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Moisture and temperature effects on the microwave dielectric behavior of soils

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Moisture and temperature effects on the microwave dielectric
behavior of soils
Curtis, John Oliver, Ph.D.
Dartmouth College, 1992
U
M
I
300 N. ZeebRd.
Ann Arbor, MI 48106
f
MOISTURE AND TEMPERATURE EFFECTS ON THE
MICROWAVE DIELECTRIC BEHAVIOR OF SOILS
John Oliver Curtis
Dartmouth College
Hanover, New Hampshire
June 1992
MOISTURE AND TEMPERATURE EFFECTS ON THE
MICROWAVE DIELECTRIC BEHAVIOR OF SOILS
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
by
John Oliver Curtis
DARTMOUTH COLLEGE
Hanover, New Hampshire
June, 1992
Examining Committee:
/^^j(efiairman)
Dean of Graduate Studies
ABSTRACT
A comprehensive experimental and analytical study of the complex dielectric
response of moist soils over the 100 MHz-18 GHz frequency range was undertaken. The
experimental setup included a square cross-section coaxial sample holder and a vector
network analyzer system for signal generation and detection as well as an external bath
for sample temperature control. Soils chosen for study included a poorly-graded Ottawa
sand, a clean well-graded tan sand, a clean poorly-graded silt, and a nearly-pure nonswelling clay mineral, kaolinite. Complex dielectric constant data for these soils were
collected at ten different temperatures ranging from -10® C to +40° C as well as at
numerous volumetric moisture contents varying from nearly saturated to nearly dry.
Two model interpretations were applied to the data. One was an equivalent circuit
that contained separate soil and water elements. This circuit simulated both low
frequency and high frequency loss mechanisms, and the elements were arranged to
represent both series and parallel electrical response of the soil constituents. The second
model assumed a fractal pore structure for small pore spaces and hypothesised that the
critical volumetric moisture content at which the soil dielectric response went through a
transition from series to parallel behavior was equivalent to the field capacity of the soil.
Testing of this hypothesis against the soils used in this study led to the conclusion that the
small-pore fractal dimension of soils tends toward 2.0 for coarse substances like the clean
sands and toward 3.0 for very fine substance like the kaolinite, with normal soils falling
in the 2.5 to 2.7 range.
i
f
ACKNOWLEDGEMENTS
I am deeply indebted to my advisor, Prof. John Walsh, for his willingness to take
me on as a student and for remaining committed to this study. In spite of the incredible
amount of time and energy consumed by teaching and the quest for research support for
his students, he has always been able to keep a better focus on what I was doing than L
I also wish to acknowledge Dr. Lewis E. Link, Technical Director at the US
Army Cold Regions Research and Engineering Laboratory in Hanover, for introducing
me to Prof. Walsh and for his continued encouragement and support.
The springboard for anything accomplished within this study was the recent work
done by Dr. Jeffrey Campbell during his doctoral and post-doctoral studies at Dartmouth.
Jeff also provided very useful and refreshing critical comments on the models presented
in this dissertation.
Only one who has undertaken a serious graduate study program as both a spouse
and parent can understand how much the family suffers from neglect. My wife, Leigh,
and children, David, Lauren, and Jensen, will never recover time lost, but they have my
commitment for the future.
I dedicate this dissertation to the memory of my father, John, a man with a good
heart who did not live to experience Jensen. He would have enjoyed the graduation
ceremony.
ii
CONTENTS
ABSTRACT
i
ACKNOWLEDGEMENTS
ii
LIST OF TABLES
vu
LIST OF FIGURES
viu
CHAPTER 1: INTRODUCTION
1
Reasons for Studying the Electrical Properties of Soils
1
Soil moisture measurements
1
Subterranean investigations
3
Remote sensing of the environment
4
Others
5
The Complex Dielectric Constant
6
Problem Statement
9
Scope of Research Effort
10
CHAPTER 2: THE ELECTRICAL PROPERTIES OF WATER
11
A Summary of Data on Pure Water
11
Microwave Frequency Loss Mechanisms
13
Breaking of hydrogen bonds
13
The Debye relaxation model
16
The Cole-Cole relaxation model
23
iii
CHAPTER 3: THE ELECTRICAL PROPERTIES OF MOIST SOILS
Data on Variations in Frequency, Moisture and Temperature
26
28
The structure of clay minerals
32
Dispersive behavior in moist soils
33
Existing Data on the Effects of Moisture Content
43
The Effects of Sample Temperature on Dielectric Properties
56
Bound vs Free Water .
61
Hydrogen bonding
62
van der Waal's forces
62
Hydration of exchangeable cations
62
Osmosis
63
Radio Frequency Loss Mechanisms
64
Free Water Relaxation
65
Bound Water Relaxation
66
Maxwell-Wagner Effect
66
Surface Conductivity
67
Charged Double Layers
67
Ionic Conductivity
68
Activation Energy Data
68
CHAPTER 4: MODELS FOR SOIL ELECTRICAL BEHAVIOR
72
Mixing Models
75
Equivalent Circuits
80
Homogeneous Materials
80
Mixtures
89
Percolation Transition or Long-Range Connectivity
92
Fractal Models of Electrical Behavior
94
Modeling of Electrodes
94
Fractal Pore-Filling Model
96
iv
CHAPTER 5: A COAXIAL APPARATUS FOR DIELECTRIC
MEASUREMENTS
99
The Measurement System
99
The Governing Equation
102
Experimental Procedure
108
Calibration
108
Sample preparation and Measurements
110
Sanity Checks
Ill
Empty Holder
112
Water
114
Ethylene Glycol
117
CHAPTER 6: EXPERIMENTAL RESULTS
120
Summary of Data Collected
120
Dispersion in Soils as a Function of Moisture Content
122
Single Frequency Observations
131
Temperature Effects
131
Moisture Effects
138
CHAPTER 7: DATA ANALYSES
143
Equivalent Circuit Representation
143
Fractal Geometry Model and Critical Water Content
157
A Fractal Model of Pore-Size Distribution
158
The Fractal Model and Pressure Plate Data
159
The Fractal Model Related to Particle-Size Distribution Data
163
Fractal Model Applied to This Study
173
CHAPTER 8: CONCLUSIONS
184
APPENDIX A; SKIN DEPTH CALCULATIONS
187
v
APPENDIX B: REFLECTION AND REFRACTION AT PLANE INTERFACES . 193
The Fresnel Coefficients
197
Reflection From Lossless Media
198
Reflection From Lossy Media
200
APPENDIX C: FRACTAL MODELS OF SOIL STRUCTURE
202
Fractals
202
Soil Structure
204
APPENDIX D: SOIL PROPERTIES
206
REFERENCES
211
vi
LIST OF TABLES
No.
Page
1. Electrical property measurements
29
2. Electrical property models
73
3. Fractal cutoff moisture content vs measured field capacity moisture
172
4. Estimating CC for the soils in this study
175
vii
LIST OF FIGURES
No.
Page
1. The complex dielectric constant for liquid water as a function of temperature
(analytical models after Ray, 1972)
12
2. Point charge model of the free water molecule (after Mitchell, 1974)
14
3. Graphical representation of Debye equations inspace
21
4. Equivalent circuit for Debye model
23
5. Representative Cole-Cole plots
25
6. Equivalent circuit for the Cole-Cole model
25
7. Radar backscatter coefficient vs. moisture content (Ulaby, et al, 1974)
27
8. Dielectric constant of clays vs frequency (Smith, 1971)
34
9. Dielectric constant of soils vs frequency (Lundien, 1971)
38
10. The dielectric constant of Goodrich clay (Hoekstra and Delaney, 1974)
40
11. Dielectric constant of Suffield silty clay (Hoekstra and Delaney, 1974)
41
12. Dielectric constant of loam vs frequency (Hallikainen, et al, 1985)
42
13. Dielectric constant of Manchester silt (Campbell, 1988)
44
14. Apparent dielectric constant of a silt loam vs gravimetric moisture (Lundien,
1966)
45
15. Apparent dielectric constant of a clay vs gravimetric moisture
(Lundien,1966)
46
16. Dielectric constant of a sand vs volumetric moisture (Lundien, 1971)
48
17. Dielectric constant of a silt vs volumetric moisture (Lundien, 1971)
49
18. Dielectric constant of a clay vs volumetric moisture content(Lundien, 1971) . .
50
19. Dielectric constant of soils vs volumetric moisture at 10°C (Hoekstra and
Delaney, 1974)
51
20. Dielectric constant of soils vs volumetric moisture (Hallikainen, et al, 1985) . .
52
21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988)
53
viii
22. Dielectric constant of soils vs temperature at three volumetric
moisture contents (Hoekstra and Delaney, 1974)
57
23. Dielectric constant of soils vs temperature at several volumetric
moisture contents and a frequency of 0.5 GHz (Delaney and
Arcone, 1982)
58
24. Dielectric constant for two soils as a function of frequency and temperature
(Hallikainen, et al, 1985)
25. Microscopic capacitive elements in clayey soils
59
63
26. Dielectric loss mechanisms for heterogeneous moist materials
(after Hasted (1973))
65
27. Dielectric loss and conductance in Na-montmorillonite as a function of
temperature (Hoekstra and Doyle, 1971)
70
28. Dielectric losses at 1 MHz (after Campbell, 1988)
71
29. Several mixing formulas plotted against real data (Wang and Smugge, 1980) . .
79
30. Simple equivalent circuits
82
31. The Debye equivalent circuit and its response
84
32. The Cole-Cole equivalent circuit and its response
85
33. An equivalent circuit for impure water (von Hippel, 1954)
87
34. Water model including DC conductivity and optical permittivity
88
35. Equivalent circuit for clay film studies (Bidadi, et al 1988)
89
36. An equivalent circuit model for saturated media (Sachs and Spiegler 1964;
Smith 1971)
90
37. Campbell's (1988) percolation model grid for moist soil
93
38. The "finite modified Sierpinski electrode" (Sapoval, et al, 1988)
95
39. Scaled critical water content vs pore size (Campbell, 1988)
98
40. Experimental Measurement System
99
41. One of the brass, coaxial sample holders
101
42. Sample geometry and voltage and current notation
103
43. Calibration and measurement planes
109
44. Empty sample holder measurements
113
ix
45. Dispersion curves for water
115
46. Dispersion measurements of ethylene glycol
118
47. Data collected at 20° C
121
48. Dispersion in a poorly-graded sand
124
49. Dispersion in a well-graded sand
125
50. Dispersion in silt (first set)
126
51. Dispersion in silt (second set)
128
52. Dispersion in kaolinite (continued)
129
53. Temperature effects for poorly-graded sand (8 GHz)
132
54. Temperature effects for well-graded sand (8 GHz)
133
55. Temperature effects for silt
134
56. Temperature effects for kaolinite
136
57. Moisture effects for non-frozen soils at 100 MHz, 20° C
139
58. Moisture effects for non-frozen soils at 800 MHz, 20° C
140
59. Moisture effects for non-frozen soils at 2 GHz, 20° C
141
60. Moisture effects for non-frozen soils at 8 GHz, 20° C
142
61. The three-path equivalent circuit used for this study
145
62. Equivalent circuit model-data comparisons for tan sand
148
63. Equivalent circuit model-data comparisons for silt
151
64. Equivalent circuit model-data comparisons for clay
154
65. Fractal model applied to real soil desorption data (after Arya and Paris, 1981) . 161
66. Comparison of measured and predicted pore-size distributions; 70% silty
clay, 30% sandy loam (Arya and Paris, 1981)
167
67. Comparison of measured and predicted pore-size distributions; loam
40-50 cm depth (Arya and Paris, 1981)
168
68. Comparison of measured and predicted pore-size distributions; 40% silty
clay, 60% sandy loam (Arya and Paris, 1981)
169
69. Comparison of measured and predicted pore-size distributions; loam
20-30 cm depth (Arya and Paris, 1981)
X
170
70. Comparison of measured and predicted pore-size distributions; 20% silty
clay, 80% sandy loam (Arya and Paris, 1981)
171
71. Predicted pore distribution curves for Ottawa sand
176
72. Predicted pore distribution curves for tan sand
177
73. Predicted pore distribution curves for tan silt
178
74. Predicted pore distribution curves for kaolinite
179
75. Inferred fractal dimension for tan sand
181
76. Inferred fractal dimension for tan silt
182
77. Inferred fractal dimension for kaolinite
183
Al. Skin depth as a function of wavelength, permittivity, and loss tangent
189
A2. Skin depth nomograph (Albrecht, 1966)
191
A3. Attenuation in moist soils (Hoekstra and Delaney, 1974)
192
Bl. Wave vectors at a plane interface
194
B2. Reflection amplitudes and phase shifts for non-magnetic lossless materials . . . 199
B3. Reflection amplitudes and phase shifts for non-magnetic lossy materials
201
CI. A fractal snowflake of dimension 1.5 (Mandelbrot, 1983)
203
C2. Fractal islands and lakes of dimension 1.6131 (Mandelbrot, 1983)
203
C3. The Menger sponge, fractal dimension 2.7268 (Mandelbrot, 1983)
205
C4. A fractal representation of soil fabric (Moore and Krepfl, 1991)
205
Dl. Gradation curve for Ottawa sand
207
D2. Gradation curve for tan sand
208
D3. Gradation curve for tan silt
209
D4. Gradation curve for kaolinite
210
xi
MOISTURE AND TEMPERATURE EFFECTS ON THE
MICROWAVE DIELECTRIC BEHAVIOR OF SOILS
CHAPTER 1: INTRODUCTION
New and improved methods of remote sensing have increased our understanding
of our planet's origins, its resources, and those processes that contribute to its dynamic
(on a large time scale) nature. Scientists and engineers from many disciplines are
constantly exploring new ways to quantify the Earth's properties for their particular
applications. Among these methods are measurements of electromagnetic energy in many
different wavelength regimes, both passive and active. The development of small
powerful sources and ultra-sensitive receivers along with data processing capabilities has
fostered renewed interest in improved measurements of the Earth within the microwave
region of the spectrum where wavelengths in air range from a few millimeters to several
meters.
Natural terrain surfaces consist of bare soils, rocks, vegetation, and water. It is
with the hope of making a meaningful contribution to the understanding of microwave
interactions with natural terrain that this dissertation proposal is offered. In particular,
this research focuses on the measurement and modeling of the electrical properties of
well-characterized soils.
Reasons for Studying the Electrical Properties of Soils
Soil moisture measurements
The dominant factor that controls the electrical behavior of soils is the presence
of water (Topp, Davis, and Annan, 1980). Obviously one would hope to take advantage
of this experimental fact to develop a means of accurately and quickly measuring the
moisture in soils without having to collect numerous samples in the field, weigh them,
dry them for extended periods of time, and weigh them again to obtain either a
1
gravimetric (weight of water/weight of dry soil) moisture content or a more useful
volumetric (volume of water/volume of soil sample) moisture content.
Numerous attempts have been made to develop a useful method for measuring
soil moisture content. All have met with varying degrees of success and none has proven
accurate under all conditions. For example, a technique that essentially amounts to
burying radar transmit and receive antennas in the soil and relating the measurements of
attenuated received signals to moisture content (Birchak, et al, 1974) is destructive to the
soil fabric (the way in which soil particles are arranged), is very much controlled by the
fabric and the size distribution of particles, and precludes the use of the same instrument
in multiple locations (Nature is not homogeneous) as well as the ability to easily repair
defective equipment.
Another approach for making field measurements of soil moisture taken by
some researchers (that also has numerous application in the biomedical field) is that of
measuring the change in fringe capacitance of an open-ended coaxial probe (Thomas,
1966; Brunfeldt, 1987; Gabriel, et al, 1986). When pressed against a soil whose
properties are unknown, the resulting change in capacitance produced by the impedance
mismatch is related to electrical properties through calibration relationships. Problems
arising from these measurements include the need to have proper contact between the
probe tip and the soil surface, the fact that the volume of material associated with the
fringe capacitance is quite small (on the order of a cubic centimeter or less), and that
calibration conditions simply cannot account for all of the dielectric loss mechanisms that
exist in natural soils. The losses in moist soils can be highly frequency dependent over a
range of several frequency decades on the electomagnetic spectrum. A recent variation
on the open-ended probe measurement scheme involves the use of a waveguide section
instead of a coaxial device (Parchomchuk, et al, 1990).
The concern over small sample volumes can be overcome with a redesign of the
open-ended coaxial probe that replaces the solid outer conductor with several pointed
tines (Campbell, 1988) that allows for the probe to be pushed into the surface of soft
soils. The volume of soil enclosed by such probes can easily be tens of cubic
centimeters. The tined coaxial probes that have been built to-date operate in a frequency
2
range that is very much subject to the material-dependent loss mechanisms alluded to
above. Research continues that is directed to better understanding these various
mechanisms and to fabricate a probe that operates in a frequency range that is not subject
to such material-dependent anomalies.
Subterranean investigations
There are many types of electrical measurements in soils (and rocks) that are
used to understand what lies beneath the Earth's surface (Telford, et al, 1984). One can
collect most of these methods under three headings; namely, those employing natural
electrical sources, those that measure soil resistivity, and those that utilize propagating
electromagnetic waves.
Resistivity measurements. Resistivity data in soils are collected by injecting
known currents (usually of a frequency less than 60 Hz; AC required to minimize effects
of charge buildup on the probe) into the ground and measuring potential differences
across pairs of nearby electrodes. Assuming homogeneous media and uniform resistivity,
it is possible to calculate from the potential differences an apparent resistivity of the earth
material. These measurements will be affected by the presence of water in the soil or
rock, by the presence of mineral compounds that could go into ionic solution with
available water, and by the physical structure of the subsurface terrain itself. This being
the case, resistivity measurements are useful as a measure of subsurface water volume,
the locations of mineral deposits, and subsurface structure.
Electromagnetic wave propagation. Another method of making electrical
subsurface measurements involves the transmission of electromagnetic waves into the soil
and detection of energy that results from waves reflected from subsurface anomalies.
Having the ability to track wave propagation in time, either by pulsing the source or
sweeping over a known frequency band in some controlled manner, means that
electromagnetic wave propagation methods of subterranean investigations are particularly
useful for locating the depth of electrical anomalies such as the water table in sandy soil
(Olhoeft, 1983; Stewart, 1982; Wright, et al, 1984), buried pipes or wires, or cavities
3
such as tunnels or caves (Ballard, 1983). Other applications include the delineation of
stratified media (Lundien, 1972) and determination of the thickness of ice and frost layers
(Jakkula, et al, 1980; O'Neill and Arcone, 1991).
There are some practical bounds on the utility of radio frequency systems to
conduct subterranean investigations due to the phenomenon of "skin depth", a measure of
the attenuation of the electromagnetic energy as it travels through the medium. For
relatively low frequency sources (on the order of 200 MHz) it can be shown that low loss
soils such as dry sands can possess a skin depth on the order of 10-15 meters, while high
loss soils such as wet silts and clays may have skin depths on the order of only a few
centimeters. Modem radio frequency receivers are extremely sensitive devices, often
having a dynamic range of 50 to 100 decibels. The signal at skin depth represents about
an 8.7 dB loss in power or a two-way loss at the receiver for reflected signals of about
17.4 dB. It is certainly not inconceivable that radio frequency receivers should be
capable of successfully detecting reflected signals from subterranean anomalies at depths
of two to three skin depths or more.
Remote sensing of the environment
Virtually all remote sensing of our environment from airborne or spacebome
platforms involves the measurement of electromagnetic radiation from the Earth's surface
and/or atmosphere. Passive surveillance involves measurements of emitted radiation and
that reflected from natural sources such as the sun, the atmosphere, and surrounding
terrain. Active remote sensing measurement systems include a source for illuminating the
target of interest. Whether passive or active, whether visual, thermal infrared,
microwave, or millimeter wave sensors are utilized, remote sensing is the collection and
interpretation of electromagnetic radiation, and as such, demands an understanding of the
dielectric properties of those materials being observed.
Environmental remote sensing applications form a list that grows yearly as
electronic components are improved and data collection and processing hardware and
software become faster, more reliable, and less costly. For example, satellites can
4
provide world-wide surveys of land-use patterns to monitor the threat of urbanization,
waste disposal, and erosion of the land (Colwell 1983). Similar systems (including those
mounted in aircraft) can monitor the health of vegetation to keep abreast of such things as
loss of forest and the potential for food shortages. Sea traffic in the far northern and
southern shipping lanes can be made safer through the use of airborne and spacebome
sensors to detect ice hazards.
One of the more obvious applications for microwave remote sensing devices is
that of conducting surface moisture surveys to help predict ground water availability and
the potential for flooding. Attempts have been made to relate soil moisture to both
laboratory reflectance data (Lundien 1966) and to airborne sensor backscatter
measurements (Ulaby 1974; John 1992). Careful airborne sensor measurements might
provide a first approximation to the complex dielectric constant of the soil near the
surface.
Because of the difference in dielectric behavior of liquid water and various
forms of ice, it may be possible to use airborne sensors to detect freezing and thawing
(Wegmuller, 1990) in remote locations that could be used to predict spring runoff
conditions and all that that encompasses for agricultural applications, the effects on the
fishing industry, and the anticipation of flooding in built-up areas. Military analysts are
concerned about soil moisture conditions because of its impact on trafficability, the ability
of vehicles to move effectively over natural terrain.
Another remote sensing application is the mapping of exposed soils and rocks in
remote areas of the world from high-flying aircraft or satellites which might prove useful
for geomorphological studies (Swanson 1988) or even mineral exploration. A less
obvious, but recent, application of microwave remote sensing in soils dealt with
archeological surveys in desert areas (Berlin, et al, 1986; McCauley et al 1986)
Others
While the above paragraphs emphasize some of the most obvious and useful
applications of a better understanding of soil electrical properties, others have been noted
5
in the literature. For example some researchers have attempted to relate electrical
property measurements to the physical properties of soils (Campbell and Ulrichs, 1969;
Hayre 1970; Arulanandan and Smith 1973; Madden 1974). And, of course, nothing has
been said about the military's need to better understand the microwave response of soils
that form the backgrounds to military targets; i.e., when and why does clutter become a
source of target-like signatures. Outside of the topic of soils, studies of the microwave
response of foodstuffs has direct application to quality control concerns in the food
industry (Nelson 1973; Nelson 1983).
Another new application of this technology that is closely related to the
discussion on soil moisture is that of detecting liquid ground contaminants, either near the
surface or at arbitrary depths using a specially-fabricated probe. If, as will be argued
later, polarizable liquids can be characterized by a unique frequency of peak losses due to
the dielectric relaxation phenomenon, then a probe could be designed to measure losses
over a frequency span that is broad enough to detect a peak loss frequency and, coupled
with the results of a thorough experimental program, to identify the particular
contaminant.
The Complex Dielectric Constant
Background information for this study would not be complete without a
definition of terms that will be used. Consider, for the moment, Ampere's law written
for linear, isotropic materials and for current density divided into a component due to the
motion of free charges such as electrons and ions and one due to other factors such a
time variation of polarization and magnetization (Gaussian units are used throughout the
text. For other units see the excellent Appendix on Units and Dimensions in Jackson
(1975)).
(1)
6
where
H = the magnetic field
Jf = the current density due to free charges
D = the electric displacement.
If one also assumes that the magnetic contribution to free charge forces is small relative
to that of the electric field, then one can rewrite (1) using Ohm' law.
Jf = oE
(2)
where a is the conductivity of the medium. Furthermore, having assumed a linear
isotropic material, one can write the electric displacement in terms of the electric field as
D = €E
(3)
where € is the electric permittivity of the material. (It is understood that if € is
frequency dependent. Equation (3) is not rigorously correct in the time domain, but rather
that the electric displacement and electric field are related through Fourier transforms of
the frequency-dependent permittivity (Jackson, 1975, p. 307). Adding this notational
complexity would only serve to cloud the qualitative development intended within this
section.) Then Ampere's law can be written as
VxH =
+ -1-^
c
(4)
cot
Thus, the flow of current in a medium is both proportional to the applied electric field
and to the time rate of change of that field. By analogy with simple electrical circuits,
permittivity is a measure of the electrical capacity or capacitance of the media, its ability
to store charge. Capacitor plates in a vacuum collect and give up charge, thus, there is a
permittivity of free space, GQ (unity in Gaussian units). Capacitor plates filled with a
polarizable material exhibit a higher capacitance due to material polarization such that the
material permittivity can be written as
7
(5)
€ = €o(l + 471%^)
where %g is the electric susceptibility for linear materials and relates the material
polarization to electric field. The value of material permittivity normalized to the
permittivity of free space is what is called the dielectric constant (or relative permittivity).
€ = -^ = 1 + 47T:Xe
(6)
From this point on in the text, 6 will refer to the relative permittivity.
As will be shown by the data, the dielectric constant of real materials is
frequency dependent (a characteristic referred to as "dispersion"). At relatively low
frequencies, polarizable particles, which are effectively electric dipoles, are able to rotate
or align themselves with the changing electric field with little loss in energy. However,
as will be discussed later, there are several other mechanisms at work that do result in
electrical energy losses, particularly at frequencies less than 100 MHz. As the frequency
goes up, the dipoles cannot keep up completely with the electric field, resulting in a
phase shift in current that peaks in the 10-20 GHz frequency range and is quite
temperature dependent. Finally, as the frequency goes even higher, the dipoles do not
even respond, resulting in a reduction of real capacitance to its optical value which is due
to electronic polarizability (Feynman, 1964). This frequency-dependent loss mechanism
can be modeled by a complex dielectric constant
€ =
where,
, reflects the phase lag in the particle motion. Finally, then. Ampere's law
could be written as
Vx5 =
+
\ c
dE
C/ d t
(8)
If one further assumes that the material is subjected to a sinusoidal electric field
8
(9)
then
(06
Vx# = 15° ^
c
C
(10)
c
The first two terms on the right hand side of Equation 10 represent the conductive nature
of the material, the flow of energy that is associated with losses. The third term is a
measure of its energy flow due to its polarizability. All of the terms within the square
brackets are collectively referred to as the admittance of the material, being the ratio of
current to voltage. The first two terms represent the conductance of the material, and the
last term represents its susceptance. It is clear, then, that (coe'V 4tt:) can be thought
of as something like a dielectric conductivity.
It appears that electrical property measurements reported in the literature
seldom, if at all, distinguish between the two physically different loss mechanisms. In
fact, the terms,
, and , 4ito/(*) , are often used interchangeably. This
observation, coupled with the fact that the engineering community uses a notational and
sign convention for dealing with dielectric properties that is different than that used by
physicists and chemists, makes reading of the literature often quite burdensome.
Problem Statement
The overall objective of this research is to measure and model the dielectric
response of moist soils in ways that will test the current understanding of loss
mechanisms over a broad range of moisture levels, frequencies, and material
temperatures and perhaps suggest new ways of thinking about losses as a function of
these variables. Of particular interest is the issue of long-range electrical connectivity of
water in the soil-water-air mixture that has been proposed as the mechanism for an abrupt
9
change in dielectric response of a given soil between low and high volumetric moisture
contents (Campbell, 1988).
What is first required to help achieve this objective is an apparatus that will
allow for the measurement of attenuation in moist soils over a broad spectrum of
frequencies while maintaining a required sample temperature. Numerous samples of the
same soil will be tested at different volumetric moisture contents to make certain that any
transition in electric response due to increasing moisture content can be observed.
Two approaches to modeling soil electrical behavior will be exercised and
compared to experimental results. One of these models will be an extension of a
previous fractal approach to explaining how long-range connectivity can occur at
saturation levels well below 100 percent (Campbell, 1988). The other will be an
equivalent circuit representation of moist soil behavior for which the circuit parameters
will be related to physical properties of the soil/water/air mixture.
Scope of Research Effort
Because so much of the electrical response of moist soils is due to the presence
of water, Chapter 2 of this dissertation describes what is known about the behavior of
pure water. Chapter 3 summarizes what past investigations have discovered about the
electrical properties of moist soils and will include several examples of others'
experimental data. Chapters 4 and 5 deal with a description of existing models and a
description of the experimental setup used in this study. Chapter 6 is a summary of data
collected, and Chapter 7 discusses model interpretations. The main body of the text is
concluded in Chapter 8 with a summary of conclusions drawn from the analysis of those
data. Several appendices are included that contain calculations of electrical skin depth, a
discussion of plane interface reflection and refraction phenomena, a brief description of
fractal geometry concepts, and a description of the properties of soils used in these
studies. The report closes with a listing of supporting journal articles and textbooks.
10
CHAPTER 2: THE ELECTRICAL PROPERTIES OF WATER
Certainly, over the range of the electromagnetic spectrum from D.C. to millimeter
waves, the single most important factor in determining the electrical behavior of soils is
the presence of water. Whether as a solvent that provides a medium and path for ionic
conduction or as particles that bond to clay changing the electrochemical characteristics of
the soil, or as a source of dipoles that lead to high frequency residual dielectric
conduction, water has a major impact on how the moist soil interacts with
electromagnetic fields. For these reasons, a study of the electrical properties of moist
soils should include a study of the electrical properties of water.
A Summary of Data on Pure Water
The available literature on electrical measurements of water show a consensus on
several points. First is that the so-called static (frequency < 100 MHz) dielectric
behavior of pure, deionized water is very well defined over a broad range of
temperatures (Hasted, 1973). Second is that there is a loss mechanism (dipole relaxation)
at higher frequencies (greater than 1 GHz) that can be modeled in a rather simple fashion
and for which data is highly repeatable (Kaatze 1986). Third is that fresh water exhibits
low-frequency conductivity losses. Finally, sea water and saline solutions have low
frequency losses that are a couple of orders of magnitude higher than fresh water. There
seems to be little, if any, information on the electrical behavior of pure, deionized water
in the frequency range between 100 MHz and 1 GHz.
Figure lis a graphical representation of the complex dielectric constant for liquid
fresh water as a function of temperature. These curves were plotted from empirical
relationships developed by Ray (1972) for which excellent correlation between the
empirical fits and actual data was clearly demonstrated. A description of the models for
losses at higher frequencies follows.
11
100
10Q •
80 •
\\
70
€
/
6
a
7
ID
9
11
LOG(F) " Hz
TEMP
0
20
40
Figure 1. The complex dielectric constant for liquid water as a function of temperature
(analytical models after Ray, 1972).
12
Microwave Frequency Loss Mechanisms
Breaking of hydrogen bonds
The precise physical structure of liquid water is unknown. We accept a physical
model of free water molecules (H2O) as consisting of two hydrogen atoms bonded to an
oxygen atom in a V-shape with an angle of 104.5 degrees formed by lines through the
hydrogen protons and meeting at the center of the oxygen nucleus. Viewing such a
molecule along the normal to the plane through the bond angle using the point charge
model of the water molecule shown in Figure 2, one can compute that the free water
molecule has a dipole moment with a value of 1.83x10"" electrostatic units. The
molecule has two positively charged "comers" at the hydrogen protons and two
negatively charged "comers" at the electron pairs which are not shared.
Hydrogen bonds are formed when one of these positive comers bonds to the
negative comer of a neighboring molecule with the resultant sharing of a hydrogen
proton. Each free molecule, therefore, has the capability of having four neighboring
molecules bonded to it. In fact, this model might predict that all of the H^O molecules
might be bonded together. Of course this can't be so because of the liquid nature of
water. This is not even true for ice (Hasted 1973). Further experimental evidence is
needed to help postulate a reasonable pure liquid water physical stmcture.
Given that the individual water molecules are dipoles (which is supported by the
fact that pure water does exhibit a relatively high static permittivity of around 80), then
one could reasonably assume that the dominant high-frequency loss mechanism shown in
Figure 1 is, in fact, associated with the dipoles not being able to keep up with the rapidly
changing electric field. Switching off a static electric field applied to a sample of water
would produce a finite, albeit small, delay in the material returning to an unpolarized
state (relaxation). Such behavior could be modeled by
P a
(11)
where
13
I
P = the polarization of the material
t = time
T = a characteristic of the material called its relaxation time.
e = electronic charge
+e
+e
-2e
Oxygen .
Nucleus
-2e
Distance from oxygen nucleus
to I
H'-H" « 0.97 .
C-C* - 0.374.
a-a* • 0.111.
Distance
z
Figure 2. Point charge model of the free water molecule (after Mitchell, 1974).
14
The rate of change of polarization is then
dt
a —P
T
(12)
Carrying this argument a bit further, assume that the relaxation process behaves as
a temperature dependent chemical reaction for which
rate of reaction = ic[S]
(13)
where
k = the rate constant, or the probability of the reaction taking place.
[S] = the concentration of reacting material
Now, Arrhenius postulated that the rate constant goes like (Grimshaw, 1971)
where
A = the activation energy, or the energy needed to overcome some equilibrium state
R = the universal gas constant
T = the absolute temperature.
Drawing parallels between the hypothesized time rate of change of material
polarization and Arrhenius' rate of chemical reaction by thinking of polarization as
concentration and the inverse of the relaxation time as the rate constant, one could write
that
(15)
•u
or
(16)
15
Then if measurements of l/t (radial frequency at peak loss) as a function of
temperature plotted semilogarithmically as a straight line, one could estimate the
activation energy required for that process to take place from the slope of the curve.
Ray's empirical fit for liquid water includes a relationship between the relaxation
wavelength and temperature that looks just like equation 15. A simple calculation results
in an estimate for activation energy of water to be about 5.4 kcal/mole which is about the
energy required to break a hydrogen bond (Cotton and Wilkinson, 1972). In other
words, hydrogen bond breaking becomes a likely candidate for describing the relaxation
process in liquid water, and therefore, an indication of water structure.
The Debye relaxation model
The experimental data on the electrical behavior of water clearly demonstrate a
frequency-dependent response or an anomalous dispersion within the 1-100 GHz
frequency range. The preceding section pointed out that the structure of liquid water is
some collection of permanent dipoles; i.e., particles subject to realignment in the
presence of an oscillating electromagnetic field. An obvious analogy found in classical
mechanics for an oscillating particle whose behavior changes with frequency is the
harmonic oscillator. What follows is the development of a simple model for the dipole
relaxation loss seen in liquid water based on the harmonic oscillator.
Using the notation and following (with some extension) the treatment of optical
dispersion in materials presented by Reitz, Milford, and Christy (1980), consider the
dipole to behave like a one-dimensional, damped, forced oscillator where the positive
charge, e, moves and the negative charge remains fixed.
16
(17)
where
X = a measure of the displacement
G = a viscosity coefficient
C = a spring resistance coefficient
e = the value of charge at each end of the dipole
Eta = the local electric field.
If the driving force was zero and the particle was given some initial displacement, Xo,
then
x{t)
=
XQG
ct
® = XQG
t
1
(18)
which is consistent with the previous description of relaxation behavior and activation
energies.
Now assume that both the driving force (the local electric field) and the resulting
displacement vary sinusoidally with amplitudes Ag and Ax, respectively, and that the
inertia forces are negligable. One can then write the equation of motion as
+ 471 vx)
(i*)
or
(20)
For the single dipole, the dipole moment is eA^. Therefore the polarization of the
material, P, which is the number of dipole moments per unit volume and is proportional
17
to the amplitude of the driving force through the susceptability, % , goes like
P = NeA^ =
(21)
where
N = the number of dipoles/unit volume.
Then one has
Ne^
(22)
which shows that the electric susceptibility of the medium is both complex and frequency
dependent. But we want to model this behavior in terms of the dielectric constant. One
must then pose the relationship between dielectric constant and susceptibility as
€ = €„ + 471%
(23)
where €„ replaces the unity term in the usual definition for linear dielectrics and
represents the high frequency limit on the real part of the dielectric constant. Combining
the last two expressions, one can now write
Ne^
(24)
1 - icOT
Referring to the static (CO = 0) dielectric constant as Eg , one then has that
18
f
or finally,
Recalling that
€ =
the above expression can be solved for
e' =
and
,
gp ~ €«
+
(27)
1 + (tot)'
g// = (€o - €.) WT
(28)
1 + (wty
These are the often-referenced Debye equations (Debye, 1929) for modeling the dielectric
behavior of materials made up of polar molecules. Plotted on semilogarithmic scales,
these equations approximate the anomalous behavior shown for liquid water on Figure 1.
At (OT = 1, €''Ms maximized, which says that the maximum loss due to the dielectric
relaxation mechanism occurs at a frequency equal to the inverse of the relaxation time.
Eliminating (*) x from these equation allows one to write the expression
J _ (ep + e.)
[e'f =
Thus, the Debye equations, when drawn in
^0 - €.
(29)
space result in a circle centered at
19
(€o + €00)
2
' e" = 0
and having a radius of {e^ - €„)/2 . Such a plot, shown in Figure 3, is known as
Cole-Cole diagram. Three different relaxation frequencies representing the three
temperatures shown in the figure were chosen based on Ray's empirical model. One
should further note that at O) =1/t (the inverse of the relaxation time)
= e
+ (ep - e.)
2
+ e.
2
and
g// = ^0 "
i.e., the peak relaxation loss is represented by the point at the top of the Cole-Cole
diagram.
20
IDD A
TBiP
D — — — 20
— 40
/
/
\
I
0
I
1
I
I
I
10
I
I
I
I
I
20
»
I
I
I
I
3D
I
I
I
>(
40
I
I
1
I
I
SO
I
I
1
I
I
60
I
1
I
I
I
1
I
I
1
70
Figure 3. Graphical representation of Debye equations in
21
I
I
90
I
I
I
I
I
I
1
I
1
90
space.
I
I
100
What has been demonstrated in the preceding paragraphs is that a simple
mechanical model can do a very good job in representing the high-frequency dispersive
dielectric behavior of liquid water. Of coarse a simple electrical analog can also be
develops. In terms of simple circuit elements, the equivalent circuit that precisely models
the Debye equations is shown in Figure 4 (Cole and Cole 1941) where
€ = e'+ie^' is taken to be the equivalent capacitance of the circuit. Simple models
like this will prove to be very helpful in analyzing the behavior of the complex dielectric
constant in moist soils.
O
a
Figure 4. Equivalent circuit for Debye model.
22
The Cole-Cole relaxation model
Cole and Cole (1941) observed that a good bit of experimental data on both polar
liquids and polar solids were not fit by the semicircular €^/
space predictions of
Debye's equations. Rather, it seemed that over large frequency intervals, the data
seemed to be best fit by circular arcs; i.e., pieces of circles in 6%
space that were
centered below the €^'=0 line in the manner shown in Figure 5. These curves on
Figure 5 were derived from Ray's empirical model for which artificially large alpha
values were selected. The vertical lines on the figure reflect the contribution of the
conductivity term added to the Cole-Cole model by Ray and were not part of the original
development. They do, however, give an indication of what real material responses may
look like.
Using a series of geometrical and analytical arguments, the authors showed that
linear materials that behaved in the manner shown in Figure 5 can be described by the
relationship
e -
= —5°——
oo)
1 - (iWT)^""
Expressions for e'' and
i-a =
found by substituting
= cosl^\ -
2 /
\ 2
are those slightly modified by Ray (1972) to produce the curves in Figure 1.
23
100 '
90
80
a =
70
-16.8129
+ 0.25
(T + 273)
60
50
TBiP
---- 0
— — — 20 —— 40
40
30
20
N \
\
10
>
\
a%/2 = 0.196 lad =
' ' ' "^1-1 'J M J'Jg
10
20
I
30
I
I
40
I
I
I
I
I
50
I
I
I
I
I
I
I
I
I
I
40" C
I
I
I
GO
Figure 5. Representative Cole-Cole plots.
24
I
1I 'I 'I 'I 'I 1I I I I I
8D
9D
Furthermore, it is clear from the form of equation 30 that a simple equivalent
circuit can still be drawn for materials that follow the behavior described above. Figure
6 shows that the resistance in the Debye equivalent circuit now becomes a complex
impedance.
O
T(i(OT)""
^0
o
Figure 6. Equivalent circuit for the Cole-Cole model.
25
CHAPTER 3: THE ELECTRICAL PROPERTIES OF MOIST SOILS
A quick review of available dielectric data on dry soils and wet soils leads to the
conclusion that the dominant factor that controls the electrical behavior of soils is the
presence of water. Over a broad frequency range, the real part of the complex relative
dielectric constant (referred to in this text as permittivity) of dry soil minerals changes
very little, covering a range of values from about 2 to about 6 (Nelson, Lindroth, and
Blake 1989; Ulaby, et al 1990). However, when considered from the perspective of the
idealized electromagnetic wave reflection phenomenon for lossless materials as described
in Appendix B, this can translate into reflection coefficients that vary from -0.17 to 0.42. In terms of power (which is proportional to radar backscatter coefficients) these
small variations in dry soil properties result in reflectances that span values from 0.029 to
0.176, or an increase of about 8 dB.
When water is added to the soil fabric, further substantial changes in reflectance at
the soil/air interface can take place. Take the permittivity of water to be about 80 (at low
frequencies). Then for normal incidence electromagnetic waves and lossless media, one
can calculate a reflection coefficient of about -0.8 and a reflectance of about 0.64. Now,
take the permittivity of a very moist soil to be about 20. Then under ideal conditions, its
reflectance at normal incidence could be as much as 0.4. In other words, a very moist
soil could theoretically reflect more than thirteen times the power (or more than 11 dB)
than can the driest soils. While airborne scatterometers may easily see this much
variation in measurements over large areas of so-called homogeneous terrain due to
surface roughness effects and interference phenomena, these differences in electrical
properties of soils could be quite significant for close-up measurements with ground-based
electromagnetic devices.
As an example of the effect of soil moisture on real-world situations, consider the
data reported by Ulaby, Cihlar, and Moore (1974) which was collected with a groundbased radar scatterometer located in an unplanted, plowed field of clay loam soil. As
shown in Figure 7, the power returned to the scatterometer while looking nearly straight
down at the soil surface (0 degree incidence) increased by 20 dB as moisture contents
increased from 4.3% to 36.3% (average values in the first five centimeters). 20 dB
means a 100-fold increase in power. The arguments above would indicate an anticipated
bound of about 11 or 12 dB increase. It is possible that surface roughness affects could
account for the larger field measurement numbers.
20
16
Frequency 4.7 GHz
Polarization VV
Incidence Angle
0°
10°
- 30°
CQ
12
TD
8
S
i 4
8
o
t o
f
-4
-8
-12
-16
4.3
15.8
24,0
30.2
Percent Moisture Content by Weight,
36.3 40.0
Figure 7. Radar backscatter coefficient vs. moisture content (Ulaby, et al, 1974).
27
Data on Variations in Frequency. Moisture and Temperature
At this point it would be prudent to assess what is already known about the
electrical behavior of moist soils. What are the conditions under which data have already
been collected? What kinds of soils have been studied? Is there general agreement on
results of previous measurements? Have the measurements led to any insight as to the
loss mechanisms involved?
While the governing equations for the propagation of electromagnetic waves in
linear, homogeneous, isotropic media have been accepted since the late 1800's, only in
the last 30 years or so has any serious attention been given to the complex dielectric
response of heterogeneous mixtures such as soils. Nevertheless, all measurements of soil
electrical properties are still interpreted in a macroscopic sense as if the soil is truly
homogeneous and isotropic. The following table identifies some of the most relevant
work on measuring the electrical response of materials that ultimately apply to this study
on the response of soils. No such list could ever hope to be complete, but it does serve
to provide an historical perspective and a proper point of departure for the work to be
conducted in this and future studies. Paragraphs that follow will highlight several of
these contributions.
28
Table 1 (continued)
Electrical Property Measurements
Authorfs")
Year
Mat'lfst
Freqfs)
Lane, Saxton
1952
water,
alcohols
9.35,24,
48 GHz
Grant, et al
1957
water
.58,1.74,3,3.65 coaxial Line
9.3,23.8 GHz
Cole-Cole model fits
Lundien
1966
sand, silt
clay
.3, 9.4
34.5 GHz
depth-of-penetration studies
de Loor
1968
moist organic
substances
1.2,3,3.8,6
8.6,9.4,11.5
16 GHz
Carroll, et al
1969
desert soils
0-3 Hz
resistivity array
.0001-100 MHz capacitance cell
Campbell, Ulrichs
1969
lunar soil
100 Hz-1 MHz capacitance cell
Saint-Amant,
Strangway
1970
dry rocks,
50 Hz-2 MHz
powdered rocks
capacitance cell
Maxwell-Wagner losses observed in
dry minerals
Lundien
1971
sand, silt
clay
free-space
transmission
established correlations between
moisture content and permittivity
1.07-1.5 GHz
Techniquefs')
waveguide
reflectance
Motivation/Interpretation
alcohols and water are polarizable
bound water vs free water studies;
develop test materials with known
complex electrical properties
Table 1 (continued)
elevated temperature behavior
Table 1 (continued)
Electrical Property Measurements
Authorfs)
Year
Mat'lCs)
Freqfs)
TechniqueCst
Motivation/Interpretation
Hoekstra,
Doyle
1971
Na-mont
100 Hz
9.8 GHz
Smith
1971
kaolinite
2-60 MHz
illite
montmorillonite
Nelson
1972
fruits
8-12 GHz
Strangway, et al
1972
lunar soil
100 Hz-1 MHz capacitance cell
impact of residual moisture
Olhoeft, et al
1974
lunar soil
100 Hz-1 MHz capacitance cell
behavior at elevated temperatures
Olhoeft
1977
illite-rich
permafrost
10 Hz-1 MHz
multiple loss mechanisms in
unfrozen water
Hall, Rose
1977
kaolinite
.0002-10 MHz capacitance cell
attributed peak losses to Debye
relaxation mechanism
Topp, et al
1980
soils,
glass beads
1 MHz-1 GHz
time-domain
reflection
moisture is the dominant factor
Delaney, Arcone
1982
silt, sand
.1-5 GHz
time-domain
reflection
looking for attenuation properties
slotted waveguide
low frequency free charge losses;
high frequency mechanisms: proton
mobility, dipole rotations,
H-bond ruptures
capacitance cell
Maxwell-Wagner loss mechanism
slotted waveguide
moisture content relationships
capacitance cell
Table 1 (concluded)
Electrical Property Measurements
Authorfsl
Year
Mat'lCs)
Freqfs)
Technique's)
Waite, et al
1984
silt
1.5, 6 GHz
Hallikainen, et al
1985
silts,
sandy loam
1-2, 4-6 GHz waveguide trans,
select freq.
free-space trans.
from 4-18 GHz
soil texture is a factor;
liquid water exists at sub-zero
temperatures
Pissis
1985
cellulose
DC
thermal
depolarization
attempts to distinguish free from
bound water response
El-Rayes, Ulaby
1987
vegetation
.2-20 GHz
open-ended coaxial rapid data collection
(fringe capacitance)
Bidadi, et al
1988
Na, Li-mont
30-100 Hz
capacitance cell
low-water content Maxwell-Wagner
behavior
Campbell
1988
sand, silt,
clay
.001-1.5 GHz
coaxial probe,
resonant cavity
low-freq ionic cond. losses;
fractal long-range connectivity
Nelson, et al
1989
mineral
powders
1,2.45,5.5,
11.7,22 GHz
slotted waveguide
observed some dispersion
Ulaby, et al
1990
dry rocks
.5-18 GHz
open-ended coaxial non-dispersive permittivity;
(&inge capacitance) loss factor decreases
bistatic
reflectance
Motivation/Interpretation
reflectance increases w/ moisture
The structure of clav minerals
Although this section may seem out of place, it is important to set the stage for
discussions of soil electrical behavior by briefly addressing the structure of the soil
elements that probably are most influential in determining their electrical response;
namely, the clay minerals.
Clay minerals are layered silicates whose fundamental building blocks are
tetrahedral sheets in which the tetrahedra are linked at their comers and octahedral sheets
in which the octahedra are linked along their edges (Moore and Reynolds, 1989). Each
tetrahedron is formed from four oxygen ions normally surrounding Si"*"^ cations (which
can also be replaced by Al^^ or Fe'^ cations). Each octahedron is formed from six
oxygen ions (or hydroxyl ions) surrounding a cation which is normally either AP^, Mg^"^,
Fe^+, or Fe^+.
Most clay minerals fit into structural classifications referred to as 1:1 or 2:1. A
1:1 structure is comprised of a tetrahedral sheet joined to an octahedral sheet. The
mechanism for this bonding is that the apical oxygens of the tetrahedra replace two out of
every three anions in the octahedral sheet. The 2:1 structure is generated by a second
tetrahedral sheet bonding to the opposite side of the octahedral sheet. Slight differences
in sheet dimensions (or anion spacing) due to various combinations of cations can cause
distortions of the layered structure, sometimes so severe as to result in tubular
geometries.
Clay mineral layers formed by the tetrahedral and octahedral sheets are sometimes
electrically neutral but most often are somewhat negatively charged due to the substitution
of lesser valence cations for Si"*"^ in the tetrahedral sheet and/or Al^+ in the octahedral
sheet. Furthermore, because of the finite lateral dimensions of clay mineral crystals,
there are unsatisfied bonds at the edges of the layers that also result in layer charge
imbalance.
The following simplified descriptions of the clay minerals that will be referred to
in this study can now be given. Kaolinite is a 1:1 structure mineral which, if in a very
pure form, will have little or no layer charge due to cation exchange but will attract ions
or polar molecules such as water to its edges. Illite, montmorillonite, and hectorite are
2:1 minerals. There seem to be many opinions as to what illite really is in terms of a
structural formula, but it is generally accepted that it is a mica material whose layers
have a half-unit-cell charge imbalance of about 0.8 and whose interlayer spaces are
occupied by cations (usually potassium) whose spatial distribution nearly balances the
layer charges. Montmorillonite and hectorite have smaller layer charge imbalances and
the interlayer cations can attract water which results in a swelling of the layered structure
and the creation of hydrogen bonds between the water molecules and the anions of the
tetrahedral sheet surfaces (Moore and Reynolds, 1989). These swelling clays have a
greater affinity for water which should cause much different electrical responses than for
the non-swelling kaolinite.
Dispersive behavior in moist soils
Smith (1971) conducted a series of tests on the electrical behavior of saturated
clays up to a maximum frequency of about 60 MHz using a capacitive bridge
measurement apparatus. He looked at three distinct clay types: a montmorillonite, an
illite, and a kaolinite.
A representative set of Smith's measurement results is given
in Figure 8 in terms of the real part of the dielectric constant and conductivity vs.
frequency. Of particular interest is the observation that clays with higher basic structural
water content have larger complex dielectric constants than those with less water. No
conclusions should be drawn from consolidation differences as density should clearly
affect the dielectric properties of the basic mineral themselves by providing varying
degrees of particle contact and, hence, varying electrical path lengths through the matrix.
Lundien (1971) reported permittivity and conductivity measurements over a range
of frequencies between 10 MHz and 1.5 GHz. At the lowest frequencies he used a
capacitive bridge setup. Data at about 300 MHz was collected from radar reflectance
measurements. The higher frequency data were collected using what was referred to as a
microwave interferometer but was, in fact, a free-space transmission measurement
apparatus with the sample faces tilted to the radar path.
IN
4 Hontnniilloflite; #/e = 457
Alllite; «/c°il
160
O KMlinile: w/e = 60
140
£•
o
d
100
A
2
3
S
7
10
U
20
30
SO
70
FREQUENCY (HHz)
UOSOO
OjOOlSO
OJOOOSO
FREQUENCY (Mb)
(a) For different clay minerals
Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (continued).
34
O Paste. #'c - 159*
0 Oj = 5.0
n
Ki 'em^ , w,e = 42*
<r^ -15.0 K( cn^ , w 'c:32t
I ft
s
*
\,
V
1
N"
\\
1
\1
\
"s
>
1N
<
V
10
15
20
1
3 0 4 0 5 0
70
100
FREQUENCY (MHz)
—r
1
1
1—r
. • c E isn
4
• "i«us Kt M?. w.e = M%
O"i * SiO Kg
. a'c:42%
o oj «15.0 Kg/oif. w. c = 32%
OMIS
&00M
0.0012
0.0010
0.0000
10
15
20
»
40
50
70
FREQUENCY (MHz)
(b) For illite with different moisture contents
Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (continued).
35
i
U
U
X
3 0 4 0 S 0
70
100
FREQUENCY (KHz)
mi
ANa* lllito; w/c = t2
fl.OOU
o K* lllita; w/t'Sl
O u*
'•
= (0
4
yA
I
1
OJNU
//
/* o
OJOU
LOOM
ojom
ir'
J
IS
'
/ y
1
/
•
y
a
30
70
FREQUENCY (HHz)
(c) For illite with different cations
Figure 8. Dielectric constant of clays vs frequency (Smith, 1971) (concluded).
36
Lundien's results are shown in Figure 9 for three types of soil: a very poorly
graded sand, a well-graded silt, and a high clay content locally-available soil (50% of
particles by weight < .0075 mm). Conclusions are difficult to draw because the data
include a broad range of moisture levels. The Long Lake clay is described by Lundien
as being composed mostly of montmorillonite particles.
A very nice set of moist soil electrical measurements over a broad range of
frequencies were conducted by Hoekstra and Delaney (1974) using a 7mm coaxial line
for frequencies up to about 3 GHz and a series of slotted waveguide devices from about 5
GHz to about 20 GHz. Copies of their reported data as a function of frequency are
shown in Figures 10 and 11 for two different clays. Figure 10 (a) shows data for one
gravimetric moisture content and two temperatures, while Figure 10 (b) shows data for
on temperature and two moisture conditions. Qualitatively they observed a Debye-like
dispersion in the moist soils but at a lower frequency than for bulk liquid water. They
also observed that increasing temperatures reduced the frequency of maximum dielectric
loss (which is inconsistent with the response of water) as did increasing water content.
They further attempted to fit a modified Debye relaxation equation to the data for one of
the clays (Figure 11) with partial success. They concluded that there was no difference
between the relaxation of water in sandy soil and that for clays which contradicts the
thesis that chemical bonding of the water dipoles to clay particles should cause a shift of
the relaxation frequency.
37
^0
3.088 0/CM^
w A(
VOLUMETR C WATER COhiTENT RANG
MOISTURE CONTENT RA MCE: 4.0 TO
o
\
0
.
-4—-
5
1—0
a. YUMA SAND
3
VOLUMETR IC WATER CO NTENT RANC E: 0.048 TO 0.077 G/CM*®
MOISTURE CONTENT RA NGE: 4.7 T() 5.8 •/.
)
1
I
1
0
\
%
8
1
8—
0
b. OPENWOOD STREET SILT
0
VOLUMETR C WATER CO NTENT RANG E: 0.103 TO D. 120 G/CM'
MOISTURE CONTENT R ANGE: 7.9 TC 10.5%
1
O
c
L
\
/
%
0
•
—
OO
— -
0.6
0.6
1
1
-1-8-
^
0
1.0
FREQUENCY, GHz
c. LONG LAKE CLAY
Figure 9. Dielectric constant of soils vs frequency (Lundien, 1971) (Continued).
38
O
O
\
1
1
O C
\ oo
0
•-
0.02
0
0
/
VOLUMETR IC WATER CC NTENT RANG E: 0.054 TO 0088 G/CM*
MOISTURE CONTENT R>INGE: 4 .0 TO 5.5 V.
)
0
g. YUMA SAND
0.06
I
D__0
O
D
)
CCD
—_ —
)
O
\
\
0.02
"-0
o2
\
8
O
X 0 04
0
0
0
VOLUMETR C WATER CO NTENT RANC,E: 0.048 TO 0.077 G/CM^
MOISTURE CONTENT RA NGE: 4.7 TO 5.8
b. OPENWOOD STREET SILT
0.20
0^^^
—
CD
\
§
OO
VOLUMETR IC WATER CC INTENT RANC E: 0.103 TO 0.120 C/CM^
MOISTURE CONTENT R/» NGE: 7.9 TO lO.S •/.
D
n
3—--8
8
8
Oy 0
3
0.6
0.6
1.0
1.2
FREQUENCY; GHz
c.
LONG LAKE CLAY
Figure 9. Dielectric constant of soils vs frequency (Lundien, 1971) (Concluded).
39
14.0
12.0
iOO
8.0
K
»
">P
6.0
Oo
o +24
-10'
10'
(a) 10% gravimetric moisture
p.io
K
0,05
0.10
•. J
A
L
0.05
—. I I !• I
i
Frequency. Hz
(b) 24°C
Figure 10. The dielectric constant of Goodrich clay (Hoekstra and Delaney, 1974)
40
12.0
10,0
6,0
2,0
.
Frequenay,
Hz
(a) 10% moisture, 24°C
14.0
12.0 •
10,0
K
*
8,0
6,0 •
2.0
K:
10"
Frequency,
Hz
(b) 10% moisture, -10°C
Figure 11. Dielectric constant of Suffield silty clay (Hoekstra and Delaney, 1974).
41
Another extensive set of experimental data was collected by Hallikainen, et al
(1985) using waveguide transmission techniques in the 1-2 and 4-6 GHz bands and a free
space transmission technique at eight selected frequencies between 4 and 18 GHz. They
conducted measurements on soil types ranging from sand loam (51% sand, 13% clay) to
a silty clay (5% sand, 47% clay). Figure 12 contains their published results for one soil
(42% sand, 49.5% silt, 8.5% clay) collected at an ambient temperature of about 23°C,
' 0.374 cm' cm'
Field 2: Loam
Temperature: 23°C
Free-Space Method:
3 - 1 8 GHz
Waveguide Method:
1.4 GHZ
* Field 2; loam
Temperature: 23®C
Free-Space Method: 3-18 GHz
m. • 0.368 cm' cm'
Waveguide Method: 1.4 GHz
m." 0.368 cm' cifl?
fflv* 0.374 cm' cnR,
6
6
8
10
12
Frequency ((GHzl.
8
10
12
Frequency ((GHzl
Figure 12. Dielectric constant of loam vs frequency (Hallikainen, et al, 1985).
42
Another set of frequency-dependent data is that collected by Campbell (1988)
using a tined coaxial probe and a device for precisely controlling sample moisture content
and sample temperature. Frequency was limited to 50 MHz for these measurements.
The objective of these studies was to examine the low frequency dielectric loss
mechanisms in moist soils.
Figure 13 is a summary of Campbell's measurements for one type of silty soil that
shows the low frequency dispersion of this soil as a function of volumetric water content.
Obviously, one is not able to ascertain the changes in Debye type relaxation because of
the frequency limitations of these data. Of particular note was the conclusion drawn by
Campbell that the predominant loss mechanism in this frequency range was that of ionic
conductivity and not the Maxwell-Wagner effect hypothesized by Smith (1971). These
mechanisms will be discussed in a later section.
Existing Data on the Effects of Moisture Content
Because of the obvious applications of remote sensing technology to the
determination of soil moisture levels, a good deal of data has been collected and reported
in the literature. An early example is that of Lundien (1966) in which he used four
different radar scatterometers and large specially prepared soil samples to measure the
apparent dielectric constant. "Apparent" in this context means that the dielectric constant
was computed from the square of the Fresnel coefficients (Appendix B) assuming no
imaginary components (no losses). Lundien's results for two soil types are shown in
Figures 14 and 15 in which he suggests an exponential fit might be in order. The data
are limited in quantity because of the difficulty of sample preparation and, as such, do
not reveal what later researchers believe to be a bilinear type of response.
43
30
!
• MAN 31.10 RE
• MAN 25.3 RE
a MAN 20.01 RE
« MAN 14.58 RE
• MAN 9.43 RE
• MANORE
A MAN 4.83 RE
20 -
BEtotonagaBBmomaBainnEEtoaBBQma I
0
1
5
10 H
*«#»
******** * * * *^***^<*****!.>***
til i k
cpc
10'
•
•
•
A
•
*>• °CL
MAN 30.1OIM
MAN 20.01 IM
MAN 9.43 IM
MAN 4.83 IM
0 IM
MAN OIM
8 10^
0
1
5
I loO
I
i^nnD I
O™'
10"
—:
10
1
1
1
20
30
FREQUENCY (MHz)
40
50
Figure 13. Dielectric constant of Manchester silt (Campbell, 1988).
44
50
A
1
0.3 GHz
0/
-
1
/°/
40
5 9 GHz
30
-
20
/
/o
/
—/
^
/
/
/
/
/
%
'
Z
s
8
20
30
40
so
60
•
A
10
20
a. P —BAND
30
40
50
MOISTURE CONTENT, %
io
MOISTURE CONTENT, %
b. C-BAND
w
I(t :
/
/
^.30
9.4 GHz
I 20
a.
/6
<
y
/
OJ
/
/
/
34. 5 GHz "
z
O
/
/
^ O
/
/
9
/
10
20
30
40
50
60
10
20
30
40
MOISTURE CONTENT, %
MOISTURE CONTENT^ %
c. X"BAND
d. Ka-BAND
50
Figure 14. Apparent dielectric constant of a silt loam vs gravimetric moisture (Lundien,
1966).
45
60
50
X o
0
40
5.9 GHz -
30
0.3 GHz _
y
0
0
o-p
iU—
>
HZ
§
10
20
30
40
50
60
MOISTURE CONTENT, %
0
S
1o
a. P-BAND
9 4 GHz -
0
10
20
30
40
50
60
MOISTURE CONTENT, %
•
b. C-BAND
k
QC
3/ # 0C GHz -
^ 30
Q.
<
O^
0—
n
/
o
0
X
0
c
%
20
30
10
40
20
30
40
MOISTURE CONTENTj %
MOISTURE CONTENT, %
d. Ka-BAND
c. X-BAND
Figure 15. Apparent dielectric constant of a clay vs gravimetric moisture
(Lundien,1966).
46
50
60
Lundien's later work (1971) using free space transmission measurement techniques
is summarized in Figures 16, 17, and 18. The moisture content reported here is
volumetric (volume of water/volume of sample) as opposed to his earlier gravimetric
(weight of water in sample/dry weight of soil in sample;
= irigP^j-yl Pw^ter )
data. Polarization labels on these figures refer to the orientation of the antennas with
regard to the horizon. The data show no polarization effects, even for the clay which
under compaction can assume a somewhat regular platelet structure or fabric. The
quantity of data collected is much more substantial than the previous results reported
above, and in some cases, shows a bilinear response. One can argue from these data that
soil fabric does not play a major role in determining the complex dielectric response of
moist soils.
Hoekstra and Delaney (1974) also found little difference in the dielectric response
of their sand, silt, and clay samples as shown in the data on Figure 19. From an
engineering perspective, their conclusion is valid. However, if one wishes to examine
the physical mechanisms behind the electrical losses in moist soils, one would have to
have more data on each soil type than is shown in Figure 19.
Hallikainen, et al (1985) reported results of measurements on five different soil
types. Figure 20 shows curves of dielectric constant vs volumetric water content drawn
from polynomial fits to the available data. One could easily argue that these results
indicate some differences in electrical behavior due to soil texture at relatively low
frequencies (< 5 GHz) but hardly any differences at higher frequencies. Their
assumption of a polynomial fit (second order in volumetric moisture content) precludes
any discussion of bilinear behavior.
47
LEGEND
COMPACTIVE effort:
0 5.74 N/CM2
0 H.83 N/CM2
9 10.47 N/CM2
EFFECT OF
WATER CONTENT ON
CONDUCTIVITY
YUMA SAND
FREQUENCY = 1.412 GHz
,o<«
0.1
O.Z
0.3
0.4
0.4
0.8
0.7
0.3
0.4
O.i
O
.e
0.7
N5
o
0.1
0.2
Figure 16. Dielectric constant of a sand vs volumetric moisture (Lundien, 1971).
48
LEGEND
EFFECT OF
WATER CONTENT ON
CONDUCTIVITY
COMPACTIVE EFFORT:
0
5.74 N/CM2
0 11.63 N/CM2
9 16.47 N/CM2
OPENWOOD STREET SILT
FREQUENCY = 1.412 GHz
o
o
f
y*
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
i
i
o
> O
•
0.1
0.2
0.3
0.4
Figure 17. Dielectric constant of a silt vs volumetric moisture (Lundien, 1971).
49
LEGEND
EFFECT OF
WATER CONTENT ON
CONDUCTIVITY
COMPACTIVE EFFORT;
O
i . l A N/C m 2
0 11.63 N/CM2
V 16.47 N/CM2
LONG LAKE CLAY
FREQUENCY = 1.412 GHZ
o
o
A
o^ <
N5
•
o "<
<
>
]
A VCP^
O
o
>
r
o
Figure 18. Dielectric constant of a clay vs volumetric moisture content(Lundien, 1971).
50
28 •
A
•
a
24 O
•i Monchesier Fmp Sand
Monclester Fine Sand
Suffjel<^ SHty Cloy
Goodricn Cloy
Foirbonfcs Silt
O.IO
Sutfield Silty Cloy
Goodrich Cloy
O Foirbonks Silt
a
p.20
Woler Content,
O.IO
0.30
M. 5 X JO* Hz.
- A
•
o
12 O
14
0.20
Woler
gHjO/cm'
.6.
Manchester Fine Sond
Suffield Silty Cloy
Goodrict) Cloy
Fairbonks Silt
14
12
6
•
o
o
0.30
Content, q HjO/cm*
4 X 10* Hz.
Monctiester Fine Sand
Suffield Silty Cloy
Goodrich Cloy
Foirbonks Silt
10
10-
/C'
K'
O.IO
0.20
O.IO
0.30
Water Content, g HjO/em'
c.
0.20
Woter Content, g HjO/cm'
10 X 10* Hz.
d.
26 V IC Hz.
Figure 19. Dielectric constant of soils vs volumetric moisture at 10°C (Hoekstra and
Delaney, 1974).
51
•
0.30
40
silt CUy
1 - Field 1 - Sandy Loam
? - Field 2 - Loam
3 - Field 3 - Silt Loam
,<( - fieia4 - Silt loam
5 - field 5 - Silly Clay
S4ndy low
lOM
Silt LoM
Silt lo*#
Silt/ Clay
47.6
47.4
30
Frequency: 5 GHz
T'2yc
Frequency: 1.4 GHz
T-zyc
Z5
o 20
10
0.2
0.3
0.4
Volumetric Moisture m,
0.2
0.3
0.4
Volumetric Moisture m,
(a)
T
ss
30
' T:—'
•
(b)
r
-I
: r
1- Field 1 - Sandy Loom
2-Field 2-Loam
3-Field 3-SIIt Loom
4-Field O-Sllt Loom
5-Field 5-SUty Cloy
1 - Field 1-Sandy Loom
2-Field 2-Loom
5-Field 3-silt Loan
4-Field 4-Sllt Loom
5-Field 5-SlItv Cloy
Frequency: J86Hz
T-23*C
25 - Frequency: 106Hz
I-23-C
% 20
ais
10
0.0
0.2
0.3
0.1
Volunetrlc Moisture n,
0.2
0.3
0.1
Volumetric Moisture n,
0.6
(d)
(C)
Figure 20. Dielectric constant of soils vs volumetric moisture (Hallikainen, et al, 1985).
52
Another source of experimental data on moisture content variation is that reported
by Campbell (1988) and shown in Figure 21 for three different soils. Clearly, at low
frequencies (<50 MHz) dielectric energy loss goes up with moisture content and is very
dependent upon soil type at the upper end of the frequency range of these experiments.
al
1 MHZREE
2MHZREE
SMHZREE
10MHZREE
20 MHZREE
SO MHZ REE
11818
tfi®
Q 10
-I
r
10
20
30
% WATER BY VOLUME
QQ
Q
1!
'• •
QQ
1-
—1
1
Qq BQ,
OBq
1
r-
10
20
30
% WATER BY VOLUME
1 MHZ LI
2MHZLT
5MHZLT
10MHZLT
20MHZLT
50MHZLT
40
(a) Hart sand
Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988)
(continued).
53
eflbr
•
•
B
•
•
a
1 MHZ REE
2MHZ REE
5 MHZREE
10MHZREE
20 MHZREE
50 MHZREE
a
•
B
•
B
a
IMHZLT
2MHZLT
SMHZLT
10MHZLT
20MHZLT
50MHZLT
mi
T
r
10
20
% WATER BY VOLUME
%
2.0-
I
BP Q
•D
°
*•••*****
IUrn
"n" "oB O BO^bb""""
1.0-
I
a
•
.j^gBisisissHBSSflsssHgiflse^^•
••
0.0 f
1
1
1
1
1
10
20
% WATER BY VOLUME
1—
30
(b) Manchester silt
Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988)
(continued).
54
1MHZREE
2MHZREE
5MHZREE
10MHZREE
20 MHZ REE
50 MHZREE
U 60
§SgSS-S
-I
1
1
r
20
30
40
% WATER BY VOLUME
12
o qo Q
10-
a
1I
8 -
• 1 MHZ LT
2MHZLT
a 5MHZLT
« 10MHZLT
• 20MHZLT
a 50 MHZLT
*
6 -
*
qD
4-
Qq
a B DO® •"'*
•
"•/••-a-
H SSSSa 5 SoSa Boo
-I
10
1
1
r
20
30
40
% WATER BY VOLUME
(d) Fort Edwards clay
Figure 21. Dielectric constant for soils vs volumetric moisture (Campbell, 1988)
(concluded).
55
The Effects of Sample Temperature on Dielectric Properties
While several of the preceding figures contain information on the effects of sample
temperature on dielectric properties, special attention should be given to the work done
by Hoekstra and Delaney (1974). While they were not able to conclude anything on the
temperature dependence of the characteristic frequency of Debye-type relaxation, they did
report on some very interesting dielectric behavior near the freezing point of water.
Figure 22 shows how frozen soils, in general, have a lower dielectric response than do
unfrozen soils, and that increasing moisture content reveals some very anomalous
behavior near 0°C. Some earlier results reported by Hoekstra and Doyle (1971) support
the same contention.
Another very informative study was carried out by Delaney and Arcone (1982)
that involved the cross-plotting of extensive time domain reflectometry data to produce
the curves shown in Figure 23 for a silt and a sand.
Hallikainen, et al (1985) also reported measurements made at four different
sample temperatures ranging from +23°C to -24°C. Their results, shown in Figure 24,
reveal the same relative insensitivity to temperature at sub-freezing conditions as were
reported by others.
56
-20
•b*"
1
10
—w
1
o-o-o-o—v ——o—.
I ''
'
0
10
Temperoiure, ®C
"o
;
o——o 0.05
1
20
30
(a) Goodrich clay
Temperoture, ®C
(b) Fairbanks silt
Figure 22. Dielectric constant of soils vs temperature at three volumetric
moisture contents (Hoekstra and Delaney, 1974).
57
32
0.55 fl- HgO/cm*
0%
24
0.29
K'
4
T
T
.0.21
O.IT
136
-25
20
-20
25
(a) Fairbanks silt
24
0.27
0.17
0.17
-25
•20
-15
-10
20
25
(b) North Slope sand
Figure 23. Dielectric constant of soils vs temperature at several volumetric
moisture contents and a frequency of 0.5 GHz (Delaney and Arcone, 1982).
58
20
18
Field 1
51.5%Sand. 35.1%Sllt. 13.4%Clay
Volumetric Wetness 0.24 cmVcm^
Bulk Density: 1.54 gcm^
16
14
S 12
23* C
<s 10
o
-18®C
nor
-18°C
8
12
Frequency i (GHz)
-24* C
16
(a) Silty sand
Figure 24. Dielectric constant for two soils as a function of frequency and temperature
(Hallikainen, et al, 1985) (continued).
59
Field 5
5%Sand, 47.6%$ilt. 47.4%Clay
Volumetric Wetness: 0.36 cm^/cm^
Bulk Density: 1.42 g cm^
5 12
c
o
<u
-II*C
23° C
-18°C
-24° C
Frequency ((GHz)
(b) Silty clay
Figure 24. Dielectric constant for two soils as a function of frequency and temperature
(Hallikainen, et al, 1985) (concluded).
60
Bound vs Free Water
In spite of the mixed bag of conclusions regarding the dependence of soil
dielectric response on soil texture many researchers contend that the dependence is there
(Wang and Smugge, 1980).
If moist soils with different texture do, in fact, possess different dielectric
properties for the same levels of moisture content, it must be because of different soilwater bonding mechanisms. Clay particles offer a relatively large amount of
electrostatically-charged surface area (van Olphen 1963; Dobson, et al 1985) that could
support numerous bonding mechanisms. The unique structure of clayey soils also present
many opportunities to alter the dielectric response of the soil as a whole by the creation
of microscopic capacitive elements. Even if the surface charge density of sand particles
is not significantly different than that of clay particles (van Olphen 1963), the rounded
shape of the sand particles and their smaller specific surface area (mVg) must lead to
different mechanical as well as electrical behavior.
It may even be possible to relate the physical behavior of moist soil to its
dielectric response. For example, soil scientists have long associated varying levels of
soil tension, the vacuum needed to draw water from a sample, to varying degrees of soil
water bonding. Wang and Smugge (1980) took the concept of soil tension a bit further
by correlating a particular level of tension, known as the wilting point, to the transition
moisture content in moist soils. The transition moisture content is the break point in the
bilinear dielectric response referred to in an earlier section.
There is no universal agreement on a single mechanism for soil-water bonding.
Rather, several hypotheses are offered, each of which can be both supported and refuted
by experimental evidence (Mitchell 1974).
61
Hydrogen bonding
The most easily supportable picture of soil-water bonding is that of hydrogen
bonding, which, as discussed previously, is the sharing of a proton by two electronegative
atoms. Mitchell (1974) points out that soil particle surfaces are usually composed of
either a layer of oxygens or a layer of hydroxyls. He argues that the shared proton is
supplied by the water molecule if the surface layer is oxygens, while the hydroxyl
supplies the proton in the other situation. In either case, activation energy measurements
do provide some supporting evidence of this mechanism. Newman (1987) further
explains that unsatisfied bonds at the edges of clay particles present further bonding
opportunities to the water molecules depending on the pH of the fluids in the moist soil.
van der Waal's fnrr.p.f:
A second proposed mechanism for soil-water bonding is that of electrostatic
attraction combined with the dipole nature of the water molecule to cause several layers
of molecules to migrate to the negatively-charged surfaces of the soil particles. In clayey
soils with platelet surfaces nearly parallel it is possible to create microscopic dielectricfilled capacitive elements through the van der Waal attraction mechanism if one also has
cations available to complete the dipole path as shown in Figure 25.
Hydration of exchangeable cations
A third soil-water bonding mechanism is a parasitic one in which cations are
attracted to the negatively-charged particle surfaces. Newman (1987) views this as the
primary water adsorption mechanism in swelling clays. This mechanism assumes that the
cations take their water of hydration along with them for the ride. Naturally one must
have cations available in solution for this model to have any validity, and the
concentration of cations determines the amount of water that can be bonded. Dobson, et
al, (1985) were motivated by such a mechanism to model the contribution of bound water
in a four-component dielectric mixing model. In reality, they used a double-layer model
for ion distribution near clay particle surfaces to modify bulk conductivity for free water
losses and simply assumed values of the complex dielectric constant for bound water.
CATION
\
/
'G)O
"G)
(G*'G^©'G>G)
G^G^OG^G)
Figure 25. Microscopic capacitive elements in clayey soils
Osmosis
The fourth mechanism (and that least supported by data) is that due to osmotic
forces. Again assuming that there are a large number of cations in solution and that their
attraction to the negatively charged particle surfaces is stronger than the attraction of the
dipole water molecules to the surfaces, this model predicts that there will be a higher
concentration of cations closer to the particle surfaces. The existence of another
concentration of cations on a nearby surface then results in a volume of lower particle
concentration between the surfaces and the resulting migration of water molecules to that
63
volume.
Whatever the mechanism is, water is attracted to soil particles, particularly clay
particles, as proven by the extraordinary vacuums required to drain soil samples. In the
immediate vicinity of the soil particle surface, the concentration of water molecules
should be higher than in the bulk, or free, water occupying the void spaces (Martin
1960). Thus, one has to believe, even if the bonding mechanism was that of hydrogen
bonds, that bound water should exhibit a different dielectric response than free water in
the soil matrix.
Radio Frequency Loss Mechanisms
Clearly, water in contact with soil particles must behave differently (electrically)
than free water. This shows up very dramatically at radio frequencies through a variety
of loss mechanisms that don't exist for the dry minerals or for pure, liquid water. Not
only does the water bind to the soil particle surfaces, but also salts are dissolved to
produce a source of conducting ions (Mitchell, 1974), and, as already mentioned,
microscopic capacitive elements can be created simply from the platelet-like structure of
clay particles. These and other phenomena lead to a number of loss mechanisms in moist
soils within the radio frequency range that are discussed in the following paragraphs.
Hasted (1973) presented a particularly useful schematic, shown in Figure 26, for
the relative magnitude and frequency range of application of each of what are believed to
be the major contributors to dielectric loss in heterogeneous moist materials. The
absolute magnitudes of these effects should not be regarded as applying to any particular
material. A very brief description of each mechanism follows. More detailed
expositions may be found in other sources (Campbell 1988; Hasted 1973; Mitchell 1974).
64
ionic conductivity
charged double layers
crystal water relaxation
ice relaxation
Maxwell-Wagner effect
surface conductivity
bound water relaxation
free water relaxation
log€ 2
0
1
2
3 4
5 6 . 7 8 9 K ) II 12 13
log V( Hz)
^
Figure 26. Dielectric loss mechanisms for heterogeneous moist materials
(after Hasted (1973)).
Free Water Relaxation
The inability of free water dipolar molecules to keep up with an applied
alternating field has already been discussed in Chapter 2 of this study. Dipole relaxation
in free water is the dominant loss mechanism at microwave frequencies.
65
Bound Water Relaxation
Experimental evidence exists (Muir, 1954; Hoekstra and Delaney, 1974) that
indicates a lowering of the critical dipole relaxation frequency in water that is bound to
solid particle surfaces. One can think of this physically in terms of the simple
mechanical analog described in Chapter 2 and imagine that, for all other factors kept
constant, an increase in the viscosity coefficient will result in an increase in the relaxation
time or a lowering of the relaxation frequency. Hall and Rose (1978) proposed bound
water relaxation as the dominant loss mechanism in carefully prepared kaolinite clays in
the 10-100 kHz range of the spectrum, but their conclusions seem to contradict the
observations of others and the implications of Figure 26.
Maxwell-Wagner Effect
Another loss mechanism often referred to in the literature as the dominant
mechanism at radio frequencies is one termed the Maxwell-Wagner effect (Campbell
1988; Bidadi, et al 1988). It originates in the experimental observation that there is
dispersion in a suspension of conducting particles in non-conducting dielectric media
much like that observed in a parallel-plate capacitor filled with two different dielectrics.
The Maxwell-Wagner effect is seen as an accumulation of charge at the interface of
dissimilar materials during the flow of current through heterogeneous material due to a
discontinuity in dielectric constant values. This charge buildup is time dependent and
results in dispersion as the frequency of the alternating current increases to a point where
the buildup and relaxation cannot keep up.
When modeled as series capacitors, one having zero conductivity and the other a
non-zero conductivity leading to a dielectric loss through equation 10 (assuming the
polarization loss is zero), the Maxwell-Wagner effect can be described by Debye-like
relationships (Hasted 1973; Campbell 1988). For example, for equal thickness
capacitors, the relaxation time expression is found to go like
66
X =
^
(31)
4no^
Thus the relaxation frequency is seen to get smaller as conductivity gets smaller.
One application of the Maxwell-Wagner model (Bidadi, et al 1988) had water as
the conductive medium and the clay platelets as the non-conductive medium. The high
capacitance of the closely separated platelets was used to argue for the observed
enhancement of clay-water permittivity above the level for pure water.
Surface Conductivity
Another radio frequency loss mechanism in moist heterogeneous materials is that
of surface conductivity. One possible scenario for this loss is a rapid intermolecular
proton exchange within the bound water layer on soil particles (Hoekstra and Doyle
1971; Fripiat, et al 1965; von Hippel 1988c). One application of this model (Schwan
1962) led to the calculation of a Debye-like relaxation time
T = —fsL4K03
where 6^ and
(32)
refer to the permittivity and conductivity of the aqueous solution in
which nonconducting particles were immersed.
Again, as with the Maxwell-Wagner
effect, reducing the conductivity of the aqueous solution reduces the relaxation frequency.
Charged Double Layers
If one accepts that cations can be attracted to negatively-charged soil particle
surfaces, then one can possibly think of the cations surrounding the charged particle like
an electron cloud surrounding a nucleus. The charged double layer loss mechanism then
has a parallel in the concept of atomic polarizability. As the electric field oscillates, so
67
too does the ion cloud about the particle. As frequency goes up, the ability of the cloud
to distort goes down with the resulting lowering of the dielectric loss.
Ionic Conductivity
The most dominant loss mechanism in moist heterogeneous materials at low
frequencies is that due to ionic conductivity, the 4itG/w term in equation (10). Of
course, this conductivity is brought about by dissolving salts in the soil and providing a
path through the pore spaces for the transport of electrical energy to take place. As a
result, low frequency ionic conductivity in moist soils would have to be related in some
way to the degree of saturation, or how the pore spaces fill with water.
Activation Energy Data
In the earlier description of the structure of water, arguments were made relating
the frequency of peak loss in a Debye-type dielectric relaxation process to the activation
energy associated with that process. The slope of a semi-logarithmic plot of that
relaxation frequency vs the inverse of the absolute temperature is proportional to the
activation energy.
Hoekstra and Doyle (1971) argued that linear plots of the log of the
dielectric loss term vs the inverse of absolute temperature can also be obtained when the
variation in dielectric loss with temperature is caused by changes in the thermal energy of
a charge carrier. In fact the dynamical theory of sorption (Hasted 1973) relates the time
of residence of a water molecule on a surface to the heat of adsorption by the same kind
of exponential relationship as used for the earlier activation energy development.
Certainly, if loss is inversely proportional to this residence time by virtue of the
molecules being free to reorient in the alternating electric field, then Hoekstra's and
Doyle's arguments are valid. Thus, if the dielectric loss at any given frequency increases
with sample temperature, the semilogarithmic plot mentioned above can lead to an
estimate of the energy required to bring about such a change, which, in turn, may help
identify the loss mechansims.
68
In their 1971 paper Hoekstra and Doyle did, in fact, measure a linear relationship
between the natural logarithm of loss terms and the inverse of absolute temperature for
Na-montmorillonite samples. Their results are reproduced in Figure 27 (a) for a
frequency of 9,8 GHz and in Figure 27 (b) for a frequency of 0.1 MHz. The activation
energy associated with the low-frequency measurements was about 12 kcal/mole, while
that associated with the high-frequency measurements was about 6 kcal/mole as long as
the temperature exceeded -52°C. The high-frequency results are consistent with the free
water dielectric relaxation mechanism discussed earlier and attributed to the breaking of
hydrogen bonds. However, the low-frequency measurements indicate another mechanism
for energy dissipation, most likely free charge carriers coupled with Maxwell-Wagner
effects.
Campbell (1988) used the same approach as did Hoekstra and Doyle to estimate
the activation energy in soils above freezing temperatures. Data he measured at 1 MHz
are reproduced in Figure 28. Campbell calculated an activation energy of about 7
kcal/mole and argued that the dominant loss mechanism at this frequency was the
Maxwell-Wagner effect.
69
10
5
6
4
3x10"
(a) 9.8 GHz, 0.69 g/g water content
E
I
1
J
4.5x10
3.6
4.0
4.8
5.2x10"'
Vt
(b) 0.1 MHz, 0.65 g/g water content
Figure 27. Dielectric loss and conductance in Na-montmorillonite as a function of
temperature (Hoekstra and Doyle, 1971)
70
y = 9.281e+4 ' 10'^(-831.089x) R = 0.99
0.0026
0.0028
I
1
1
r
0.0030
0.0032
0.0034
0.00313
1/KELT
Figure 28. Dielectric losses at 1 MHz (after Campbell, 1988).
71
0.0038
CHAPTER 4: MODELS FOR SOIL ELECTRICAL BEHAVIOR
Moist soils are a heterogeneous mixture of mineral particles, water, air, and
possibly any number of organic substances. The mineral particles cover a broad range of
sizes, varying from as small as tens of microns to several centimeters. Water is not pure
and, as discussed in the previous section, in combination with the structural features of
the mineral particles, can show anomalous dielectric behavior at radio frequencies unlike
that of pure water. Nevertheless, the only practical approach to predicting soil response
to either physical or electrical driving forces is that a real soil must be modeled as a
continuum with homogeneous properties.
As with the review of related electrical property measurements in the previous
chapter, a review of electrical property models for soils and water would be remiss
without a summary table of significant contributions. The table follows on the next two
pages, and many of the entries in the table will be expounded on in succeeding sections.
72
Table 2 (continued)
Electrical Property Models
AuthorCs")
Year
Mat'lfs)
Debye
1929
liquids
Cole, Cole
1941
liquids,
solids
FreqCs^
—
Description
classical anomalous dispersion model with single
peak loss frequency (or relaxation time)
modification to Debye's model to account for a
distribution of relaxation times
Malmberg, Maryott 1956
water
< 1000 Hz
polynomial function relating permittivity to temp.
Reynolds, Hough
1957
het. mixtures
—
mixture model incorporating ratio of field strength
within particles to the macroscopic average
Looyenga
1965
het. mixtures
—
symmetrical weighted mixture formula
de Loor
1968
het. mixtures
—
begins to focus on water
Stogryn
1971
saline water
—
Debye model with empirical relationships for static
permittivity and relaxation time
Ray
1972
ice, water
Cole-Cole model plus DC conductivity and temperature
dependence
Arulanandun, Smith 1973
clays
2-60 MHz
equivalent circuit applications
Olhoeft, Strangway
lunar soils
> 100 MHz
permittivity is a nonlinear function of density
1975
Table 2 (Concluded)
Electrical Property Models
Authorfs")
Year
Mat'lfs)
Freqfs)
Description
Klein, Swift
1977
sea water
1.4, 2.65 GHz similar to Stogryn
Wang, Smugge
1980
clays, sands
1.4, 5 GHz
Troitsldi,
Stepanov
1980
soils
.6, 1, 10 MHz dependency of permittivity on specific surface and
moisture at fixed frequencies
Shutko, Reutov
1982
soils
1-3 GHz
Katz, Thompson
1985
sandstones
Manabe, et al
1987
water
< 100 GHz
single Debye relaxation model with temperature
dependance
Ulaby, El-Rayes
1987
vegetation
.2-20 GHz
mixture model distinguishing free and bound water
Sapoval, et al
1988
theoretical
an attempt to relate the observed response of porous
electrodes to a fractal description of its surface
Dissado, Hill
1989
theoretical
frequency-dependent fractal dimensions of electrodes
mixture model that accounts for moisture content, a
transition moisture content, and soil porosity
surveyed existing models to obtain best fits under
selected conditions
electrical conductivity modeled with fractal geometry
concepts
Mixing Models
The most common approach to modeling the dielectric response of moist soils that
attempts to draw upon a combination of the physical structure of the soil and
experimental observations of the electrical response of laboratory samples is that of
mixture theory in which the effective dielectric constant for the material takes the
functional form
(33)
where
= frequency-dependent dielectric constant of the mixture
€^(f) = frequency-dependent dielectric constant of the ith component of the mixture
In fact, the approach that is often taken is that the mixture dielectric lies somewhere
between that of a parallel plate capacitor that is filled with a mix of fibers extending from
one plate to the other and that of a capacitor filled with sheets of material whose
interfaces are parallel to the plates (Hasted 1973). The former is called parallel mixing,
and the latter is called series mixing. For example, if one took the soil to consist of air,
water, and some homogeneous soil mineral, the two mixing models would take the form
PARALLEL MIXING
(34)
75
SERIES MIXING
A = 5 + ^ + 5
(35,
where the subscripts are self-explanatory and the W; represent weighting factors. The
weighting factors are most commonly taken to be the volume fraction occupied by that
substance.
One reference to an early generalization of the mixture model for three components
was given by Ansoult, deBacker, and DeClercq (1984) as
In this case, k = +1, represents parallel mixing, and, k = -1, represents series mixing.
From an even more physical basis many researchers have developed dielectric
constant mixture models that consider the effects of local electric fields on the induced
internal fields of specially shaped homogeneous dielectric particles. Reynolds and Hough
(1957) argued that a general formulation for a three component system could be written
as either
Gm =
+ €262^2 +
(37)
or
(e»,-«l)8lfl + (€.-€2)624 + (€„-€3)83f3 = 0
where the 6^ are the volume fractions of material "i", and the
(38)
are the local field
ratios; that is, the ratio of the field within the dielectric component to the average field in
the sample. From Stratton (1941) an homogeneous electric field induces a field inside a
spheroid in such a way that
76
3
COS^ttj.
fi = E
J=1
(39)
1
where the a^. are angles between the field and the axes of the spheroid, the Aj are
depolarization factors related to the shape of the spheroid, and €* is the dielectric
constant of the homogeneous material into which the spheroid is inserted. DeLoor (1968)
published the same result but less general in the sense that he assumed a random
orientation of spheroids for which cos^a^-=l/3 .
One of the more well-known mixture formulae is that associated with Boettcher
which, for a two-component mixture, can be written as
^2 _ g
(^1 ^2)
' («i+2€„)
(40)
which comes from Equations (37) and (39) by setting Aj = 1/3 (spheres) and
€* =
. In this formula,
is the dielectric constant of the particles and Eg is
the dielectric constant for the host medium.
Another popular formula comes from Equations (38) and (42) and €*=€3 , Aj =
1/3 (spheres). This yields Rayleigh's formula.
^2 _ § (^1 ^2)
i^+262
(€1+262)
(41)
Bruggeman took the differential form of Rayleigh's formula and integrated it
between the limits of €% and
to arrive at the formula
- (€i-€2)(1-6I) -\^2j
77
(42)
Looyenga (1965) converted Boettcher's formula into a differential equation on
volume fraction 6 ^ and integrated between the limits of 0 and 1 to give the symmetrical
form (similar to Equation (36))
Birchak, et al (1974) utilized a symmetrical form much like Equation (43) except that the
powers were taken to be 1/2 instead of 1/3.
One positive feature of the mixture models is that they all come from the same
origin; namely, consideration of the average electric field iand electric displacement in a
mixture of otherwise homogeneous materials in which the variables are physically
measurable quantities of volume fractions and the dielectric constants of the components.
Thus there is a good bit of physics involved in developing these formulae. The
approximations involved in obtaining different formulae include the particle shapes and
orientation distribution and what value gets assigned to the host medium dielectric
constant when computing the field ratios.
All of these mixture formulae, as well as many others (Reynolds and Hough 1957)
have been applied to particular data sets with satisfactory results. It would, however be
impossible to say that one formula applies to all situations and materials.
The converse is also true; i.e., one cannot expect all of these formulae to fit a
limited set of data. Wang and Smugge (1980) exercised several two-component mixture
models (probably resulting in slight overestimates at low moisture contents due to
ignoring air voids) to calculate the real part of the complex dielectric constant as a
function of moisture content and to compare the span of model predictions to real data
collected at 1.4 GHz. Figure 29 shows the results of those calculations. Curve 3 is the
bounding simple parallel mixing formula (Equation 34); curve 1 is Rayleigh's formula
78
(Equation 41); curve 2 is Boettcher's formula (Equation 40); and curve 4, which gives
the best fit is Birchak's model (Equation 41). Looyenga's results (Equation 43) have also
been added as'L' symbols.
O YUMA SAND
lUNOIEN. 1971
+ VERNON ClAY LOAM
• MILLER ClAY
NEWTON. 1977
80
0.2
0.3
0.4
VOLUMETRIC WATER CONTENT (em'/cm'l
0.5
0.6
Figure 29. Several mixing formulas plotted against real data (Wang and Smugge, 1980).
79
Campbell (1988) has demonstrated that the dielectric response of real materials is
bounded by the series and parallel models. It is not surprising that Looyenga's model for
spherical particles is a good fit to much of the data (Hasted 1973) as it clearly lies
between these limits. Dobson, et al (1985) have carried mixture models to the fourcomponent level by introducing different parameters for soil particles, air, free water,
and water tightly bound to the soil particles.
Equivalent Circuits
It is often advantageous to think of mixtures in terms of their electrical analogs,
particularly when one wants to consider different electrical paths through the medium and
some semiempirical way to account for the weighting of those paths. The following
paragraphs take a careful look at equivalent circuits for both individual components of
mixtures and the mixtures themselves. They range from quite simple models to quite
complex circuits. However, with today's readily available computational power,
exercising these models is very straightforward.
Homogeneous Materials
The complex dielectric constant and conductivity of a linear material relate an
applied electric field to current flow within the material through Ampere's law (Equation
10). Current flow in the material can be modeled by a two terminal network of lumped
circuit elements. Furthermore, because real materials do exhibit both energy storage
(capacitive) and energy loss (free charge movement and dielectric relaxation) mechanisms
it is natural to imagine that simple equivalent circuits will have to include both capacitive
and resistive elements. The following paragraphs describe models of the shunt response
of two-terminal equivalent circuits which represent the flow of current (both conduction
and displacement) between the inner and outer conducting surfaces and through the
dielectric mixture being measured. The series response, or current flow along the inner
and outer conducting surfaces, which would be represented by series inductive and
resistive elements, is not modeled.
As examples of simple equivalent circuits, consider the two circuits shown in
Figure 31, one having constant value elements in series and the other with elements in
parallel. Drawing upon the very useful concept of equivalent capacitances for each of the
= i/(x>R ;
elements in these and the following circuits(
= -l/co^L ), the
frequency-dependent dielectric response of the circuits in Figure 31 can be shown to be
SIMPLE SERIES MODEL
e/
C
€ //
(Oi?C2
(45)
SIMPLE PARALLEL MODEL
(46)
81
(a) Series
W
(b) Parallel
Figure 30. Simple equivalent circuits
82
The physical interpretation of the parameters contained in these equations is that
the capacitor accounts for the dielectric polarization of the material and the resistor is
representative of ionic conduction or other losses that decrease with increasing frequency.
Both parameters are adjustable to provide the best fit the amplitude and phase response of
the material as it is measured over a broad range of frequencies.
However, neither of these circuits adequately model the high frequency behavior of
water ith regard to the real part of the dielectric constant. The parallel model, when
placed in series with another parallel model, has been used to model the Maxwell-Wagner
losses in saturated soils (Smith 1971; Bidadi, et al 1988).
As reported in an earlier section, Cole and Cole (1941) used equivalent circuits to
describe both the ideal Debye relaxation model and their own interpretation of real data.
To help make this section of the study self-contained, their circuits are reproduced in
Figures 31 and 32.
83
Q
LOG[F) - Hz
Figure 31. The Debye equivalent circuit and its response.
84
o
€o - €«
T(iCi)T)"^
^0 ""
100
-1D0
O
90
90
T = 20® C
70
70
a =0.125
60
GO
40
40
30
20
\
%
/
/
2^
20
%
10
; I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
B
9
10
11
LOG[F) - Hz
Figure 32. The Cole-Cole equivalent circuit and its response.
85
The corresponding governing equations for the Debye and Cole-Cole models are
DEBYE MODEL
./ _
(47)
Co
Co[i+(wac2)2]
g// _
Co[l+((Oi?C2)2]
COLE-COLE MODEL
1 +
^2
""sin-
an
2
(48)
air ^
((01.^^2(1-.)
1 + 2((OTo)^""sin
2
e" =
^2
((OTo)^""COS
an'
2
1 + 2(cox o)'""sin-^
where 1/Tq is the radial frequency at the peak loss.
It is even possible to approximate the dielectric response of impure water through
the use of equivalent circuits, von Hippel (1954) showed the following circuit (Figure
33) as an approximation to the frequency response of water exhibiting low frequency
ionic conductivity. The frequency response of the dielectric constant in von Hippel's
water model is governed by the following set of equations.
86
VON HIPPEL'S WATER MODEL (with ionic conductivity)
e/=
^
Co[i +
^// ^ [1 +
c'l
CoO)i?Ji + ((oi?2q^]
85
75
tan 8
100 /i/if
53000
ohms
0.085 ohm
0.1
0.01
RC Circuit
0.001
Frequency in cycles per second
Figure 33. An equivalent circuit for impure water (von Hippel, 1954).
In fact, there is no reason not to include another parallel capacitor to von Hippel's
water model to account for the high frequency permittivity, ending up with a four
element model shown in Figure 34 for which the governing equations are
87
e' =
ii +
(54)
C„[l + (wKzC,):]
€" =
[1 +
C^oS^[l +
-r100
10Q •
90
90
Cj/Co — 4
Cj/Co = 76
80
Ri = 5.888x10® s/cm
70
70
Rj = 1.25x10"" s/cm
GO
60
50
40
40
A
30
30
\\
%
\
I
r\ -
20
10
20
-r
-r
"T"
nr
11
12
13
14
"T
IS
LOG(F) « Hz
Figure 34. Water model including DC conductivity and optical permittivity
88
.//
Mixtures
It certainly seems reasonable to use combinations of homogeneous material
equivalent circuits to model the frequency dependent dielectric response of mixtures such
as moist soils. In fact, such models would be particularly useful for studying the effects
of water content in nonsaturated soils if there could be a sensible weighting of the
equivalent circuit elements to account for varying moisture contents.
Campbell (1988) demonstrated quite clearly that if one considers the behavior of
the real part of the complex dielectric constant as a function of water content for soils, it
falls between the limits for a three component system (soil, water, and air) created by
assuming, on the one hand, that the three components can be represented by equivalent
capacitances in series and, on the other hand, by equivalent capacitances in parallel.
Bidadi, et al (1988) extended this concept a bit further to successfully model the
complex dielectric response of clay films as a Maxwell-Wagner mechanism using the
equivalent circuit shown in Figure 35. The upper frequency limit of their study was only
about 100 MHz.
R
R
1
2
AAA
AAA
•o
o
c
c
Figure 35. Equivalent circuit for clay film studies (Bidadi, et al 1988)
89
Their reasoning for choosing such an arrangement of elements is the following. They
wanted to model the physical situation of easy conducting paths through the clay
interstices that could be interrupted by thin insulating barriers (clay platelets, for
example), this being their interpretation of the Maxwell-Wagner effect. Small values of
R, and Cj allow for easy current flow, while large values of R; and C2 account for the
barriers. A similar model was proposed by Eicke, et al (1986) in which the barriers
(actually, the solvent) were taken as lossless (R; = 0).
Backing up further in time, Sachs and Spiegler (1964), and later, Smith (1971),
applied even more complicated equivalent circuits to the study of the complex dielectric
response of saturated materials. Theirs was a very physically satisfying approach to the
problem in which they assumed a combination of equivalent circuits in series representing
a discontinuous electrical path through the soil-water system (Bidadi's reasoning for
interrupted current flow) and of equivalent circuit elements in parallel representing
continuous paths for effective current flow. The resulting combined circuit is shown in
Figure 36 and was applied to data that did not exceed 100 MHz.
Solution
1-d
w
Figure 36. An equivalent circuit model for saturated media (Sachs and Spiegler 1964;
Smith 1971).
90
The geometrical parameters a, b, c, and d were empirically established by Sachs
and Spiegler for their work by obtaining a best fit to low frequency conductivity data.
One has to believe, however, that these parameters must be related to physical quantities.
For example, it seems likely that d could be associated with porosity. For saturated
media, the higher the porosity, the less chance for series-like interruptions in effective
current flow. Likewise, b should be related to dry density. The greater the sample dry
density, the more opportunity there is for conductive paths through the soil matrix. And
finally, c should somehow relate to free water content in the soil sample. The greater the
amount of free water, the greater would be the opportunity for current to flow through,
unimpeded by the solid matrix. The governing equations for the circuit representation in
Figure 36 are
THE THREE-PATH EQUIVALENT CIRCUIT
./ _
+ N2D2) ^
3cf(l
Co(DI2 +
g// ^
^
Co
(51)
Co
ad{l-d)(N2Dj_ - N^Dz) ^
b
^
c
CQ{D^^ +
Cq(^Rs
Co(oi?
where
N, =
N, =
D,
=
w2
- 1
+ ®»rC„)
+ dcj
D, - 0>[<1R„ + (l-C^i?J
It certainly seems reasonable to include another set of equivalent circuits to those
of Figure 36 to account for air in non-saturated soils. Perhaps even more simply, allow
a+b-hc < 1 and, 1-d, to be replaced by x-d, where x < 1.
91
Furthermore, it is undoubtedly possible to arrange a limited number of electrical
circuit elements in such a way as to approximate almost any dielectric response in a
single homogeneous material. What is really needed is an equivalent circuit for which
the elements not only allow a good reproduction of spectral response over a broad range
of frequencies, but also have sensible physical meaning or whose values can be measured
in the laboratory.
Percolation Transition or Long-Ranpe Connectivity
As pointed out in Chapter 3, experimental data on moisture content variations in
soils shows that the real part of the dielectric constant at low frequencies increases more
rapidly with increasing moisture at higher levels of moisture content than it does at lower
moisture levels. Some of the data can almost be viewed as bilinear, where below some
critical value of moisture content, the rate of increase in dielectric constant is nearly
constant. Above that critical value, the rate is again nearly constant but somewhat higher.
Wang and Smugge (1980) argued that this transition point was a function of soil
texture; i.e., a function of the amount of sand and clay. They also related the wilting
point of the soil (the volumetric water content under a pressure of 15 bars at standard
temperature) to its texture, and , in so doing, were able to show a strong correlation
between the transition point and wilting point. In fact, Wang and Smugge went on to
develop these observed soil texture effects into empirical relationships for the electrical
properties of soils.
Campbell (1988) viewed this phenomenon by electrical analogy as a transition
between series mixing (Equation 35) at lower water contents and parallel mixing
(Equation 34) at moisture levels above the critical value. He borrowed from the
percolation theory the concept of a percolation transition in moisture (Zallen, 1983) at
which level one or more continuous, low resistance electrical paths through the porous
media suddenly appear. He referred to this transition as the appearance of "long-range
connectivity".
92
Campbell attempted to create a computerized model of moist soil passing through
the transition point using the grid shown in Figure 37. The grid was initially set up with
a random distribution of soil and air squares to provide a given porosity. The air squares
were then randomly replaced with water squares as water content was allowed to
increase. A dielectric constant was then calculated for each column of soil, air , and
water squares extending from one capacitor plate to the other by the series model
(Equation 35) and, finally, an effective dielectric constant for that moisture content was
calculated as the average value for all of the columns.
+ + + + + + + + + + + + + + + + + +
mm SEPARATION
I
WATER
LENGTH
I
Figure 39. Campbell's (1988) percolation model grid for moist soil.
93
While this model simulated the increasing rate of change in real dielectric constant
with increasing water content, it did not, in fact, simulate long-range connectivity because
it was only a one-dimensional model. Unless the dry soil grid contained several columns
of only air spaces, there was no way to simulate long-range connectivity. Campbell also
pointed out that another shortcoming of this model was its inability to scale properly.
Doubling capacitor dimensions gave a different result.
Fractal Models of Electrical Behavior
Modeling of Electrodes
In recent years fractals (see Appendix C for definitions) have been utilized to study
the electrical behavior of fluid-saturated porous electrodes. The fundamental thought
behind the use of fractal concepts is that the number of elements used to construct a
fractal geometry scales as a power law of the element length or relative size (Mandelbrot,
1983). It seems to have become fashionable to also talk about a fractal geometry as one
for which "physical quantities scale as a power law of the length scale....on which they
are measured" (Brouers and Ramsamugh, 1988). For example, if there is a region of the
electromagnetic spectrum where conductivity of a sample of material is observed to vary
as a power law in frequency (hence, wavelength), then that sample is thought to possess a
fractal geometry. In fact, Brouers and Ramsamugh (1988) combined low frequency
permittivity and conductivity data for brine-saturated alumina ceramics with some
statistical models to calculate fractal dimensions of 2.65 ± 0.10 for five different
ceramic samples, independent of sample porosity.
As part of a study on the frequency response of porous battery electrodes, Sapoval,
Chazalviel, and Peyriere (1988) also observed that the impedance of rough or porous
electrodes often goes like a power law in frequency (constant phase angle behavior) and
is, in fact, related to the fractal dimension of the electrode surface in some cases. The
model they used to associate fractal dimension to electrode structure was that called a
"finite modified Sierpinski electrode" as shown in Figure 38. The Sierpinski electrode is
something like a one-dimensional Menger sponge that is discussed in Appendix C.
Sapoval, et al, obtained an expression for the admittance of a single pore within the
electrode and used the fact that the total pore admittance is the sum of the admittances of
all the pores to analytically study the effect of fractal dimensions on the electrode
response.
Figure 38. The "finite modified Sierpinski electrode" (Sapoval, et al, 1988).
Dissado and Hill (1989) used the same model to simulate analytical response
functions and to attempt to prove that there is one form of self-similarity at high
frequencies (within individual fractal pores) and another at low frequencies (sum of all
pores).
95
Fractal Pore-Filling Model
Campbell (1988) observed that, up to the time of his publication, fractal concepts
had been applied to the bound water and saturated regimes for porous media, but not to
the broad regime of moisture contents below saturation. He was searching for a way to
model the dielectric response of moist soils at moisture" contents less than saturation and
decided that it might be possible to do so using fractal techniques. His assumptions
were:
(a) Water within pore spaces accumulates in uniformly-sized fractal clusters.
(b) The pore spaces are all of equal characteristic size, 1 , which is equal to the
average particle size of the soil.
(c) The dielectric constant of the soil-water-air mixture increases with increasing
moisture content in a series mixing fashion until the size of the water clusters
reaches the size limitation on pores at which time long-range water connectivity in
the soil mixture is established; i.e., the percolation threshold is reached.
(d) Once long-range connectivity is established, additional moisture enters in a
non-fractal manner and the dielectric constant rises in a parallel mixing fashion.
(e) The soil is neither a clean gravel (which precludes water cluster spanning of
the pore spaces) nor a pure clay (dominated by bound water effects).
The key assumptions made by Campbell that allowed a rather simple expression
between critical water content (the moisture content at which long-range connectivity is
established) and soil porosity to be developed were assumptions (a) and (b). From (a),
Campbell wrote that the volume of water contained in a cluster of size, r, and fractal
dimension, D, was
96
{cluster water volume) a r^
(5%)
Thus the volumetric water content also goes as r®. Mandelbrot, himself, found no
argument against the mass density relationship that stated that mass within a fractal
cluster of size, R, went as (Mandelbrot, 1983)
(cluster mass) a
and, if one accepts that individual water molecules all have the same size and mass, then
Equation 52 is very believable.
At such time as the clusters just span the pores of size, 1 , to produce long-range
connectivity, the volumetric water content can be written as
^critical « -I"
Campbell took the soil sample porosity, (|) , to be defined in three-space as
(J) a i3
(55)
He finally concluded from the ratio of these last two quantities that
^ a
4)
(56)
and that a log-log plot of 6^/<j) vs i should be a straight line with slope equal to D3. If there is no difference in how the fractal water clusters grow in different soils, then
D is the same for any type of soil that fits within the bounds presented by assumption (e).
In fact, Campbell plotted critical water content data for five soils at a frequency of 50
MHz (shown in Figure 39) and found a fractal dimension of 2.4 that is similar to others'
calculations of about 2.5 for aggregate clustering. He made no attempt, nor is there
sufficient data, to draw any conclusions about the effects of soil texture or signal
frequency on critical water content.
97
SCALED CRITICAL WATER CONTENT VS. PORE SIZE
10'
y = 0.9818 * x'^-0.6051 R = 0.92
N.
i
Ig 10'
I
Q
•
N
i
10^
•| q-3q+0
•J Q-2e+0
ijQ-le+0
10Oe+O
PORE SIZE (mm)
Figure 39, Scaled critical water content vs pore size (Campbell, 1988).
Campbell's fractal pore filling model is very simplistic but, in his own words,
"..the similar dielectric behavior of soils with huge ranges in particle size and shape of
particles argues against a complicated model which considers many different soil
parameters.." (Campbell, 1988). He went on to admit that the weakest point in his
model was, in fact, assumption (b); i.e., that the characteristic pore size equals the
median particle size of the soil. This point is examined closely in a future section.
98
CHAPTER 5: A COAXIAL APPARATUS FOR DIELECTRIC MEASUREMENTS
The Measurement System
A schematic of the measurement system used to conduct these experiments is
shown in Figure 40. The electronic equipment that was available is not that which would
normally be used for making material electrical property measurements, but it was
successfully adapted for this use.
HP
B5iac
•
HP 9000
COMPUTER
NETWORK ANALYZER
HP 8340B
HP 8511A
SYNTHESIZED SWEEPER
FREQUENCY CONVERTER
REFKace
RB'LECTED
DUAL DIRECTIONAL
COUPLER
' TmNSMinH)
TRANSMISSION CELL
CARON 20B5
CIRCULATING
BATH
Figure 40. Experimental Measurement System
99
'SOURCE
The Hewlett-Packard 85 IOC Vector Network Analyzer (Hewlett-Packard, 1991) is
a laboratory device designed to measure the magnitude and phase characteristics of
electronic networks and/or components. It consists of a detector unit that measures
signals in up to four different channels at an intermediate frequency of 100 kHz and a
data processing unit that provides a display of the desired quantities. The HP 85 IOC
directs the signal source (in this case, an HP 8340B Synthesized Sweeper) to generate a
sinusoidal signal at a specific frequency and directs a test set ( an HP 8511A Frequency
Converter) to act as the receiver for up to four channels of information which undergoes
a first frequency conversion to about 20 MHz through the process of harmonic mixing.
The combined dual-directional coupler and frequency converter serve as a
substitute for a special test set, called the S-Parameter Test Set, which is usually used for
such measurements (Hewlett-Packard, 1985). The S-Parameter Test Set contains power
splitters, coupling devices, and means of establishing equal circuit electrical lengths
where appropriate that are transparent to the user. In truth, the S-Parameter Test Set is a
bit of overkill, as it provides information on measurements made in the reverse direction.
As long as one believes that the material response is not dependent on which direction
current flows through the sample, then the reverse measurements should be redundant.
Figure 41 is a photograph of one of the two nearly-identical sample holders that
were fabricated for these measurements. It was made from 4.5 mm brass stock and
provides a square cross-section coaxial test volume that is 7.5 mm square and 40 mm
long. A piece of copper bus wire was soldered into place as the center conductor,
connecting the two 50 ohm SMA-type connectors at each end of the sample holder. The
value of sample volume for each holder was estimated, by length measurements, to be
2.23 cm\
100
Figure 41. One of the brass, coaxial sample holders.
The choices for laboratory techniques for making material electrical property
measurements include resonant cavity, waveguide, and coaxial with some new work
being done on hybrids of the latter two (Taherian, et al, 1991). Resonant cavity
applications are limited to small volumes of material which could present a problem with
trying to measure the homogeneous response of substances like sands which are highly
non-homogeneous at small scales. Waveguide measurements are very sensitive to how
the sections are mated (and how repeatable are those connections). While they involve
adequate volumes of material, sample preparation is not likely to produce a uniform
density for granular materials, as the substance must be poured into one end and tamped
to fill the volume. Dielectric seals would be required to provide a plane interface and
must be properly accounted for. The literature also indicates that any phase shifts due to
non-ideal calibrations are non-linear functions of frequency and must be accounted for
during processing of the data. Waveguards also possess a cutoff frequency, below which
101
plane waves cannot propagate. As is true for waveguides, the usual coaxial devices
(headless air lines) suffer from the problems of uniform sample preparation, and, in fact,
are best suited to measuring solid cylinders of material that must be precisely machined to
provide a snug fit within the measurement cavity. However, coaxial lines allow plane
waves to propagate at all frequencies. Therefore, the sample holder used in these
experiments provides a practical tradeoff among sample volume, repeatability of
connections, and the ability to prepare uniform samples.
One area of concern with this sample holder deals with the development of higherorder transverse modes. Theory says that a coaxial device should support only the
transverse electromagnetic (TEM) mode as long as the wavelength in the material is
greater than twice the cavity width (Stratton, 1941). Thus, if the cavity is empty, the
highest frequency allowed for these measurements would be
(•^max) • = ^ = -^ =
3x10^° cm/sec
^ 20 GHz (57)
A
nX
^/Txl.5 cm/cycle
while for pure water
3 X 1 0 ^ °. = 2 . 2 4 GHz
(58)
y/WOxl.5
Plane wave conditions for measurements of moist soils should, therefore, be allowed to
some intermediate frequency.
The Governing Equation
Consider the geometry of the coaxial measurement device and the voltages in all
regions as shown schematically in Figure 42. A relatively simple governing equation that
relates the sample properties and dimension to quantities that can be measured by the
network analyzer system can be developed by proper consideration of the boundary
conditions.
102
OUTER CONDUCTOR
2"
«*CABLE J 2^
V, , I,
Vj . Ig SAMPLE , Zg
CENTER CONDUCTOR
^3 , I3
CABLE ^
Figure 42. Sample geometry and voltage and current notation.
Using the customary notation for spatial variation of transverse voltages within the
cables and sample and considering a snapshot in time (without loss of generality, let time
be zero), one can write:
^2 =
0 <
V3 =
X
< d
(58)
X > d
where the k's are the complex propagation constants and contain all of the material
electrical property information (see Appendix A). Similar expressions for the center
conductor current can be written by taking advantage of the definition of the
characteristic impedance of the coaxial line as being the ratio of transverse voltage to the
conduction current along the center conductor (Stratton, 1941).
103
X < 0
I2
0 < X < d J
=
X > d
where the Z's are the characteristic impedances within each section of the line.
The boundary conditions that can be applied to the geometry shown in Figure 51
are that the tangential field components at the interfaces are continuous and, from the
steady state continuity relationship, that the currents are also continuous across the
interfaces. In equation form these boundary conditions reduce to:
= V2 ® x=0
= V'3 @ x=d
Vinp +
(60)
(61)
;
+ ^'2
= I2 @ x=0
J2 = J3 ® x=d
= y+2 + y-2
:
104
(62)
Not being able to measure
or V~2 but certainly being able to obtain a reading on
the other voltages, the two voltage amplitudes inside the material can be eliminated by
adding together Equations 60 and 62 and by taking their difference:
V\ = ^
+
-^(Vinp - V-i)
(64)
- y-i)
(65)
^0
Substituting these results into Equation 61, one has
+ V\ +
Vinp +
Normalizing to
- y-J
(66)
- -i{V^
and collecting terms,
l + -^)cos(i:^d) - i
V"
1--
:sin(jCgd)
(67)
^inp;
A similar relationship can be written from Equation 62.
y1-.
^inpj
!cos(kgd)
-
1 l+-|^]sin(ic^d)=
^inpj
^0 ^inp
105
<">
It is further convenient to eliminate the characteristic impedances by solving each of the
last two equations for Z/Zg and setting the two expressions equal:
(69)
Simplifying,
inp/
(70)
Now, following an appropriate calibration of the measurement apparatus, the
network analyzer effectively measures V'lfVinp and
• They are
designated, using standard 'S', or scattering parameter notation, as
and S21,
respectively. The definition of S parameters requires that there be no current traveling
into the sample from the right side of Figure 42. This implies that all circuitry and
connectors on the transmission side of the sample are perfectly matched so that no energy
is reflected back to the sample. Although this is an assumption for this derivation, the
fact that proper equipment was not available for these measurements and that a less-than
desirable calibration technique was used (described in the next section) makes this
idealized configuration quite unlikely.
106
Using the relationships between the propagation constant and material properties
(Appendix A) and assuming non-magnetic materials, one can then write down the final
form of the governing equation:
COS
V I I = 1 + F21)' - Fii)' =
(71)
This form of the governing equation was reported by Campbell (1990). An earlier
derivation, although arriving at a different solution technique, has also been recently
discovered (Hewlett-Packard, 1985).
The process for calculating values of the complex dielectric constant can now be
outlined:
a) At a given frequency, measure the complex values of Si, and Sji.
b) Compute cos"^(IV^) =
.
c) Square both sides of this expression and solve for € .
The procedure is not as straightforward as it seems because of a problem with
principal values of the cos"' function. One method for solving for the cos ' function
yields (Zucker, 1972):
cos"^(IV^) = -i ln(re^®) = 0 -i ln(r)
(72)
where
re^® =
+ ^{W^f - 1
A great deal of time was spent developing a calculation procedure for Equation 72 that
would yield the correct values of
and
. However, it was later discovered that
the calculation routine for the inverse complex cosine function that resides in the HP
BASIC compiler within the HP9000 computer that serves as the overall system controller
gave exactly the same results. The only difficulty is that, depending on where W, lies in
107
complex space, the cos'^(WJ value computed using the available routines returns real and
imaginary components that are both positive, both negative, or having opposite signs. A
set of rules had to be developed to select the sensible results; namely, those that would
yield positive values of e' and
.
The real part of (0 dsfe/c is a measure of the number of wavelengths
(expressed in radians) of signal that can be contained within the sample holder at the
given radial frequency. The value of
0
that is returned by the above solution algorithm
is a principal value, being bounded by ±7r . However, any multiple of 2TZ can be
added to
0
and give the same result. It turns out that there is a sensible way of
determining how many multiples of 2TZ to add. One can either estimate the properties
of the material to establish the correct value of 0 , or one can use the value of "group
delay" which is calculated by the network analyzer to establish the correct value of 0 .
Group delay is defined as the signal transit time through a test device (Hewlett-Packard,
1991). What the designers of the network analyzer assumed is that the phase for the
wave in the material is a term like " (O t ". Therefore, group delay, or the derivative
of phase with respect to radial frequency is a number with dimensions of time and
interpreted as the time required to transit the sample. An estimate of the initial value of
the number of cycles (multiples of 2 Tt ) to add to 0 was taken as the truncated
product of group delay and frequency.
Experimental Procedure
Calibration
Because of the arrangement and type of measurement apparatus used for these
studies, the most appropriate calibration was found to be the One-Path, 2-Port calibration
combined with the adapter-switching technique (Hewlett-Packard, 1991). During this
procedure reflection and transmission cables were first connected with a thru adapter
from an HP85052B, 3.5mm Calibration Kit and forward transmission and forward match
measurements were conducted. Next, an equivalent adapter (male-female) from the same
108
Kit was connected to the input (reflection) cable and three reflection standards were
measured. These standards were an offset short, an offset open, and a broadband load,
all from the same Calibration Kit. The net result of the calibration procedure is that the
S,i measurement effectively takes place at the bead interface in the SMA connector on the
reflection side of the sample holder, and the S21 measurement effectively takes place at
the bead interface in the connector on the transmission side of the holder.
Due to the physical separation of the calibration planes and the interface between
the sample and the connector on the end of the holder, an additional phase shift needed to
be added to the S,, parameter and subtracted from the S21 parameter to make the
measured quantities consistent with what was defined in the earlier derivation of the
govering equation. Referring to the sketch in Figure 43, the reflected signal must travel
an additional distance of 2a beyond the calibration plane to be recorded in the frequency
converter. The transmitted signal travels an additional distance of a+b. For these
measurements, a=b and the required phase shift was found to be about 80 picoseconds
from measurements of the empty holder.
REFLECTION
TRANSMISSION
CALIBRATION PLANE
MEASUREMENT PLANES
\
X
y
CALIBRATION PLANE
CABLE
CABLE
SAMPLE HOLDER
Figure 43. Calibration and measurement planes.
109
Sample preparation and Measurements
Soil samples were prepared from one of four distinctly different soils. These
consisted of a poorly-graded sand, a well-graded sand, a poorly-graded clean silt, and
almost pure kaolinite, a non-swelling clay mineral. The soils and their associated
properties were obtained from the Geotechnical Laboratory at the US Army Engineer
Waterways Experiment Station. Each soil type batch was made as homogeneous as
possible by drying, pulverizing and mixing thoroughly. Samples were then taken to
determine the usual engineering properties of grain size distribution and plasticity indices.
The results of those measurements are contained in Appendix D, along with the results of
specific surface measurements on each sample that were conducted by a commercial
laboratory.
The normal measurement procedure was the following:
(a) The dry, empty sample holder was carefully weighed.
(b) A nearly-saturated sample of soil was prepared using distilled, deionized
water and, following a reasonable amount of time for the silt and clay to reach
some sort of equilibrium, was placed in the sample holder. The holder was
tapped against a flat, hard surface to uniformly settle the sample.
(c) The sample and holder were carefully weighed, then sealed and connected to
the measurement system.
(d) The temperature of the bath was set at -10°C. For each new temperature
setting, the sample was given ten minutes to reach equilibrium once the bath had
reached the desired setting. The ten minute wait was found experimentally to be
more than adequate for sample equilibrium to be achieved.
110
(e) The frequency range for the dual directional coupler being used was stepped
through and calculations of 6^ and
were made.
(f) The temperature of the sample was changed and measurements repeated in the
following sequence: -10, -5, -2, 0, 2, 5, 10, 20, 30, and 40°C.
(g) The holder was unsealed and its cover loosened. The sample and holder were
periodically weighed to determine approximate values of moisture content. When
the sample was thought to be at about the desired volumetric moisture content, the
holder was resealed and reconnected, the sample temperature was taken back to 10°C and the whole measurement sequence repeated.
(h) After all measurements were conducted, the sample was carefully removed,
dried and weighed to determine its original dry density and to facilitate
calculations of volumetric moisture.
As explained earlier, measurements had to be conducted over two different frequency
ranges because of the unavailability of a single dual directional coupler that could cover
the entire range desired.
Sanity Checks
In order to develop confidence in the experimental procedure materials were
measured whose responses are known experimentally or can be modelled very accurately.
These measurements also served to confirm the amount of phase shift required to collapse
the calibration and measurement planes as discussed in last section. Data were collected
for empty sample holders, and samples of distilled water and ethylene glycol (ethanediol).
Each is discussed in a following section.
Ill
Empty Holder
The first question to answer is 'How well does this experimental procedure
measure the properties of air?'. Ideally, the real part of the complex dielectric constant
for dry air should be unity, and the imaginary part should be zero. Figure 44 shows the
results of measuring empty sample holders. The vertical dashed line at 2 GHz separates
the low-frequency apparatus measurements from the high-frequency measurements
alluded to earlier in this chapter. Considering all of the experimental variables that could
have an impact on any measurement such as measurement system drift, connector and
connection variability, outer conductor discontinuities created by the sample holder cover,
etc., the agreement of empty holder (air) measurements with theoretical predictions is
quite remarkable.
The anomalous behavior centered about 3.75 GHz and 7.5 GHz is associated with
the mathematical solution technique. At 3.75 GHz the empty sample holder should be
spanned by about a half of a wavelength, at 7.5 GHz it should hold about a full
wavelength, and so on. The number of wavelengths contained within the sample holder
for a material with no loss is proportional to {(XID\F^/ C , or the cosine function
argument in Equation 71. A half wavelength in the sample holder will result in the
complex vector representing the right-hand-side of Equation 71 approaching the negative
real axis in complex space.
A full wavelength corresponds to the vector having made a
complete revolution in complex space and approaching the positive real axis. But these
two conditions are branch cuts for the
cos"'(z) function and are, therefore, inherently unstable.
Much of the data that follow in this chapter show the same anomalous behavior
whenever integer multiples of half-wavelengths exist within the sample holders. The
severity of these peaks decreases with increasing frequency because at low frequencies
the computation of
€ from Equation 71 involves the division of the right-hand-side of
the equation by very small numbers. Any inaccuracies are amplified at low frequencies
(< 1 GHz). This fact also had a very definite negative impact on low-frequency
measurement calibrations. Instrumentation drift was often enough to ruin a calibration at
112
the lowest frequencies in just a few hours. Better measurement equipment; namely, an Sparameter test set and a high-quality set of cables and connectors, would help alleviate
this low-frequency measurement problem.
2.0o
1
1
I
1
1
10
ru
0
TEMP
I
•2.0
1.8
-1.8
1.6;
D
I
1.4;
1.4
1.2-
1.2
1
e'
1.0
ryv^
1
*
0.0-
0.8
0.6-
0.6
0.4
0.4
0.2
•0.2
0.01
0.01
•
•
1 1 1 1 1 11
0.10
1
1 1 1 1 1 1 11
1
1.00
1 1 1 1 1 11
10.00
FREQUENCY ~ GHz
Figure 44. Empty sample holder measurements.
113
1
1
1 1 1 1 111
•0.0
100.00
Water
Measured values of the complex dielectric constant for samples of distilled,
deionized water are shown in Figure 45. For comparison, Ray's (1972) empirical fits
previously shown in Figure 1 are duplicated here. Several observations are in order.
First of all, the comparison of measured data to the accepted standard is quite good (less
then 4% difference), especially for the imaginary term. The sample holder volume was
evidently not completely filled with water due to the fact that the holder cover must be
pressed down upon the water meniscus, causing water to be squeezed from the holder in
an unpredictable way. It is also possible that some evaporation of the sample took place
during the 2-3 hour measurement period. Both of these phenomena could account for the
lower measured values of e' . The hypothesis that the sample holder was never
completely full was supported by weight measurements that indicate the holders were
never more than 95% full during these tests.
Another observation is that because of the good correlation with idealized data, the
impact of higher order modes must be relatively small as one would expect (Stratton,
1941). As pointed out at Equation 58, higher-order transverse modes should appear
above 2.3 GHz for pure water, but these data that extend to 4-5 GHz continue to behave
well. (Water data are cut off at 4-5 GHz because the noise floor of the network analyzer
measurement system was reached.). The same anomalies observed at half-wavelength
intervals in the empty holders also occur in the water with very little impact being
observed beyond one and a half wavelengths.
The low frequency tail on
may possibly be associated with the ionic
conductivity loss mechanism as a result of either trace quantities of ions that escape the
deionization process or as a result of ions that originate from unclean and/or oxydized
outer or inner conductor surfaces. The magnitude of low-frequency losses are
approximately those associated with fresh water conductivity as shown in Figure 1, but
the actual loss mechanism has to be viewed as purely speculative.
114
TEMP
5
40
10
50
20
60
30
90-
•90
80
40
20-
10-
I
0.01
I
I I II I
0.10
"T
1.00
10.00
FREQUENCY ~ GHz
(a) these measurements
Figure 45. Dispersion curves for water (continued).
115
100.00
TEMP
10
40
30
60
•90
90-
60
\\
/\
/ / /
//,
tttt
0.01
0.10
T
1
I I I I I
T
T
1.00
1
I
I I I I
10.00
FREQUENCY ~ GHz
(b) Ray's (1972) fit
Figure 45. Dispersion curves for water (concluded).
T—r
100.00
Ethylene Glycol
Data were collected on several different alcohols: ethylene glycol (ethanediol),
methanol, and ethanol. However, due to surface tension problems and rapid evaporation
rates, only the results for ethylene glycol (HOCH2CH2OH, molecular weight 62.07,
specific gravity 1.1088 at 20° C relative to water at 4° C, < .05% water) are reproduced
here (Figure 46). Measured values of both e/ and
are consistently lower than
Jordan's (1978) results for frequencies greater than 1 GHz. This is not a surprising
observation in that the experimental procedure does suffer from the shortcoming that it is
difficult to completely fill the sample holder volume with liquids because of surface
tension problems, and it is difficult to maintain the volume of liquid during the 2-3 hour
measurement period due to evaporation. Although ethanediol was least affected by these
problems, sample weights for the high frequency measurements indicate that the holder
was 90-95% full during the experiments. At low frequencies, measurements revealed
another loss mechanism, most likely ionic conductivity.
In summary, results of measurements of air, water, and ethylene glycol are quite
good, resulting in a high degree of confidence in the experimental procedure.
Differences between these measurements and more precise measurements reported in the
literature can be explained by physical and/or chemical factors. One must conclude,
however, that improvements would have to be made to the experimental procedure before
it could be used to accurately measure the electrical properties of a variety of fluids. In
particular it would be necessary to find a way to completely fill the sample holder and to
maintain that fluid level throughout the measurements. The only current concerns with
measurements of heterogeneous mixtures of soil and water are cleanliness of the sample
holder and uneven drying of the sample throughout its volume.
117
10
20
TEMP
0
—
10
30
—
AO
40-
1%
rrrr
0.01
prq-
10.00
0.10
I
100.00
FREQUENCY " GHZ
(a) these measurements
Figure 46. Dispersion measurements of ethylene glycol (continued).
118
TEMP
————• JO
——— 20
30
''
40
40 •
35
30
25
20
15
10
\
\ \\\
5
0
I 11111|
TTvr
I I I 1 iif
40
35
30
25
.//
SO
15
10
//
y
-r-r-rrn|
0.01
0.10
i
i i i rrii|—-i—i"r-i-rriij
1.00
10.00
100.00
FREQUENCY " GHz
(b) Jordan's data (1978)
Figure 46. Dispersion measurements of ethylene glycol (concluded).
119
CHAPTER 6: EXPERIMENTAL RESULTS
Summary of Data Collected
As was mentioned in the previous chapter, attention in this study was focused on
four different soils, two sand, a silt, and a pure non-swelling clay mineral. Nearly
saturated samples were prepared, measured, then incrementally dried and remeasured
until the samples could be dried no further by conventional methods. For each moisture
content, measurements were normally made at ten different temperatures, which took
between three and four hours to complete. There was no control over sample density.
Figure 47 serves as an indication of both data quantity and quality. On this figure
are symbols that represent data collected for the four primary soil materials at one
temperature, 20° C. For most of these data points, data were collected at nine other
sample temperatures. Although they are not shown on this figure, data were also
collected for the empty holder, for water, alcohols, and a swelling clay mineral called
hectorite. Results for hectorite are not reported because physical property information is
not currently available for that material. The total number of data sweeps (a sweep of
frequencies) collected during these experiments easily exceeds five hundred.
The reference above to data quality is in terms of the repeatability of sample dry
density as a result of the crude sample preparation technique described in the last chapter.
The strong clustering of the data points for each material type indicates how closely
sample dry density matched for all samples. Only one string of data is significantly
shifted from the others, and that is the data for kaolinite. The four data points that lie to
the right of the others represent a sample with a dry density approximately 20% greater
than the other clay mineral samples. The starting point for this sample was a batch of
kaolinite that had been allowed to dry somewhat before the sample was inserted into the
sample holder.
120
D
'o
0
to
00
0
•
^00
0
a 0
0
*
, n
n
«
°
^
0
/
0
0
8
I
0
0 ^
0
:n
n
D
0
0
-4—I—I—I—I—I—I—I—I—I—I—I—1—I—I—I—I—I—I—I—I—r4—I—I—I—I—I—I—I—I—I—I—r2
3
4
SAMPLE MASS
^ ^ Ottsws Sind
nnn
Sand
000
(g)
Sill
Figure 47. Data collected at 20° C,
121
000
Kaollnlte
Dispersion in Soils as a Function of Moisture Content
If the electrical behavior of moist soils is controlled by the amount of and purity
of water and how it fills the interstices, then one should be able to observe highfrequency dielectric relaxation and low frequency losses due to conductivity or MaxwellWagner effects or bound water, and these phenomena should be more pronounced as
moisture content increases. One should also observe different responses in different soils
at the same moisture content because of such factors as specific surface and the ways in
which water molecules are attracted to the solid surfaces.
Figures 48 through 52 summarize dispersion measurements for the four soil types.
Each figure contains data for sample temperatures of -10°C and -f-20°C. These two
temperatures were chosen simply to compare soils with frozen water and soils with liquid
water. Remember that the vertical dashed line at 2 GHz represents the break point
between measurements made with the low-frequency apparatus and those made with the
high-frequency setup. Two figures contain silt data, the second one representing results
for a slightly different experimental procedure. In the latter case, both frequency range
sweeps were conducted before the sample was incrementally dried. It is very satisfying
the results for the two setups overlap almost exactly.
Conclusions that may be drawn from these data include the following:
(a) Frozen soils do not exhibit high-frequency dielectric relaxation, but do reveal
some low-frequency losses.
(b) Non-frozen soils do demonstrate high-frequency relaxation which does
increase with increasing moisture content. These results are quite consistent with
those published by Hallikainen, et al (1985).
(c) Low-frequency losses in non-frozen soils are also proportional to moisture
content. Values are consistent with those of Campbell (1988). Minimum losses
in non-frozen soils occur in the 1-2 GHz range which agrees with Hallikainen's
122
(1985) observations.
(d) Comparison of sand and silt high-frequency losses at comparable moisture
contents demonstrates that there is less free water in the silt because of its larger
specific surface (84 mVg versus 10 mVg).
(e) If the relative magnitudes and frequency ranges of different loss mechanisms
published by Hasted (1973) and reproduced in Figure 29 of this report are correct,
then bound water relaxation is a good candidate for some of the low frequency
losses seen in these data. One cannot, however, discount ionic conductivity or
Maxwell-Wagner losses, or both, as contributing loss mechanisms.
(c) The electrical response of frozen soils appears to be relatively insensitive to
moisture content within the frequency range measured in this report, but does
show some dependence on soil type at low frequencies.
From the results of their excellent measurements on real soils ranging from sandy
loams to heavy clays, Topp, Davis, and Annan (1980) reported, that in the frequency
range of 20 MHz to 1 GHz, the apparent permittivity "was strongly dependent on
(moisture content) and only weakly on soil type, density, temperature, and frequency".
While the measurements reported in this study have not dealt with density effects, and
they do support the contention that temperature effects are minimal for unfrozen soils,
they clearly demonstrate that soil type and frequency are important variables.
123
OTTAWA SAND
. Temp=20
/|0
MOIS
34.1
31.2
19.3
35
30
25
20
15
10
I
0.01
I
I
^—
0.10
I I
111 ll—
-1—1—I
I I I
"1—
mnj
100.00
10.00
1.00
20-
18
16
14
12
.//
10
8
6
4
2
0
1—I I I I ii|
-I—I—I I 1111
0.01
1.00
0.10
1—I—I I I 1111
10.00
FREQUENCY " GHz
Figure 48. Dispersion in a poorly-graded sand.
124
100.00
SAND
, Temp=20
40HOIS
14.3
27.2
11.2
24.6
7.3
20.0
35
17.7
29.2
—
30
25-
20
15
10-
TTTI
0.01
0.10
1 1—I I I I ll|
I I III
r-
100.00
10.00
1.00
20-
18-
16-
14-
12
.//
10
8
6
4
2
A
0-
-1—I
1
).01
0.10
1.00
1.00
I I 1 1 1 1
1
1 — I
10.00
FREQUENCY " GHZ
Figure 49. Dispersion in a well-graded sand.
125
I
I 1 1
T
100.00
SILT
Temp=-10
16.8
40.6
I I I I I I
I
I I I 1 1 1
10.00
100.00
10,00
100.00
.//
0.01
0.10
1.00
FREQUENCY " GHz
Figure 50. Dispersion in silt (first set) (continued).
126
23.3
• — • — 45.7
29.8
46.2
SILT . Temp=20
MOIS
I "
0.01
I
I
I
1' I I 1 1 1
"
0.10
T " ' I
I ' l l l l l J
I
I
l l l ' I I I J
1.00
10.00
I
l ' l l l l l l | '
100.00
I
0.01
0.10
10.00
1.00
100.00
FREQUENCY ~ GHz
Figure 50. Dispersion in silt (first set) (concluded).
127
SILT
, Temp=20
HOIS
FREQUENCY * GHz
Figure 51. Dispersion in silt (second set).
128
KAOLINITE Temp=-10
40
35
30
0.0
; 49.0
• 4.4
• 55.0
—
• •
•34.7
• 62.B
•— •
41.6
64.3
• 45.3
• 65.6
25
J
€
20
MOIS
20.50
56.10
31.80
63.90
' 11.80
- 20.10
• 40.60
- 45.20
15
10
I I l l |
0.01
I
TTTJ
I I 1 1l l
-1—I
I I 1111
0.10
1.00
10.00
100.00
0.10
1.00
10.00
100.00
20 H
10-
1614-
12
.//
10
8
6
4
2
00.
FREQUENCY " GHz
Figure 52. Dispersion in kaolinite (continued).
129
KAOLINITE Temp=20
0.0
•
^9.0
4.A
• 55.0
'34.7
•62.0
——- 41.6
' — - 64.3
45.3
65.6
— 1.90
— 20.50
- 56.10
—— 3.84
—— 31.00
63.90
— 11.00
40.60
40 *
MOTS
:\-4.
35
\
30
\
25
20
15
10
I
I—
I—
I—
KII
I
111n
0.01
0.10
I
I
I I M i l
I 1111
10.00
1.00
100.00
so­
le
\
16
\
14
,\
\\
12
.//
10
//
//
/.
8
.//. /
6
//. /
4
2
0
I
0.01
I I 1 1 1 1
0.10
111'j
i i 1 i i 111|™r • "i"t i i 1111
1.00
10.00
FREQUENCY ~ GHz
Figure 52, Dispersion in kaolinite (concluded).
130
100.00
— 20.10
• • 45.20
Single Frequency Observations
Temperature Effects
Data collected at -10, -5, -2, 0, 2, 5, 10, 20, 30, and 40°C is quite adequate for
visualizing the temperature-dependent behavior of the complex dielectric constant at
selected frequencies. These results are shown on Figures 53 through 56 and lend
themselves to several observations:
(a) Liquid water appears at temperatures less than 0°C, probably somewhere
between -2°C and -5°C. This is consistent with results reported by Delaney and
Arcone (1982).
(b) The real part of the dielectric constant for non-frozen soils is approximately
independent of temperature for all soil types.
(c) The imaginary part of the dielectric constant for non-frozen soils is not
independent of temperature, showing a rise with temperature at low frequencies
and a drop with temperature at high frequencies.
Because the low frequency data show a rise in the loss term with increasing
temperature, an estimate of the activation energy involved in the loss mechanism can be
made following the method of Hoekstra and Doyle (1971) and Campbell (1988) which
was discussed earlier in Chapter 3. Using their formula
(73)
Kelvin
and the temperature effects data at 100 MHz yields an average (over moisture contents)
activation energy of 3.1 Kcal/mole for silt and 3.3 Kcal/mole for kaolinite. These
131
numbers are within the range of values attributed to hydrogen bond breakage.
40
35
30
25
15
I I I I I I
-10
0
I I I I I I I I I I I I I I I I I I I I I I I I I I I I'
10
20
30
40
TEMPERATURE'DEG C
Figure 53. Temperature effects for poorly-graded sand (8 GHz).
132
40 •
35MC
13.4
31.3
30
25
g!
20
15-
10-
I I I I I I I I I I I I I I I I I I I I I II 1 I I I I I I I I•'I I I I • I ' I
-10
0
10
20
30
40
40-
35-
30-
25-
e"
20
15-
10-
1111111111»111111111111111111111111111111111111
-10
0
10
20
30
40
TEMPERATURE " DEG C
Figure 54. Temperature effects for well-graded sand (8 GHz).
133
40 H
MC
14.0
35.4
.
26.6
42.3
31.1
47.8
32.1
35
30
35
30
25
.//
20
15
10-
04
i"i r
-10
0
r'l' I• I 'I I"I" m ~ n
10
20
• I'I''
30
40
TEMPERATURE " DE6 C
(a) 100 MHz
Figure 55. Temperature effects for silt (continued).
134
40 •
MC
35
7.2
.35.9
9.3
36,0
IB.a
40.6
23.3
• — •"•45.7
29.8
— —46.2
30
25
e
20
15
10
0-•j-t-n
-10
I I I I I I
< I 1 I I I I I I I I I
20
10
30
40
40
35
30
25
e."
20
15
10
0-
I I I I I I I I I I I I I I
-10
10
20
30
40
TEMPERATURE " DEG C
(b) 8 GHz
Figure 55. Temperature effects for silt (concluded).
135
25
20
15
10
5
0
-n-r
10
I I I I I I I I I
20
10
30
nr
40
40
MC
35
'
0.0
49.0
4.4
55.0
——— 34.7
• • . . 62.B
——
— 45.3
65.6
64.3
30
25
20
15
r
10
5
0
10
0
10
20
30
40
TEMPERATURE " OEG C
(a) 100 MHz
igure 56. Temperature effects for kaolinite (continued).
136
MC
40
1.90
20.50
56.10
3.84
31.80
63.90
11.80
40.60
20.10
• • • • 45.20
35
30
25
20
15
J
10
I II I I I I I I I I I I II II II I I I 1 I I
-10
I II III I
20
10
I II I I I I I I I I I I
30
40
40
35
30
25
20
15
10
0-10
I I I I II IN
10
20
30
I I I II
40
TEMPERATURE ~ DEG C
(b) 8 GHz
Figure 56. Temperature effects for kaolinite (concluded).
137
Moisture Effects
Figures 57-60 summarize data at selected frequencies as a function of volumetric
moisture content for each of the materials tested and at +20° C. What can said
immediately from an examination of the data on moist soils presented in this format is:
(a) The real part of the dielectric constant for all non-frozen soils is best
described by some non-linear relationship with volumetric moisture. Bilinear fits
such as those sketched in on each figure would break at critical values of moisture
content in the range of 30-35% for the silt and clay minerals and in the 10-15%
range for the sands. Wang and Schmugge (1980) reported critical moisture
contents in the 20-30% range for the real soils that they studied.
(b) The capacitive nature of non-frozen soils as reflected in the real part of the
dielectric constant varies with soil type.
138
Silt
000
* A * Kaolinite
VOLUMETRIC MOISTURE CONTENT " X
Figure 57. Moisture effects for non-frozen soils at 100 MHz, 20° C.
139
* * * Kaolinite
000
10
"
oU^:
°°° °n
*
*
**
*
Y « ' « « I i • I I I' l 'i' i ' i I I I I I ' l ' i 1 1 " : I I I 1 1 I I I I 1 1 1 I I I I 1 1 I I ' l r i 1 1 I 1 1 I I ' l ' i i T i I I I I I I'
0
5
10
15
20
25
30
35
40
45
50
55
60
65
VOLUMETRIC MOISTURE CONTENT " %
Figure 58. Moisture effects for non-frozen soils at 800 MHz, 20° C.
140
40
35
30
25
20
15
10
5
0
40
35
n n n Ottawa Sand
000
Sand
o oo
***
Kaolinlte
30
Silt
25
20
15
10
5
Q,
g
*
0
I • 1 1 1 1 r i 1 1 I 1 1 n 1 1 I • I r i 1 1 1 1 1 1 1 1 1 1 1 • 1 1I I n I • I n I I 1 1 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1
5
10
15
20
25
30
35
40
45
50
55
60
65
VOLUMETRIC MOISTURE CONTENT " %
Moisture effects for non-frozen soils at 2 GHz, 20° C.
141
40-
35
30
25
20
15-
10
11111111111111 H111111111111111111111111111111111 n 111111111111111
0
5
10
15
20
25
30
35
40
15
50
55
60
65
40-
35
n n D Ottawa Sand
^
Sand
Q oo
***
30
Silt
Kaollnlte
25
20
15
10
Oiff
no
>»
A
I I I I I I I I I I I I I IT |"l I I I'jTI'l 'I'l II II I I I I I 'I IIII I llll|llll|llll|'
0
5
10
15
20
25
30
35
40
45
50
55
60
65
VOLUMETRIC MOISTURE CONTENT " %
Figure 60. Moisture effects for non-frozen soils at 8 GHz, 20° C.
142
CHAPTER 7: DATA ANALYSES
Contained within the following paragraphs are two approaches to interpreting the
data shown in the previous chapter. One is an adaptation of the classical technique of
representing the electrical response of an heterogeneous mixture by an equivalent circuit
that incorporates the concept of series and parallel behavior discussed earlier as well as a
model for water behavior that accounts for dielectric relaxation. The new features of this
model are the broad frequency band of simulations through a modification of the
equivalent circuit for water and the attempt to fix the values of as many circuit elements
as possible while giving physical interpretations to the remaining parameters.
The other analysis technique involves the application of fractal geometry to model
the structure of the porous media and the association of a change in the fractal dimension
with the onset of long-range connectivity. While the model has been used in basic
research on the properties of coal, this is its first application to soils. A new hypothesis
that the soil moisture content (or cumulative pore volume) that bounds the fractal
behavior is equivalent to the critical moisture that defines a transition from series to
parallel electrical response is tested and yields fractal dimensions for the soils tested
within this study that are supported by other references.
Equivalent Circuit Representation
If one accepts that the electrical response of soil-water mixtures is bounded on the
lower end by a "series-like" behavior and on the upper end by a "parallel-like" behavior
(Campbell, 1988), then the most sensible simulation should include both elements. With
the requirement for both series and parallel response, one very sound approach to
modeling the response of moist soil by the method of equivalent circuits is the three-path
model first suggested by Sachs and Spiegler (1964) and adapted by Smith (1971).
However, the specific representation used by Smith and shown on Figure 36 is not
adequate to simulate both low-frequency loss mechanisms and losses at higher frequencies
due to dielectric relaxation. A variation of the three-path model was adopted for this
143
study using the following reasoning.
Data from the last chapter and from other sources show clearly that the response
of moist soils is controlled by the presence of water in the samples. Most of the data
show a high-frequency anomalous loss highly correlated with moisture content and a
conductivity-like low frequency loss mechanism. In fact the data do not say whether the
low frequency losses are due to a conductivity effect in free water or a Maxwell-Wagner
mechanism caused by the presence of the soil particles as suggested by many. Lower
frequency measurements coupled with a much more thorough understanding or control of
the soil chemistry would help answer this last concern. Nevertheless, the best equivalent
circuit for the liquid paths has to be that suggested in Figure 38 that includes a highfrequency permittivity, a DC conductivity, and relaxation losses in the 10r20 GHz range
of frequency.
As for the contribution by the soil, it is felt that a good representation for its
electrical response would be one that accounts for a constant wave speed at all
frequencies while including a low-frequency loss term. Therefore, the parallel circuit
shown in Figure 30 (b) was chosen as the most appropriate representation for the soil
component of the mixture. One implication of this selection is that one cannot model
anomalous losses within the soil that result from the interaction of the soil and water
particles. However, in the range of frequencies covered by these experiments it is
probably not possible to detect the difference in losses due to one mechanism or the
other.
Combining these two sets of elements, the final three-path equivalent circuit used
to simulate the response of the soils tested in this study is that shown in Figure 61, where
the parameters a, b, c, and d have the same meaning as discussed previously. A thought
process for selecting the model parameters is discussed later, and the results of a number
of simulations will be shown, but first, a brief presentation of the governing equation for
this circuit is given.
As was done in previous sections, the macroscopic complex dielectric constant of
the moist soil material is modeled as the equivalent capacitance of the circuit shown in
Figure 61. Using the w subscript for water and the s subscript for soil, the macroscopic
144
dielectric properties can be written
€ =
+ jb€g +
(71)
where
e
g
(74)
e
sol I
SOI I
water
c,
vz
water
c.
J
*1
1-d
J
Figure 61. The three-path equivalent circuit used for this study
145
In terms of the simple circuit capacitances and resistances,
€3 = e/g +
+ i
1
(75)
COi?
and
(76)
1 +
1 + [^R„2^w2)^
A reasonable set of circuit parameters begins with the same values chosen for
water that are shown on Figure 34. They allow for a dielectric relaxation peak at about
16.75 GHz and for the finite values of conductivity losses at frequencies less than 1
MHz. Although the data measured in this study did not go as low as 1 MHz, it was
discovered by trial and error that the low frequency conductivity term helped improve the
model's fit to data through its interaction with the soil element response.
As for the soil parameters, a static permittivity, C„ was taken to be about 2. That
only leaves the soil resistivity, R,, to be considered. It so happens that I allowed soil
resistivity to be a variable; however, it is rather easy to estimate its range in the
following way. The model has associated the low frequency losses with the soil element.
They, in turn, go like l/(cojRg) . A typical value of the loss term is 10 at 2 GHz for
wet silt. This leads to an estimate of the resistivity as about SxlO^'^s/cm.
To obtain values of the weighting parameters, a, b, c, and d, as well as the value
of R„ a simple iterative code was developed that minimized the difference between circuit
predictions and real data. Smith (1971) used a similar approach to calculate the
parameters associated with his three-path model and hinted that it was a rather
sophisticated calculation process. I did not find it necessary to do anything more
complicated than to increment each parameter in both directions and to calculated the
difference between all data points and predicted values for each parameter change taken
146
one at a time. That parameter increment that caused the greatest reduction in difference
was applied at each step.
Smith (1971) also chose to allow all of the parameters in his three-path model to
vary except for the permittivity of water. I chose to hold most of the electrical
parameters fixed because they have physical meaning. The permittivities selected for
both the water element and the soil element reflect observed values that are associated
with wave velocity in each media. The value of resistivity in the water element that
results in the anomalous behavior is necessary to locate the peak losses due to dielectric
relaxation. In the absence of data at higher frequencies, the frequency of peak loss was
taken to be constant for all moisture contents. The soil resistivity was allowed to be
variable to help optimize the fits to low frequency loss data.
Figures 62-64 contain a comparison of model fits to real data for tan sand, silt,
and kaolinite samples at three different volumetric moisture conditions. Besides
demonstrating the ability to simulate the real response of the soil samples quite well, it is
gratifying to note sensible trends in the weighting parameters. Obviously, as moisture
levels rise, the contribution to sample response from the parallel (or long-range
connectivity) water element should increase as reflected in an increase in the parameter,
c. Because water dominates the electrical response, the soil-related parameter, b, should
drop as moisture increases. Similarly, if the series elements can account for MaxwellWagner like behavior in the soil-water-air mixture, then it is also sensible for the losses
contributed by the soil particles in the presence of an increasing amount of water to
increase as well. This would be reflected in an increase in the parameter, d, which, in
general, is observed.
147
40
35
30
25
20
15
IB
Doamooo an cipcpBi>BoaoD° ouoawwaRQ
5
Q
I
.1
I
_J
L_
1
10
100
FREQUENCY (GHz)
40
as
C w l=
2
4
Cw2"=
76
Rs
5.07110793561E-9
Cs
30
25
=•
=
Rwl =
Ruj2=
20
15
10 I-
PMMllrfHT mg gea*WWtjliL n
.1
5.888888E-8
1.25E-13
B
=•
.951482080462
C
=
.0393254985868
D
=
.0970299
,
(0
100
(a) 7 percent moisture
Figure 62. Equivalent circuit model-data comparisons for tan sand (continued).
148
4B
35
30
25
20
IS
IB
owimuo aauJijjuuixtfiPu uoqi
5
a
_J
.1
40
1
FREQUENCY (GHz)
10
im
as
=
2
C w l=
4
Cw2=
76
Cs
30
25
Rs
28
Rwl =
Rw2:
B =
IS
IB
au nn
-J
=
2.1550865132lE-S
5.888888E-8
1.25E-13
.914355577452
C
'
.0835669675074
D
=
.09801
RwrtiUP"<
IQQ
IQ
(b) 14 percent moisture
Figure 62. Equivalent circuit model-data comparisons for tan sand (continued).
149
.]
4B
1
100
10
FREQUENCY (GHz)
as
"
2
Cwi «=
4
Cw2=
76
Cs
30 h
•=
25
Rs
2B
Rwl"
Rw2=
5.888888E-8
1.25E-13
B
=
.795455493409
C
=
.198338903211
D
=
.099
15
IB
1.0217900642GE-
5
B
(c) 27 percent moisture
Figure 62. Equivalent circuit model-data comparisons for tan sand (concluded).
150
4B
35
30
25 •
28
-
15 •
IB -
"(tfOawjUQ Dat,cc(Sa%%
5 •
B -
J
.1
40
I
I
I
]
10
FREQUENCY (GHz)
100
35 .
Cs
=
2
Cwl=
4
Cw2=
76
25 •
Rs
4.01007541109E-10
28
RwlRw2=
5.888888E-8
1.25E-13
B
=
.869456101922
C
=
.0194597770088
D
=
.0351609206558
30 •
•
IS •
IB •
=
5 •
B -
.]
]
im
iQ
(a) 10 percent moisture
Figure 63. Equivalent circuit model-data comparisons for silt (continued).
151
4B -
35 30 -
Z5 -
20 -
]5 -
IB 5 •
0 -
.]
1
100
10
FREQUENCY (GHz)
4B -
as •
Cs
30 •
25 •
20
-
15 •
IB
5
\
=
2
C w l=
4
Cw2=
76
Rs
2,14242679558E-10
=
R w l=
5.888888E-8
Rui2 =
1.25E-13
B
«
.77923619408
C
=
.121208323022
D
=
.0501166414118
\
'•"*^0 0 ri
B
_J
__L
IQ
IQQ
(b) 25 percent moisture
Figure 63. Equivalent circuit model-data comparisons for silt (continued).
152
40 r-
35
A
38
25
20
J5
IB
S
_J
B
.1
1
100
10
FREQUENCY (GHz)
40
35
=
2
Cu)l =
4
Cs
30
25
J!
€
Cw2=
76
Rs
9.10474624347E-11
=
Ru)l =
20
Rw2=
B =
15
18
5.888888E-8
1.25E-13
.620209064457
C
=
.293510390969
D
=
,09801
_J
.1
Km
IQ
(c) 39 percent moisture
Figure 63. Equivalent circuit model-data comparisons for silt (concluded).
153
40
35
38
25
e'
20
15
10
aaoewoDMmeu u'daortuciHBgO
I
l_
.]
J
10
FREQUENCY (GHz)
I
4EI
_1
100
as
'
2
Cwl=
4
Cw2=
76
Rs
3.10090155884E-9
Cs
30
25
=
Rwl=
2B
Rw2=
B =
15
IB
5.888888E-8
1.25E-13
.988829689491
C
=
.00676569027829
D
=
.0730840153457
5
0
.]
a •ndgtunomuJo
1
10
IQQ
(a) 3.5 percent moisture
Figure 64. Equivalent circuit model-data comparisons for clay (continued).
154
4B •
35 •
3B 25 •
2B •flu mujuiygJ5 •
10 -
5
D
J
.1
I
I
}
10
.
I
100
FREQUENCY (GHz)
4B
as
Cs
=
2
C w 1• =
4
Cw2=
76
25
Rs
7.2538305G.4GE-10
2B •
Rwl=
Rw2=
5.888888E-8
i.25E-13
B
=
.712059299585
C
=
.157215804939
D
•=
.0738222377229
3D
15
IB
=
S
B
.1
1
(b) 42 percent moisture
Figure 64. Equivalent circuit model-data comparisons for clay (continued).
155
40 35 SB -
Z5 -
20 -
15 -
IB -
5 •
_J
B -
.1
40
1
FREQUENCY (GHz)
100
10
35 -
=
2
C w l=
4
Cw2=
76
Rs
5.19282102695E-
Cs
30 •
25 •
20 •
IS •
la -
=
Ruj1"=
5.888888E-8
Rw2=
I.25E-13
B
=
.52898037522
C
=
.451648706606
D
=
.120798813955
5 •
0
IQ
100
(c) 64 percent moisture
Figure 64. Equivalent circuit model-data comparisons for clay (concluded).
156
While the arrangement of elements in the three-path model is physically
reasonable and the simulations are very accurate, a number of questions concerning its
application remain unanswered. For example, when does one hold circuit element
parameter values constant and, if not, how should those parameters be allowed to vary?
It might seem preferable, at first thought, to hold all circuit element values fixed and
allow only the weighting constants to vary. In fact, this approach was tried, but with
only limited success. If only the weighting coefficients could vary, then they would have
to account for all of the dramatic changes that take place chemically within the mixture,
and they don't appear to be able to do so. In other words, by allowing only weighting
coefficients to change implies that the properties of the basic constituents do not change
as the mixture changes. Even if the selected parameters for dry soil particles and
unbound liquid water are quite accurate, weighting parameters, by themselves, cannot be
expected to account for the loss mechanisms due to bound water, or due to MaxwellWagner type effects, or enhanced conductivity due to salts going into solution, or
enhanced capacitive response set up by the parallel plate-like structure of the clay
particles being filled by water in the interlayer spaces or simply between particles in nonswelling clays.
If, then, some of the circuit parameters are allowed to vary, which ones should?
Why let the resistivity of the soil element vary and not the water?
A number of
simulations were tried with R, fixed and R^i free. These results were unsatisfactory for
the mixtures that showed increasing
simulation values of
values with decreasing frequency, with the
becoming constant with decreasing frequency.
In summary, the three-path equivalent circuit is a promising tool for exploring the
complex dielectric behavior of heterogeneous mixtures. Simulations done in this study
very accurately reproduce measured responses. The model accounts for both series and
parallel electrical responses and possesses some parameters that can be fixed to values
measured in other experiments. There are still, however, unresolved issues such as what
physical or chemical properties can be associated with the parameters that do vary that
should provide a rich opportunity for further research.
157
Fractal Geometry Model and Critical Water Content
Campbell (1988) suggested that water may fill the pore spaces of a soil sample in
a fractal manner. Pursuit of this concept, particularly as it relates to the distribution of
pore sizes in a soil sample, has led to the discovery of related research in pore size
distributions (Sridharan, et al 1971; Arya and Paris, 1981) and in the modeling of
porous media as fractal geometries (Friesen and Mikula, 1987) that lends itself to a
relatively simple fractal interpretation of pore size distribution data. The difference
between the following study and Campbell's model is that rather than assuming porefilling water clusters have a fractal geometry, this new approach models the soil
structure, itself, using fractal concepts and makes certain assumptions to relate volumetric
soil moisture content to the fractal model behavior.
A Fractal Model of Pore-Size Distribution
The porous soil-water-air structure is taken to be described by a sponge-like
structure similar to that shown in Figure C3. Much of what follows has been adopted
from Friesen and Mikula (1987), where additional references are given to some earlier
work. Their application was toward a better way of quantifying the porous structure of
coal.
Consider, first, a unit solid cube of soil minerals. Divide the cube equally into m^
subcubes of size d=l/m. Next remove the subcubes necessary to give the basic structure
generator shape. (In the case of the Menger sponge shown in Figure C3, m=3, and the
number of subcubes removed is 7.) That leaves a number of remaining cubes = N^. But
by the definition of fractals given in equation C2,
= (d)-^
(77)
where D is the fractal "dimension" of the porous media. Now repeat the entire process
on each subcube. After this second step, the new subcubes have a dimension of d^ and
the number of remaining subcubes is N^^. Continuing to the kth subdivision, we have a
particle size of d"^ and the number of remaining subcubes is
In other words, the
volume of remaining solid material at step k is
Vk =
(78)
Noting that at each step, the particle size is identical to the smallest pore size, and
renaming the particle size (d)"' as 1^,
and the pore volume is just
Vpor, = 1 - {hf"
What I now have is a fractal model for a porous medium that relates pore volume
(which has to be related to moisture content) and pore size. If I had such a pore-size
distribution relationship, then I would have a graphical technique for determining the
fractal dimension by plotting the slope of the distribution vs pore size on a log-log plot:
/
log
V
= constant - (2-D)log(i)
The Fractal Model and Pressure Plate Data
Unfortunately, simple pore-size distribution data do not exist. What is available is
a limited amount of porosimetry and/or pressure plate data which can be interpreted as
pore distribution data through the capillary relationship (Bear and Verruijt, 1987)
p = --^COS(0)
159
(82)
where
p =
equilibrium pressure
6 =
surface tension of non-wetting fluid
6 =
contact angle between fluid meniscus and pore surface
D =
capillary diameter
Substituting the capillary relationship into the fractal model for pore distribution,
Vpoxa = 1 - (-f)"" = 1
where K and K' respresent a lumping of constants. Then
^^pore _ -Kfm-2)p^-'^
dp
(84)
from which
/
log
j = constant - (D-4)log(p)
Therefore, the slope of a log-log plot of the derivative of a porosimetry or pressure plate
curve is proportional to D-4.
As a test of this relationship, the actual pressure plate data from Arya and Paris
(1981) for five different soil samples were digitized, their slopes estimated by difference
methods, and the results plotted on Figure 65. In general these curves reveal a bilinear
response with the break point occurring at approximately the field capacity of 0.333 bars.
At pressures higher than field capacity but less than 15 bars, which soil scientists refer to
as the wilting point (Wang and Schmugge, 1980), a straight line fit to the data on Figure
72 would result in a fractal dimension for the soil structure of approximately 2.67. The
break in the slope of these data roughly corresponds to volumetric moisture contents of
25-40 percent.
160
Arya and Paris Desorption Data
Q•
-1
-a •
I Field Capacity
O O O
dV.m
dP
Wilting Point
-5
-6
-7
Capillary
Gravitational
Forces
j
Forces
-8 -
"T I I I I I I I I j 'I•I—I—I—I—I—I—I—I—I—I—I—I—I n I 'm I I r T
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
4
log(P)
* * * 70% s-c, 30% sO O O loom, 20-30 cm d
I
p
5
(cm
•Q n l O B n i ' 40-50 cm d
20% s-c, 80% s-
tr ir ir
^^^ 40% s-c, 60% B- I
Figure 65. Fractal model applied to real soil desorption data (after Arya and Paris, 1981)
161
An interesting question is whether or not one might be able to predict these
observed moisture contents from ancillary data. For example, if the fractal behavior of
the soil is, in fact, associated with water being held by capillary forces, then the field
capacity can be identified with a certain pore diameter through the capillary equation.
Taking conditions for water and clean glass (contact angle equal zero degrees and surface
tension equal to 72 dynes/cm at 25°C) as an approximation for the elements in moist soil,
one can calculate that the range of pressures from infinity to field capacity cover a range
of pore diameters up to about 8 microns. Furthermore, if one assumes as did Campbell
(1988) that the smallest pores in soil fill first with water, then one could always estimate
the moisture content below which the soil structure can be modeled as fractal in the
following way. First of all, we have by definition
(86)
where
Wv = volumetiic moisture content
V„ = volume of water in a sample
Vt = total volume of the sample
Vv = volume of voids in a sample
n = sample porosity
S = degree of saturation of sample
But the sample porosity can be rewritten in terms of sample dry density and particle
density, and the degree of saturation is (under the pore-filling assumptions) just the
integral of the pore distribution curve from zero volume to the cumulative volume
associated with the particular pore diameter. The cutoff value of volumetric moisture
content for fractal soil sti*ucture response can then be written simply as
162
.5,
(87)
where Sg refers to the degree of saturation (or the decimal value of cummulative pore
volume) at the 8 micron mark on a pore-size distribution curve.
In summary, then, if one assumes that soil fills with water beginning with the
smallest pores, and if one accepts the small-pore fractal model, then the moisture content
below which the soil possesses a fractal structure can be estimated from the sample dry
density and cumulative pore volume given by the 8 micron pore diameter mark on the
pore-size distribution curve.
The Fractal Model Related to Particle-Size Distribution Data
In the absence of porosimetry and pressure plate data (which is the case for this
research), it would be advantageous to estimate pore volume distibutions from other
measurements that are more readily available.
At least one simple model for pore distributions does exist; namely, the
physicoemperical model offered by Arya and Paris (1981). Their research centered on
the need for a model to predict soil moisture characteristics from simple data like
particle-size distributions and bulk (dry) density, and their key assumptions were:
(a) The particle-size distribution curve can be broken into segments covering a
small range of diameters and that the solid fraction within each segment, or
particle-size range, has a bulk density equal to that of the natural-structure sample.
(b) Each solid fraction consists of uniform-size spheres having the mean radius of
the fraction.
163
(c) The pores in each segment are uniform-size cylindrical capillary tubes whose
radii are related to the mean particle radius for that segment.
If there are n, spherical particles of size R, in the ith particle-size segment of the
distribution curve, then
Kn
where
Vp^ = solid volume of the ith fraction per unit sample mass
IVj
= solid mass/unit sample mass in the ith fraction which comes from the particlesize distribution curve
Pp
= particle density
Arya and Paris then let the total pore volume for the ith fraction be represented by a
single pore of radius r, and length h; that threads the volume occupied by the particle-size
fraction. Thus
V„ = TCZ/j]; =
W,
(89)
P/
where
= void volume of the ith particle-size fraction per unit sample mass
e = void ratio = volume voids/volume solids
Not knowing exactly how this single pore threads the mass of spherical particles,
Arya and Paris assumed that the total pore length would be something greater than n;2R;
and introduced an empirical factor, a , defined by
164
Aj =
(90)
Then dividing Equation 89 by Equation 88 and substituting Equation 90, one arrives at
their final relationship between pore radius and particle radius in the ith particle-size
fraction
1/2
=
(91)
Ri
where % comes from Equation 88 and the manner in which the particle-size distribution
curve gets subdivided.
By assuming that the pores fill with water beginning with the smallest size and
accumulating the total pore volume as one progresses with the calculation, a volumetric
water content can be calculated
= E ^VjPb
j=l
At the ith particle-size fraction, the capillary suction comes from Equation 82. Thus one
can track a soil tension vs volumetric water content. Arya and Paris did this for several
soils and adjusted the empirical parameter, a , for each to give the best fit to the
experimental data. For five different soils, the best fit values of a ranged from 1.35 to
1.39.
In summary, given some knowledge of the a parameter that first appears in
Equation 90, we have a model for converting a particle-size distribution curve into a
pore-size distribution curve and an opportunity to apply the fractal geometry model and
compute a limiting volumetric moisture content, below which the soil structure can be
taken as a fractal geometry.
As a test of this model, consider its reapplication to the data that Arya and Paris
provided in their paper. Figures 66 to 70 contain a reproduction of their particle-size
distribution curves for five of the soils and soil mixtures that they studied along with a
165
pore size distribution deduced from their soil tension data using the standard capillary
relationship. The contact angle was taken to be zero degrees and the surface tension for
pure water at 25°C was used to convert pressure data to pore diameters. Maximum
volumetric moisture (100% saturation) was taken to be defined by the dry density of the
sample as before. The solid line on each figure is the particle-size distribution curve, and
the dashed line is the pressure-plate derived pore-size distribution. The open circles on
the curves represent the results of calculating a pore-size distribution curve using the
model above.
166
100
90
80
H
«
60
§
CO
a
w
40
PM
100
1
0
1
2
-3
-4
-5
log (Diameter) ~ millimeters
Figure 66. Comparison of measured and predicted pore-size distributions; 70% silty
clay, 30% sandy loam (Arya and Paris, 1981).
167
100
CO
20-
O,
0.87
OO CD
1
0
100
1
2
-3
4
-5
log (Diameter) ~ millimeters
Figure 67. Comparison of measured and predicted pore-size distributions; loam
40-50 cm depth (Arya and Paris, 1981).
168
100
I
H
o/
o.
3
50
C/}
CO
O/'
i
o /•
100
1
0
-1
2
-3
-4
-5
log (Diameter) " millimeters
Figure 68. Comparison of measured and predicted pore-size distributions; 40% silty
clay, 60% sandy loam (Arya and Paris, 1981).
169
100
c
90
o
o
80
o
I
H
70
^
60
o
/
/
/
/
o
o
^
^
/
H
7^
T-.°
40
1
1
/
50
\
Sg = 0 . 5 8
\ o'. *
1
1
30
/A\
/
20
^
6
1
X.
10
1
1
oP
• 11 11 1 11 1 1
1 1 1 11 11 111
-1
-2
-3
-A
log (Diameter) ~ millimeters
Figure 69. Comparison of measured and predicted pore-size distributions; loam
20-30 cm depth (Arya and Paris, 1981).
170
-5
100
i
B
I
Sg = 0 . 5 6
1
Pi
100
-1
-2
-3
-4
log (Diameter) ~ millimeters
Figure 70. Comparison of measured and predicted pore-size distributions; 20% silty
clay, 80% sandy loam (Arya and Paris, 1981).
171
Other than the large pore size discrepancy for the sample shown in Figure 70, the
two techniques for calculating pore-size distributions (one from pressure-plate data and
one from pore-size distribution data) produce quite similar results. As a brief aside, it is
interesting to note that Campbell's earlier assumption of median pore size equalling
median particle size does not hold for these data. One might even say that based on this
limited model exercise, the median pore size is approximately an order of magnitude
smaller than the median particle size.
Now, how well does the fractal model for small pore structure apply to the Arya
and Paris data? From their data I can obtain the moisture content for each sample at field
capacity, and from the previous figures, I can read Sg, the degree of saturation at a pore
diameter of 8 microns. Equation 87 can then be used to compute the moisture content at
the limits of fractal soil structure (particle density is taken to be 2.65 g/cc). These results
are summarized in the following table and show a remarkable correlation.
Table 3
Fractal Cutoff Moisture Content vs Measured Field Capacity Moisture
(data from Arya and Paris, 1981)
Sample
Dry Density Peg Sat-8 mic
Mvlww {Mvlndd cpaci^
B
1.400
0.81
0.38
0.39
C
1.416
0.87
0.41
0.42
D
1.480
0.67
0.30
0.33
E
1.456
0.58
0.26
0.28
F
1.517
0.56
0.24
0.25
172
Fractal Model Applied to This Study
Is the fractal structure model compatible with the data collected in this study?
Most importantly, can I correlate the electrical response of the test soils with the fractal
model? First of all, let me state the hypothesis:
(a) Soil water fills the smallest pores first in a dry sample.
(b) Soil structure can be modeled by a fractal geometry in the range of pore sizes
in which water is held by capillary forces.
(c) Within the small-pore fractal range, the fractal dimension can be calculated
from knowledge of either the pore volume-pressure relationship (Equation 85) or
the pore volume-pore size relationship (Equation 81).
(d) The fractal model fails at pressures less than field capacity (1/3 bar), which,
using the capillary equation and assumed values of surface tension and contact
angle for water, is equivalent to a pore diameter of about 8 microns (in other
words, the fractal model applies for pore diameters less than 8 microns).
(e) The upper bound on moisture content associated with the small-pore fractal
model is the field capacity, which can be estimated from the product of sample
porosity (which is calculated from dry density and particle density) and the degree
of saturation at the 8 micron mark on the pore volume-pore size curve.
(f) The critical moisture content at which there is a transition from series to
parallel-like electrical behavior is identical to the field capacity.
173
Items (a) through (e) were examined in the previous sections. It is item (f) that
contains the physics of the relationship between soil moisture levels and the electrical
response of the soil. What it says is that long-range connectivity becomes achievable
when the pores governed by the capillary equation become filled with water. At moisture
contents higher than this value, free water exists throughout the sample; i.e., the sample
is no longer just "moist," it is now "wet".
The ideal set of data required to test this hypothesis would include enough
electrical response measurements as a function of moisture content to determine the
critical moisture content and field capacity measurements taken from pressure plate data
at the same sample densities. Unfortunately, these data do not exist. The best that I can
do is to use the electrical property measurements in this study and estimate the field
capacity of my soil samples from the particle-size distribution curves using the Arya-Paris
model to generate a pore-size distribution. This requires values of the fitting
parameter, oc .
Although Arya and Paris (1981) were able to use values of the model parameter
that fell within the range of 1.35 to 1.39 for the five soils discussed in the previous
section, I would argue that these values may not be suitable for all soil samples. Their
soils were well-graded and all contained some sand as well as some clay. The soils used
in this study are, relative to the Arya-Paris soils, poorly graded. Recalling that
the a parameter was a way of accounting for the fact that the length of a pore
associated with an assumed spherical particle is something more than twice the particle
radius, it seems likely that it could vary considerably for soils that are poorly graded and
for which the basic particle shapes are distinctly different.
For example, I see no reason for not believing that for the flat plate-like structure
of kaolinite, that the pore length contribution from each particle should be about the
particle diameter. In other words, a for kaolinite should be on the order of 1.0. At
the other extreme, Ottawa sand in a close-pack geometry should possess pore channels
that hug the surface of the almost-spherical particles which means each particle
contributes a path length much greater than its diameter. This would imply the largest
values of a .
174
Since I have no knowledge of the a parameter for my soils, I will take the
reverse approach to testing the above hypothesis. I will assume that the critical moisture
content is the field capacity, which, in turn, is represented by the degree of saturation
shown on the pore volume vs pore size curve at the 8 micron diameter mark. Following
the generation of several pore-size distribution curves from my particle-size distributions,
I can estimate a
for each soil by the value that yields a pore-size distribution that
matches the critical moisture and the 8 micron intersection. With this value of a I can
generate the most likely pore-size distribution curves for each of my soils and graphically
determine the soil fractal dimension for each soil type using the relationship in Equation
81. The final test of the hypothesis will be to compare those values for firactal
dimensions with data for similar soils that are published in the open literature.
First of all, Figures 71 to 74 show the results of applying Arya and Paris' model
to each soil used in this study with a allowed to vary from 1.0 to 2.0 in increments of
0.1, Secondly, values of critical moisture were assigned to the four soils based on the
moisture-related data shown on Figures 57-60. These values, along with soil densities
are used to calculate the required degrees of saturation shown in Table 4. Armed with
this information, one can return to Figures 71 to 74 and estimate the value of a
associated with each soil that are also shown in the table.
Table 4
Estimating tt
Soil
(Mv)criUcal
Pdry
for the Soils in This Study
n
P part
a J]
a
00
Ottawa sand
0.0
1.7
2.67
0.36
0.0
tan sand
0.10
1.7
2.66
0.36
0.28
1.8
tan silt
0.35
1.4
2.71
0.48
0.73
1.3
kaolinite
0.35
0.9
2.61
0.66
0.53
0.9
* assumed to be like that of the tan sand
175
100
90
80
§
70
60
50
40
30
1 /mmp•
^iiiiiiiiil
iiiijiiiii
iiiiii i
iiiij#
illlnllili
I
a = 1.0 • 1 *
« I
i «
a = 2.0
liM/M...
20
10
> 1'*1'' 11 i * w \ \\
i#7'
f
I I I I 11 TT-rr
-2
-3
IIIII
IIIIIII
-4
log (Diameter) ~ millimeters
Figure 71. Predicted pore distribution curves for Ottawa sand.
176
-5
PERCENT SMALLER BY WEIGHT
PERCENT SMALLER BY VOLUME
100
1.0
90
V'
g
H
<±
30
2.0
100
1
0
1
2
3
4
log (Diameter) ~ millimeters
Figure 73. Predicted pore distribution curves for tan silt.
178
5
100
\ « = 10 /
90
,.V,/
\
•\
\
'
•
\
... A
80
/
/•
i•
/ .
f
,
i
/
/
i
'
y'
' •
r
\
\ ''l I' i 'L 'l
70
<
.
/
60
%
50
:
/
/
:
.
&
/
40
i
Y
£
'/
•
/
•'V
/
/
30
/ A/ / : / /
20
10
,
1 « 4
/ / A/ /://
/ / l /y'
• •
, <
«
I . I « «
\ 1 1
1
1 • « /
* • 1 ^1 1
0
-1
-2
-3
-4
log (Diameter) ~ millimeters
Figure 74. Predicted pore distribution curves for kaolinite.
179
-5
Finally, I can now take the slopes of the calculated pore-size distribution curve for
each soil (using the estimated a value) and infer from those results a fractal dimension
for the soil structure in the manner described in a previous section. Figures 75 to 77
show the results of this exercise and include the estimate of the soil fractal dimensions.
Ottawa sand is not reported because it appears to have no critical moisture electrical
response.
Although very little data exist in the literature on fractal dimensions of minerals
and/or soils, the extremes are supported by at least a couple of references. For example,
Pfeifer (1984) calculated a fractal dimension for kaolinite of 2.92 using a dye adsorption
and photometric measurement technique. On the other hand, he and other colleagues
(Farin, Avnir, and Pfeifer, 1984) report from another source a dimension of about 2.15
for quartz sands using a nitrogen adsorption technique.
In summary, the above exercise does not prove, conclusively, that the critical
moisture content of a soil that identifies a transition from series to parallel electrical
response is equivalent to its field capacity, but it does demonstrate that assuming this
hypothesis is true results in a fractal structure model for the soil that is supported by a
limited amount of data reported in the literature.
180
8
a
= 1.8
=
SLOPE == 2-D = 0, D = 2.0
/ ...
i
o
o °o
0
' o
9,,.o.
O o'
oo
OcA)
o
<
io
o
1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 II 1 rr ri-i-i
-6
-5
_4
_3
_2
log (Diameter) " millimeters
Figure 75. Inferred fractal dimension for tan sand.
181
—J
8
a = 1.3
SLOPE = 2-D = -.5, D = 2.5
Q&
I I I IIII I I I I I II I I I I I I II I II I I I II I I I I I I I I I I I I j I I I III I I
-6
-5
-4
-3
-2
log (Diameter) ~ millimeters
Figure 76. Inferred fractal dimension for tan silt.
182
-1
8
a = 0.9
SLOP z = 2-D = -.9, D = 2.9
4
3
o
o
o
o
Oo
o
9o
o
o
o
11111111111111 1111111 1111111111111111111 111111111111111111
-6
-5
-A
-3
-2
log (Diameter) " millimeters
Figure 77. Inferred fractal dimension for kaolinite.
183
—1
CHAPTERS: CONCLUSIONS
The response of moist soils (or any other moist heterogeneous mixture) to active
microwave sensors is controlled by surface geometry and the dielectric properties of the
medium. A review of the literature reveals that what data do exist on dielectric
properties suffer from a number of shortcomings. In most cases, the soils are simply not
properly characterized in terms of their physical parameters such as the distribution of
particle sizes, the dry density under which tests were conducted, or the temperature of the
soil, all of which have some impact on the dielectric properties. The volumetric moisture
content of the soil, which is the usually the most important factor controlling dielectric
response, is often not specified or is not computable. And, finally, the data are usually
collected at a limited number of frequency values and over a very limited range of
frequencies.
Attempts to model the dielectric response of these soils are also limited in several
respects. Many models are strictly empirical and, therefore, apply only to that data set
with its few variations. Equivalent circuit models have been successfully applied to data
over small frequency range, but the model parameters seldom have any physical meaning.
This study began with an attempt to supplement existing data on the complex
dielectric properties of moist soils. A coaxial measurement apparatus for which
measurements are controlled by a vector network analyzer was fabricated to allow
property measurements in nominally moist soils to frequencies above 10 GHz which
allowed one to see the beginning of the dipole loss mechanism. Sample temperatures
were controlled by an external bath, and sample moisture contents were varied by
incremental drying of the samples. Actual test moisture values were determined after
each series of measurements were completed by weighing the completely dry sample.
Distilled, deionized water was used to saturate dry soil samples.
Soil samples were chosen to reflect a broad span of responses. They varied from
a poorly-graded clean sand to a very pure non-swelling clay mineral. The physical
properties of the soil samples were characterized by the development of grain-size
distribution curves and measurements of specific gravity and specific surface. With
184
hindsight either mercury porosimetry measurements or pressure plate measurements
should have been conducted to obtain a relationship between pore volume and pore size.
As expected, the soil dielectric response was found to be a strong function of
volumetric moisture content for non-frozen soils. Low frequency losses due to either
ionic conductivity or Maxwell-Wagner effects, or both, were quite apparent, as were high
frequency losses due to the free-water dipole relaxation loss mechanism. Minimum
losses occur in the 1-2 GHz range.
Sample temperature did not have a significant effect on the dielectric response of
non-frozen soils over a range of temperatures from 0° C to 40° C at any frequency. Data
collected on frozen soils at -10°C and -5° C also did not show any significant difference.
The pronounced transition from frozen to nonfrozen soils took place within the -5° C to
0° C range, but the spacing of data within this range was not enough to completely define
that transition.
Data collected in the study supported the claim of previous researchers that there
is a definite transition in normal soils from a series-like capacitive response to a parallel­
like response with increasing volumetric moisture content. For the silt and clay samples,
that transition occurred in the 30-35 % moisture content range, v/hile for the well-graded
sand, it occurred in the 10-15% range. For equal moisture contents, then, the sands will
always have a higher permittivity than silts and non-swelling clays.
Two approaches were taken to modeling the dielectric response of these soils.
One was an equivalent circuit that allowed for both series-like and parallel-like responses
of soil and water elements. An iterative solution technique was used to arrive at the
weighting parameters for each element that varied in a sensible way with moisture
content. The permittivities assigned to the circuit elements were held fixed at acceptable
values and excellent fits to the data over the entire range of frequencies were achieved for
all of the soil types.
The second model was one that assumed the small pore geometry of a porous
medium like soils could be described by fractal methods. The result was an expression
for pore volume as a function of pore size or capillary pressure that included a fractal
dimension of the soil structure. When applied to a set of pressure plate data taken from
the open literature, this model showed that the fractal geometry assumption held for the
small-pore portion of the data and collapsed at the field capacity of the soils (1/3 bar
pressure, or 8 micron pore diameter). The model goes on to hypothesize that the critical
moisture content of the soils at which its electrical response makes a transition from
series to parallel behavior is, in fact, the moisture content at field capacity. Lacking a
measured pore volume distribution for these soils in terms of either pressures or pore
size, a pore volume-pore size distribution was estimated from the measured particle size
distribution curves and used to evaluate bounds on the small-pore fractal dimension of the
idealized soils used in this study. The results indicate that while real soils are likely to
have fractal dimensions on the order of 2.5 to 2.7, the value for coarse sands tends
toward 2,0 and the value for fine clay minerals tend toward 3.0.
186
APPENDIX A: SKIN DEPTH CALCULATIONS
Obviously, skin depth effects preclude the utility of practical electromagnetic
devices to penetrate to distances comparable to acoustic devices, but nevertheless, low
frequency devices have some utility (Berlin, et al, 1986; Farr, et al, 1986; Schaber, et al,
1986). As a means of quantifying the penetration of electromagnetic waves into earth
media in terms of the complex dielectric constant of the media, consider the following
arguments. Let the one-dimensional propagation of an electromagnetic wave into some
medium be described by an amplitude function like
g i(kx-w t)
(Al)
where
i = the symbol designating the imaginary quantity \/-l
k = the complex wave number
(0 = the radial frequency.
Furthermore, let
(A2)
where p is called the phase constant and governs how a lossless medium propagates a
wave, and a is called the attenuation factor and controls how the amplitude of the wave
decreases with distance traveled through the medium. Then the amplitude of the wave
goes like
(A3)
i.e., a traveling wave combined with an attenuation factor.
187
If skin depth is defined as that distance over which the wave amplitude decreases
by a factor of 1/e, then the skin depth, 8 , is
5=—
a
(A4)
But the wave number can also be written
(A5)
V
c
where
v = the phase velocity in the medium,
c = the speed of light
N = the complex index of refraction,
Thus
2 _
^
C
(OH //
(A6)
or the skin depth is inversely proportional to the imaginary part of the complex index of
refraction.
However, this study focuses on the complex dielectric constant, € , as the
measure of soil electrical characteristics.
e=e^+ie^^=N^
(A7)
Following several substitutions and simplifications, one can show that
_1
2
6=
271
188
(A8)
where
is the free-space wavelength. A visualization of this equation for the range of
values of permittivity (
) and loss tangent ( €^V
) normally found in soils is
provided in Figure Al. This figure does not represent any new science, as the same
information has been presented in other forms (von Hippel, 1954). It is, however, a
more condensed way of visualizing the skin depth that has not been observed elsewhere
in the literature. Figure Al is valid for any interpretation of loss tangent; i.e., whether
one is speaking of a loss mechanism due to actual charge migration or one due to
dielectric relaxation.
100.
10.
1.
.01
.1
1.
10.
SKIN DEPTH / WAVELENGTH
Figure Al. Skin depth as a function of wavelength, permittivity, and loss tangent.
189
Another useful skin depth relationship found in the literature is the nomograph
generated by Albrecht (1966) and reproduced on Figure A2. Albrecht first developed an
empirical relationship that associated ground conductivity to gravimetric moisture content
and ground temperature at low frequencies. He then related skin depth to wavelength and
conductivity by the usual equation for highly conductive materials (assumes that the
conductivity phenomenon dominates the loss mechanism in soils at low frequencies):
6
(A9)
Ttjxca
where the units used are MKS, and the skin depth is in meters and the ground
conductivity is in units of mho/meter. The magnetic permeability was taken to
be 4 %X10
Henrys/meter. The way to use this nomograph is to select a moisture
content and move on a horizontal line to the appropriate temperature curve. Then move
on a vertical line to the appropriate frequency curve. From that a point a horizontal line
to the right hand scale reveals the desired value of skin depth. There is no way to relate
Albrecht's results to those of Figure Al, because it begins with an unknown empirical
relationship among moisture, temperature, and conductivity.
Hoekstra and Delaney (1974) also published some useful data on plane wave
attenuation in moist soils at higher frequencies. These data are shown in Figure A3.
They concluded that attenuation was relatively independent of soil fabric and that either
passive or active microwave sensors could obtain a measureable response from only about
the first five centimeters of ground below the surface.
190
>-4
2
10"^ 2
^
lOT" 2
CONDUCTIVITY
*
10"
(mho/m)
Figure A2. Skin depth nomograph (Albrecht, 1966).
191
Freq
4.0 «10".
5.0 «10'
.30
.10.
WATER CONTENT
(g/cc)
Figure A3. Attenuation in moist soils (Hoekstra and Delaney, 1974).
192
APPENDIX B; REFLECTION AND REFRACTION AT PLANE INTERFACES
Remote sensing of the environment usually involves the interaction of
electromagnetic energy waves with solid media at some air-media interface. For
example, an active radar transmits waves toward a surface (usually at some oblique
angle) and measures what is reflected back toward itself (backscatter) or toward some
other receiver (bistatic reflection). Passive devices receive energy emitted by the surface
(and the atmosphere within the antenna's beamwidth) due to internal sources as well as
what is reflected from natural sources such as the sun, the atmosphere, and surrounding
terrain.
An idealization of the wave interaction phenomenon is shown in Figure B1 where
an incident plane wave, defined by its wave vector, kj , gives rise to a reflected
wave, iCjj , and a refracted (or transmitted) wave, k.^ . (Nearly every textbook on
electromagnetic theory has a development on reflected and refracted plane waves. What
follows was taken primarily from Jackson (1975) with a slight change in notation. Units
are cgs.) The wave vectors are related to the dielectric properties of both media by
l^ll = I-^rI = -fc = — =
141 = ic, = ^
^
where
<*) = radial frequency
N = complex index of refraction
€ = complex dielectric constant
= magnetic permeability (hereafter taken to be unity)
c = speed of light in a vacuum (3x10' m/sec)
193
(»!)
2^-2
(B2)
The plane of incidence is taken to be that plane formed by the k j and i5 vectors;
i.e., the plane of the paper.
MATERIAL 2
MATERIAL 1
Figure Bl. Wave vectors at a plane interface.
Under these ideal conditions of plane waves at a smooth interface, one can write
the following expressions for the three sets of electromagnetic energy fields.
194
INCIDENT
1
J.Ci
REFRACTED (TRANSMITTED)
£„ = E,
'r
•"To
Br =
= -yW
REFLECTED
4 =
4=
= ^4x4
195
By insisting that the phase factors ( k'X terms) are all equal at z=0, one is led to
Snell's law and the fact that 62=8^ .
sinGj _ icj. _
sinSy
k
P2E2 _
\
(B9)
N
What one desires to have are expressions for the amplitudes of the reflected and
refracted waves, for it is the wave amplitude (or voltage amplitude) that is measured by a
receiver unit. Amplitude relationships can be found by applying the electromagnetic
boundary conditions; namely, that the normal components of electric displacement, D ,
and magnetic induction, B , are continuous across the interface as are the tangential
components of the electric field, E , and the magnetic field, H . These conditions
may be written as:
(BIO)
(BID
= 0
(B12)
xi5
196
0
(B13)
The Fresnel Coefficients
At this point two further idealizations are made before finally writing down the
expressions for reflected and refracted wave amplitudes. One is to assume that the
incident electric field vector is perpendicular to the plane of incidence (into the paper
with respect to Figure Bl), and the other is to assume it is parallel to the plane of
incidence (lying within the paper with respect to Figure Bl). In radar parlance, the terms
are "horizontal polarization" and "vertical polarization", respectively.
Due to redundancies generated by Snell's law, one only needs the third and fourth
boundary conditions expressed in equations B6 to arrive at the final expressions for
amplitude ratios, which are known as the Fresnel reflection coefficients.
HORIZONTAL POLARIZATION
E.
'0
_
2COS0J
E,
(B14)
cosGj. + —.
E .^0 _
E,
COS0J - —.
1^2 \
COS0J + — .
197
- sin^0,
- sin^e.
HlGi
(B15)
- Sin20,
VERTICAL POLARIZATION
'0
2COS0J
_
(B16)
E,
COS0J +
\^2^1
COS0J ^0 _
E,
\
1 - lifisin^Sj.
V-2^2
\^2^1 _ f i l l
[^1^2
COS0J + ^ j4fi
(B17)
sin^0j
Reflection From Lossless Media
The previous equations apply to lossy as well as lossless media. For lossless
media, the dielectric constant and index of refraction are real quantities. Refracted waves
propagate without attenuation, and both refracted and reflected waves retain the
polarization of the incident wave.
Of practical value to the remote sensing arena is the variation with incidence angle
of the reflection coefficient in a lossless media. Because received power in a sensor goes
as the square of the amplitude of the signal and is proportional to the radar cross sections
of the object being illuminated, scientists and engineers have a simple model between
electrical properties of the media and its response to electromagnetic energy. What is
deduced from the sensor measurements and a knowledge of the surface slopes is an
apparent dielectric constant or apparent permittivity of the medium.
Figure B2 is a representation of how the apparent permittivity of a medium affects
the reflected amplitude ratio as a function of incidence angle. The angle at which the
reflected wave is totally eliminated in the vertical polarization case is referred to as the
Brewster angle ( tan0g =
= y(H2^2)/(M'i^i) )• For angles less than the
198
Brewster angle, the reflected wave suffers a phase reversal. The values of dielectric
constant were chosen to span the range of values from dry soil to pure water.
HORIZ POL
VERT POL
e =80+10
e =10+10
e
0
=2+10
10
20
30
40
50
E0
78
8B
90
INCIDENCE ANGLE, DEGREES
a. Normalized reflected voltage amplitudes.
Hpol ; Vpoi
X" for angles >
+ 180.
+ 160.
Brewster
0
+ 140.
+ 120.
1
«—I
II
+ 100.
+80.
II !
+G0.
+40.
1 i
II
Vj
: vu
+20.
+0.
-20.
-40.
-60.
-80.
-100.
-120.
-140.
-160.
I
-180.
10
20
30
40
50
G0
70
60
•
I
90
INCIDENCE ANGLE, DEGREES
b. Phase shifts.
Figure B2. Reflection amplitudes and phase shifts for non-magnetic lossless materials.
199
Reflection From Lossy Media
Appendix A gave some indication of what happens to electromagnetic waves
incident on a lossy media (
> 0 . The refracted, or transmitted, wave is
attenuated. Furthermore, for oblique angles of incidence, surfaces of constant amplitude
are no longer parallel to the surfaces of constant phase, resulting in what is referred to as
an inhomogeneous wave (Stratton 1941). In fact, the index of refraction for lossy media
becomes a function of the angle of incidence of the incoming wave. As lossiness goes up
the refracted wave vectors get closer and closer to normal to the interface.
Similarly, it is possible to rationalize the second expressions in equations B7 and
B8 to obtain ratios of reflected amplitudes to incident amplitudes. One representation of
these relationships may be found in Stratton (1941). The effects of adding a loss term to
the dielectric constant are to enhance reflectance and to reduce the destructive
interference mechanism that causes the total loss of reflected energy at the Brewster
angle. A visual comparison of the effects of loss factors in materials is shown in Figure
B3.
It is not inconceivable that if one had good reflectance data at a single frequency
over a large range of incidence angles (enough to include the minimum on the vertical
polarization curve) and at the wavelength of the sensor could assume smooth interface
conditions, then one could iteratively obtain a first approximation to the complex
dielectric constant for the material near the surface.
200
HORIZ POL
VERT POL
e =10+110
5
§
M_i
Q.
e =10+10
U
b
a.
_L
I
'
_L
I
e 10 20 30 40 50
B0
IVi.,1
70
80 96
INCIDENCE ANGLE, DEGREES
a. Normalized reflected voltage amplitudes.
HORIZ POL
+ 180.
+ 160.
+ 140.
VERT POL
e =10+10
+ 120.
+ 100.
+80.
+60.
+40.
£ =10+110
+20.
+0.
-20.
-40.
e =10+10
-60.
-80,
-100.
-120.
-140.
6 =10+110
-160.
-180.
0 10 20 30 40 50 BQ 70 80 90
INCIDENCE ANGLE, DEGREES
b. Phase shifts.
Figure B3. Reflection amplitudes and phase shifts for non-magnetic lossy materials.
201
APPENDIX C: FRACTAL MODELS OF SOIL STRUCTURE
Fractals
In recent years there has been some very interesting work done on the conductive
behavior of porous media using the concept of self-similar geometries, or fractals, to
model the behavior of certain physical quantities of that media. Mandelbrot, the guru of
fractals, defines self-similar shapes as those in which a certain part of the shape can be
broken up into N smaller parts, each looking like the original, but reduced in size by a
fractional factor labelled, r (Mandelbrot, 1983). The fractal dimension, D, of this selfsimilar shape can be written as
D
=
(Gl)
or
N
= 1
(C2)
Mandelbrot insisted that the exponent, D, be thought of as a dimension because it arose
from a method of measuring the perimeter of an object having an irregular boundary.
Fractals are often used to generate complex, or textured, curves or surfaces from
very simple initiator geometries. For example consider Figure CI in which the initiator
(or the initial segment of the geometry which will be fractalized) is a solid square
(precisely stated, each side of the square is an initiator), and the generator (or the desired
geometry) is the broken line segment as shown. Moving clockwise from the initiator, the
first two applications of the generator are shown at full scale, while the next two are
enlarged to show the edge detail. Figure C2 shows another construction initiated with a
solid square in which the generator results in the formation of both "lakes" and "islands".
202
Figure CI. A fractal snowflake of dimension 1.5 (Mandelbrot, 1983).
#-
•
rn
rn
I UU . niU' UV
W ftft
N=18
r=l/6
D~1.6131
OR
BK .
"ow
s [R s "
.»«
Bh>
BK
#
BR , BA
#
*
Ub
MM
S
#1
t]
BM E
#- #'
* *
*•
«
Figure C2. Fractal islands and lakes of dimension 1.6131 (Mandelbrot, 1983).
203
Soil Structure
A third example of a fractal construction that is useful for studying porous media,
and may be particularly useful for studying soil structure, is that shown in Figure C3 in
which the initiator is a cube with square holes centered on each face and joining in the
center. The generator is the same object but reduced by a factor of 1/3. Therefore, in
the first stage, the initiator is divided into 20 smaller cubes, resulting in a fractal
dimension of 2.7268. Mandelbrot referred to this construction as a Menger sponge.
Friesen and Mikula (1987) used the Menger sponge to model the fractal dimension of the
surface area of coal in such a way that the slope of the isotherm (fractional volume
change vs pressure for porosimetry measurements) plotted on a log-log scale is linearly
related to the fractal dimension. At least one recent study attempted to measure the
fractal dimensions of sandstone solid-void interfaces (Katz and Thompson, 1985).
Measured values of interface fractal dimensions on five different samples using electron
microscope techniques varied from 2.57 to 2.87.
Work has been done very recently on modeling the microstructure of soil using
fractals (Krepfl, Moore, and Lee, 1989; Moore and Krepfl, 1991). A two dimensional
representation of a three-dimensional model using hexagonal-shaped flakes as the
generator is shown in Figure C4, Such models may be useful as a representation of pure
clays. This particular fractal structure is strongly reminiscent of microphotographs of the
pure kaolinite mineral structure.
204
Figure C3. The Menger sponge, fractal dimension 2.7268 (Mandelbrot, 1983).
generator
Figure C4. A fractal representation of soil fabric (Moore and Krepfl, 1991).
205
APPENDIX D: SOIL PROPERTIES
Particle size distribution curves for each of the four soils used in these
experiments were developed by technicans in the Geotechnical Laboratory at the US
Army Engineer Waterways Experiment Station, Vicksburg, MS, and are shown on the
following four figures.
Also included on each figure is the specific surface for that soil
which was measured by laboratory personnel at Soil Analytical Services, Inc., in College
Station, TX, using an ethylene glycol adsorption technique.
206
U,S. SWIDWD SIEVE OPENING « INCHES
I
J
1 1 1 i 1
1
100
US. SBNOARD SIQE NUieEHS
1 1 y 1
1
f
HYDROMETER
1 1 1 1
10
90
I
i
D
N)
O
-0
fi
80
20
70
30
0'
40 I
^ 50
50 E
1
i
i
u.
60
t 40
ca
^30
70
20
80
0
10
1
c«
90
1
0
100
500
COBBLES
u.
50
10
avRSE
FL
GRAVEL
1
GS
nuE
2.67
5
0.1
I
0.5
GRAIN SIZE IN MUJMEIERS
CONGE 1
NATW^
SAND
ItCIMM
1
0.05
0.01
100
0.001
0.005
SILT or CLAY
FINE
OROS!
PROJECT
CLASSFICAHON
SAND (SP). WHfTE
GRADATION CURVE
specific area = 10 m^/g
I
LABORATORY USAE WES - STF/GL
BORING NO.
OEPTH/ELEV
SAND (WHITE)
SAMPLE NO.
DATE
19 AUG 91
U.S. SIWOARO SIEVE NUVEERS
US. srmm soe opening n inocs
6
100
43
" "' '
2 1 15 1 43 ~1 ~3
3
4
6
810
HYDROMETER
IB 20 30 40 50 70 100 140 200
T—I—nTTT
II I I If
90
80
70
i
s
\
0'
1
I 50
Q
s
00
i
3
^ 40
a! 30
s
20
CD
S
10
0
1
CL
100
500
COARSE
COBBLES
PL
LL
10
50
PI
GRAVEL
S
0.1
1
05
GRAIN SIZE IN MlilMETERS
1
fNE
CQWSE 1
^
2.66
NATW.X
SAND
KOIUN
1
0.05
0.01
0.005
0.001
SILT or CLAY
FINE
ORG.%
PROJECT
OASSFCATION
SAND (SP), BROWN
m
7/
specific area = 10 m^/g
SORING NO.
DEPTH/ELEV
GRADATION CURVE
LABORATORY USAE WES - STF/GL
SAND CTAN)
SAMPLE NO.
DATE
19 AUG 91
U.S. swrow® SIEVE OPENINS N WCHES
100
1 3 - 1 3
1 1 1 r 1 1 1 1 1
1
U& 5ONDAR0 SIEVE NUkCERS
HYDROMETER
n 1 T- 1
T 1
90
i
80
1
70
I
I 60
%
t
^ 50
\
i
c
>
\
g 40
\
S 30
\
20
\
10
100
500
COBBLES
LL
PL
27
10
50
5
22
1
GS
0.05
SAND
GRAVEL
tXWS
0.1
I
0.5
CRAM SIZE N MILUMETERS
FWE
2.71
CQWSE 1
NWW.X
IfSIUM
1
0.01
0.001
0.005
SILT or CWY
FINE
CROS
PROJECT
O/SSFICAHON
CLAYEY SILT (ML). BROWN; TRACE OF SAND
specific area = 84 m^/g
GRADATION CURVE
LABORATORY USAE WES - STF/GL
SORING NO.
DEPTH/ELEV
SILT (TAN)
SAMPLE NO.
DATE
19 AUG 91
U5. SWNOARO SEVE MAEERS
VS. snwcw© 9E\€ OPENING NINOCS
1
100
1
1
3
1
1 1
r
mDROMETER
3
1 1 1 1 —r t
—T -I" 1 K -v
N
90
\
\
80
It
A
\
\
70
5 60
\:
\
'i
\
° 50
L.
g 40
L
20
10
'
100
500
COBBLES
LL
PL
47
10
50
CQ«SE
30
17
GRAVEL
1 TOE
GS
2.61
5
0.1
1
0J5
GRAIN SIZE N MIUMEIERS
CCWBE 1
NATW.S
SAND
MEDIUM
1
0.05
0.01
0.001
0.005
SLT or CLAY
FINE
CRG.J!
PROJECT
CUVSSFIOMION
SLT (ML). WHITE
specific area = 15 m^/g
BORING NO.
OEPTH/ELEV
GRADATION CURVE
U\BORATORY USAE WES - STF/GL
OAY (WHITE)
SAMPLE NO.
DATE
19 AUG 91
REFERENCES
Adldns, B.D. and Davis, Burtron H. 1986. "Particle Packings and the Computation of
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