidal amplitude occurs at 0.5 miles 

inland when Le = 25 percent, at 2.2 miles inland when 

Le = 50 percent, and at 3.2 miles inland when Le = 

75 percent. A 99-percent reduction in tidal amplitude 

occurs at 6.1 miles inland when Le = 25 percent, at 

7.8 miles inland when Le = 50 percent, and at 8.8 miles 

inland when Le = 75 percent. Because the period of the 

S, component of the tide is similar to that of the M, 

component, the required distances for S, would be 

similar. 

Assuming a hydraulic diffusivity of 1 x 10+9 ft-'/d 

for the aquifers used as monitor zones in the present 

study. lunar tide components induced by the ocean tide 

at well C-1111, 4.9 miles from the coast, would probably be reduced to less than 10 percent of the ocean-tide 

variation. The M.2 components would likely be reduced 

= 25 

= 50 

= 75 

= 50 

ratio = 0.10 

= 25 

2 3 4 5 6 2 3 4 5 6 9 10 

10 10 1011

HYDRAULIC DIFFUSIVITY, IN FEET SQUARED PER DAY 

Figure 29. Distance from the coastline at which 90 and 99 percent reductions of the amplitude of the O, component of 

the ocean tide would be observed in wells in a confined aquifer, based on the equation of Van der Kamp (1972). 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 41 
		 		

	 	

50 

DISTANCE

FRO

MCOASTLINE, IN MILES 

EXPLANATION 

40 

M2 component (t = 0.5175 days) 

r is the period of the M2 component 

Le is loading efficiency in percent 

Le =75 

30 

Ratio is well water-level amplitude 

at the given distance inland divided 

by the ocean-tide amplitude 

at the coast 

= 50 

20 

ratio = 0.01 

Le = 50 10 

LP = 25 

ratio = 0.10 

0 

107 

2 3 4 5 6 

108 2 3 4 5 6 109 2 3 4 5 6 1010 2 3 4 5 6 1011 

HYDRAULIC DIFFUSIVITY, IN FEET SQUARED PER DAY 

Figure 30. Distance from the coastline at which 90 and 99 percent reductions of the amplitude of the M2 component of 

the ocean tide would be observed in wells in a confined aquifer, based on the equation of Van der Kamp (1972). 

to a greater extent than the 0, components. Wells C1112 (28 miles inland) and HE-1087 (57 miles inland) 

are sufficiently far from the coast that the lunar component amplitudes induced by the ocean tide would be 

less than 1 percent of the ocean-tide variation—negligible for purposes of this study. Lunar tidal components measured at C-1112 and HE-1087 would chiefly 

be the result of the earth-tide influence. 

Choice of method for estimating hydraulic 

diffusivity 

Several solutions have been presented for estimating aquifer properties from ocean-tide data that 

differ with respect to the assumed nature of the aquifer-sea interface. In this section, the characteristics of 

the aquifer and confining layer in the vicinity of well 

C-1111 near Naples, Florida, and the tidal Gulf of 

Mexico are evaluated to determine the best solution for 

making aquifer parameter estimates based on the head 

data from this well. 

Formations open to the monitor zones in 

C-1111 are overlain by numerous confining beds of 

the intermediate aquifer system (fig. 2). MZ-1 is at 

the base of the Hawthorn Formation, MZ-2 is within 

the Suwannee Formation, and MZ-3 is within the 

undifferentiated Avon Park-Oldsmar Formation 

(Reese, 2000). Regional studies (Meyer, 1989; Ryder, 

1985; Miller, 1986) show that the Hawthorn Formation 

is extensive across the Floridan Peninsula to the east 

and southeast. To the north, the Hawthorn Formation, 

Suwannee and Ocala Limestones, and Avon Park Formation rise and thin until the Ocala Limestone is at 

land surface in central Florida. The confining beds 

of the Hawthorn Formation pinch out more than 

100 miles to the north of the site of C-1111. 

Because the aquifers and confining beds are of 

regional extent to the north, east, and southeast, and 

are assumed not to vary appreciably in thickness near 

the study area, theoretical assumptions of uniform 

aquifer thickness and confinement are assumed to be 

approximately satisfied in those directions. To the 

south and west, the Gulf of Mexico extends over a 

submerged part of the Florida Platform, an extension 

of the Continental Shelf. The edge of the shelf (indicated by the 6,000-ft depth contour in fig. 31) is about 

200 miles west of Naples and about 160 miles south of 

Naples (Uchupi, 1966), where it bottoms in the shallower Straits of Florida at depths ranging between 

3,000 and 6,000 ft. The top of the platform (bottom of 

the Gulf) is about 650 feet below present-day sea level 

about 165 miles west of Naples (fig. 31), a depth 

42 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
		 		

higher than the top of any monitor zones providing 

data for this study. The top of the platform is about 

2,300 ft deep, lower than any of the monitor zones, 

about 25 miles farther to the west. To the south of 

Naples, depths of 650 ft or greater are found only 

south of Key West, about 125 miles south of Naples. 

To the east, the shelf drops steeply east of Ft. Lauderdale, about 110 miles east of Naples, but bottoms at 

about 2,400 ft. From south and west of Key West, to 

the east and north around the southern tip of Florida, 

the bottom of the Straits of Florida rises from a depth 

of over 6,000 ft until it reaches a depth of 2,400 ft east 

of Ft. Lauderdale. These distances make the location 

of the edge of the Continental Shelf at Ft. Lauderdale, 

to the east of Naples, the closest to Naples. 

Information is not known to exist on the stratigraphy of the Floridan aquifer system beneath the sub87° 86° 85° 84° 

29° 

28° ,Poo O

O 

27° 

c\-\ 

— 0 25 50 75 100 MILES 

I I I I II I " 

0 25 50 75 100 KILOMETERS 

26° 

EXPLANATION 25° 650 CONTOURS OF DEPTH TO 

SEA BOTTOM, IN FEET 

24° 

merged part of the Florida Platform to the west and 

south of present-day peninsular Florida. Oil test wells 

have been drilled in the eastern Gulf of Mexico and 

near the Florida Keys, but normally, cuttings are not 

collected for examination at shallow depths. For this 

study, it is assumed that aquifer and confining zone 

depths and thicknesses found beneath present-day 

Florida continue uniformly westward and southward 

until the zones crop out at the edge of the Florida Platform. This assumption, together with previous 

assumptions, is equivalent to a hypothesis that the 

confinement of the monitor zones in C-1111 is extensive over a broad region, and that theoretical solutions 

assuming a confining zone of infinite extent (Van der 

Kamp, 1972; the special case of Li and Jiao, 2001a) 

are approximately valid for the analysis of tidally 

influenced head data inland of Naples. 

83° 82° 81° 80°

O 

NAPLES L FT. LAUDERDALE 

000 

Figure 31. Peninsular Florida and contour lines showing the extent and slope of the Florida Platform. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 43 
The intermediate confining unit, comprised 

largely of elastics of low permeability of the Hawthorn 

Formation, has a low vertical conductivity in southern 

Florida. Heads in the Upper Floridan aquifer at 

C-1111 range from 34 to 35 ft above NGVD 29. 

Because land surface is only 10 ft above sea level at 

the C-1111 wellhead (Bennett, 2001b), the high heads 

maintained in the upper Floridan demonstrate the 

effectiveness of the confining layer. Depending upon 

geographical location, the thickness of the overlying 

surficial aquifer system, and the presence or absence 

of an intermediate aquifer system, the thickness of 

confining materials overlying the Upper Floridan 

aquifer could range from 500 to 1,000 ft. 

In a highly generalized model of the aquifer systems underlying the southern part of the Floridan peninsula (Merritt, 1997), a realistic head distribution in 

the Upper Floridan aquifer was obtained by using vertical hydraulic conductivity values of 0.01 ft/d or less 

for the intermediate confining unit. Leakance values 

for 500 and 1,000 ft, respectively, would be 1 x 10-5 to 

2 x 10-5 d- '. Such values were needed to confine the 

aquifer under pressure and prevent substantial loss of 

head in the aquifer due to upward leakage. These values were applied regionally. If upward leakage occurs 

at discrete locations, vertical hydraulic conductivity 

values would be lower at a distance from the locations 

of leakage. 

In the "mega-model" constructed by Sepulveda 

(2002), the intermediate aquifer system in northern 

Charlotte County, about 60 miles north of C-1111, 

consists of an aquifer bounded above and below by 

confining units, to which ranges of leakance values are 

assigned (1 x 10-6 to 1 x 10-5 d-I for the lower and 

1.1 x 10-5 to 6 x 10-5 d- ' for the higher). Using the two 

lower values, and using the harmonic mean, a single 

combined leakance value of 0.9 x 10-6 d-' is obtained. 

Using the two higher values, a combined value of 

0.86 x 10-5 d-' is obtained. In a model of the Floridan 

aquifer system in west-central Florida, Ryder (1985) 

specified leakance values ranging from 1 x 10-7 to 

7 x 10-5 d- ' for what was termed "the lowermost 

intermediate confining bed." For purposes of this 

study, the leakance of the intermediate confining unit 

(intermediate aquifer system) is considered to lie 

within a range of values from 1 x 10-6 to 2 x 10-5 d-I. 

The cited analytical methods all assume an 

impermeable layer beneath the confined aquifer for 

which parameter estimates are made. Merritt (1997) 

assumed a vertical hydraulic conductivity of 1 foot 

per day (ft/d) or greater for the middle confining 

unit underlying a layer representing the Upper Floridan aquifer, but this value was assumed to apply 

regionally, and could have represented leakage at discrete locations. The 1-ft/d value was chosen because 

few if any vertical hydraulic gradients in the Floridan 

aquifer system have been documented, and vertical 

pressure gradients within the aquifer system generally 

are related to temperature and density variations. 

Locally, zones of massive limestone underlying solution-riddled flow zones might have lower vertical 

hydraulic conductivity, but still might not be completely confining. 

As a first approximation for the present study, 

equations 36-37, based on the solution of Van der 

Kamp (1972) that assumes a completely non-leaky 

confining unit of infinite extent, were applied to data 

from the three monitor zones. As noted above, however, few if any confining layers in Florida, including 

the intermediate confining unit and middle confining 

unit of southern peninsular Florida, are entirely nonleaky, and it remains of interest to determine whether 

the estimated low leakance values could have an 

appreciable effect on estimates of hydraulic diffusivity 

derived from Van der Kamp's solution. Therefore, data 

from the two higher monitor zones (MZ-1 and MZ-2) 

were reanalyzed using the formulas of Li and Jiao 

(2001a) (equations 41-49), assuming a finite leakance. 

Leakance from below the monitor zone providing the 

tidally influenced data was assumed to be represented, 

together with leakance from overlying layers, by the 

leakance term used in the solution. 

Estimates of hydraulic diffusivity 

In applying the method of Van der Kamp (1972) 

for estimates of hydraulic diffusivity, only the lunar 

diurnal (Os ), lunar semidiurnal (M2), and lunar elliptic 

(N2) data were used, because the solar components (K, 

and S2) contained both atmospheric-pressure and 

ocean-tide influences. A nominal value for S, 1 x 10-5, 

was assumed for computational purposes. Lohman 

(1979, p. 8) states that the storage coefficient (S) of 

many aquifers is about 1 x 10-6 per foot of thickness 

multiplied by the thickness. The cited storage 

coefficient value might, therefore, represent an aquifer 

10 ft thick, which is characteristic of the thin, discrete 

flow zones commonly found in the Floridan aquifer 

system. Because estimates of transmissivity (T) are 

linear in the value of S, T values for thicker aquifers 

would be proportional multiples of the values 

44 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
computed for thinner aquifers. For instance, if the 

aquifer were 100 ft thick, 1 x 10' would be a typical S 

value for the aquifer, and T estimates would be 10 

times those shown in the figures. A value of 4.9 miles 

is used for xo. 

The phase lag relation (equation 37) is applied 

by using the periods for the O,, M,, and N, harmonic 

components (table 4) and the measured differences in 

phase of the components (table 5). For instance, the 

computation for T of MZ-1 based on the 0, component uses a period value of 1.0758 days and the difference (41.26 degrees) between phases (42.13 degrees 

for MZ-1 and 0.87 degrees for the ocean tide). The 

phase difference must be converted to radians (divide 

by 57.2957795). 

The amplitude ratio relation (equation 36) 

requires specification of the dimensionless measured 

ratio of the amplitudes of the corresponding harmonic 

components of the heads in the wells and the ocean 

tide. In the example above (0, component, MZ-1), the 

amplitudes are 0.03981 and 0.38326, respectively, and 

the ratio is 0.103872. For the M, component in MZ-1, 

the amplitude ratio is 0.070323. (As previously predicted, the O, component was reduced to about 

10 percent of the ocean-tide amplitude, and the M, 

component amplitude reduction was even greater.) 

Equation 36 also requires specification of a value for 

loading efficiency (Le). In this study, Le was varied 

from 0 to 100 percent in increments of 5 percent to 

obtain a series of estimates of T based on the series of 

values of Le. However, no estimates were made for values of Le less than twice the amplitude ratio. As previously noted, such values of L, were invalid. t

The two relations (equations 36-37) for the three 

components (0,, M,, and N,) give six independent 

estimates of T as a function of the specified value of S 

for each monitor zone. Values are plotted (figs. 32-34) 

as a function of valid values of Le. Because the phase 

difference relation is not a function of Le, these estimates plot as horizontal lines. T values computed 

based on the M, phase lags are about half those 

computed based on the 0, phase lags. From 

equation 37, this occurs because the measured M, 

phase lags are about twice the 0, phase lags and the 

M, period is about half the O, period. This difference 

in estimates may not substantially detract from their 

usefulness, however, as T estimates from aquifer tests 

in solution-riddled carbonate aquifers can differ by as 

much as an order of magnitude. The period of N, is 

slightly longer than that of M,. The N, phase lag estimates of T (figs. 32-34) are slightly higher than the M, 

estimates. 

A more serious problem is manifest in the 

results of the estimates based on the amplitude ratio. It 

is observed that the previously estimated value of loading efficiency for MZ-1 (table 3, 17 percent) is less 

100,000 

a 80,000 

cc 

0 

0

>- 00z 

o

w° 

01amplitude ratio EXPLANATION 

For other values of storage coefficient (S), 

values in figure should be multiplied by S/10-5 

17 - ° 607 000 <HuLL 

E75 >- u) w M2 and N2 

< 

u) 

W 

cc Z 40,000 

W 

z 20,000 

amplitude ratios 

N2 phase lag (23,700) 

M2 phase lag (20,060) 

01 phase lag (37,700) 

0 20 40 60 

LOADING EFFICIENCY, IN PERCENT 

80 100 

Figure 32. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage 

coefficient of 1 x 10-5 and zero leakance, by the method of Van der Kamp (1972). 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 45 
	 	 	 	

	 	

100,000 

EXPLANATION 

M2 and N2 For other values of storage coefficient (S), 01amplitude ratio amplitude ratios values in figure should be multiplied by S/10-5 

0 80,000 

w

I 

>- D i— 

> u) (7) >- u) 60,000 

CD 

z 

a

<>- 0°_,z 

wo 

u) cc 

z

< 0- 40,000 

Lu <u _ 

cc 

1— w 

LLI 

Li_ 

01 phase lag (27,050) 

z _ 20,000 N2 phase lag (18,050) 

M2 phase lag (15,250) 

20 40 60 80 100 

LOADING EFFICIENCY, IN PERCENT 

Figure 33. Estimates of transmissivity based on total-head data from MZ-2, well C-1111, assuming a storage 

coefficient of 1 x 10-sand zero leakance, by the method of Van der Kamp (1972). 

150,000 

01 phase lag (139,725) 

EXPLANATION 

For other values of storage coefficient (S), 

values in figure should be multiplied by S/10-5 

amplitude ratio M2

cr 01 amplitude ratio 100,000 

N2 phase lag (88,850) > 

>- M2 phase lag (76,700) C/) 

° 

(n CC

Z W 

< z 

CC I— N2 amplitude ratio 

F— w 50,000 

00 

_J z 

o— LL 

z w() 

< 

2w 

H 

w 

20 40 60 80 100 

LOADING EFFICIENCY, IN PERCENT 

Figure 34. Estimates of transmissivity based on total-head data from MZ-3, well C-1111, assuming a storage coefficient 

of 1 x 10 s and zero leakance, by the method of Van der Kamp (1972). 

46 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
than the range of loading efficiency estimates considered valid according to equation 32. Given a loading 

efficiency value of 17 percent, equation 34 predicts 

that the amplitude of tidal oscillation in the monitored 

formation at the coast (x = 0) should be 8.5 percent of 

the tidal amplitude in the ocean. However, the amplitude ratios of tidal components O, and M, measured in 

well C-1111, 4.9 miles inland, are 10 and 7 percent, 

respectively, of the ocean-tide range. Thus, the loading 

efficiency estimates (table 3) are not consistent with 

results of the amplitude ratio analysis. A re-examination and reanalysis of the atmospheric-pressure and 

head data did not resolve this issue. The estimated 

component amplitudes (table 5) are consistent with the 

tidal range observed in the ocean and in the total-head 

time series for MZ-1, in which the daily range of variation reaches a maximum of about 0.34 ft in early 

July 1999. 

Resolution of this problem would require additional analysis beyond the scope of this study, and 

possibly additional well drilling and data collection. It 

might be possible that loading efficiency is greater 

under the sea at the coast than at the location of well 

C-1111, 4.9 miles inland, or that loading efficiency 

estimates based on atmospheric-pressure loading data 

are not valid for ocean-tide analysis. It is unlikely that 

the unsaturated zone separating the atmosphere from 

the water table can reduce the loading efficiency of a 

confined layer, considering the results of Rojstaczer 

(1988). The unsaturated zone is probably less than 

10 ft thick at each of the wells (C-1111, C-1112, 

HE-1087) providing tidally influenced water-level data 

for the study. 

Amplitude ratio analyses based on the N, components also were made. For MZ-1 and MZ-2 data, 

computed values were almost exactly the same as 

those of the M, component, and results from the 

analyses based on the two components are plotted as a 

single line (figs. 32-33). Estimates of T for MZ-3 

based on the N, component are slightly higher than M, 

estimates (fig. 34). 

Depending mainly on results of the phase lag 

analysis, the results shown in figs. 32-34 can be interpreted to suggest that estimates of T for the three monitor zones range between 20,000 and 38,000 ft2/d 

(MZ-1), between 15,000 and 27,000 ft2/d (MZ-2), 

and between 75,000 and 140,000 ft2/d (MZ-3). As previously noted, if the storage coefficient (S) is different 

from 1 x 10-5, these estimates should be modified by 

the factor S/10-5. 

Aquifer performance tests in MZ-1 and MZ-2 

were performed by the SFWMD in 1996-1997 

(Bennett, 2001b). The data were analyzed using a 

procedure developed by Moench (1985) for large 

diameter wells in leaky semiconfined aquifers with 

storative semiconfining layers and a wellbore skin. 

For MZ-1 and MZ-2, estimated T values were 15,560 

and 7,018 ft2/d, respectively; estimated storage coefficients were 1.7 x 10-5 and 2.3 x 10-5, respectively; 

and estimated leakance values were 3.1 x 10-4 and 

2.2 x 10' d- ', respectively. Using the SFWMD values 

for storage coefficient, the estimated T values determined by the Van der Kamp method would range 

between 34,000 and 65,000 ft2/d for MZ-1 and 

between 34,000 and 62,000 ft-/d for MZ-2. 

The measured amplitude ratios and phase shifts 

of the 01 and M, components determined from the 

head data from the three well monitor zones were 

reanalyzed using the analytical method of Li and Jiao 

(2001a) and previously cited leakance values (1 x 10- `' 

to 2 x 10-5 d-1 ) that were obtained from the literature. 

As before, the leaky semiconfining layer was 

considered to be of infinite extent (L = c.). This 

means that R, and I (equations 46-47) are zero. 

The procedures involved in the calculation, therefore, 

are: (1) to calculate the dimensionless leakage a (equation 42); (2) to calculate the intermediate variables p 

and q from the value of u (equation 48); (3) to calculate the intermediate variables X and μ (equation 45) 

from the values of u and loading efficiency Le; (4) to 

calculate the value of Ce related to loading efficiency 

and leakage (equation 44); (5) to calculate the leakagerelated phase shift (f) (equation 49) from the values of 

X and 1.1; and (6) to compute T as a function of S by 

using the amplitude ratio and phase shift relations 

(equations 52-53). 

Using a value for S of 1 x 10-5, as before, and a 

leakance value of 1 x 10 6 d-' (fig. 35), valid amplitude 

ratio estimates are obtained only for L, values much 

larger than estimated, as before. The estimate based 

on the phase lag is now a weak function of loading 

efficiency (Le) and no longer plots as a straight line. 

Phase lag estimates are slightly lower than when a zero 

leakance was used, especially for low values of Le. For 

a Le of 50 percent, the range of estimates of T from the 

phase lag analysis is now 20,000 to 35,000 ft'-/d. When 

the leakance is increased to 2 x 10-5 d-1 (fig. 36), 

estimates based on both amplitude ratio and phase lag 

are lower. For Le = 50 percent, the amplitude ratio 

estimates are in a narrow range of 26,500-28,000 ft2/d 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 47 
	 	 	 	 	

	 	

100,000 

w 80,000 

1 

EXPLANATION 

For other values of storage coefficient (S), 

values in figure should be multiplied by S/10-5 

01amplitude ratio >- D 

H 0 

> 

Fr) >- 60,000

< 

cn cc 

Z 

< aCC I— 40,000 

ESTIMATED LOADING EFFICIENCY

M2 

amplitude ratio 01phase lag 

w

U_ 

20,000 M2 phase lag 

0 20 40 60 80 100 

LOADING EFFICIENCY, IN PERCENT 

Figure 35. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage 

coefficient of 1 x 10-5 and a leakance of 1 x 10-6 d- I, by the method of Li and Jiao (2001a). 

100,000 

EXPLANATION 

For other values of storage coefficient (S), 

a values in figure should be multiplied by S/10-5

80,000 

< >- D 

}— 0 

>(f) 

C- < 7) >- u) 60,000 

cn cc 

Z 1-1-1 < 

CC I— I— w 40,000 

w 

ESTIMATED LOADINGEFFICIEN

CY 

M2 amplitude ratio 

01phase lag 

01 amplitude ratio 

20,000 

M2 phase lag 

20 40 60 80 100 

LOADING EFFICIENCY, IN PERCENT 

Figure 36. Estimates of transmissivity based on total-head data from MZ-1, well C-1111, assuming a storage coefficient 

of 1 x 10 5 and a leakance of 2 x 10-5 d-I, by the method of Li and Jiao (2001a). 

48 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
	 	

and the phase lag estimates are almost identical at Using the method of Li and Jiao (2001a) with a 

14,000 ft2/d. For the estimated L(, value of 17 percent, 

the phase lag estimates range from 6,000 to 8,000 ft2/d. 

Using the leakance value of 2 x 10' d-1 to analyze 

the data from MZ-2 (fig. 37), and assuming Le to be 

50 percent, the amplitude ratio analysis provides T 

estimates of 23,000-24,000 ft2/d, and the phase lag 

analysis provides estimates of 11,000 ft2 /d for both 0, 

and M, components. When the estimated loading efficiency of 26 percent is used, valid amplitude ratio estimates of 51,000-59,000 ft2/d are obtained, but the 

phase lag estimates are only 7,000-8,000 ft2/d. When 

the leakance coefficient is increased further by an 

order of magnitude (to 2x10' d-1 ), the 0, and M, 

amplitude ratio curves for MZ-1 flatten, with T values 

over an Le range of 5 to 100 percent ranging from 

about 44,500 to 37,750 ft2/d (M,) and 58,300 to 

55,300 ft2/d (0). The phase lag curves also flatten, but 

at values less than one-tenth of the amplitude ratio 

curves. Values for an Le range of 5 to 100 percent are 

about 3,000 to 5,600 ft2/d (M2 ) and 2,800 to 5,400 ft2/d 

(0,). Although valid amplitude ratio estimates are 

obtained for the estimated Le value, there is no value of 

Le for which there is even near agreement between the 

amplitude ratio and phase lag estimates. 

100,000 

0 

80,000 

>- D 

I— 0 

leakance value large enough to permit a small degree 

of leakage into the aquifer provides T estimates based 

on the phase lag analysis that are mutually consistent 

and appreciably lower than those from the non-leaky 

Van der Kamp method, and closer to the SFWMD estimates. Application of this method appears promising; 

however, its application requires an estimate of the rate 

of leakage into the aquifer through confining layers, a 

property for which accurate estimates are typically 

scarce or unavailable. 

The results of the various analyses using the 

phase lag calculation methods of Van der Kamp (1972) 

and Li and Jiao (2001a), rounded to the nearest thousands, are summarized in table 6. Also included, in 

parentheses, are the values of loading efficiency for 

which the amplitude ratio calculations would have provided approximately the same estimates. The listed 

loading efficiencies fall in a narrow range (0.42 to 

0.60) for the non-leaky method of Van der Kamp 

(1972) and for the low value of leakance using the 

method of Li and Jiao (2001a). When the higher 

leakance value is used in calculations with the method 

of Li and Jiao (2001a), the estimated transmissivities 

are substantially lower and the listed loading efficiencies are substantially higher. 

EXPLANATION 

For other values of storage coefficient (S), 

values in figure should be multiplied by S/10-5 

> (/) M2 amplitude ratio 

CT) g >-

(-0 60,000 

amplitude ratio 

co cc 

z w 

< a- >-

0 

m 

w 

LLI 

40,000 wz

w() LL < z 2L.Li 01phase lag 

20,000 wz w — 

a0 

M2 phase lag 

0 20 40 60 80 

LOADING EFFICIENCY, IN PERCENT 

Figure 37. Estimates of transmissivity based on total-head data from MZ-2, well C-1111, assuming a storage coefficient 

of 1 x 10-5 and a leakance of 2 x 10-5 d-1, by the method of Li and Jiao (2001a). 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 49 

100 
Table 6: Summary of transmissivities computed for the 

three monitor zones in well C-1111 using the phase lag 

calculation methods of Van der Kamp (1972) and Li and 

Jiao (2001a), and the values of loading efficiency at which 

the amplitude ratio calculation would yield the same 

estimate, in parentheses 

Id, day; ft, feet; s, second; --, not calculated] 

Estimated Monitor Transmissivity in ft2/s x 10' Method leakance zone (loading efficiency in percent) of designa- analysis (d-1) 01 M2 N2 tion 

Van der 0.0 MZ-1 38 (0.42) 20 (0.60) 24 (0.53) 

Kamp (1972) 

Van der 0.0 MZ-2 27 (0.43) 15 (0.60) 18 (0.55) 

Kamp (1972) 

Van der 0.0 MZ-3 140 (0.42) 77 (0.44) 89 (0.47) 

Kamp (1972) 

Li and Jiao 1.0 x MZ-I 35 (0.45) 20 (0.60) --

(2001a) 

Li and Jiao 2.0 x 10-5 MZ-1 20 (0.66) 17 (0.78) --

(2001a) 

Li and Jiao 2.0 x 10-5 MZ-2 16 (0.72) 12 (0.84) 

(2001a) 

It is of interest to examine the amplitudes developed from the least-squares analysis (table 5) to determine the accuracy of the assertion of Gregg (1966) 

that ocean and earth tides are 180 degrees out of phase 

at the coast. It is noted that inland wells C-1112 and 

HE-1087 are 0.24 and 0.44 degrees west of C-1111 at 

Naples. (The longitude dependence of tidal amplitude 

is of the form cos(A-1-(1)) or cos(A+24)), where 4) is the 

longitude and A is the sum of other arguments (Doodson and Warburg, 1941).) Because of the proximity of 

the latitudes, and because the inland wells are not 

appreciably affected by ocean tides, the lunar components in those wells are caused solely by earth tides, 

and Gregg's hypothesis would imply that lunar tidal 

components at the inland wells should be approximately 180 degrees out of phase with the ocean tides. 

The M2 tidal components in the two shallower monitor 

zones in well C-1112 are 171 degrees out of phase 

with the ocean tide at Naples; however, the 0, components are only 114-115 degrees out of phase. The M, 

and 0, components in well HE-1087 are 150 and 

86 degrees out of phase, respectively, with the oceantide components. The data do not appear to support 

Gregg's hypothesis. Why the amplitudes and phases of 

the lunar components in wells C-1112 and HE-1087 

are not approximately the same remains unexplained. 

Additional accuracy could be obtained in the 

estimates of hydraulic diffusivity if the effects of earth 

tide were filtered from the C-1111 head data. This was 

attempted by Meyer (1974), using a method developed 

for that study by H. H. Cooper (U.S. Geological 

Survey, written commun., March 15, 1973) but not 

described in detail. The method was specialized to 

their application and did not appear to be useful for 

this study. Amplitudes of the lunar components of 

tides in the monitor zones of inland wells C-1112 and 

HE-1087 were of the order of 0.010-0.013 ft (0,) and 

0.020-0.030 ft (M2) (table 5). These lunar components 

are the result of earth tides and are not caused by 

semidiurnal atmospheric-pressure oscillations. The 

0, component amplitudes generally are about 

17-34 percent of those in the upper monitor zones of 

well C-1111, and the M2 components generally are 

about 20-55 percent of those in the upper monitor 

zones of well C-1111. These percentages suggest that 

earth tides could have an appreciable influence on the 

water-level data used for the ocean-tide analysis. Substantial effort was expended during this study to 

develop a method to correct the C-1111 well data for 

the earth-tide effect; however, this objective was not 

achievable within the timeframe of the study. 

Analysis with numerical model 

A numerical model was applied for the purposes 

of: (1) using another means to illustrate the response 

of a coastal aquifer to ocean-tide loading; (2) investigating additional aspects of the effects of tidal forces 

on pressure heads in the monitor zones; and (3) demonstrating an application of a numerical model to the 

problem of aquifer response to ocean-tide loading. The 

model application was not intended to verify the analytic methods nor to replicate the values obtained from 

application of those methods. 

The model code applied was the version of 

MODFLOW released by the USGS in 1996 (Harbaugh 

and McDonald, 1996a and 1996b). A cross-sectional 

grid design was devised in which ocean-tide oscillations were represented as specified-head boundary 

conditions on the left-hand side of the grid. The 

remainder of the grid represented a system of aquifers 

and confining layers inland of the coast. The simulation of observed data was not attempted. Rather, comparison runs (sensitivity analyses) were made to test 

the theoretical response of the system to variations in 

hydraulic characteristics represented by model parameters. Computed heads were relative to a fixed value of 

0 ft at the upper vertical surface of the model. 

A conceptual model was developed to represent 

a system of flow zones and less-permeable layers that 

generally resemble the system of aquifers and 

50 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
confining layers present in the vicinity of the C-1111 1.5 miles wide, and 5 columns 10 miles wide. Confintest well, 4.9 miles inland of the coast of the Gulf of ing zones associated with the Hawthorn Formation 

Mexico near Naples, Florida. This conceptual model (layers 2, 4, 6, and 8), known to consist of elastics of 

was based on the geologic description (section titled low permeability, were assigned an isotropic hydraulic 

Hydrogeologic Framework) provided by Bennett conductivity of 0.001 ft/d. This value was sufficiently 

(2001b) for well C-1111. Apparent differences in low to maintain a hydraulic head difference of 35 ft 

lithology and permeability were used to delineate between the surficial layer and flow zones (layers 9, 

thicknesses and depths of 20 distinct layers (fig. 38). 11, 13, 15, 17, and 19) within the Floridan aquifer 

Hydraulic conductivities for the layers were hypothe- system. A single value of isotropic hydraulic conducsized based on knowledge of hydraulic characteristics tivity was assigned to confining zones (layers 10. 12. 

of similar lithologies in other parts of southern Florida. 14, 16, 18, and 20) associated with the Floridan aquiThe model was discretized horizontally, pro- fer system. This value was varied in model tests to test 

ceeding from the left side (assumed to correspond to the influence of this parameter on the propagation of 

the coast) into 10 columns 1 mile wide, a column oceanic-tide oscillations into the aquifer flow zones. 

E 

CELLS: GULF 

1 MILE WIDE COAST WELL 

CELL: CELLS: ft2 C-1111 

T /1.5 MILES WIDE 10 MILES WIDE

LAYER 1 r-`7 

10,000 1 

6,000 3 

5,000 5 

6,000 7 

45,000 9 

72,500 11 

3,000 13 

2,000 15 

5,000 17 

35,000 19 

2,595 FEET DEEP 

EXPLANATION 

FLOW ZONES -- values of transmissivity 

(T) listed in column to left of grid 

HAWTHORN CONFINING LAYERS --

vertical hydraulic conductivity = 0.001 1-dt--

MID-FLORIDAN CONFINING LAYERS --

vertical hydraulic conductivity varies in model analyses 

Figure 38. Model grid design used for testing concepts related to the inland influence of ocean tides on pressure heads in 

wells. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 51 
Layers 9 and 11 were associated with monitor 

zones MZ-1 and MZ-2, respectively, and were assigned 

a horizontal hydraulic conductivity of 500 ft/d, which 

was equivalent to transmissivities of 45,000 and 

72,500 ft2/d, respectively. Layer 19 was associated with 

MZ-3 and was assigned a horizontal hydraulic conductivity of 1,000 ft/d (transmissivity of 35,000 ft2/d). Other 

flow zones were assigned a horizontal hydraulic conductivity value of 100 ft/d. Vertical hydraulic conductivities for the flow zones were a tenth of the horizontal 

values. The input to MODFLOW-96 required the specification of vertical leakances; these were computed 

from the specified layer thicknesses and vertical 

hydraulic conductivities. Confined storage coefficients 

were estimated by using layer thicknesses and the ruleof-thumb value, 0.000001 per foot of thickness, suggested by Lohman (1979). 

MODFLOW does not simulate pressure loading 

from the surface. Tidal oscillations specified as a surface boundary condition in layer 1 would not have 

been transmitted hydraulically across Hawthorn confining layers tight enough to maintain a pressure-head 

difference of 35 ft. Therefore, the tidal-pressure oscillations were represented by specified-head conditions 

for all layers along the vertical ocean boundary in column 1 using module FHB (Leake and Lilly, 1997). 

This was exactly analogous to the conceptual models 

of Ferris (1951) and Van der Kamp (1972) that postulate that pressure loading from the surface occurs 

beneath the ocean seaward of the shoreline, but is not 

transmitted vertically inland of the shoreline. The 

specified tide was a simple harmonic of amplitude 

0.38 ft and with period 25.819341 hours. Both 

specifications are parameters of the main lunar diurnal 

(Os ) in the ocean tide at Naples. The tidal oscillations 

were normalized about 0 ft in layers 1-8, but were 

increased by an additive constant of 35 ft in layers 

9-20 to correspond to the presumed head difference 

across the Hawthorn. There is actually a head gradient 

through the vertical thickness of the Hawthorn, but this 

assumption simplified the model design without 

affecting the objectives of the exercise. 

The model was run for 100 simulation days with 

240 time steps per day. Some aperiodic hydraulic 

trends were evident in early time steps as the model 

simulated the hydraulic adjustment to some degree of 

vertical leakage in parts of the grid. Simulated water 

levels at horizontal and vertical grid locations corresponding to monitor zones in well C-1111 were downloaded to special files for graphical representation. 

The first experiments performed with the model 

were to test the response of water levels at the monitorzone locations to variations in the vertical hydraulic 

conductivity of the confining layers in the layers 

(9-20) representing the Floridan aquifer system. The 

uniform specification of this parameter value in these 

layers was given values of 0.1, 0.05, 0.01, and 

0.001 ft/d (fig. 39). Given the lowest specified value, a 

strong tidal response occurs at the position (column 6, 

layer 19) of the lowest monitor zone (MZ-3) approximately normalized about the value of 35 ft used as a 

base value to the boundary tidal specification. With 

increasing specified values for the parameter, however, 

the tidal response lessens and the average pressurehead value declines. When the specified vertical 

hydraulic conductivity is 0.1 ft/d, the tidal variation is 

barely detectable, and the average pressure-head value 

is about 33.4 ft. 

The explanation for these results is that higher 

values of vertical hydraulic conductivity represent a 

loss of confinement for the permeable layer (aquifer). 

Oscillations in head caused by ocean tides are negligible in vertically adjacent confining layers because of 

their low transmissivity. If these confining layers are 

slightly permeable vertically, pressure oscillations 

propagating into the aquifers from the ocean dissipate 

vertically into them. This result highlights the importance of aquifer confinement in the theory of Ferris 

(1951) and Van der Kamp (1972), and the potential 

usefulness of a solution that accounts for vertical leakage (Li and Jiao, 2001a). 

A second set of model runs was made to test the 

effect of aquifer transmissivity on the response of the 

water level at the location of MZ-3. Transmissivity 

values of 3,500, 35,000, and 350,000 ft2/d were specified (35,000 ft2/d was used in the previous set of model 

runs). The effect of the parameter variation is appreciable (fig. 40); the model illustrates that tidal 

response at inland locations is positively correlated to 

the aquifer transmissivity, as shown by Ferris (1951) 

and Van der Kamp (1972). When the transmissivity 

specified in the model run is 350,000 ft2/d, the amplitude of simulated head oscillations at the monitor well 

location is more than half of that specified at the ocean 

boundary. But when the specified transmissivity is 

reduced to 3,500 ft2/s, the head series is nearly flat, 

and tidal oscillations are barely discernible (fig. 40). 

52 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
						

		

35.0 

35.5 

35.0 

35.5 i 1 1 1 i 1 i 1 i 

34.5 

H SPECIFIED OCEAN TIDE VHC = 0.001

Li.1 

W

LL 

VHC = 0.010 

34.0 EXPLANATION 

VHC is vertical hydraulic 

VHC = 0.050 conductivity in feet per day 

33.5 

VHC . 0.100 

33.0 1 i I I I I I I I 

90 91 92 93 94 95 96 97 98 99 100 

DAYS 

Figure 39. Simulated ocean-tide induced head oscillations in MZ-3 at the approximate inland distance of well C-1111 

(5 miles), assuming various values of vertical hydraulic conductivity for overlying and underlying confining layers. 

EXPLANATION 

SPECIFIED OCEAN TIDE T is transmissivity in 

square feet per day 35.000 

34.5 1 1 1 I 1 L I 1 

90 91 92 93 94 95 96 97 98 99 100 

DAYS 

Figure 40. Simulated ocean-tide induced head oscillations in MZ-3 at the approximate inland distance of well C-1111 

(5 miles), assuming various values of transmissivity for the aquifer. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 53 
Using Heads Influenced by Earth Tides for 

Estimates of Specific Storage, Porosity, 

and Bulk Modulus of the Aquifer 

The second cited category of research is the use 

of inland head oscillations caused by earth-tide and 

periodic atmospheric-pressure oscillations to derive 

estimates of the elastic storage capacity of the aquifer. 

Because both forces are effectively uniform in magnitude over large lateral distances, the effect on the confined aquifer also would be uniform over large 

distances (assuming no areal variation of barometric 

efficiency) and would, therefore, not induce additional 

lateral flow in the aquifer. (The possibility of inducing 

vertical flow is discussed later.) Because lateral flow is 

not affected by the periodic stresses, no information 

can be derived regarding the horizontal transmissivity 

of the aquifer. 

Jacob (1940) and Lohman (1979, p. 9) cite 

Theis (1938) for presenting the expression for the unitless storage coefficient (S) of an aquifer: 

S = pri/3(1i + (21, (55) 

where 

p is weight density (pound-force/L3), 

is porosity (unitless), 

B is aquifer thickness (L), and 

a and 13 are as defined for equation 1. 

Dividing equation 55 by the thickness B results in the 

definition of specific storage (S5 ) (L-') of the aquifer. 

By considering the dilatation of the aquifer 

material and relating it to the specific storage of the 

aquifer (SS ) and to the tide-generating potential (W2 ), 

Bredehoeft (1967) derived the following relation 

between specific storage, the tide-generating potential, 

and the head (h) oscillation observed in a well open to 

the aquifer: 

-2vrh-6/ dIt', 

Ss = — (56) - v A (/; dh ' 

where 

v is the Poisson ratio of the aquifer material 

(dimensionless), 

and / are Love numbers at the surface of the earth 

(dimensionless), 

a is the radius of the earth (L), and 

g is the gravitational acceleration at the surface 

of the earth (L/T2). 

The Love numbers are used to relate the theoretical 

tidal potential to actual displacements at the earth's 

surface (Bredehoeft, 1967).The Love numbers have a 

theoretical basis but usually are determined experimentally. 

The expression for the vertical tidal strain developed by Bredehoeft (1967) is re-derived by Hsieh and 

others (1988) using two approaches: (1) that of Bredehoeft and (2) one that follows the development of Van 

der Kamp and Gale (1983), but with the solid grains 

assumed incompressible. The functional form of the 

equation for the theoretical tidal potential (Munk and 

MacDonald, 1960) is obtained from a manual published by the British Navy (Doodson and Warburg, 

1941): 

W2 (0, 4, t) = gKm bf(0)cos[(3(:1), 1)1 , (57) 

where 

Km is the general lunar coefficient, taking into 

account the masses of the earth and moon, 

the distance to the moon, and the earth's 

radius, and is equal to 53.7 cm (1,7618 ft); 

b is an amplitude factor (unitless) that has a distinct value for each tidal component with 

period , 

f(8) is the latitude function (unitless); and 

13(0, t) is a phase term that depends on the longitude •131 

and the Greenwich Mean Time (GMT) t. 

The terms W2 and ag each have units of L2/T2. The 

dimensionless terms b, f(6), and 13(4,t) were obtained 

for individual harmonic components of the tide from 

Munk and McDonald (1960) and Doodson and Warburg (1941) and are shown in table 7 for seven major 

tidal components (PI and K2 are treated separately 

from K1 and S2, respectively, in the cited references 

and also are used in some of the subsequent calculations). The Munk and MacDonald reference contains 

several errors in the specifications of the functionsf(0) 

and 13((,t). Because neither reference is organized for 

ease of use by hydrologists, it was found useful in this 

study to use both sources and to correlate their 

information. 

54 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
Table 7. Some parameters for diurnal and semidiurnal 

equilibrium tides 

(Periods, frequencies, and common names for tidal components are given 

in table 4. Symbols: 8. latitude: q, angular velocity of the Earth relative to 

the mean sun (15 degrees per mean solar hour): CO, longitude of the mean 

sun (increasing by 0.0411 per mean solar hour): (LW. mean longitude of 

the moon (increasing by 0.549 degrees per mean solar hour): 41,(t). mean 

longitude of lunar perigee (increasing by 0.(X)46 degrees per mean solar 

hour): and 43., longitude of the observation point. Further definitions of 

terms are provided in Munk and MacDonald (1960) and Doodson and 

Warburg (194 I )1 

IMPPPIIMPPIIMPIPMITIPIPP Tidal 

com- b f (0) MoM 

ponent 

0, 0.377 sinOcos0 yr + c),( t) - 2c)„,(t) - 169.8 degrees +4) 

13, 0.176 sitiOcos9 yt - 4),(t) - 10.2 degrees + 

K, 0.531 sinews() yt + + 10.2 degrees + 

N, 0.174 0.5 cos213 2(qt +4),(t) - 1.54),„(t) + 0.54)p(t) 

- 79.8 degrees + 4)) 

0.908 0.5 cos-0 2(yt +4),(t) -4:•np) - 79.8 degrees +4)) 

s, 0.423 0.5 cos20 2(qt + 4)) 

K, 0.115 0.5 cos'e 2(qt +4),(t) - 79.8 degrees + 4)) 

The minus sign in equation 56 indicates that 

the head in the aquifer decreases as the tide-generating 

potential increases. The entire term in brackets is a 

constant. Values for the coefficients (n = 0.27, h = 

0.60, 7= 0.07, a = 6.371 x 108 cm (centimeters) = 

2.0902 x 107 ft) were obtained from Marine (1975). 

The value of the gravitational constant was g = 979 

cm/s- (centimeters per second squared) = 32.12 ft/s2 at 

the latitudes (26-27 degrees) of the observation wells 

analyzed during this study. Although Marine's application was to sand, clay, crystalline basement rock, sandstone, and mudstone, P.A. Hsieh (U.S. Geological 

Survey, oral commun., 2002) states that the Poisson 

ratio does not vary much from 0.25 in any rocks in 

which it has been measured. Using the cited values, the 

value of the term in brackets is 0.788 x 101' (cm2/s2 )-1, 

or 0.7321 x 10-9 (ft'/s-)-1. 

Although equation 56 is written as a derivative, 

mathematically the instantaneous change of W, as a 

function of the instantaneous change of aquifer head 

(h), the derivative can be approximated by a finite differential, the change in W„ (OW,) when the aquifer 

head changes by an amount Ah. In the present study, 

the ratio of small finite changes OW, and Ah in a small 

time period At are considered to be proportional to the 

ratio of the corresponding amplitudes of the theoretical 

and observed tides. 

Bredehoeft (1967) applies equation 56 to individual harmonic components of the head oscillations 

caused by earth-tide forcing. As previously noted, the 

solar components of the periodic head oscillations are 

caused partly by periodic atmospheric-pressure oscillations, so that their use in the equation 56 could lead 

to inaccurate results. Again, only the lunar components 

of the head oscillations (0, and M,), considered to be 

caused wholly by earth tides, are used for the analysis. 

Heads from C-1111 are excluded from use in the analysis because the lunar components are primarily the 

result of the ocean tide. 

If the amplitude of a harmonic component of W, 

is denoted A2(T,0), where t is the period of the harmonic component, and the amplitude of a component 

of the head change of period t is denoted A„(t ), then 

an approximation of equation 56 is: 

0) 

Ss = 0.7321x10 where (58) Ah()t 

A 2(t , 0) = gKmbl(0) . (59) 

A,(T,0) is a function of latitude (0), but not of longitude (4)). Using equations 58-59, estimates of S, range 

from 0.49 x 10-6 to 0.76 x 10.6 ft-1 and average 

0.60 x 10.6 ft-1 (table 8). These values are lower than 

the rule-of-thumb estimate (1.0 x 10' ft-I ) suggested 

by Lohman (1979, p. 8). According to E.P. Weeks 

(U.S. Geological Survey, written commun., 2003), 

Lohman's estimate is based on data from unconsolidated aquifers, and specific storage values for consolidated rock should be lower. The estimates based on the 

0, and M, harmonic components of the measured 

earth-tide effect appear to be mutually consistent in 

each monitor zone. 

Use of the theoretical tidal potential as a standard of comparison for well pressure-head observations is not without some uncertainty. Beaumont and 

Berger (1975) analyzed data from wells and determined that the actual tidal potential could differ appreciably from the predicted value as a result of 

topographical variations and geologic nonuniformities. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 55 
Table 8. Summary of calculations for specific storage (Se) and porosity based on head data from 

monitor zones in wells C-1112 and HE-1087 

Latitude 

Moni- (0) 

tor (degrees, 40) (days) b 

zone minutes, 

A2(t ) 

(ft2/s2) 

Alhtt ) 

(ft) 

Ss Be Porosity 

(ft-1 x 10-6) (percent) (percent) 

seconds) 

Well C-1112 

MZ-2 26° 24' 48" 0.3984 1.0758 (0,) 0.377 8.4994 0.01074 0.58 63 25 

MZ-2 26" 24' 48" .4011 .5175 (M,) .908 20.6095 .02476 .61 63 27 

MZ-3 26° 24' 48" .3984 1.0758 (0,) .377 8.4994 .01278 .49 71 24 

MZ-3 26° 24' 48" .4011 .5175 (M2 ) .908 20.6095 .03022 .50 71 25 

MZ-4 26° 24' 48" .3984 1.0758 (0,) .377 8.4994 .00978 .64 49 22 

MZ-4 26° 24' 48" .4011 .5175 (M2 ) .908 20.6095 .02679 .56 49 19 

Well HE-1087 

MZ 26° 36' 30" .4004 1.0758 (0,) .377 8.5421 .00961 .65 75 34 

MZ 26° 36' 30" .3997 .5175 (M2) .908 20.5377 .01991 .76 75 39 

Jacob (1940) derived an equation that relates 

porosity and specific storage: 

11= (60) 

PP 

Equation 60 is obtained by combining equations 1 

and 6 with equation 55. As before, Be refers to the 

barometric efficiency measured from observations of 

the water level in a well open to the atmosphere. 

Parameter values are i3 (the compressibility of water) = 

3.34 x 0-6 psi- ', or 2.32 x 10.8 ft2/pound-force, and p 

(the weight density of water) = 62.4 pound-force/ft3. 

Using previously estimated values for barometric efficiency (table 3), a series of estimates of porosity can 

be made based on the computed values of specific 

storage (table 8). The values of these porosity estimates, ranging from 19 to 39 percent, appear to be 

mutually consistent within each monitor zone, and are 

consistent with typical values of porosity measured on 

cores and used in modeling applications. Numerous 

measurements of porosity on cores from the monitor 

zone in HE-1087 are cited by Bennett (2001a), and 

range from 20 to 43.5 percent, with the majority of 

values falling between 30 and 40 percent. 

Recent research into methods of estimating 

specific storage, porosity, and bulk modulus 

Another relation between specific storage, 

porosity, and bulk modulus that can be implemented 

by the acquisition of the appropriate field data was 

derived by Robinson and Bell (1971). Their theory 

"extends the development of Bredehoeft (1967) by 

considering the separate aquifer dilatations caused by 

the barometric tide and the ocean tide (in coastal 

areas) as well as the solid earth tide" (Rhoads and 

Robinson, 1979). Rhoads and Robinson (1979) refine 

the derivation. Using a definition of barometric efficiency due to Bear (1972), Rhoads and Robinson 

(1979) provide additional equations for estimating 

porosity and the aquifer matrix bulk modulus (inverse 

of compressibility). 

Cooper and others (1965) stated that "when the 

aquifer is compressed, the change in volume of the 

solid material due to deformation of the individual particles is small in comparison with the change in volume of the water." Thus, compressibility of the solid 

fraction is assumed to be negligible (1/K, = 0) in the 

development of equation 56 and "volume changes of 

the formation are taken as equal to changes of the pore 

volume" (Van der Kamp and Gale, 1983). Cooper 

and others (1965) do note, however, that "This 

assumption is apparently valid for granular aquifers, 

56 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
but, depending on the pore geometry and Poisson's 

ratio, it may not be valid for such aquifers as limestone 

and basalt." 

Van der Kamp and Gale (1983) developed 

expressions for specific storage (Ss ) and its relation to 

earth-tide strain that include the effect of the compressibility of the solids because "the compressibility 

of solids may not be negligible for many common formations, especially if they have low compressibility 

and low porosity." Their "expression for the coefficient 

of specific storage with vertical deformation only, 

including the effect of compressibility of solids" is: 

1 1 1 Ss = pg[(- — 1)(1 — X.) + ri(- — — , where (61) K K K f Ks 

2(1 —2v)(i _ IC) (62) 3(1 —v) Ks) • 

As before, 

K is the bulk modulus of elasticity of the formation, 

Kf and K, are the bulk moduli of elasticity of water 

and "the solid fraction including the effect 

of unconnected pores" (Van der Kamp and 

Gale, 1983), 

v is Poisson's ratio, and 

ri is the interconnected porosity of the formation. 

If the compressibility of solids is considered to be zero 

( 1/K, = 0), equation 61 reduces to a form that Van der 

Kamp and Gale (1983) consider equivalent to the storage coefficient derived by Jacob (1940), when the confined modulus of elasticity (K) (equation 5) is used in 

place of the bulk modulus (K). 

Finally, Van der Kamp and Gale (1983) derive 

the relation between aquifer-pressure oscillations and 

the earth-tide strain. Inserting the relation between 

earth-tide strain (dilatation) and the theoretical tidal 

potential (W2) assumed by Bredehoeft (1967), and 

solving for Ss in terms of the corresponding changes in 

W2 and pressure-head in the aquifer (h), a more precise 

expression for S, results: 

K —2v)ph 61)1 d2 , W (63) Ss = —[(1 — —Ks A - v ag dh 

where all terms have been previously defined. 

As the bulk modulus of the formation (K) is less than 

the bulk modulus of the solid fraction (K,) (bulk modulus is the inverse of compressibility and increases as 

compressibility decreases), the first term in parentheses in equation 63 is always between 0 and 1. If the 

solid fraction is incompressible, as assumed by Bredehoeft (1967), this term is unity and equation 63 

reduces to equation 56. No attempt was made to apply 

this relation in the present study because of the lack of 

information available to support estimates of K,. However, assuming some compressibility of the solid fraction of the carbonate aquifers in the study area, using 

equation 63 would have resulted in some reduction of 

the previously calculated estimates of specific storage. 

Rojstaczer and Agnew (1989) followed Van der 

Kamp and Gale (1983) in assuming that the solid portion of the aquifer matrix was compressible to some 

extent. Rojstaczer and Agnew (1989) also removed the 

assumption of previous papers that uniform surface 

loading caused only vertical deformation of the aquifer 

material. The theory analyzes water levels in an open 

well penetrating a confined aquifer for responses to 

earth-tide and atmospheric loading. Accounting for 

lateral deformation leads to an iterative procedure for 

determining matrix compressibility, after which the 

porosity can be determined. Given these parameters, 

specific storage can also be determined. 

Dual porosity in aquifers, where a permeable 

fracture network is embedded in a less permeable 

porous matrix, and where the aquifer is isotropic and 

homogeneous at the continuum scale, was studied by 

Desbarats and others (1999). Water-level and atmospheric-pressure fluctuations are analyzed for barometric efficiency. Independent estimates are required 

for the porosities of the fractured and porous parts of 

the aquifer. Distinct pressure fields are generated in the 

fractures and the pore space by changes in atmospheric 

pressure that may take an appreciable amount of time 

to equilibrate. The theory is used for estimates of specific storage. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 57 
Using Heads in Open Wells Penetrating 

Confined Aquifers for Estimates of the 

Transmissivity and Storage Coefficient of 

an Aquifer 

The well-aquifer system has been a productive 

scenario for theoretical and applied hydrologic studies 

from the early days of scientific hydrology. In most 

theoretical developments, the well is assumed to be 

open to the atmosphere, and the well is assumed to be 

either screened (or open) throughout the thickness of a 

confined aquifer and cased to the top of the aquifer 

(fig. 41). When oscillations of pressure head in the 

aquifer occur, water moves into or out of the screened 

or open part of the well. This causes an oscillation of 

the water surface in the open well, which, depending 

on aquifer and well characteristics, might be damped 

or lagged in time relative to the pressure-head oscillations, although the frequency will remain the same. 

Characteristics of the oscillation of the water 

level in the open well in response to a pressure-head 

disturbance in the aquifer are a function of the periodicity of the disturbance, the aquifer parameters (transmissivity and storativity), as well as inertial and 

storage effects affecting water movement in the well 

and the air pressure at the water surface in the well 

(atmospheric pressure). Given adequate field data, 

mathematical solutions for the well-aquifer interaction 

can be used to estimate values of aquifer coefficients. 

Jacob (1940) used the open artesian well scenario 

to define concepts of barometric and tidal efficiency 

and the response of well water levels to periodic pressure-head variations originating at a remote line source 

within the aquifer (the ocean-tide theory previously 

described). A relation between the motion of water 

within an open wellbore (taking into account the inertia of water within the wellbore) and pressure-head 

oscillations in the confined aquifer was developed by 

Cooper and others (1965) with the objective of 

describing mathematically the water-level response to 

seismic waves. The resulting equation established a 

relation between the amplitude and phase lag of oscillations of the water level in the well to the amplitude 

of oscillations of pressure-head in the aquifer. 

Bredehoeft (1967) showed that the inertial effects 

were negligible when the transmissivity of the aquifer 

was above a low value (0.001 ft'-/s), and his relation 

(equation 56) for estimating specific storage as a function of the change in water level within a well depends 

on this result. 

Cooper and others (1967) considered the problem of the response of the head in an aquifer and in a 

well following an instantaneous charge of a known 

volume of water to the well. Inertial effects related to 

the movement of water in the well are ignored. Their 

results are used to relate the time rate of change of the 

water level in the well to the aquifer properties (transmissivity and storativity), leading to an analytical 

method for determining these properties from field 

data acquired in slug testing of the well. The results of 

Cooper and others (1967) were used as an element of 

at least one solution (Furbish, 1991) to the problem of 

determining aquifer parameters from periodic or tidal 

data. 

The well-aquifer scenario was used by Weeks 

(1979) in an analysis of the effect of atmospheric-pressure fluctuations on the water levels in open wells 

screened below the water table in deep unconfined 

aquifers. Because the unsaturated zone resists air 

movement, a temporary imbalance occurs between 

pressure below the water table and that at the water 

surface in the well. The theory of Weeks (1979) provides for the estimation of pneumatic diffusivity of the 

unsaturated zone, knowledge of which enables well 

water levels to be corrected for the effects of atmospheric-pressure fluctuations. According to E. P. 

Weeks (U.S. Geological Survey, written commun., 

2003), this approach may not be applicable in southern 

Florida because of the shallow depth to the water 

table. 

The response to earth tides of an open well 

screened in a confined aquifer was studied by 

Narasimhan and others (1984), who recognized the 

importance of wellbore storage effects, the period of 

the tidal pulses, and the aquifer properties (permeability and specific storage). They applied a numerical 

model of saturated flow to demonstrate the qualitative 

importance of these factors. These relations were presented analytically by Hsieh and others (1987). Ignoring inertial effects of water in the wellbore, Hsieh and 

others (1987) determined the time-varying drawdown 

in the open well as a function of periodic pressurehead oscillations in the aquifer (considered to be a 

simple harmonic function). Expressions were derived 

for the amplitude response (A) and the phase shift cri): 

o -1/2

A = x = (1.7 + F-) and (6-1) 

h0 

58 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
WATER LEVEL 

IN WELL 

X ='Coin 0.)t 

CONFINING 

LAYER 

INITIAL PIEZOMETRIC SURFACE WELL 

CASING H 

• 

pf=pg(H+Z) 

+p0 sin ((01-11) 

AQUIFER 

WATER MOVEMENT 

TO / FROM WELL 4 

WELL 

SCREEN 

/7 7 7 7 7 7 / / / 7 / 7 7 /// 

EXPLANATION 

x is displacement of water in well ( L ) 

xc, is amplitude of water-level fluctuation ( L ) 

CO is angular frequency of tidal disturbance (radians / T ) 

1 is time ( T ) 

H is height of water column in well casing ( L ) 

p1is fluctuating pressure in aquifer ( :19:'Li 2 ) 

p is pressure ( M/LT 2 ) 

g is gravitational acceleration ( L / T ) 

Z is depth below bottom of casing ( L ) 

po is amplitude of pressure fluctuation ( MIL T2 ) 

p is density of water ( Al / L3) 

11 is phase angle (radians) 

M is mass unit 

L is length unit 

T is time unit 

Figure 41. Idealized representation of an open well in which water-level oscillations are caused by oscillation of 

artesian pressure (modified from Cooper and others, 1965). 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 59 
= —atan (F/ E) , where (65) 

E = 1 0)r2 

2 T Kei(oc , and (66) 

(ors. 

2T Ker „,), and (67) 

vt. 

( 0)S) I/'

r (68) 

where 

xo is the complex amplitude of the water-level 

oscillation in the well; 

h0 is the fluctuating pressure head in the aquifer; 

rw is the radius of the screened or open portion of 

the well; 

re is the radius of the well casing; 

w is the frequency of the oscillations; 

T is the aquifer transmissivity; 

S is the storage coefficient; and 

Ker and Kei are Kelvin functions of order zero. 

The expressions for E and F are simplified expressions 

for values of oc, < 0.1, which is generally the case for 

realistic values of r„,T, and S in earth-tide analysis 

(Hsieh and others, 1987). The theory was developed 

for "a single, laterally extensive confined aquifer that 

is homogeneous and isotropic." 

The required data analysis is similar to that done 

in the ocean-tide application or for the computation of 

specific storage using the method of Bredehoeft. For 

the earth-tide analysis, two sets of data are needed to 

determine the amplitude ratio or the phase shift: 

(1) water-level oscillation in the well and (2) pressurehead measurements of the tidally generated oscillations in the aquifer. Pressure-head measurements can 

be obtained from a pressure transducer installed in a 

shut-in or packed-off well. Dilatometer instruments 

have been used in a few cases. If measurements of 

pressure heads or the dilatation of the aquifer are not 

available, the theoretical tidal potential (W2) may be 

used to derive dilatation estimates, subject to reservations concerning departure from theoretical values 

owing to local conditions. 

The well water-level and aquifer pressure-head 

data are detrended; then, the regression analysis is 

applied to determine the amplitude and phase of the 

five major tidal components of each data set. The 

analysis proceeds with a comparison of the lunar (01 

and M2) components of the two data sets, known to not 

be affected by atmospheric-pressure oscillations. The 

comparison can be done iteratively on a trial-and-error 

basis using equations 64-68, by introducing various 

estimates of T and S. Hsieh and others (1987), however, applied a different approach. Converting from the 

variables oc, and wre2/T to an alternative set of dimensionless parameters, Ti/r,2 and Sr„,21re2, where ti is the 

period ('r = 21r/w), they prepared graphs of the amplitude ratio and phase shift computed from equations 

64-68 as a function of 7t/r,.2 for selected values of 

Sr„,21r,2. These illustrations have been expanded and 

replicated (figs. 42-43) for inclusion in this report. 

While these are not referred to as type curves by 

Hsieh and others (1987), they could serve a similar 

purpose. It is clear that, given estimates of the amplitude ratio and phase lag based on field data and an estimate of the storage coefficient, values of the 

dimensionless transmissivity can be obtained from the 

graphs in figures 42-43 and translated into common 

transmissivity units (ft2/d) by using the period of the 

harmonic cycle (r) and the casing radius (re). Use of 

the phase lag graph (fig. 43) can lead to two interpretations; some pre-knowledge of the likely range of transmissivity of the aquifer could be useful in this case. 

Because of the closeness of the graphs for the 

various values of storage coefficient in figures 42-43, 

it is noted that the amplitude ratio and phase lag are 

each weak functions of storage coefficient, as noted by 

Hsieh and others (1987). If a reasonable estimate of 

the storage coefficient is available from the literature 

or previous aquifer testing, then the relations shown in 

these figures can be used to provide reasonably precise 

estimates of the transmissivity. 

The graphs (figs. 42-43) also indicate a substantial limitation to the applicability of these methods to 

data from some hydrologic environments. It is noted 

that, for values of the dimensionless transmissivity 

approaching 1 x 103, the amplitude ratio curves flatten 

and converge as they approach the value of 1.0 (no 

reduction in amplitude) and the phase lag curves 

flatten and converge as they approach the value of 0.0 

(no phase lag). The physical interpretation is that, for 

sufficiently high values of aquifer transmissivity, the 

amplitude of earth-tide oscillations is not reduced in 

the well water levels, nor do the water-level oscillations in the well exhibit a phase lag. 

60 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
	 	

-100 

1.0 

AMPLITUDERATIO 

EXPLANATION 

S - storage coefficient (unitless) 

0.8 S'- redefined storage coefficient (unitless) 

T - transmissivity (L2/ T) 

T'- dimensionless transmissivity 

r - period (T) 

rw - radius of well (L) 

0.6 rc - radius of casing (L) 

0.4 

0.2 

0.0 

0.01 0.10 1.00 

= Srw2 

o-7 

10.00 100.00 1000.00 

T' = T /r2 

Figure 42. The ratio of the amplitudes of the well water-level and aquifer pressure-head oscillations as a function of 

dimensionless transmissivity, for various values of redefined storage coefficient. 

111111 1 11111 

PHAS

ELAG, INDEGREES

-80 , ------

10-5 

-60 

= Srw2irc2 = 10 

3 10-6 

-40 

-20 

EXPLANATION 

S - storage coefficient (unitless) 

S'- redefined storage coefficient (unitless) 

T - transmissivity (L2/ T) 

T'- dimensionless transmissivity 

- period (T) 

rw - radius of well (L) 

rc - radius of casing (L) 

10-4 

10-

0.01 

0 

0.10 1.00 10.00 100.00 1000.00 

T' = T /rc2 

Figure 43. The phase lag between the well water-level and aquifer pressure-head oscillations as a function of 

dimensionless transmissivity, for various values of redefined storage coefficient. 

Analytical Methods for Computing Aquifer Properties from Naturally Forced Periodic Data 61 
If 0.95 is accepted as the limit at which the 

amplitude ratio can be defined by field data, then the 

highest transmissivity that can be estimated by the 

amplitude ratio method, given a well casing radius of 

0.5 ft and a tidal harmonic period of 0.5 day, is 37 ft2/d 

for a dimensionless storage coefficient of 10' and 

59 ft2/d for a dimensionless storage coefficient of 10'. 

If the period is assumed to be 1 day, these estimates 

are halved. If 5 degrees is accepted as the limit for 

which the phase lag can be defined by field data, then 

the highest transmissivity that can be estimated by the 

phase lag method, given a well casing radius of 0.5 ft 

and a tidal harmonic period of 0.5 day, is 93 ft'-/d for a 

dimensionless storage coefficient of 10' and 182 ft2/d 

for a dimensionless storage coefficient of 10-7. Again, 

if the period is assumed to be 1 day, these estimates are 

halved. 

The conclusion is that the method of Hsieh and 

others (1987) probably is applicable only to aquifers 

having transmissivities less than 500 ft2/d. In fact, the 

funding for development of similar methods has 

come from sources interested in testing the degree 

of confinement of formations vertically adjacent to 

hazardous waste repositories (P. A. Hsieh, U.S. Geological Survey, oral commun., 2002), where normal 

aquifer testing procedures are considered hazardous. 

Estimates of transmissivity of the upper Floridan aquifer developed as part of the present study, and the earlier estimates by SFWMD based on aquifer testing, 

range above 7,000 ft2/d. Upper Floridan transmissivity 

estimates less than 500 ft2/d are not known to have 

been reported; thus, the usefulness of the method 

described here in the upper Floridan aquifer is quite 

limited. It also is noted that no open wells are constructed into the upper Floridan aquifer in the study 

area. Such wells would need standpipes several tens of 

feet above land surface because of the appreciable 

artesian head in the upper Floridan aquifer. Shut-in 

wells might not produce useful information regardless 

of the type of pressure transducer used, because the 

water level may not exhibit a response to the storage 

properties of the well. Shut-in wells could, however, 

provide useful comparison data if open-well data were 

available from another location. 

Solutions to the response of the well-aquifer 

system to periodic pressure oscillations that take into 

account additional aspects of the hydrologic system 

include the work of Rojstaczer (1988). Rojstaczer considers the effect of periodic atmospheric-pressure 

oscillations on an aquifer system that includes a 

partially confined aquifer separated from a surficial 

aquifer by a partially confining layer. The solution 

takes into account "groundwater flow between the 

borehole and the aquifer, groundwater flow between 

the aquifer and the water table, and air flow between 

the land surface and the water table through the unsaturated zone." The water-level response in the well may 

show a phase lag because of some of these processes. 

For convenience, it is assumed that no pressure differential occurs between the aquifer and partially confining layer. Measured data are used for an estimate of 

barometric efficiency. Parameters that can be estimated include the pneumatic diffusivity of the unsaturated zone, the vertical hydraulic diffusivity of the 

partially confining layer, and the lateral hydraulic diffusivity of the partially confined aquifer. Rojstaczer 

identifies three frequency bands (of the perturbing 

atmospheric pressure), in each of which a distinct set 

of parameters determines the response. 

The analysis of wells in an unconfined aquifer 

was continued by Rojstaczer and Riley (1990), who 

considered the response to earth tides and atmospheric 

loading and reported methods of determining both the 

pneumatic diffusivity of the unsaturated zone and the 

vertical hydraulic conductivity of the aquifer. A vertical diffusion equation was stated and solved for the 

variable pressure within the wellbore. Lateral permeability was assumed to be high enough that wellbore 

storage effects could be ignored. 

Another solution to the well-aquifer problem 

was documented by Furbish (1991), who suggested 

that it could "be used to filter raw loading and waterlevel records" to remove atmospheric-pressure influences. The solution assumes that "the aquifer transmissivity is large relative to the permeability of the 

confining layer, the confining layer is thick, and 

oscillations in atmospheric load are of moderately 

high dimensionless frequency." The solution is a convolution approach. A change in atmospheric pressure, 

converted to its equivalent in feet of water, is considered to be equivalent to a slug or bail test of the well 

(the immediate injection or withdrawal of a volume of 

water in the wellbore), and the solution sums the 

responses to a discrete series of atmospheric-pressure 

changes, treated formally as a series of slug or bail 

tests. 

Two alternatives are suggested for the response 

function. One is precise but computationally intensive, 

based on the solution of Cooper and others (1967) for 

the aquifer response to a slug test. The other, due to 

62 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
Hvorslev (1951) is computationally simple but 

applicable only to aquifers with small storativity. 

Application of the solution requires making prior estimates of aquifer parameters and then trying to replicate observed water-level oscillations. Like Hsieh and 

others (1987), Furbish (1991) notes that "for large 

transmissivities . . . little attenuation or lag of the signal occurs," and the method is of little use in these 

cases because any large value of transmissivity will 

replicate the observed well data. 

Using Heads or Water Levels to Estimate 

Vertical Hydraulic Conductivity 

This category of research is the newest discussed in this report. Theoretical developments are 

still tentative and few field applications have been 

made., however, the distinctiveness and promise of the 

approach warrant its being highlighted. 

Earth tides and atmospheric-pressure fluctuations can cause vertical flow in the subsurface in certain cases, and measurements that reveal the degree or 

rate of flow in response to a known degree of forcing 

can, in theory, be used to deduce the values of hydraulic properties related to vertical flow. The theory of 

Weeks (1979) leads to a diffusion equation for head 

fluctuations as a function of pneumatic diffusivity of 

the unsaturated zone. 

Van der Kamp and Gale (1983) note that, under 

tidal strain or atmospheric loading, "vertical flow may 

occur in a formation if the overlying and underlying 

formations have different elastic or hydraulic properties," or under water-table conditions. These authors 

present a diffusion equation that represents their concept of the process. Rojstaczer and Agnew (1989) 

present a vertical diffusion equation that incorporates 

assumptions that the solid fraction is compressible and 

the deformation caused by surface loading is lateral as 

well as vertical. Rojstaczer and Riley (1990) implement this equation in an analysis of the response of 

water levels in an unconfined aquifer to earth tides. An 

attempt was made by Seo and others (2000) to apply 

vertical diffusion theory to data from an unconfined 

aquifer, ignoring effects of air flow in the unsaturated 

zone. Pressure-head data were acquired from buried 

transducers, eliminating the need for any consideration 

of well-aquifer interaction. 

CONCLUSIONS 

Various methods for using well water-level 

fluctuations caused by tidal or aperiodic natural forces 

to estimate aquifer properties are described and evaluated in the preceding pages. Table 9 is a summary of 

the results of the evaluation. Fifteen methods are 

described, of which four (nos. 2-5 in table 9) were 

used in the course of this study. The other methods 

were not used either because they were not suitable for 

the hydrologic environment from which the data sets 

provided for the study were obtained or because they 

required independent estimates of petrophysical properties that were not available. Only limited statements 

can be made about the usefulness or validity of the 

solutions not applied in the present study. A brief 

discussion of these methods is provided as a guide to 

readers who desire to investigate them further. 

The analyses for transmissivity (based on a prior 

estimate of storage coefficient) that used data from 

near-ocean monitor zones using methods 2 and 3 provided answers generally in agreement with results of 

aquifer performance tests performed by the SFWMD. 

More precise estimates, however, were difficult to 

obtain because of the imprecision of estimates of the 

storage coefficients of the individual monitor zones. In 

addition, independent estimates of loading efficiency 

were too low to be consistent with tidal amplitudes 

measured in the monitor zones, so that amplitude ratio 

calculations that required loading efficiency estimates 

could not be completed. Additional research and field 

work to determine loading efficiencies and the pattern 

of variation of tidal amplitudes in heads would be useful in providing additional insight into this problem. 

The phase lag estimates obtained based on separate 

tidal components using method 2 showed a difference 

of somewhat less than 100 percent. 

Although estimates using method 2 were 

imprecise, transmissivity estimates based on aquifer 

testing in carbonate aquifers can show even greater 

differences. Phase lag estimates using method 3 

(assuming some leakance to or from adjacent formations) were in better agreement, and both phase lag 

and amplitude ratio estimates were lower and in better agreement with the SFWMD estimates. A technique of correcting the near-ocean data for the effects 

of earth tides is needed and might substantially 

improve estimates. 

Conclusions 63 
		

0 a) 

o Table 9. Summary of methods evaluated, properties determined by each method, and applicability in southern Florida cD 

z 

CD 

Method 

.2 

0•fi 0 

c c 

= 

o Si) -13 -n a 

I ca 

as 

U) 

1 

2 

Analysis of water-level oscillations caused by ocean tide in 

surficial or fully confined aquifers (confinement ends at 

the coast). Jacob (1940;1950), Ferris (1951). 

Analysis of water-level oscillations caused by ocean tide in 

fully confined aquifers. Confinement is of infinite extent 

beneath ocean. Van der Kamp (1972). 

5 

CD 

m

0 

a 

a) 

3 Analysis of water-level oscillations caused by ocean tide in 

fully confined aquifers or semiconfined aquifers separated 

from the water table by a leaky semiconfining layer. Confinement under ocean is of either finite or infinite extent. 

Li and Jiao (2001a). 

.o 

(/) 

U) 

CD 

3 

0 

4 

5 

Analysis of water-level oscillations caused by earth tide in 

fully confined aquifers. Assumes incompressible solid 

fraction. Bredehoeft (1967), Hsieh and others (1988). 

Computation of porosity of a fully confined aquifer. Jacob 

(1940). 

a) 

to 

m 0) 

a 

6 Analysis of water-level oscillations caused by earth tide, 

ocean tide, and diurnal atmospheric-pressure oscillations 

in fully confined aquifers. Robinson and Bell (1971), 

Rhoads and Robinson (1979). 

ca 

a 

C) 

caa) 

7 Analysis of water-level oscillations caused by earth tide in 

fully confined aquifers with a compressible solid fraction. 

Van der Kamp and Gale (1983). 

03 

a 

8 Analysis of water-level oscillations in an open well penetrating a confined aquifer. Assumes solid portion compressible, also makes other new theoretical assumptions. 

Rojstaczer and Agnew (1989). w 

a) 

0 

3 

9 Analysis of water-level oscillations caused by atmosphericpressure fluctuations in an aquifer characterized by dual 

porosity. Desbarats and others (1999). 

m 

CD 

C) 

U) 

Applied in 

Properties determined present 

study 

Lateral hydraulic diffusivity No 

Lateral hydraulic diffusivity Yes 

Lateral hydraulic diffusivity Yes 

Specific storage Yes 

Porosity Yes 

Specific storage No 

Specific storage No 

Matrix compressibility, poros- No 

ity, and specific storage 

Specific storage No 

Applicability in southern Florida 

Only to data from wells very near coast in surficial aquifers, 

or near the coast of Palm Beach, Broward, or northern 

Dade Counties in Floridan aquifer system. 

Useful for data from wells in confined aquifers near any part 

of the coast of southern Florida except in Palm Beach, 

Broward, or northern Dade Counties. 

Useful for data from wells near coast in southern Florida in 

any aquifer. In surficial aquifers, or (possibly) in the 

Floridan aquifer system in Palm Beach, Broward, and 

northern Dade Counties, reduces to Jacob/Ferris method. 

In fully confined aquifers extensive under ocean, reduces 

to Van der Kamp method. 

Useful for data from wells in confined aquifers at any location. However, data must be free of ocean-tide effects, 

either by location or by filtering. 

Anywhere that locally valid estimates of specific storage 

and barometric efficiency are available. 

Designed for use at inland locations free of ocean-tide influence where barometric efficiency, amplitude, and phase 

of atmospheric-pressure oscillations are known, or near 

ocean where tidal efficiency, amplitude, and phase of 

ocean tides are also known. 

Potentially useful for data from wells in confined aquifers 

where an estimate of the compressibility of the solid 

fraction is available. Data must be free of ocean-tide 

effects, either by location or by filtering. 

Potentially useful for data from open wells penetrating 

confined aquifers affected by earth tide and atmosphericpressure fluctuations. Assumes the availability of an 

estimate of the compressibility of the solid fraction. 

No apparent geographical restriction. However, assumes 

availability of independent estimates of barometric 

efficiency, porosities of the fractured and porous 

fractions, and other rock properties. 
Table 9. Summary of methods evaluated, properties determined by each method, and applicability in southern Florida (Continued) 

Method 

10 Analysis of water-level fluctuations caused by atmosphericpressure fluctuations in open wells screened below the 

water table in unconfined aquifers. Leads to method of 

correcting water levels for effects of fluctuations in 

atmospheric pressure. Weeks (1979). 

11 Analysis of water-level oscillations caused by earth tide in 

an open well penetrating a confined aquifer. Hsieh and 

others (1987). 

12 Analysis of water-level fluctuations caused by atmosphericpressure fluctuations in a partially confined aquifer 

separated from a surficial aquifer by a leaky semiconfining layer. Rojstaczer (1988). 

13 Analysis of the response of the water level in a well in an 

unconfined aquifer to earth tide and atmospheric loading. 

The lateral permeability is high enough that wellbore 

storage effects can be ignored. A vertical diffusion equation is solved for variable pressure in the water column. 

Rojstaczer and Riley (1990). 

14 Analysis of water-level fluctuations caused by atmosphericpressure fluctuations in a well open to a confined aquifer. 

A convolution method is used to replicate the water-level 

response. Furbish (1991). 

15 Analysis of pore-pressure fluctuations caused by atmospheric-pressure fluctuations in an unconfined aquifer. 

A vertical diffusion equation is used to determine the 

vertical hydraulic diffusivity by a trial-and-error process. 

Seo and Choe (2000). 

Properties determined 

Pneumatic diffusivity of the 

unsaturated zone 

Transmissivity and storage coefficient 

Barometric efficiency, pneumatic diffusivity of the unsaturated zone, vertical hydraulic 

diffusivity of the semiconfining layer, and lateral hydraulic 

diffusivity of the partially 

confined aquifer 

Pneumatic diffusivity of the 

unsaturated zone and vertical 

hydraulic conductivity of the 

aquifer 

Transmissivity and storage coefficient 

Vertical hydraulic diffusivity 

Applied in 

present 

study 

No 

No 

No 

No 

No 

No 

Applicability in southern Florida 

Potentially useful for data from open wells in unconfined 

aquifers overlain by a thick unsaturated zone. 

Potentially useful for data from open wells in confined aquifers having a transmissivity less than 500 square feet per 

day. Minimal opportunity for use in southern Florida. 

Potentially useful for water-level data from an open well 

penetrating a partially confined aquifer of low transmissivity. Minimal opportunity for use in southern Florida. 

Potentially useful for data from open wells in unconfined 

aquifers. 

Potentially useful for data from an open well penetrating a 

confined aquifer of low transmissivity. Minimal opportunity for use in southern Florida. 

Potentially useful for pore-pressure data from unconfined 

aquifers. Data must be free of effects on water-table 

altitude caused by recharge and drying. 

SUO!Srli3U00 

rn 

C.71 
The well-aquifer methods (nos. 11-13) for determining transmissivity based on data from inland wells 

are not suitable for use in the study area because these 

methods are based on physical processes that are measurable only in formations of low transmissivity. Permeable zones in the Floridan aquifer system 

commonly have transmissivities much higher than 

those considered sufficiently low for these methods. 

The inland head data, however, was used successfully 

in method 4 to estimate specific storage. Independent 

estimates based on separate harmonic components in 

each monitor zone showed a high degree of consistency, and were of reasonable magnitude. Estimates of 

porosity using method 5 were entirely consistent with 

results from petrophysical analyses performed by the 

SFWMD. 

Using naturally forced data to obtain estimates 

of aquifer properties has been found to provide generally useful transmissivity estimates and realistic estimates of specific storage and porosity. However, the 

methods used have not provided a completely satisfactory answer to the problem of determining properties 

that describe flow and storage in the subsurface. This 

field of research has concentrated on theoretical development supported by "theoretical applications" or 

application to synthetic data sets. While the research is 

useful in providing techniques that have the potential 

to be used by hydrologists, few attempts have been 

made to apply the theory to real-world data sets. 

Although field studies are costly, the most productive 

approach in future efforts might be to combine theoretical development with a fully committed field application with a view to resolving apparent inconsistencies 

between them. A fully developed and tested field technique consistent with theory would be highly useful in 

future studies. 

SUMMARY 

Aquifers are subjected to mechanical stresses 

from natural, non-anthropogenic, processes. These 

include pressure loading or mechanical forcing of the 

aquifer by ocean tides, earth tides, and pressure fluctuations in the atmosphere. The resulting head fluctuations are evident in deep confined aquifers that 

maintain a substantial head difference from surface 

formations across tight confining layers. The present 

study: (I) reviews the research that has been done on 

the use of tidally influenced and other naturally forced 

head fluctuations for estimating the values of aquifer 

properties, (2) determines which of these methods 

would be useful in the hydrologic environment of 

southern Florida, and (3) applies selected algorithms 

to a data set of heads from wells in Collier and Hendry 

Counties. 

The Upper Floridan aquifer in southwestern 

Florida consists of several thin water-bearing zones of 

relatively high permeability interlayered with thick 

zones of much lower permeability. Water in the upper 

Floridan producing zones is under artesian pressure 

and flows from wells open at land surface. Tidally 

influenced head data for this study were acquired from 

monitor zones in three wells drilled by the South Florida Water Management District (SFWMD) as part of a 

program to investigate the Floridan aquifer system in 

southwestern Florida as a possible future source of 

water for public supply. Monitor-zone depth intervals 

extend from 690 feet (ft) in the basal Hawthorn unit to 

2,350 ft in the dolomite-evaporite unit. 

Barometric efficiency is the ratio of the change 

in water level in a well to the atmospheric- pressure 

change at the surface. Barometric efficiency is needed 

in some analytic methods to determine aquifer diffusivity and porosity. Clark (1967) devised a method to 

determine the barometric efficiency based on aperiodic, long-term pressure variations arising from the 

movement of air masses and corresponding head 

changes in the well. Using Clark's method, estimates 

that appeared valid based on data from certain monitor 

zones ranged from 63 to 86 percent. It was concluded 

that the Clark method can be an effective tool for 

determining barometric efficiency when the head data 

are of good quality. It seems essential that a process of 

correcting head data sets for atmospheric-pressure 

effects be applied to determine whether barometric 

efficiency values determined by the Clark method are 

correct. Where they are not or where results of the 

method are ambiguous, barometric efficiency must be 

determined by a trial-and-error approach. 

A modified version of Clark's method was 

applied to total-head data from the same monitor 

zones. Resulting estimates of loading efficiency that 

appeared valid ranged from 15 to 37 percent, and are 

subject to the same reservations as those for barometric efficiency estimates. Barometric and tidal efficiencies theoretically sum to 1.0, as did estimates made in 

this study, so that an estimate of one is equivalent to an 

estimate of the other. 

66 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 
Being of solar and lunar origin, ocean and earth 

tides have a periodicity that can be described as a combination of "harmonic" components, sinusoidal functions of given amplitude and frequency and phase 

relation to some common time reference. Of all the 

harmonic components, only five are of importance 

geophysically, together accounting for approximately 

95 percent of the tidal potential. Analysis of timeseries data collected for this study using the discrete 

Fourier transform technique for spectral analysis 

demonstrated the presence of these five principal 

harmonics. 

The analysis of tidally influenced data sets for 

estimates of aquifer parameters depends largely on 

being able to resolve the amplitude and phase of the 

various harmonic components of those data sets. In 

particular, the amplitude and phase of the five principal harmonic components must be determined. A 

means for doing this is by regression analysis. A computer program to perform this analysis was applied to 

ocean tide data at Naples and to the tidally influenced 

head data in each monitor zone in wells C-1111, C1112, and HE-1087. Before tidally influenced data 

could be analyzed, it was necessary to detrend the data 

so that fluctuations would be strictly tidal in origin. 

The detrending procedure was generally done in two 

steps: (1) estimating a trend line by using a low-pass 

filter, and (2) computing the detrended data set by subtracting the trend line from the raw data. 

Research into the analytical use of tidally influenced and other naturally forced head fluctuations to 

estimate values of aquifer properties can be segregated 

into four principal categories: (1) use of heads influenced by ocean tides to estimate aquifer diffusivity 

(transmissivity divided by storage coefficient); (2)' use 

of heads influenced by earth tides or atmosphericpressure fluctuations for estimating specific storage 

and porosity; (3) use of water levels from open wells 

penetrating a confined aquifer and influenced by earth 

tides or atmospheric-pressure fluctuations to estimate 

aquifer transmissivity and storage coefficient; and 

(4) use of water levels or pressure heads to determine 

vertical hydraulic conductivity. 

The analytical solution of Van der Kamp (1972) 

to the problem of calculating aquifer diffusivity from 

aquifer-head oscillations at a location near enough to 

the ocean to be influenced by ocean tides, and which 

assumes a non-leaky confining unit of infinite extent, 

was applied to data from the three monitor zones in 

well C-1111. In applying this method, only the lunar 

diurnal (0,) and lunar semidiurnal (M,) data were 

used, because the solar components (K1 and S,) were 

influenced by atmospheric pressure. It was observed 

that T values computed based on the M, phase lags are 

about half those computed based on the 01 phase lags. 

Furthermore, loading efficiency estimates were not 

consistent with results of the amplitude ratio analysis, 

as they indicated that tidal oscillations in monitor 

zones should have been much smaller than observed. 

Depending mainly on results of the phase lag 

analysis and assuming a storage coefficient value of 

1 x 10-5, results can be interpreted to suggest estimates 

of T for the three monitor zones to range between 

20,000 and 38,000 feet squared per day (ft2/d) (MZ-1), 

between 15,000 and 27,000 ft2/d (MZ-2), and between 

75,000 and 140,000 ft2/d (MZ-3). These values are 

higher, but in general agreement with the results of 

aquifer performance tests in MZ-1 and MZ-2 performed by the SFWMD in 1996-1997 (Bennett. 

2001b). Transmissivity values estimated from these 

tests were 15,560 and 7,018 ft2/d, respectively, for 

MZ-1 and MZ-2. 

Li and Jiao (2001a) derived a more general solution in which the coastal aquifer system consists of a 

leaky semiconfining layer separating an unconfined 

aquifer from an underlying semiconfined aquifer. 

Assuming that the aquifer and semiconfining layer 

extend for an infinite distance beneath the ocean, and 

assuming a leakance value large enough to permit a 

small degree of leakage into the aquifer, application of 

this method yielded estimates based on the phase lag 

analysis that are mutually consistent, appreciably 

lower than those from the non-leaky Van der Kamp 

method, and closer to the SFWMD estimates. When 

the leakance is 2 x 10-5 d-I , and loading efficiency (Le) 

equals 50 percent, the amplitude ratio estimates for 

MZ-1 are in a narrow range of 26,500-28,000 ft2/d and 

the phase lag estimates are almost identical at 

14,000 ft2/d. Using the leakance value of 2 x 10-5 d- ' to 

analyze the data from MZ-2, and assuming Le is 50 

percent, the amplitude ratio analysis provides T estimates of 23,000-24,000 ft2/d, and the phase lag analysis provides estimates of 11,000 ft'-/d for both O, and 

M, components. 

A numerical model was applied to: (1) use an 

alternative means to illustrate the response of a coastal 

aquifer to ocean-tide loading; (2) investigate additional 

aspects of the effects of tidal forces on pressure heads 

in the monitor zones; and (3) demonstrate an application of a numerical model to the problem of aquifer 

Summary 67 
response to ocean-tide loading. The model code 

applied was MODFLOW (1996 version). A crosssectional grid design was devised in which ocean-tide 

oscillations were represented by specified-head 

boundary conditions on the left-hand side of the grid. 

A single value of horizontal and vertical hydraulic 

conductivity was assigned to confining zones associated with the Floridan aquifer system. With increasing 

specified values for this parameter, the tidal response 

lessens. Tidal oscillations in head are negligible in vertically adjacent layers because of their low transmissivity. If the adjacent layers are slightly permeable 

vertically, then pressure oscillations in the aquifers 

dissipate vertically into them. This result highlights 

the importance of aquifer confinement in the theory of 

Ferris (1952) and Van der Kamp (1972), and the 

potential usefulness of a solution that accounts for 

vertical leakage (Li and Jiao, 2001a). 

The second cited category of research was the 

use of inland head oscillations caused by earth-tide 

and periodic atmospheric-pressure oscillations to 

derive estimates of the elastic storage capability of the 

aquifer. By considering the dilatation of the aquifer 

material and relating it mathematically to the specific 

storage of the aquifer and to the tide-generating potential, Bredehoeft (1967) derived a relation between specific storage and the head oscillation observed in a 

well. Applying this relation, estimates of specific storage range from 0.49 x 10' to 0.76 x 10' fu' and average 0.60 x 10-6 ft-I . The relation between porosity, 

barometric efficiency, and specific storage reported by 

Jacob (1940) and based on the work by Theis (1938) 

was used for estimates of porosity in the monitor 

zones of the inland wells. Estimated values range from 

19 to 39 percent. 

The third cited category of research is based on 

a theoretical consideration of the well-aquifer system. 

In most theoretical developments, the well is assumed 

to be open to the atmosphere, and the well is assumed 

to be either screened or open throughout the thickness 

of a confined aquifer, and cased to the top of the aquifer. When oscillations of pressure-head in the aquifer 

occur, water moves into or out of the screened or open 

part of the well. This causes an oscillation of the water 

surface in the open well, which, depending on aquifer 

and well characteristics, might be damped or lagged in 

time relative to the pressure-head oscillations, 

although the frequency will be the same. The response 

to earth tides of an open well screened in a confined 

aquifer was studied by Hsieh and others (1987). The 

limitation to the applicability of their methods in some 

hydrologic environments is that, for sufficiently high 

values of aquifer transmissivity (more than 500 ft'/d), 

the amplitude of earth tide oscillations is not reduced 

in the well, nor is there a phase lag of oscillations in 

the well. Thus, the applicability of these methods 

might be limited in studies of the upper Floridan 

aquifer. 

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70 Estimating Hydraulic Properties of the Floridan Aquifer System by Analysis of Earth-Tide, Ocean-Tide, and Barometric Effects, 

Collier and Hendry Counties, Florida 

Estimating Hydraulic Properties of the Floridan Aquifer ystem by Analysis of Earth-Tide, Michael L. Merritt — —US

Ocean-Tide,and Barometric Effects, Collier an endry Counties, Florida 

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