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Waves, scale, sand, and water : dielectric constant of unconsolidated sediments Chan, Christina Ye 1999

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W A V E S , S C A L E , SAND, A N D WATER: DIELECTRIC CONSTANT OF UNCONSOLIDATED SEDIMENTS by CHRISTINA Y E C H A N A . B . , Princeton University, 1992 M.S., Stanford University, 1994 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Earth and Ocean Sciences School of Science Programme in Geophysics We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A July 1999 © Christina Ye Chan, 1999  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  available for  copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and study. scholarly  or for  her  I further  purposes  gain  permission.  Department of  £a.r4-k + Oct**. Sciences  The University of British Vancouver, Canada  Date  DE-6 (2/88)  3*  Columbia  H^Wl  shall  requirements that  agree  may  representatives.  financial  the  It not  be is  that  the  permission  granted  allowed  an  advanced  Library shall  by  understood be  for  for  the that  without  make  it  extensive  head  of  my  copying  or  my  written  Abstract Field dielectric measurements are used to estimate the water content of the subsurface. In order to estimate water content accurately, the Earth's heterogeneity should be taken into account. Layering is a simple form of heterogeneity which is a close approximation in many sedimentary environments. The interplay between the average layer thickness of a sedimentary system and the wavelength of the E M wave used for the dielectric measurement is important in determining average dielectric constant. When the layers are thick compared to the wavelength, the system falls under the ray theory regime; when the layers are thin, the system falls under the effective medium theory regime. Using numerical and experimental techniques, I confirm these two regimes. I also investigate the transition zone between the regimes and find that it falls at a wavelength to layer thickness ratio of around 4. The breadth of the zone is affected by the dielectric constants of the components, the proportions of the components, and the distribution of the layers, but not the conductivity of the soils. Because many sedimentary environments have layering, the presence of these layers must be accounted for when using the average dielectric constant measured in the field to estimate water content. I compare relationships between dielectric constant and water content which take into account the presence of layering with relationships which assume homogeneity. Modeling dielectric constant as a function of lithology and water content, I find differences among the dielectric constants predicted from the different relationships. I show the potential error in water content estimation if a layered system is assumed to be homogeneous. I also present a flow chart for more accurately estimating water content and saturation from field measurements. This method not only gives the global water content of the whole system but also gives the water contents of the individual sedimentary layers if they are present. In this thesis, I present research which can provide more accurate estimates of water content from dielectric measurements. These investigations advance the knowledge of E M  ii  wave propagation and increase the accuracy of estimating water content from field dielectric measurements of the subsurface.  iii  T A B L E OF CONTENTS Abstract  ii  Table of Contents  iv  List of Tables  vi  List  of  Figures  vii  Acknowledgments  x  Chapter 1  Introduction  1  Chapter 2  Numerical Confirmation for the Transition Zone between Effective Medium Theory and Ray Theory for Electromagnetic Waves  5  Chapter 3  Chapter 4  Introduction The Effect of Layering on Seismic Waves Averaging E M Properties in a Layered System Numerical Analysis for the Transition Zone for E M Waves Results and Discussion Effect of Dielectric Contrast Effect of Volumetric Proportions Effect of Layer Thickness Distribution Effect of Electrical Conductivity Conclusions and Implications for Interpretation of Field Data  5 6 9 10 13 15 16 18 19 22  Laboratory Confirmation of the Ray Theory and Effective Medium Theory Limits for Electromagnetic Waves  39  Introduction Experimental Method Results and Discussion Conclusions  39 40 43 45  Determining Water Content and Saturation from Dielectric Measurements in Layered Materials  53  Introduction Models of Homogeneous Sand-Fine Systems Description of the Homogeneous System Modeling the Dielectric Constant of Homogeneous Systems Models of Layered Sand-Fine Systems Description of the Layered Systems Modeling the Dielectric Constant of Layered Systems Comparison Of Model Results For Homogeneous Systems And Layered Systems Estimating Water Content from Dielectric Constant Estimating Water Saturation from Dielectric Constant Conclusions Chapter 5  Accounting for Saturation Heterogeneity in Obtaining Estimates of Water Content From Dielectric Data iv  53 55 55 56 58 58 60 62 63 65 66 77  Introduction 77 Determining Water Content And Water Saturation From Dielectric Field Measurements 79 Step 1: Taking the Measurements 80 Step 2: Determining the Geometry ..81 Homogeneous Systems 81 Layered Systems 81 Step 3: Determining Water Content and Saturation 82 Homogeneous Systems 82 Thinly Layered Systems 83 Thickly Layered Systems 84 Example 85 Generating the Sample 85 Determining Water Saturation and Water Content 86 Discussion 87 Conclusions 89 Chapter 6  Conclusions  96  Bibliography Appendix A  100 Dependence of Velocity and Attenuation on Frequency and Conductivity  102  Appendix B  Typical Wavelengths for Selected Materials  103  Appendix C  Results from Numerical Simulations  104  Appendix D  Characteristics of Sands and Clay  106  Appendix E  Details for Coaxial Cell Measurements  108  Appendix F  Results from Measured Laboratory Experiments  Ill  Appendix G  Results from Modeled Laboratory Experiments  113  Appendix H  Details of Calculations for Theoretical Study  114  Appendix I  Details of Theoretical Error Calculations  116  Appendix J  Results from Interpretation Examples  125  v  List of Tables Table 4-1  Properties of components  vi  Figure 2-1 Figure 2-2  Figure 2-3  Figure 2-4  Figure 2-5  Figure 2-6  Figure 2-7  Figure 2-8  Figure 2-9 Figure 2-10  Figure 2-11  Figure 2-12  Figure 2-13  Figure 2-14  List of Figures A plane E M wave normally incident to a series of layers a) The amplitudes recorded with time at the bottom of a series of layers showing a typical calculated E M waveform, b) The above waveform transformed into the frequency domain, interpolated, and transformed back into the time domain, c) The frequency spectrum for the above waveform Waveforms for an E M wave propagated at 750 M H z through a set of layers. Half the layers have a dielectric constant of 15, and half have a dielectric constant of 5. The waveforms of the ray theory and E M T limits are shown in bold Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 MHz, and 750 M H z through a set of layers. Half the layers have a dielectric constant of 15, and half have a dielectric constant of 5 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 M H z , and 750 M H z through a set of layers. Half the layers have a dielectric constant of 24, and half have a dielectric constant of 2 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 M H z , and 750 M H z through a set of layers. Half the layers have a dielectric constant of 11, and half have a dielectric constant of 9 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 MHz, 200 MHz, and 750 M H z through a set of layers. 75% of the layers have a dielectric constant of 15, and 25% have a dielectric constant of 5 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 M H z , and 750 M H z through a set of layers. 25% of the layers have a dielectric constant of 15, and 75% have a dielectric constant of 5 Sketch of different layer thickness distributions Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 MHz, and 750 M H z through a set of layers. Half the layeres are three times thicker than the other half of the layers. Half the layers have a dielectric constant of 15, and half have a dielectric constant of 5 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 M H z , and 750 M H z through a set of layers. Half the layers are nine times thicker than the other half of the layers. Half the layers have a dielectric constant of 15, and half have a dielectric constant of 5 Waveforms for an E M wave propagated at 750 M H z through a set of layers. Half the layers have a dielectric constant of 15 and a conductivity of 0.7 mS/m, and half have a dielectric constant of 5 and a conductivity of 0.2 |iS/m. The waveforms of the ray theory and E M T limits are shown in bold Waveforms for an E M wave propagated at 750 M H z through a set of layers. Half the layers have a dielectric constant of 15 and a conductivity of 35 mS/m, and half have a dielectric constant of 5 and a conductivity of 10 |j,S/m. The waveforms of the ray theory and E M T limits are shown in bold Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 50 M H z , 200 MHz, and 750 M H z through a set of vii  24  25  26  27  28  29  30  31 32  33  34  35  36  Figure 2-15  Figure 3-1 Figure 3-2 Figure 3-3 Figure 3-4 Figure 3-5 Figure 3-6 Figure 4-1 Figure 4-2  Figure 4-3  Figure 4-4  Figure 4-5  Figure 4-6  Figure 4-7  layers. Half the layers have a dielectric constant of 15 and a conductivity of 0.7 mS/m, and half have a dielectric constant of 5 and a conductivity of 0.2 uS/m 37 Normalized velocity versus wavelength to period thickness ratio for E M waves propagated at 200 MHz, 500 MHz, and 750 M H z through a set of layers. Half the layers have a dielectric constant of 15 and a conductivity of 35 mS/m, and half have a dielectric constant of 5 and a conductivity of 10 |a,S/m 38 The effect of matched impedance, open circuit, closed circuit, and conductivity on an E M wave pulse traveling along a TDR cell 47 a) Picture of TDR set-up. b) Close-up of cable tester, c) Close-up of coaxial cell 48 Schematic of coaxial TDR cell and experimental setup 49 Typical T D R measurement 50 Normalized E M wave velocity versus 7Jt for the measured laboratory data 51 Normalized E M wave velocity versus 7Jt for the numerically modeled data 52 Total porosity versus fine fraction for a homogeneous mixture of sand and fines 68 Illustration of layer thickness to wavelength ratio for the effective medium theory and ray theory regimes. The layer thickness is much less than the wavelength in the E M T regime and much more than the wavelength in the ray theory regime 69 Dielectric constant as a function of fine fraction and global water saturation calculated using the E M T parallel relationship. The dielectric constants of the individual layers were calculated using the TP model. This plot is for a sand-clay system. Similar plots can be generated for other relationships and other soil systems 70 A comparison of dielectric constant calculated for the sand-clay system using E M T for layering perpendicular to the propagation direction ( A ) , ray theory for layering perpendicular to the propagation direction (•), E M T for layering parallel to the propagation direction (•), the TP model (•), and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50 . 71 A comparison of dielectric constant calculated for the sand-silty clay system using E M T perpendicular to the propagation direction ( A ) , the ray theory perpendicular to the propagation direction (•), the E M T parallel to the propagation direction (•), the TP model (•), and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50 72 A comparison of dielectric constant calculated for the sand-silt loam system using the E M T perpendicular to the propagation directio(A), the ray theory perpendicular to the propagation direction(#), the E M T parallel to the propagation direction (•), the TP model (•), and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50 73 The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) and viii  Figure 4-8  Figure 4-9  Figure 5-1 Figure 5-2 Figure 5-3  Figure 5-4  Figure 5-5  Figure 5-6  global water saturation (bottom). These plots are for thin layers parallel to the E M wave propagation direction at a fine fraction of 0.50. The solid symbols show the error produced when the TP model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares (•) are for the sand-clay system; the circles (•) are for the sand-silty clay system; and the triangles ( A ) are for the sand-silt loam system 74 The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) and global water saturation (bottom). These plots are for thin layers perpendicular to the E M wave propagation direction at a fine fraction of 0.50. The solid symbols show the error produced when the TP model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares (•) are for the sand-clay system; the circles (•) are for the sand-silty clay system; and the triangles ( A ) are for the sand-silt loam system 75 The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) and global water saturation (bottom). These plots are for thick layers perpendicular to the E M wave propagation direction at a fine fraction of 0.50. The solid symbols show the error produced when the TP model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares (•) are for the sand-clay system; the circles ( • ) are for the sand-silty clay system; arid the triangles ( A ) are for the sand-silt loam system 76 Schematic of E M wave propagation through homogeneous and layered systems 90 Flow Chart for Determining Water Saturation and Content from Field Measurements 91 Simulated logs for thin-layered system a) Lithology b) True saturation c) Blocked true saturation for each interval d) Estimated saturation assuming homogeneous system e) Estimated saturation using flowchart method f) Saturation error assuming homogeneous system g) Saturation error using flowchart method 92 Simulated logs for thick-layered system a) Lithology b) True saturation c) Blocked true saturation for each interval d) Estimated saturation assuming homogeneous system e) Estimated saturation using flowchart method f) Saturation error assuming homogeneous system g) Saturation error using flowchart method 93 Simulated logs for thin-layered system a) Lithology b) True water content c) Blocked true water content for each interval d) Estimated water content assuming homogeneous system e) Estimated water content using flowchart method f) Water content error assuming homogeneous system g) Water content error using flowchart method — 94 Simulated logs for thick-layered system a) Lithology b) True water content c) Blocked true water content for each interval d) Estimated water content assuming homogeneous system e) Estimated water content using flowchart method f) Water content error assuming homogeneous system g) Water content error using flowchart method .... 95  ix  ACKNOWLEDGMENTS I would like to thank my supervisor Rosemary Knight for her guidance and encouragement throughout my four years at U B C I also thank the other members of my supervisory committee, Roger Beckie and Matthew Yedlin, for their input. I would never had started or finished this Ph.D. without encouragement from my parents and friends. I would also like to express my gratitude to Steve Cardimona of the University of Missouri-Rolla for so generously giving me his program which I used for the numerical modeling of E M wave propagation in Chapter 2. I also thank David Redman of the University of Waterloo for the program which I used to collect the experimental data in Chapter 3. Chapter 4 is published in modified form as "Chan, C.Y., Knight, R.J., 1999, Determining Water Content and Saturation from Dielectric Measurements in Layered Materials: Water Resources Research, 35, 85-93." (copyright by the American Geophysical Union). Chapter 5 is published in modified form as "Chan, C.Y., Knight, R.J., 1999, Accounting for Saturation Heterogeneity in Obtaining Estimates of Water Content from Dielectric Data: Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems, 435-444." (copyright by the Environmental and Engineering Geophysical Society). Both chapters are used with permission. This research is sponsored in part by the Air Force Office of Scientific Research, Air Force Material Command, U S A F , under grant number F49620-95-I-0166. The U . S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U . S. Government. I was also supported by a University of British Columbia University Graduate Fellowship and a Killam Predoctoral Fellowship.  x  Chapter 1 Introduction "There is more than meets the eye." The earth is largely invisible to the naked eye. On the surface one can only sample and study a small part of the earth. Geophysical methods and techniques are necessary to sample and study the earth's subsurface. However, geophysics only provides information about geophysical parameters such as shear and compressional wave velocity, electrical conductivity, electromagnetic (EM) wave velocity, and dielectric constant when the information of interest is actually about geological properties such as lithology, porosity, water content or saturation, permeability, and spatial heterogeneity. In this thesis, I address some of the issues governing the relationships between geological properties and the geophysical parameter of dielectric constant. Measurements of the dielectric constant in regions of the subsurface are commonly made to determine water content. The volume of a typical field survey is often on the order of tens to hundreds of cubic meters while the volume of the sampled system for which a single dielectric constant is measured is on the order of tens of cubic centimeters to tens of cubic meters. The accuracy of the estimated water content of the sample volume obtained from dielectric data is very dependent upon the model used to relate dielectric constant to water content. Most models assume that the sampled system can be described as a homogeneous mixture of solids and fluids. While this may be valid in the interpretation of some data, there is no doubt that the earth, in general, is spatially heterogeneous. Given the heterogeneous nature of the subsurface, the assumption of homogeneity can often be invalid. In this thesis, I focus on the simplest form of heterogeneity: layered systems. Although quite a simple model of the subsurface, it can be an appropriate approximation for many depositional systems where sediments are layered. As simple as layers may be, not all layered systems behave in the same manner when one measures an average dielectric  1  constant for these systems. The differences occur because dielectric measurements in the field are made using E M wave propagation. The interaction of the layers with a propagating E M wave can affect the average dielectric constant of the whole volume being sampled. The interaction of layers with seismic wave propagation is well known, but little is currently known on the details of the interaction of layers with E M wave propagation. Of interest in this thesis is to investigate this interaction between layers and E M wave propagation and to assess the error involved when estimating water content from dielectric constant values measured using E M wave propagation. Propagating E M waves have a characteristic wavelength X which is related to the frequency of the E M wave. The interplay of this characteristic wavelength with the average thickness of the layers t becomes important in determining the velocity of the E M wave and the average dielectric constant. The average dielectric constant for a layered system is related to the total time a propagating E M wave takes to travel through all the different layers. However, this total travel time of the E M wave is affected by the ratio of the characteristic wavelength to the average layer thickness. For some layered systems, the E M wave which is used to measure dielectric constant can have a wavelength that is larger than the average thickness of the layers. Systems under these measurement conditions fall under the effective medium theory regime. For other layered systems, the E M wave which is used to measure dielectric constant can have a wavelength that is smaller than the average layer thickness. Systems under these measurement conditions fall under the ray theory regime. Do these two extreme regimes exist? If so, how much larger does the wavelength have to be compared to the average layer thickness for the system to fall under the effective medium theory regime? How much smaller does the wavelength have to be compared to the average layer thickness for the system to fall under the ray theory regime? What happens when the wavelength is about the same size as the average layer thickness? Does some sort of transition zone exist between the two regimes? What are the characteristics of  2  this transition zone? Chapter two addresses these questions in a numerical study while chapter three addresses them in an experimental study. However, I am not simply interested in the abstract interaction of propagating E M waves and layers. I am also interested in how this knowledge can be used for more accurate interpretation of dielectric field data. Because the dielectric constant of water is high compared to earth materials and air, field measurements of dielectric constant are used to estimate the amount of water in the subsurface. Obtaining accurate estimates of the water content of the subsurface, particularly in sedimentary environments, is important in many applied disciplines such as agriculture, environmental engineering, and geotechnical engineering. Current practice in estimating water content from dielectric constant is to assume homogeneity of the subsurface. The results of investigating the connection between dielectric constant and layers becomes important when searching for the correct relationship between dielectric constant and water content. Because many sedimentary environments are layered, the interplay between the wavelength of the E M wave used to measure dielectric constant and the average layer thickness of the sediments plays an important role in determining the average dielectric constant measured in the field. Therefore, when estimating water content from the measured dielectric constant, the ratio of the E M wavelength to the average layer thickness must be taken into account. A layered sedimentary system can consist of different soil types with different individual saturation characteristics. Because of these different saturation characteristics, each layer of soil can have a different value of dielectric constant. The dielectric constant of each of these layers contributes to the measured average dielectric constant of the whole system. In the field, layers can exist that are both much smaller and much larger than the characteristic E M wavelength used to measure dielectric constant. Therefore, the ratio of the characteristic wavelength to the average thickness of the layers can become important when measuring the average dielectric constant of the whole system of layers.  3  So what are the relationships between water saturation, lithology, and dielectric constant? What is the relationship for homogeneous systems? How is this relationship different for systems under the effective medium theory regime? Or for systems under the ray theory regime? What kind of errors in estimating water saturation can occur if a system under the effective medium theory regime is assumed to homogeneous? What kind of errors in estimating water saturation can occur if a system under the ray theory regime is assumed to be homogeneous? Is there a way to ascertain whether a system is homogeneous or layered? Is there a way to ascertain whether a layered system falls under the effective medium theory regime or ray theory regime? Chapter four examines the theoretical relationships among water saturation, lithology, and dielectric constant while chapter five presents a method for more accurately interpreting field data. One of the strengths of geophysics lies in its ability to image below the earth's surface. Geophysical measurements of dielectric constant are used to estimate the amount of water in the subsurface. However in order to estimate water content accurately, one must take into account the heterogeneity of the earth. In this thesis, I focus on how to interpret dielectric measurements more accurately under the simple heterogeneous case of layering. First, I investigate waves and scale. In chapters two and three I look at how the interplay between the characteristic wave length of the propagating E M wave and the average thickness of layers influences the average dielectric constant of the system. Then I explore sand and water. In chapters four and five, I examine the relationships among lithology, water content, and dielectric constant for layered and homogeneous systems and the effects on estimates of water content if there are incorrect assumptions. These investigations advance the theory of E M wave propagation and increase the accuracy of estimating water content from field dielectric measurements of the subsurface. "Have you comprehended the vast expanses of the earth? Tell me, if you know all this." Job 38:18  4  Chapter 2 Numerical Confirmation for the Transition Zone between Effective Medium Theory and Ray Theory for Electromagnetic Waves  INTRODUCTION Material properties such as lithology, porosity, and water saturation affect the velocity at which an electromagnetic (EM) wave travels through geologic materials. This has led to the use of in situ measurements of E M wave velocity for determining these material properties of soils, sediments, and rocks in the subsurface. An understanding of the relationship between E M wave properties and material properties is a critical part of using E M wave measurements to characterize the subsurface. Of interest in this chapter is the effect of the spatial heterogeneity of the subsurface on the interpretation of E M wave measurements. There are two methods commonly used to determine the E M wave velocity of a region in the subsurface. In time domain reflectometry (TDR), the E M velocity is calculated from the time it takes an E M wave to travel along the TDR probe, a wave guide, inserted into the ground. The sampled material over which the velocity is determined is the volume of material immediately adjacent to and along the full length of the TDR probe. E M wave velocity can also be obtained using a ground penetrating radar (GPR) system. In using GPR, two antennae are moved across the earth's surface: one to transmit E M energy and the other to receive energy that has been reflected back to the surface from interfaces across which changes in E M wave velocity occur. Assuming the magnetic permeability to be equal to that of free space, the dielectric constant and electrical conductivity of a region give the velocity (v) of the electromagnetic wave as  V  _c_ I 2cos((5) - V l + cos(<5)-  5  n { Z  n  '  [ )  where <5 = arctan(a), a =  °  ,  (2-2)  c is the speed of light in a vacuum (2.99xl0 m/s), /c is the dielectric constant of the 8  material, o is conductivity, co is frequency, and Eo is the permittivity of free space (8.85xl0~ C /Nm ). However, it is common to assume that a~0: measurements are 12  2  2  often taken at high frequencies and/or in low conductivity environments. (See Appendix A for the effect of a on velocity and the effect of conductivity on a.) When ce=0, E M velocity is defined by  The region over which an E M wave velocity survey is taken can be on the order of hundreds of cubic meters, and the volume over which a single E M wave velocity is measured can be on the order of tens of cubic meters. Within this large volume of earth, there can be many layers through which one measures an average E M velocity or dielectric constant. Each of the layers can have individual E M velocities or dielectric constants that differ from its neighbors. This chapter reports how the average dielectric constant of the whole volume is affected by the dielectric constant and by other characteristics of the individual layers. The Effect of Layering  on Elastic  Waves  The motivation for this study comes from results of studies on elastic waves. Elastic waves have a characteristic wavelength which depends on the frequency of the propagating wave. It has been shown for elastic waves that the interplay between the characteristic wavelength of the elastic wave X and the average thickness of the layers in the system t can affect the average elastic velocity of the system. When the X/t ratio is small or the layers are relatively thick, the system of layers is said to fall under the ray theory regime. When the X/t ratio is large or the layers are relatively thin, the system of layers is said to fall under the effective medium theory (EMT) regime.  6  For elastic waves, numerical studies have been used to determine the transition zone between the ray theory and E M T regimes for elastic waves traveling through a system of layers. The systems studied are binary systems made of two components. The components within these systems are distributed in alternating layers. Values for the transition zone between ray theory and E M T from numerical analysis of an elastic planewave traveling normally incident through a system of layers are X/t =10-16 (Carcione et al., 1991; Marion et al., 1994). The transition zone is quite narrow, and the X/t ratio need not be far from the transition zone to be in either the ray theory or E M T regimes. A n interesting result of these studies is the fact that the transition zone occurs when X/t is greater than one, rather than at one, as is commonly thought. The exact X/t ratio for the transition appears to depend weakly on the relative proportions of the two components and on the contrast between the components. Marion et al. (1994) show that the transition zone occurs at a slightly higher X/t ratio for systems with relatively equal amounts of each of the two components. Carcione et al. (1991) show that the transition zone also occurs at a slightly higher X/t ratio for systems with highly contrasting components. Mavko et al. (1998) report that at X/t~2nox just after the transition zone Rayleigh scattering due to diffraction can occur resulting in velocities that are lower than the E M T limit. They also report that at X/t~l or just before the transition zone resonant or Mie scattering can occur resulting in velocities that are greater than the ray theory limit. The full transition from E M T to ray theory can be seen schematically in Figure 3.12.1 in Mavko et al. (1998). However, in the case of normal incident plane waves through a set of elastic (i.e. non-attenuating) layers, the terms Rayleigh-type and Mie-type scattering are a better description of the phenomena that occur around the transition zone because the same physical processes may not be operating in both the 1-D and 3-D cases. Therefore, throughout this thesis I will use the terms Rayleigh-type and Mie-type scattering to describe these phenomena around the transition zone.  7  Rayleigh-type scattering can be thought of as a wave dispersion which causes pulse to broaden and flatten. Because there is no attenuation, no energy is lost from the traveling pulse although the peak of the pulse may move. However, the redistribution of the energy due to the dispersion provides a pseudo-attenuation effect (Christensen, 1979). As shown in Christensen (1979) for 1-dimensional periodic layers, velocity in the long wavelength limit (i.e. E M T regime) can be approximated by V = V  emt  +  ri  v  2  oh  2  v, v  (2-4)  or  2 2  where p,G, + p G  2  2,/p,P G,G  2  2  X  2  v  emt  (2-5)  is the velocity in the E M T limit, h is distance, h\ and hi are the thicknesses of the two  layers, v\ and V2 are the velocities of the two layers, p i and P2 are the densities of the two layers, G i and G2 are the moduli of the two layers, and CO is the angular frequency. As one can see, if _£>1, velocity decreases with increasing frequency or with decreasing wavelength (Christensen, 1979). This dispersion can account for the low velocities just after the transition zone. For true Rayleigh scattering due to inclusions, velocity is found to be co 01A Q ^p, +(l-0,)p  (2-6)  2  2  where M, Vn  0  \ , 3(3-5v )0, 2  =  e^+{\-e,)p,  (4-5v )  9p  Q = 2  f  (2-7)  2  (2-8)  2  \  +1 \  V  K Tj  L J  V  the index 1 refers to the inclusions, index 2 refers to the matrix, M2 is the p-wave modulus of the matrix, 6\ is the volume fraction of material 1 or the inclusions, a is the radius of the inclusions, V L is p-wave velocity of the matrix, and v j is shear wave velocity of the matrix  8  (Christensen, 1979). Although the dependence of v on frequency is different for the Rayleigh-type and Rayleigh scattering, in both cases velocity decreases with increasing frequency or decreasing wavelength. Mie developed a general theory of scattering due to scattering centers which take into account size, shape, refractive index, and absorptivity (Pedrotti and Pedrotti, 1993). Rayleigh scattering is a special case of Mie scattering. What is commonly called the Mie effect is the phenomenon of forward scattering of energy at normal incidence as the diameter of the scattering sphere approaches the size of the wavelength (Born and Wolf, 1980). This Mie effect is much less pronounced than Rayleigh scattering (Pedrotti and Pedrotti, 1993). I conjecture that the Mie-type scattering as seen in the case of waves propagating through layers is also due to forward scattering of energy. When the thicknesses of the layers is on par with the wavelength (i.e. X/t~l), the energy of the normal incident wave is scattered forward. When the pulse or wave consists of multiple frequencies, the shape of the pulse will change due to Mie-type scattering. This redistribution of energy manifests itself as an apparent faster travel time resulting in a higher velocity than predicted by ray theory. Because both E M waves and elastic waves can be described using the wave equation, these two waves can be seen as mathematically analogous (Ursin, 1983). As seen above, numerical studies have already been used to determine the transition zone between ray theory and effective medium theory for elastic waves. Therefore, numerical methods similar to those used for elastic waves will be used in this chapter to determine the X/t ratio for the transition zone between the ray theory regime and E M T regime for E M waves. Averaging  EM Properties  in Layered  Systems  As with elastic wave propagation, a critical parameter in the development of a model for a layered system is the scale of the heterogeneity. I describe the scale of heterogeneity by X/t, the ratio between the wavelength of the E M wave and the average layer thickness of  9  the system t. When the wavelength is much smaller than the average thickness of the layers, ray theory is used to calculate the average (Brown, 1956): ^ where v  avg  =(!><T  (2-9)  is the average E M velocity, 61 is the volumetric fraction of each layer, and v/ is  the E M velocity of the layer. When the wavelength is much larger than the average thickness of the layers, effective medium theory (EMT) is used to calculate the average E M velocity of the layers (Brown, 1956):  Sedimentary environments range in average layer thicknesses from a few millimeters to many meters. Typical TDR frequencies are around 750 MHz, and typical GPR frequencies range from 50 to 200 MHz. At a typical soil dielectric constant of 12, X/t can range from 0.1 to 200000. (See Appendix B for more details on typical wavelengths for different materials at different frequencies.) Because field measurements can fall within both the ray theory regime and the E M T regime, it is vital to confirm the existence of the two regimes and study the transition zone between them. In this chapter, I wish to determine the exact region as defined in terms of the ratio X/t over which each regime is valid and to characterize the transition zone between the ray theory and E M T regimes.  N U M E R I C A L ANALYSIS OF THE TRANSITION ZONE FOR E M W A V E S For my analysis, I numerically propagated an E M wave through a system of layers. Each system was binary and consisted of only two components. The two components were arranged in alternating layers. I ran a total of nine series of numerical simulations. For each series of numerical simulations, I kept the proportion of each component and the properties of each component constant. However, I did vary both the frequency and the average thickness of the layers. In order to vary the average layer thickness, I kept the total thickness of the system constant and increased the number of layers. Thus, the average thickness of the layers decreased as the number of layers increased. I examined the effect  10  of changing the dielectric contrast between the two components, the proportion of the two components, the variability of the layer thicknesses within the system, and the electrical conductivity of the two components. A numerical wave propagation model is used to assess the valid regions of ray theory and E M T for E M waves traveling through layered systems. The wave propagation model is based on an exact solution to the 1-dimensional wave equation for a wave traveling through a series of layers. As described by Ward and Hohmann (1994), the electric (Ey) and magnetic (Hx) fields of the z layer as seen in Figure 2-1 can be th  represented as a uniform plane wave: (2-11)  E = E e- *-* +-E e *- ' +  vi  ikll  i  )  Jk,l  i  z )  J^\^ ~M^z )_-  H  i  EE  m  MZ-^  £  1  (  2  .  1  2  )  J  0  where kj is the complex wave number of the z' layer, th  K=i& i  e (a2  -m°fi>  (2- 3) 1  co is the angular frequency, \\Q is the permeability of free space, m is the permeability of the j' layer, e, is the permittivity of the i layer, zi is the vertical distance to the bottom of the th  th  i layer, z is any vertical distance within a layer at which the field is measured, E is the t h  +  amplitude of the electric wave traveling in the positive direction in the z layer, and ~E is tn  the amplitude of the electric wave traveling in the negative direction in the z' layer. th  Because both tangential E M fields must be continuous across layer boundaries, the E M fields immediately above a boundary can be rewritten in terms of the E M fields immediately below the boundary: E ^_i^= E i  cosh(ikihi)-  x(i-l)= xi  coshiikihi)  +  y  H  y  +H  Z{H  xi  - y E  y  i  sinh^fy)  (2-14)  sinh(^/i )  (2-15)  ;  where Z, is the impedance of the z' layer th  A  (2-16) fri-  ll  and hi is the thickness of the i layer. Since Equations 2-14 and 2-15 are true at every (h  boundary and since they can be written in matrix form, the E M fields at any given layer is then the product of all the layers below it: cosh( jkfr) n F  - Z . sinh( jk h ) t  t  coshOfyfy)  - ^-sinh( jkfr)  H x(n+\)  (2-17)  These equations have been programmed by Steven Cardimona at the University of Missouri-Rolla to calculate the waveform of an E M wave through a system of layers. A Ricker wavelet with a specified dominant frequency is numerically propagated through a series of layers with given thicknesses, dielectric constants, and conductivities. This program, although based on propagator matrices, uses the Kennett method (Kennett and Kerry, 1979) for stability. The program calculates the amplitude of the E M wave as it arrives at the end of a series of layers with given dielectric constants and conductivities (see Figure 2-2a). From the amplitudes I pick the first arrival time of the E M wave. In order to more accurately pick the first arrival, I transform the waveform into the frequency domain, interpolate, and then transform the wave back into the time domain (see Figure 2-2b) before picking the first arrival. Using the arrival time and the total thickness of layers, I can measure an average E M velocity for each system of layers. In order to compare the E M velocities measured from the different trials, I determine a normalized velocity (v ) for each measured velocity v emt  (v ): meas  —v  meas  (2-18)  For each trial, the measured velocity is normalized with respect to the two limiting cases of ray theory (v ) and E M T (y ). ray  emt  These values for these two limiting cases are determined  numerically using the same program described above. In order to determine the ray theory limit, I use Equation 2-9 to calculate the theoretical ray theory velocity. I then use the program described above to measure the arrival time of an E M wave traveling through a material with this theoretical velocity. The velocity calculated from this procedure is used 12  as v . Similarly, in order to determine the E M T limit, I use Equation 2-10 to calculate the ray  theoretical E M T velocity. To compensate for a small discrepancy between the measured values for v  ray  and v  emt  and the theoretical values due to the discretization of the numerical  program, I numerically propagate an E M wave through a material with this theoretical velocity and set v  emt  when v =v , meas  ray  to this calculated velocity. As a result of this normalization, v=l  and v=0 when  v =v . meas  emt  In order to determine the X/t for each numerical simulation, I must also calculate the wavelength for each simulation. Because the dominant frequency of a propagating E M wave can change after it travels through the different layers, the characteristic wavelength of the E M wave will also change: A = v//.  (2-19)  In order to calculate the characteristic wavelength of the E M wave, I take a Fourier transform of the waveform (see Figure 2-2c). I then pick the dominant frequency/of the waveform. Because the dielectric constant and velocity of each component is different, the wavelength within each component is also different even though the dominant frequency of the E M wave remains the same in every layer. Therefore, I use the measured or average velocity of the whole system along with the measured dominant frequency of the propagated waveform to characterize the wavelength of the propagating E M wave: A = v Jf. mea  (2-20)  RESULTS A N D DISCUSSION I performed nine different series of simulations. For each series of simulations I numerically propagated an E M wave through a system of layers at three different dominant frequencies. For each frequency, I used twelve different average layer thicknesses. Individual results of all simulations are given in Appendix C. The base series of simulations consisted of a system composed of equal amounts of two materials. The first material has a dielectric constant of 15, and the second material has a dielectric constant of  13  5. I measured velocities at 50MHz, 200MHz, and 750MHz. I set the average layer thickness to twelve different values for each frequency. I compared to this base series of simulations eight other series of simulations where I varied the dielectric contrast between the two components, the proportion of the two components, the variability of the layer thicknesses within the system, and the electrical conductivity of the two components. Figure 2-3 shows waveforms for an E M wave propagated at 750MHz through the base system of layers. The system is composed of equal amounts of two materials—one with a dielectric constant of 15 and the other with a dielectric constant of 5. The thicknesses of the layers of each component are equal. The measured v  ray  used for the normalization is  determined from the bold waveform at the top of the figure. The measured v  emt  used for the  normalization is determined from the bold waveform at the bottom of the figure. The waveforms in between these two limits cover a range X/t ratios: the waveforms near the top of the figure have small X/t ratios while the waveforms near the bottom have large X/t ratios. In this figure, the peak of the waveform indicates the arrival time of the E M wave. As X/t increases, the arrival time of the E M wave increases indicating slower velocities with higher X/t ratios. When X/t is small, the arrival times approximate those predicted by ray theory; and when X/t is large, the arrival times approximate those predicted by EMT. In the transition zone when X/t~A, the waveforms become very attenuated and broad due to scattering and it becomes difficult to pick the arrival time. Figure 2-4 is a compilation of normalized velocities versus X/t from measured arrival times determined for the 50 MHz, 200 MHz, and 750 M H z examples. The true range in velocities falls between 9.38xl0 m/s and 9.92xl0 m/s. The error bars on the 7  7  data are calculated from the errors in picking the measured arrival times and frequencies. The thin solid line indicating the transition zone is drawn in simply to guide the eye. One of the marked characteristics of the plot is the number of points that fall below the effective medium limit. There are also a few data points which lie above the ray theory limit.  14  As for the numerically propagated seismic waves described above, Rayleigh-type and Mie-type scattering can account for these velocities that fall outside my predicted limits. With exception of the data affected by scattering, the velocities tend to be near either the ray theory or the E M T limit with only one point falling in between these two limits. I define the transition zone to be the range of X/t between the Rayleigh-type scattering and the Mietype scattering. The data at all three frequencies show the same trend: the transition zone between ray theory and E M T occurs at a X/t value close to 4. This result indicates that the transition is independent of frequency but is dependent on X/t. One should also notice that there is a shift in the dominant frequency in the systems when the initial X/t falls within the transition zone.. In the waveforms for these systems, the dominant frequency shifts either higher or lower changing the wavelength so that the measured X/t no longer falls within the transition zone. Note that there are many data with X/t~3 and X/t~5, but no data in the range between 3 and 5. Effect of Dielectric  Contrast  In order to study the transition zone in more detail, I varied the dielectric contrast between my layers. Figures 2-5 and 2-6 show the results from the simulations at 50 MHz, 200 MHz, and 750 MHz for layers with large and small dielectric constant contrasts, respectively. Both figures show normalized velocities versus X/t. In Figure 2-5 half the layers of the system have a dielectric constant of 24, while half have a dielectric constant of 2. In Figure 2-6 half the layers of the system have a dielectric constant of 11, while half have a dielectric constant of 9. Once again on these figures, the error bars are calculated from the errors in picking the arrival times and velocities, and the line indicating the transition zone is drawn in simply to guide one's eye. For the layers with a large dielectric contrast (Figure 2-5), one can see that both the ray theory and E M T limits are reached. The true velocities range from 8.07xl0 m/s to 7  1.01x10 m/s. One can see evidence of Rayleigh-type scattering at X/t between 5 and 10. s  There is also evidence of Mie-type scattering at X/t between 0.2 and 0.8. For this series of  15  simulations, the transition zone is very broad. The transition zone expands the whole decade from A/f~0.8 and continues until X/t~%. A few data points fall within the transition zone and have velocities that are between the two limits. For the layers with a small dielectric contrast (Figure 2-6), both the ray theory and E M T limits are reached. The true velocities range from 9.46xl0 m/s to 9.51xl0 m/s. 7  7  Only a few data points indicate Mie-type or Rayleigh-type scattering. The transition zone occurs at X/t~6 and is quite sharp. There are no measured velocities falling between the two limits. The difference in character of the transition zone can be explained in terms of actual X/t ratios within each layer as opposed to average X/t ratio for the whole system of layers. I plot on the figures the average X/t ratios calculated from the measured dominant frequency and measured average velocity for an E M wave traveling through the entire system of layers. Remember that measured average velocity for the entire system of layers differs from the velocities of the individual layers. When there is little dielectric contrast between the components, the assumption that this average X is approximately the same as the X within each layer is valid. However, when there is a large dielectric contrast, this assumption breaks down. Therefore, as the wave propagates through the layers with a large dielectric contrast, it can be in the effective medium regime when traveling through layers with low dielectric constant and high E M velocity and in the ray theory regime when traveling through layers with high dielectric constant and low E M velocity. Because the E M wave can span both regimes as it travels through layers with a large dielectric contrast, the transition zone becomes broader. Effect of Volumetric Proportions Another two series of simulations were conducted to assess the sensitivity of the transition zone to changes in the volumetric proportions of the components. Once again the E M wave was propagated at 50 MHz, 200 MHz, and 750 MHz. Figure 2-7 shows data from simulations where 75% of the system has a dielectric constant of 15 and 25% has a  16  dielectric constant of 5. Figure 2-8 shows data from simulations where 25% of the system has a dielectric constant of 15 and 75% has a dielectric constant of 5. In both figures I plot normalized velocity versus X/t. As before, the error bars are calculated from the errors in measuring the velocity and dominant frequency, and the line indicating the transition zone is only to guide the eye. The transition zone is broad for the simulation where 75% of the system has a dielectric constant of 15 (Figure 2-7). The range of velocities for this system is between 8.44xl0 m/s and 8.72xl0 m/s. The transition zone lies at X/t between 1 and 8. There 7  7  are a number of data points within the transition zone which have velocities that are between the ray theory and E M T limits. There is evidence of Mie-type scattering at ^=0.6 and of Rayleigh-type scattering at X/t between 6 and 20. The transition zone is sharp for the simulation where 25% of the system has a dielectric constant of 15 (Figure 2-8). The range of velocities for this system falls between 1.08xl0 m/s and 1.14xl0 m/s. The transition zone occurs at X/t~l, and no data points 8  8  fall within the transition zone. There is evidence of Mie-type scattering at X/t~3 and of Rayleigh-type scattering at X/t between 5 and 20. Once again, the difference in character of the transition zone can be explained in terms of actual X/t ratios as opposed to average X/t ratios. I plot on the figures the average X/t ratios calculated from the dominant frequency, average velocity, and the average layer thickness. However, because the proportion of the components is not equal, the thicknesses of the layers are also not equal. When the system consists of thick layers of a low dielectric constant and thin layers of a high dielectric constant, the E M wave travels quickly through the thick layers and slowly through the thin layers. In this situation the wavelength is large when the E M wave travels through the thick layers and small when the E M wave travels through the thin layers. Thus, the X/t for one layer of this system is approximately equal to the X/t for the other layers of the system. As a result the transition zone is sharp. When the system consists of thin layers of a low dielectric constant and  17  thick layers of a high dielectric constant, the E M wave travels quickly through the thin layers and slowly through the thick layers. In this situation the wavelength is large when the E M wave travels through the thin layers and small when the E M wave travels through the thick layers. Thus, the X/t for the thin layers of this system is much larger than the X/t for the thick layers of the system. Because the E M wave can span both regimes as it travels through thin layers of low dielectric constant and thick layers of high dielectric constant, the transition zone becomes broad. Effect of Layer Thickness  Distribution  I next assessed the effect of the distribution of layer thicknesses on the location and characteristics of the transition zone. For all these simulations, half of the layers have a dielectric constant of 15 and half have a dielectric constant of 5. Each series of simulations contains data from an E M wave propagated at 50 MHz, 200 MHz, and 750 M H z . Until now, all layer thicknesses have simply reflected the proportion of the components and the number of layers in the system. However, I now change the distribution of the layer thicknesses as shown in Figure 2-9. I compare the base system of evenly distributed layer thicknesses with two systems of unevenly distributed layer thicknesses. The data in Figure 2-10 are from E M waves propagated through layers where the "thick" layers are 3 times thicker than the "thin" layers. The data in Figure 2-11 are from E M waves propagated through layers where "thick" layers are 9 times thicker than the "thin" layers. For both examples, one set of thick layers alternate with a set of thin layers. In Figures 2-10 and 211 I plot normalized velocity versus X/t where t in these cases is the average layer thickness. The error bars are calculated from the errors in measuring the velocity and dominant frequency, and the line showing the transition zone is only there to guide the eye. When the layer thickness distribution is slightly uneven and the system has half the layers three times thicker than the rest of the layers (Figure 2-10), the transition zone remains quite sharp and occurs around X/t~6. Unlike the base system with even layers (Figure 2-4), many of the simulations have average velocities that fall between the ray  18  theory and E M T limits. The range of velocities falls between 9.39xl0 m/s and 9.90xl0 7  7  m/s. There is also evidence of Mie-type scattering at X/t~l.5 and of Rayleigh-type scattering at X/t between 6 and 24. When the layer thickness distribution is very uneven and the system has half the layers nine times thicker than the others (Figure 2-11), the transition zone is very broad and occurs at X/t between 0.6 and 12. Once again, many of the data have velocities that fall between the ray theory and E M T limits. The range of velocities is between 9.38xl0 m/s 7  and 9.89xl0 m/s. There is no evidence of Mie-type scattering, and the range of Rayleigh7  type scattering is between at X/t 10 and 22. When there is a distribution of layer thicknesses, the transition zone between ray theory and E M T is difficult to define. Because I use average layer thickness to calculate X/t, the plotted X/t may not be representative of the actual values for each layer. When the average X/t falls near the transition zone, the E M wave travels through some thick layers where E M T is valid and other thin layers where ray theory is valid. This results in more data points having velocities between these two limits. As the layer thickness distribution becomes more uneven or as the ratio between the thick and thin layers increase, the average X/t becomes less representative of the individual X/fs for each layer. Therefore, the transition zone broadens. However, as the average X/t becomes very large (i.e. X/t>20), all the layers fall under the E M T regime; and as the average X/t becomes very small (i.e. X/t<0.6), all the layers fall under the ray theory regime. Effect of Electrical  Conductivity  The final effect examined was that of electrical conductivity. I compare the base series of simulations that have no conductivity with simulations of low conductivity and high conductivity systems. For the low conductivity system, half of the layers have a dielectric constant of 15 and an electrical conductivity of 0.7 mS/m, and half of the layers have a dielectric constant of 5 and an electrical conductivity of 0.2 u.S/m. These conductivities and dielectric constants are typical for clay and sand minerals, respectively.  19  E M waves for these simulations were propagated at 50 MHz, 200 MHz, and 750 MHz. For the high conductivity system, half the system has a dielectric constant of 15 and a conductivity of 35 mS/m, and half the system has a dielectric constant of 5 and a conductivity of 10 |j,S/m. These values are typical for slightly saturated clay and sand soils. E M waves for these simulations were propagated at 200 MHz, 500 MHz, and 750 MHz. Figures 2-12 and 2-13 show waveforms for an E M wave propagated at 750 MHz for a low conductivity case and a high conductivity case, respectively. The main effect of conductivity is to decrease the amplitude of the E M wave. This can be seen by comparing the smaller peaks in Figures 2-12 and 2-13 with those in Figure 2-3. As conductivity increases, amplitude and skin depth (d) decrease. The skin depth is the distance an E M wave can propagate before its amplitude falls to lie, or 0.37, of its original value and is given by (2-21) where 8 is given by Equation 2-2. When a=0, the skin depth is infinite; however, when a=0.3, the skin depth decreases quickly to approximately one wavelength (see Appendix A for the effects of a on skin depth). The parameter a is proportional to conductivity and inversely proportional to frequency. Because of the inverse dependence of a on co, the effect of conductivity on amplitude is more pronounced at lower frequencies. At low frequencies, conductivity is the major contributor to attenuation; while at high frequencies, frequency is the major contributor to attenuation and conductivity is only a minor contributor. Conductivity appears to have only a minor effect on velocity at the frequencies used in this study. Using Equation 2-1,1 can calculate velocity when conductivity is present. As with skin depth, the inverse dependence of a onft)causes the effect of conductivity on velocity to be the most pronounced at lower frequencies. However, my simulations are run at relatively high frequencies so velocity is not affected by conductivity. For example, at  20  750 MHz, the velocity of a material with a dielectric constant of 15 has a velocity of 7.7xl0 m/s regardless of whether the electrical conductivity is 0, 0.7, or 25 mS/m. In this 7  example, conductivity does not begin to affect velocity until it reaches at least 200 mS/m. Likewise, at 750 Hz, the velocity of a material with a dielectric constant of 5 has a velocity of 1.0x10 m/s regardless of whether the electrical conductivity is 0, 0.2, or 10 ixS/m. In s  this example, conductivity does not begin to affect velocity until it reaches at least 60 mS/m. In general, the E M wave arrives at approximately the same time regardless of the conductivity. For most of the waveforms, there is not a great difference in arrival times. However, I do notice a slight phase shift in some of the waveforms with increasing conductivity. The change in the dominant frequency of the waveforms is noticeable and affects the wavelength to layer thickness ratio of the data. When the dominant frequency decreases while the average velocity stays the same, wavelength increases and X/t for that system increases. Likewise, when the dominant frequency increases and the average velocity remains the same, wavelength decreases and X/t for that system decreases. This phenomenon is more readily seen by comparing plots of normalized velocity versus X/t. I plot normalized velocity versus X/t for the low and high conductivity cases in Figures 2-14 and 2-15, respectively. Figure 2-4 for the base series of simulations is generally the same as Figure 2-14 for the simulations with the low conductivities. The range of velocities for the low conductivity simulations is 9.32xl0 m/s to 9.81xl0 m/s. 7  7  For each simulation, the measured velocity is about the same for both the zero and low conductivity cases. However, for a few simulations, the dominant frequency of the E M waveform for the low conductivity case is lower than for the zero conductivity case. Thus, those data for the low conductivity case plot at a higher X/t than for the zero conductivity case. This phenomenon is especially evident in the few points which lie on the edge of the transition zone. The data at v=l.l are plotted at a higher X/t in the low conductivity case  21  than in the zero conductivity case. This shift causes the transition zone to become very sharp and to occur at X/t~S. The situation in the high conductivity case is different from that for the low conductivity case. The low amplitudes of the waveforms makes it difficult to pick the arrival times, but the range of velocities for the high conductivity case range from 9.17xl0  7  m/s to 9.86xl0 m/s. At the lower frequencies, the conductivities are high enough to begin 7  affecting the average velocities. However, both the ray theory and E M T limits are reached. The transition zone is sharp and falls at X/t~5. There is no evidence of Mie-type scattering, but Rayleigh-type scattering occurs at X/t between 5 and 20. Although high conductivity does affect velocity to a small extent, the main effect of conductivity is to attenuate the waveforms.  CONCLUSIONS A N D IMPLICATIONS FOR INTERPRETATION OF FIELD D A T A Numerical simulations for 1-D layered media reveal a wavelength dependence of velocity which resembles closely that theoretically predicted for a heterogeneous distribution of 3-D scatterers. In particular, the transition between ray theory and effective medium theory exhibits a behaviour similar to the resonance phenomena of Rayleigh-type and Mie-type scattering. It should be noted, however, that although these terms are therefore used throughout this thesis, the correspondence between wave propagation in 1-D finely layered and 3-D heterogeneous media is not a proof that the same physical processes are operative. The results from these simulations clearly demonstrate the transition of E M wave velocities from the ray theory to E M T limits as X/t increases. The ray theory relationship was found to be valid when X/t is small while the E M T relationship is valid when X/t is large. Mie-type scattering causes some data before the transition zone to have velocities higher than the ray theory limit. Rayleigh-type scattering occurs in every series of simulations, covers a broad range of X/t just after than the transition zone, and causes the  22  measured velocities to fall below the E M T limit. In general, the transition zone between the two limits is narrow and occurs between 4 and 6. In my simulations not many data points had velocities between the ray theory and E M T limits or had X/t within the range of the transition zone. The transition zone is sharper when the dielectric contrast between layers is small and broader when the contrast is large. The transition zone is also sharper when a larger proportion of the system is of the material with the higher dielectric constant. When a larger proportion of the system has a lower dielectric constant, the transition zone is broader and more data points fall within the transition zone. These effects on the transition zone can be explained through the local X/t variation. Uneven layer thickness distributions also lead to broad transition zones. These effects on the transition zone can also be explained through the local X/t variation. The main effect of conductivity on E M wave propagation is to lower the amplitude of the waveform. Electrical conductivity also has a slight effect on both velocity and frequency. In this study, data have been presented in terms of X/t and normalized velocity. I conclude by considering these results in the context of field measurement of E M velocities. At typical ground penetrating radar (GPR) frequencies and conditions, I find that the transition zone occurs at average layer thicknesses between 0.5 and 3 meters. (See Appendix B for more details on typical wavelengths for different materials at different frequencies.) At typical time domain reflectometry (TDR) frequencies and conditions, the transition zone occurs between 0.05 and 0.3 meters. The layers in sedimentary environments can range from millimeters to meters. Therefore, typical site conditions for both methods can include average layer thicknesses on both sides of the transition zone so care must be taken when interpreting field data over layered sediments.  "The men were amazed and asked, '"What kind of man is this? Even the wind and the waves obey him!'" Matthew 8:27  23  0 Zl  Zi Zi-1  Zn  A plane E M wave normally incident to a series of layers. Ey, Hx, and k are a right-handed orthogonal set.  Figure 2-1: A plane E M wave normally incident to a series of layers. 24  a)  49  b)  Time (ns)  49.5  52.5  Time (ns)  2000  2500  3000  Frequency (MHz)  Figure 2-2:  a) The amplitudes recorded with time at the bottom of a series of layers showing a typical calculated E M waveform. b) The waveform in a) transformed into the frequency domain, interpolated, and transformed back into the time domain. c) The frequency spectrum for the waveform in a).  25  CD  _>  =  on  o3 -a o 'EL u <u C3 6 T3 P C ca IT) 2 < + H "o o c c ca  OO  cons sho wn  in  cu 'C* o _cu "3 '-a ca cu  H  av  c mo  cu id ca •a 3  an  ye  >-,  ca CU  in  -o  ay heo;  JS  1-  •5 o  c  i n a> E  ca of  in C\J  ID  crt C3  CU  >  ca CU  ao X3 p o O 3  o _> m 3 rc3 "CO U o o c3 S3  opag and  nd  o in  MHz ity o  <+H  co  in o > ca c ccfla  avef ave e da  co  C  o o cu *C CA CU u c U eft "3 •o •3 cu .o ca -3 w c ca o  !>  J=  4H  .g  "S3  J= CN T3  ed  ;>  Figui of  cu ca ca o" u •o a  26  J3 3  O N  X o IT)  -oc S3  S JS  H  Ui  e u >• «8  (^a-xivaMpsjnSfeapv-xpva)  27  S? >.  X! 60 3 O  N  X o m r-  •o c 03  CU  S  u I.  cu  C  cu  "3 >•  (^ a-xiAia)/(p3-insB3iAj-xiAia) B  AJIDOpA  p3ZI|BUIJ0{^  28  3 60  03  tL.  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E o  of •old.  &  E —  CD  .SP c o oo xs  36  eg  Q  -a & o .5 M o N  "O  s§ °« •o * e >o ca  IX  c ca  C  u  cu  >%  ox  c mm  "a*  >• ca  CN  (^H-IP\ia)/(P3JnsB3px[-xi\[a)  CO t-l  1)  3  so  u  ca  4-1 PL, O  37  e  '3c  M ° O •~  1 J £ £ £ s H H S S S ^  5  o o b 5Bb 3U © o ui o h w K5 N  s§  o  in  ©  a u  cd  •° s  <  C cd  |»  cd ^  o ^ S cj  CO  ©  Ml  a cu "3 >• cs  (^H-XI\[a)/(P3JnsB3iAT-iM[a) yf}pO|3A p 3 Z I | B U I J 0 J ^ [  38  tN <D l-c 3 •  M  S£ « >, cd  ~ O  c*-(  Chapter 3 Laboratory Confirmation of the Ray Theory and Effective Medium Theory Limits for Electromagnetic Waves  INTRODUCTION Because of the high value of dielectric constant for water compared to minerals and air, measurements of dielectric constant using electromagnetic (EM) wave propagation in geologic materials are used to estimate water saturation and water content. One common method of measuring dielectric constant is time domain reflectometry (TDR). In the field, dielectric constant is calculated from the time it takes an E M wave to travel along the TDR probe, a wave guide, inserted into the ground. The sampled material over which dielectric constant is determined is the volume of material immediately adjacent to and along the full length of the TDR probe. In the laboratory, dielectric constant is calculated from the time it takes an E M wave to travel along the TDR cell. The material within the cell determines the average velocity of the E M wave. Most laboratory TDR investigations to date have focused on homogeneous samples (for example Topp et al., 1980; Hokett et al., 1992; Jacobsen and Schj0nning, 1993b). From these laboratory investigations, relationships for estimating water content from dielectric constant for homogeneous mixtures have been established and confirmed. As a result, a common assumption when interpreting field data is that the material being sampled is also a homogeneous mixture of its components (air, water, and minerals). However, in chapter two, I showed that heterogeneity, specifically layering, can also influence E M wave velocity. In this chapter, I am specifically interested in confirming with laboratory time domain reflectometry data the findings of chapter two. Let us consider the propagation of an E M wave through a system of layers. The wave travels perpendicular to the layers. The layered system consists of two components  39  of equal proportion. The components are arranged in alternating layers. The geometry of the system can be defined in terms of the average thickness of the layers t. As in the numerical study in chapter two, the critical parameter in determining E M wave velocity in a layered system is A/t, the ratio between the sampling wavelength A of the E M wave and the layer thickness t. When the wavelength is much smaller than the layer thickness, ray theory is used to calculate the average E M velocity of the layered system:  « ={L '^T  v  0  t  where v  avg  (3-1)  is the average E M velocity, 0/ is the volumetric percent of each layer, and v/ is  the E M velocity of the layer. When the wavelength is much larger than the period thickness, effective medium theory (EMT) is used to calculate the average velocity:  The region over which each of these relationships is valid is defined in terms of the ratio ?Jt. The numerical modeling in chapter two confirms that both the E M T and the ray theory limits do exist. The transition zone between these two limits depends on a number of different factors: the dielectric contrast between the two components, the proportion of each component, and the layer thickness distribution of the system. Mie-type scattering just before the transition zone can cause velocities slightly higher than the ray theory prediction, and Rayleigh-type scattering just after the transition zone can cause velocities slightly lower than the E M T prediction. In general for a layered system with moderate dielectric contrast and equal proportions, the transition zone between these two theoretical limits occurs in a region at X/t=\ to 10. The purpose of this chapter is to obtain experimental verification of these numerical results. Laboratory TDR measurements are used to measure E M velocities in samples composed of layers of dry coarse sand and saturated fine sand.  EXPERIMENTAL METHOD  40  A set of laboratory experiments was conducted to measure the E M velocity of samples composed of homogeneous layers of dry coarse sand and saturated fine sand. The samples were composed of 50% coarse sand and 50% fine sand by volume; and the experimental variable was the layer thickness. The coarse sand has a median grain size of approximately 0.7 mm, a porosity (0) of 0.37, and a saturation (S ) of 0. The fine sand w  has a median grain size of approximately 0.13 mm, a porosity of 0.38, and was filled with water to a saturation of 0.95. (See Appendix D for details on the characteristics of the soils.) The laboratory measurements were made using TDR. In TDR, E M wave velocity is measured using a coaxial TDR cell and a cable tester. This method of measuring dielectric constant has been used extensively by others (for example, Davis and Chudobiak, 1974; Davis and Annan, 1977; Topp et al., 1980; Topp et al., 1988). The E M wave velocity is calculated from the time it takes an E M wave to travel along the TDR cell. Under high frequency and/or low conductivity conditions, the dielectric constant of the material with which the cell is filled determines the velocity v of the E M wave:  where c is the speed of light in a vacuum (2.99x10 m/s) and K"is the dielectric constant of 8  the material. As the E M wave travels as a pulse down the TDR cell, changes in amplitude occur as it encounters different materials. Figure 3-1 is a schematic diagram of four different scenarios which the E M wave propagating down a TDR probe or cell can encounter. In the top example, the initial pulse does not encounter a material change or an impedance contrast; therefore, there is no change in amplitude. In the second example, the pulse encounters an open circuit, and there is a jump in amplitude. In the third example, when the pulse encounters a short circuit, there is a drop in amplitude. In the last example, the presence of conductivity in the material along the probe causes a decay in amplitude. Figure 3-2 is a picture of the experimental method used to take the laboratory dielectric constant measurements. The measurements were taken using a coaxial TDR cell  41  connected to a cable tester (see Figure 3-2a). The cable tester is a Tektronix 1502C metallic cable tester (see Figure 3-2b) which is commonly used to measure dielectric constant in both the field and the laboratory. The cylindrical TDR cell consists of a center rod and a concentric outer shield made from aluminum as both must be electrically conductive (see Figure 3-2c). Figure 3-3 shows a schematic of the TDR cell. The center rod is 1 cm in diameter and the inner radius of the outer shield is 10 cm. Both the center rod and the outer shield are segmented into twelve 10 cm pieces. The center rod pieces are threaded and screw together. The outer shield is sealed with O-rings. The inside of the outer shield segments are also scored every half centimeter. The bottom segments of both the center rod and outer shield are fixed to a base where there is a coaxial cable connection between the cell and the cable tester. Also in the base is a mechanical switch which is used to short the center rod and the outer shield. The cable tester is controlled by a 486 PC using an RS232 interface and the program TDR-Main written by David Redman at the University of Waterloo. This program records the amplitude of the TDR trace between two specified times. As seen in Figure 3-4, three TDR traces are collected for each measurement—an open trace, a trace where the TDR is shorted using the switch at the base of the cell, and a trace where the center rod and the outer sheath are shorted using a metal rod placed immediately above the top layer of soil. The three traces are plotted together, and the times when the shorted traces deviate from the open trace are used to calculate the travel time through the sample and in turn the average E M velocity. To pack the TDR cell with the sample material, a premeasured weight of sand is placed in the bottom segment of the cylinder and tamped down until it fills the desired layer thickness and compacts to the desired porosity. For the saturated layers, a calculated amount of water is poured into the sand as it is tamped. This process is repeated until the desired number of layers is reached. A piece of wax paper is placed between the layers to prevent water from flowing from the saturated to the unsaturated layers. Cylinder  42  segments can be added to the apparatus as they are needed. In order to fulfill the requirements for E M wave transmission, the length of the cell needs to be larger than the width of the cell. Therefore, a minimum height of 30 cm is necessary for each experiment. This constraint was confirmed experimentally. See Appendix E for details on the experimental set-up and preliminary calibration experiments. In order to ensure that the average dielectric constant of the whole system is measured, I also use at least 3 sets of layers per measurements.  RESULTS A N D DISCUSSION I measured the E M wave velocity of six different layered systems of dry coarse sand and saturated fine sand. Each system consisted of half dry coarse sand and half saturated fine sand by volume. I used a dry coarse sand and saturated fine sand to maximize the dielectric contrast between the layers and to ensure that there was a sufficient measurable difference between the theoretical E M T average and ray theory average given the accuracy of the measurement system. The layer thicknesses of each system were 20 cm, 10 cm, 5 cm, 2 cm, 1 cm, and 0.5 cm, respectively. The frequency of the measurements as given by Tektronix was approximately 750 MHz. In order to compare the measured velocities from the six different systems, I normalized the measured E M velocities by the theoretical E M T and ray theory values. In order to determine the E M T and ray theory limits, I first calculated the dielectric constant of each homogeneous layer using the time propagation (TP) model for homogeneous materials: < - = 0(1 - S )K%+4>sy 5  w  +(i  5 w ater  -  where K / is the dielectric constant of the layer, Kai is the dielectric constant of air, K  (3-4)  -  r  the dielectric constant of water, and Kquartz  IS  m  e  water  is  dielectric constant of the solid sand grain.  The values of the dielectric constants at room temperature are 1, 80.36, and 4.5, respectively (Gueguen and Palciasuskas, 1994). I then calculated the E M wave velocity for  43  each layer using Equation 3-3. The theoretical ray theory and E M T velocities were then calculated using Equations 3-1 and 3-2, respectively. Finally, I normalized my measured data using - v —v v= v  (3-5)  — V  emt ray  where v is the normalized E M wave velocity, v  meas  v  emt  is the measured E M wave velocity,  is the theoretical effective medium velocity, and v  ray  is the theoretical ray theory  velocity. Figure 3-5 shows the normalized E M velocity versus X/t for the measured data. Each point is an average of three to twenty-eight measurements taken for specific average layer thickness. The error bars show the standard deviation for each set of data. The recorded waveforms are recorded in Appendix F. Also included in Appendix F are the measured dielectric constant, normalized E M velocity, and X/t ratio for each experiment. The range of measured dielectric constant is from 8.9 to 13.1. As expected, the ray theory limit is reached at low values of X/t, and the E M T limit is reached at high values of X/t. The transition zone is hard to define because of the spread of the data, but occurs at X/t between 1 and 6. For each laboratory trial, I also generated a synthetic model First, I calculated the dielectric constant of each layer using the TP model. I then input the dielectric constant of each layer into the numerical wave propagation program described in chapter two. I next reduced the data as described in chapter two to arrive at the average E M velocity and an average X/t for each trial. In Figure 3-6,1 plot normalized E M velocity versus X/t of numerically modeled data. The error bars indicate the standard deviation of the modeled data. The modeled waveforms are recorded in Appendix G. Also included in Appendix G are the measured dielectric constant, normalized E M velocity, measured and frequency, and  X/t ratio for each experiment. The range of modeled dielectric constant is from 9.2 to 12.7.  44  Once again, the ray theory limit is reached at low values of X/t, and the E M T limit is reached at high values of X/t. The transition zone occurs at X/t between 3 and 6. At low values of X/t (A/f<3), the numerical velocities are close to the ray theory values. The numerical and measured data for the 20 cm layers are equal to the ray theory limit. The numerical velocity for the 10 cm layers fall along the ray theory limit, but the experimental data fall below but near the ray theory limit considering the scatter in the experimental data. The numerical data for the 5 cm layers shows some evidence of Mietype scattering (Mavko et al, 1998) and fall slightly above the ray theory limit. The measured data within experimental error equal the ray theory limit. The measured data in this region are close to the ray theory values. At high values of X/t (X/t>6), the numerical data approach the E M T value. The velocities are slower than the E M T limit but begin to approach the limit at higher X/t. These low velocities are due to Rayleigh-type scattering (Mavko et al., 1998). The trend of the measured data is quite different from that of the modeled data. The measured velocity for the 2 cm layers lies near the E M T limit; the measured velocity for the 1 cm layers lies below the E M T limit; and the measured velocity for 0.5 cm layers also lies near the E M T limit. The modeled data and the measured data for the 1 cm and 0.5 cm layers agree quite closely within experimental error while those for the 2 cm layers do not agree. The effect of scattering is more pronounced in the numerical simulation because of the discrete nature of the modeling.  CONCLUSIONS These laboratory data confirm the numerical simulations of chapter two. The experiments demonstrate the transition of E M wave velocities from the ray theory to E M T limits as X/t increases. The thickness of layering, relative to the wavelength and not the absolute thickness, is an important contributor in determining E M wave velocities. I have confirmed the E M T and ray theory limits through both laboratory measurements and  45  numerical simulations of the experiments. The transition zone from the measured data is broader than that from the numerical simulations, but both center around X/t=4. Because experimental conditions are difficult to control the measured data show more scatter than the numerical simulations. The simulations show a strong presence of Rayleigh-type scattering. Four of the six measured and modeled data points agree within experimental error. " A truthful witness gives honest testimony, but a false witness tells lies." Proverbs 12:17  46  perfectly matched impedance CD  T3 3  open circuit -a 3  E  short circuit 3  ex S  3  Time Figure 3-1: The effect of matched impedance, open circuit, closed circuit, and conductivity on an E M wave pulse traveling along a TDR cell  47  48  Assembled Coaxial Cell  Single Segment of the Outer Shield Scored line every 0.5 cm -o-ring E E E E EE EE EE E EE _EE_  Outer Shield  Single Segment of the Center Pin  Screw for assembling center pin  each segment 10 cm high 1 cm diameter rod 10 cm inner diameter shield  Cable Tester  -y/  Coaxial Cable  Mechanical Shorting Switch Figure 3-3 Schematic of coaxial TDR cell  49  50  o  S3  1  * — I  co CO CO  c u 0>  >. CO  _l  in ©  o> ca> CD  >  4  (tey-±|/\l3)/(p9Jnse9|fl-ii/\i3) Aijoo|3A pazjieiujON  51  o  3  o  -H-  w CO  ccu .£ u CD  (0  in  d  oc> a> > CO  nsit cm cm CO  «3  ca  cc  B  l1  1  1 1  o CM  1• I  o  E E E o o o in  CNJ  •  cm  c o  o cu  i— a era  (Aey-ll/\l3)/(p9Jnsea|/\|-ii/\i3) PL,  52  Chapter 4 Determining Water Content and Saturation from Dielectric Measurements in Layered Materials*  INTRODUCTION The presence of water in porous geological materials can have a large effect on the dielectric constant (K) of the material due primarily to the large contrast between the dielectric constant of water (-80) and that of air (~1) and the solid components (-4-15). This has led to the use of in situ measurements of the dielectric constant for determining the volumetric water content (6 ) in soils, sediments, and rocks in the subsurface (for example W  Birchak et al., 1974; Topp et al., 1980; Hokett et al., 1992; Zegelin et al., 1992; Jacobsen and Schj0nning, 1993a; Selker et al., 1993) where water content is the volume fraction of the total solid/fluid system that is composed of water. Other studies (Alharthi and Lange, 1987; Hubbard et al., 1997) have extended the interpretation of dielectric constant to extract water saturation (Sw) where water saturation is the volume fraction of the pore space filled with water. Of specific interest in this chapter are the relationships that are used to relate the measured dielectric constants to water content or saturation. These relationships are often based upon laboratory studies of small homogeneous samples. In the interpretation of field data it is then commonly assumed that the dielectric constant is measured over a volume of the subsurface that can also be treated as a homogeneous mixture of solids and fluids. In this chapter, I question the validity of this assumption and present relationships between dielectric constant and water content or saturation that can be used in geologic systems where there is interlayering of different lithologies.  * A modified form of this chapter is published as Chan, C.Y., Knight, R.J., 1999, Determining Water Content and Saturation from Dielectric Measurements in Layered Materials: Water Resources Research, 35, 8593. Used with permission. Copyright by the American Geophysical Union.  53  There are two methods commonly used to determine the dielectric constant of a region in the subsurface. In time domain reflectometry (TDR) the dielectric constant is calculated from the time it takes an electromagnetic (EM) wave to travel along the TDR probe, a wave guide, inserted into the ground. The sampled material over which the dielectric constant is determined is the volume of material immediately adjacent to and along the full length of the TDR probe. Measurements of dielectric constant can also be obtained using a ground penetrating radar (GPR) system. In using GPR, two antennae are moved across the earth's surface: one to transmit electromagnetic energy and the other to receive energy that has been reflected back to the surface from interfaces across which there have been changes in dielectric properties. The dielectric constant of a region determines the velocity of the electromagnetic wave such that GPR velocity data can be used to obtain a model of the dielectric properties of the subsurface. Most TDR measurements and many GPR measurements are made in sedimentary environments where it is expected that the soils and sediments are a complex interlayering of different lithologies due to changes in the depositional environment. The layers can range in thickness from millimeters to many meters. Dasberg and Hopmans (1992) investigated the relationship between dielectric constant and water content in layered soils but limited their study to testing relationships that assume a homogeneous system. In a study of TDR measurements in layered soils Topp et al. (1982) account for the presence of layers by averaging the travel time through the layers, an approach that is valid for certain cases. I expand on this previous research by considering the scale and orientation of the layering, the role of lithology, and presenting the valid theoretical descriptions of these layered systems. In this chapter I explore the theoretical differences between the dielectric constantwater content and dielectric constant-water saturation relationships for homogeneous systems and layered systems. Specifically, I am interested in determining the level of  54  inaccuracy introduced in water content or saturation determinations when a sampled layered system is incorrectly assumed to be homogeneous. I find that the assumption of homogeneity, when layering is present, can introduce significant error into the determined water content or saturation levels. In my calculations I ignore the effect of conductivity because of the high frequencies at which TDR and GPR measurements are made. At these frequencies, conductivity affects only the amplitude of a signal and not the velocity of the E M wave propagation.  MODELS OF HOMOGENEOUS SAND-FINE SYSTEMS I consider in my theoretical study three systems composed of a sand component and a fine component. The fine component can be clay, silty clay, or silt loam. I refer to the three systems as sand-clay, sand-silty clay, and sand-silt loam. In each of these systems I vary the composition of the solid phase and the water/air content. The properties of all components (sand, clay, silty clay, silt loam, water, air) used in this study are listed in Table 4-1. Description of the Homogeneous System I model a specific form of binary mixtures, previously studied by Nur et al. (1991) and Knoll and Knight (1999). This form of mixture is a valid representation of binary mixtures where the two solid components each have an approximately constant grain size, and the grain size of one component is significantly smaller than the grain size of the other. In constructing these mixtures (mathematically or in the laboratory) I start with the larger grain size solid component, in this study pure sand which has a porosity of 0.49. The second solid component (clay, silty clay, or silt loam), referred to as the fine fraction, is then added to the sand in such a way that it fills the pore space between the sand grains, thus lowering the porosity until it reaches a minimum. Once the pore space between the sand grains has been filled, I continue to increase the volume fraction of the fines by replacing the sand grains.  55  I describe these mixtures in terms of the initial porosity of the pure sand sample (<t> l)', the volume fraction of the fine fraction (dfi ) that includes both the solid and the sanc  ne  interstitial pore space; and the porosity of the fine fraction ((j)fi ). The porosity of the ne  mixture (0 ) can then be given by the following relationship (Nur et al., 1991). mjx  Qmix = <t>sand ~ W f ~ Qfi* ) • 6  0m« = 0fine*fine >  1  lf  i f  fine < <t>sand  6  fine> <Psand •  In modeling this homogeneous system, I assume that the water and air will be distributed evenly such that I can consider the global water content or saturation of the sand-fine mixture to be uniform throughout the system. Modeling the Dielectric Constant of Homogeneous Systems A number of different methods can be used to model the relationship between the dielectric constant of a material and the level of water content or saturation. These include effective medium theories, volumetric averaging models, and empirical relationships. While the effective medium theories such as the Hanai-Bruggeman-Sen (Sen et al. 1981) provide a more rigorous treatment of the physics governing dielectric properties, it has been found that the volumetric averaging models can provide reasonably accurate estimates of the dielectric properties of porous materials (Greaves et al., 1996; Knight and Endres, 1990; Knoll and Knight, 1999). Due to the simpler form of these averaging models, they are more commonly used to derive water content or saturation from field measurements of dielectric constant, and I will use one example in our study to model the behavior of the homogeneous systems. In addition to these averaging models, which require some knowledge of the properties of the material, I will also use the universally accepted empirical relationship introduced by Topp et al. (1980) for modeling homogeneous systems. Volumetric averaging models are semi-empirical and take the general form  56  4  1  (4"2)  Q  A plot illustrating the change of porosity with fine fraction is shown in Figure 4-1.  ^/=2>,<  (" )  (4-3)  where K  avg  is the average dielectric constant,  is the volume fraction of a component,  is the dielectric constant of that component, and n is a constant used to describe the geometrical arrangement of the components (Lichtenecker and Rother, 1937). When n = - 1 , the components lie in layers parallel to the propagation direction of the E M wave; and when n =1, the components lie in layers perpendicular to the propagation direction of the E M wave (Brown, 1956). In sediments, the value of n has been measured from 0.46 (Roth et al., 1990) to 0.65 (Dobson et al., 1985). However, the value of n most commonly used is 0.5 (for example Birchak, 1974; Alharthi and Lange, 1987). When n equals 0.5, the volumetric averaging equation is often referred to as the complex refractive index model (CRIM) if the values of dielectric constant are complex or the time propagation (TP) model if the values of dielectric constant are real. In this chapter I use the TP model to describe the dielectric constant of the homogeneous mixtures: K ft K° ft tC°sand-solid *avg = ftwK° w + ^ ft a K-° a 4^ °fine-solid* fine-solid4^°sand-solid* 0 5  5  u  x  5  U  5  5  X  (A-A\ V* > H  where the subscript w refers to water, a to air, fine-solid to the solid grain component in the fine fraction, and sand-solid to the solid grain component in the sand. The TP model has been shown to agree well with laboratory measurements of the dielectric properties of dry and saturated sand-clay mixtures (Knoll and Knight, 1999; Knoll et al., 1999) and is thus taken to be a valid model for the homogeneous systems. Empirical models for determining the dielectric constant of sediments are based either on a logarithmic relationship or a polynomial relationship. The most widely-used polynomial empirical formula relating dielectric constant to water content is known as the Topp equation (Topp et al., 1980) and takes the form of K  = 3.03 + 9.3O(0 ) + 146.00(0* ) - 76.7O(0 ) 2  mg  W  lv  3  (4-5)  This equation was determined from a number of soils with different water contents and is one of the standard methods for extracting water content from dielectric measurements because the dielectric constant of the mineral grains are not required.  57  These two models, which are commonly used to interpret measured dielectric constants, assume a homogeneous mixture of air, water, and solid. This assumption is accurate under laboratory conditions where these methods are often tested using small samples. However, the use of these equations for the interpretation of field data may not be appropriate for extracting information about water content or saturation from a dielectric constant measured over a large volume of material. It is most likely that the distribution of the components (air, water, solids) in a large volume will not be homogeneous. This leads one to consider models of the dielectric constant that take into account the presence of layering.  MODELS OF L A Y E R E D SAND-FINE SYSTEMS Description of the Layered Systems The layered systems which I model are composed of the sand and fine fraction used in the homogeneous mixtures but arranged in distinct layers. Each layer is composed of a single sediment type. As before I vary the proportion of the sand and fine fraction as well as the water/air content. The thickness of the layers and the orientation of the layering are additional variables in our modeling. A critical issue in the layered model is the heterogeneity that will exist in the distribution of water in the system. In contrast to the homogeneous case, where I assume a uniform fluid distribution, I must consider the differences in water content that exist between sand and fine layers under natural conditions which in turn translate into differences in water saturation for the layers. These differences in water content and saturation from layer to layer can affect the dielectric constant more than the differences in mineralogy. In my modeling I treat the layered system as being in a state of capillary equilibrium and determine the water saturation and water content of each individual layer from the  58  capillary pressure-saturation relationship for the material comprising the layer. In this study I use the simple relationship given by Clapp and Horaberger (1978): Sw =  ($f  (4-6)  where P is the threshold pressure, P is the capillary pressure, and X is the pore-size t  c  distribution index. These predicted water saturation values are converted to water content using  6  w  (4-7)  = (pS . w  The threshold pressure is defined as the maximum value the capillary pressure can reach before the material begins to drain. When the capillary pressure reaches a high enough level, the material is assumed to be completely drained. The pore-size distribution index is a measure of the connectivity of the pore space. Both these parameters will be different for each material; and the values used for the sand and fines in our models are given in Table 41. As a result, if I assume that a set of layered lithologies is at capillary equilibrium, the water content or saturation in one layer can differ from those in the adjoining layers. The average volumetric water content, for which I use the term global water content (dwg) of a layered system is given by 9w = ^d,9w,  (4-8)  g  where 0/ is the volume fraction of a layer, and 6wi is the water content of the layer. In turn the average water saturation, for which I used the term global water saturation (Sw ) of the g  layered system is given by Sw = x< , Q  g  (4-9)  where 0/ is the porosity of a layer, and Sw/ is the saturation of a layer. A single value of global water content or saturation can correspond to a myriad of combinations of layers with different individual water contents and saturations. Because global water content, global water saturation, and average dielectric constant depend so strongly on the water distribution in individual layers, there cannot be a unique relationship  59  between global water content or saturation and average dielectric constant. I therefore consider a hypothetical case for varying water distribution. The sand layers begin to drain first. Then before the sand layers have completely drained, the fine layers begin to drain. As time progresses, the sand layers have completely drained, but the fine layers still contain water. Eventually, the fine layers are also completely drained. This scenario is close to what one would find in nature. Modeling the Dielectric Constant of Layered Systems Of interest in this chapter is the interpretation of the dielectric constant measured using GPR or TDR methods. In both techniques an E M wave is propagated through the geological materials, and the dielectric constant is determined from the velocity of the wave. In developing my models I must therefore take into account the wavelength and the propagation direction of the E M wave as these two parameters determine the way in which the subsurface is sampled. I define three methods for averaging the dielectric constants of the layers based on the relationship between the thickness of the layers (t) and the wavelength (X) of the E M wave and on the direction the E M wave travels relative to the layering. When the wavelength is much larger than thickness of the layers, I use effective medium theory (EMT) to calculate the average dielectric constant of the layers. When the wavelength is much smaller than the thickness of the layers, I use ray theory to calculate the average. The transition zone is generally assumed to occur in a narrow band where l<Mf<10. In addition, measurements in the transition zone are often difficult to make because of scattering. As Figure 4-2 illustrates, it is important to note that the parameter of interest is the ratio of wavelength to layer thickness and not the absolute thickness of the layer. For example, a layer thickness of 40 cm can fall in the E M T regime under certain circumstances and in the ray theory regime under other circumstances. At a GPR frequency of 25 M H z the wavelength of an E M wave in a dry soil (?c=4) will have a wavelength of 600 cm  60  resulting in X/t=\5 placing the system in the E M T regime. At a TDR frequency of 750 MHz, the wavelength in a wet soil (?e=25) will be 8 cm resulting in X/t=0.2 placing the system in the ray theory regime. In this chapter, I consider propagation of the E M wave either perpendicular or parallel to the layers. If the layering in the subsurface is parallel to the surface, the sampling of the subsurface with surface GPR or with TDR probes inserted vertically into the ground would correspond to the perpendicular case. A TDR probe inserted horizontally into the ground from a well or trench or borehole-to-borehole GPR measurements would correspond to the parallel case. Let one first consider the case where the E M wave travels perpendicular to the layers. For a layered system where X » t, E M T is used to average the dielectric constant of the layers and is an arithmetic average (Brown, 1956): *W = Xc9,K-;  (4-10)  where K"; is the dielectric constant of the layer. When X « t, we calculate the average dielectric constant using ray theory which is calculated from the geometric average of the E M wave velocities:  where v  avg  is the average E M wave velocity, and v/ the E M wave velocity of the layer. The  following expression, valid at high frequencies and/or low loss conditions, relates the velocity (v) to the speed of light in a vacuum (c) and the dielectric constant (K): v=-£-  (4-12)  This results in the following equation for calculating the average dielectric constant: ^KaV =YJQI4KI g  (4-13)  When the E M wave travels parallel to the layering, the layering is only sampled when the wavelength is much greater than the thickness of the layers; i.e. in this case ray theory reduces to the result for a single layer. Therefore, I only consider the E M T average  61  which is a harmonic average of the dielectric constants of the individual layers (Brown, 1956): (4-14)  COMPARISON OF M O D E L RESULTS FOR HOMOGENEOUS SYSTEMS A N D L A Y E R E D SYSTEMS Of primary interest in this chapter is an assessment of the potential inaccuracy that can occur in estimates of water content or saturation if measurements of the dielectric constant of layered systems are interpreted assuming a homogeneous system. This is commonly done in the interpretation of most field data. In order to look at the dependence of dielectric constant on global water content or saturation in a layered sand-fine system, I generate dielectric constant and water content and water saturation data. I vary the volume fraction of fines from 0 to 1 and the global water content from 0 to the value obtained when the pore space of the entire system is filled with water. For the three systems (sand-clay, sand-silty clay, sand-silt loam), I calculate the dielectric constant predicted by E M T and ray theory for E M wave propagation perpendicular to the layers and by E M T theory for E M wave propagation parallel to the layers using my hypothetical saturation scenario (see Appendix H for details on all the theoretical calculations). First, for a given fine fraction and capillary pressure, I calculate the individual saturations of the sand and fine layers and compute a global saturation. Then I use the TP model or Topp equation to find the individual dielectric constants of these layers which are then used in the different averages of dielectric constant. Finally, I compare the dielectric constants predicted by the relationships for layered media to those predicted by the TP model and Topp equation for homogeneous media. The results of this numerical modeling can be displayed as a series of 3D plots of the dielectric constant of each sand-fine system with variation in the volume fraction of the  62  fine component and in water content or saturation. One such plot is given in Figure 4-3 where I show the variation in dielectric constant predicted by E M T relationship for E M wave propagation parallel to the layering for the sand-clay system where the dielectric constants of the individual layers were calculated using the TP model. In order to simplify the comparison of the results, rather than present the individual 3D plots, I show in Figures 4-4 to 4-6 the models obtained for the three systems, each at a fine volume fraction of 0.5. I only present data where the dielectric constants of individual layers were calculated using the TP model. When the Topp equation is used to calculate the dielectric constants of the individual layers, I note that the five models produce slightly lower dielectric constants at lower water contents and much higher values at high water contents. As shown in these figures, I consider both the dielectric constant-water content relationship (upper plots) and the dielectric constant-water saturation relationship (lower plots). Estimating Water Content from Dielectric Constant The upper plots of Figures 4-4 to 4-6 show the variation in dielectric constant with water content predicted by the five models. For the layered systems, water content ranges from 0.0 to approximately 0.50. For the homogeneous systems with 50% fines, the nature of the binary mixture results in a lower total porosity, which means a maximum water content of approximately 0.3. As a result, the predictions made with the TP model are over this limited range in water content. Given that the Topp equation is an empirical relationship, I extend predictions with this model to higher water contents. My first objective in this study is to gain some insight into the errors that can be introduced in the determination of water content if the presence of layering in the natural system is not accounted for (see Appendix I for details on all the error calculations). As seen in Figures 4-4a, 4-5a, and 4-6a, the results from the modeling of the three systems are qualitatively very similar. I note that each sand-fine system has a unique dependence on water content and saturation because of the different saturation scenarios. I use as a baseline for our comparisons the dielectric constant-water content relationships predicted  63  for layered systems. I then determine the error that would result if the TP model or the Topp equation relationship is used to extract water content from a dielectric constant measured for these layered systems, that is the extracted water content if the presence of the layering is neglected. In all three soil systems, the dielectric constant-water content relationship predicted for the case where layering exists parallel to the propagation direction of the E M wave is distinctly different from those obtained using the other models. This indicates that the use of the TP model or the Topp equation can yield highly inaccurate estimates of water content if the system actually contains such layering. The magnitude of the error in the determined value of water content varies with the actual water content and is shown in Figure 4-7a where I show the difference between water content predicted using the TP model or Topp equation and the true water content of the layered material. The error in using the TP model to extract water content is greatest for the sand-clay system. The value of water content for the sand-clay system can be overestimated by as high as 0.104 while the highest error for the sand-silty clay system is less than 0.084 and that for the sand-silt loam system is less than 0.080. At all actual water contents, the Topp equation predicts a more accurate water content than the TP model. The largest error predicted by the Topp equation is 0.046 for the sand-clay system. As seen in Figure 4-8a, if a homogeneous system is assumed when thin layering exists perpendicular to the wave propagation direction, the errors in extracting water content can be as high as 0.064. For such a layered sand-clay system, using the TP model to extract water content yields smaller errors (up to 0.026) than using the Topp equation (up to 0.064). The errors using the TP model (up to 0.028) for the sand-silty clay system are also smaller than those using the Topp equation (up to 0.036). However, for the sandsilt loam system, the errors from the TP model are slightly worse (up to 0.028) than those from the Topp equation (up to 0.026).  64  Figure 4-9a shows the errors incurred if relationships for homogeneous systems are used to estimate water content from a system that has thick layers perpendicular to the wave propagation direction. For all three sand-fine systems, the TP model produces an error of 0.028 at all water contents. These errors are all equal for three reasons: the sand properties are the same for all the sand-fine systems; the TP model and the ray theory model are similar in form; and we use the TP model to calculate the dielectric constants of the individual layers. The error varies with both water content and soil type when the Topp equation is used to calculate the dielectric constants of the individual layers. At most water contents, the error produced when the Topp equation is used to interpret the sand-clay system is high (up to 0.053). For the sand-silty clay system, the Topp equation performs better than the TP model at most water contents (error up to 0.035). The Topp equation gives a more accurate prediction (error up to 0.025) than the TP model at all water contents. In summary, if sand-fine layers exist, using either the TP model or Topp equation to extract water content will produce errors. Additionally, the extracted water contents can untenably be higher than the porosity of the sample. For all three layered scenarios, the TP model underestimates water content. In general, the Topp equation underestimates water content when thin layers exist parallel to the E M wave propagation but overestimates water content when either thin or thick layers are perpendicular to the propagation direction. Estimating Water Saturation from Dielectric Constant The lower plots of Figures 4-4 to 4-6 show the variation in dielectric constant with water saturation predicted by the five models. Water saturation ranges from 0.0 to 1.0. Because the nature of the binary mixture for homogeneous systems results in a lower total porosity, the dielectric constant-water saturation relationship for the TP model and Topp equation are quite different from those for layered systems. Figures 4-7b, 4-8b, and 4-9b show the error in water saturation for thin layers parallel, thin layers perpendicular, and thick layers perpendicular to the E M wave propagation. Each plot shows the difference between actual water saturation and extracted  65  water saturation using both the TP model and the Topp equation versus actual water content for all three sand-fine systems. In most cases, if layers are present, errors will arise if a relationship for a homogeneous system such as the TP model or the Topp equation is used to estimate water saturation. In general the relationships for homogeneous systems will overestimate water saturation of layered systems. The errors become enormous at higher water saturations and can even predict water saturations that are unrealistically over 1. These results show that care must be taken in using dielectric constant measurements for interpreting water content or saturation in areas where layering may be present. In particular, large errors in estimating water content can occur when using the TP model instead of a more accurate relationship for thin layers parallel to the E M wave propagation for all sand-fine systems. In addition, large errors can occur for a sand-fine system when the layers are perpendicular to the propagation direction.  CONCLUSIONS The measured dielectric constant of a system contains information about both water content and water saturation. The accuracy with which water content or saturation can be determined depends upon the relationship used to relate the measured dielectric constant to water content or saturation. Most of the theoretical and laboratory studies to date have addressed these relationships in homogeneous materials. In many geologic systems of interest it is very likely that the assumption of a homogeneous system is invalid—with the result that any prediction of water content or saturation is likely to be inaccurate. I show that very large errors in estimates of water content and saturation can occur if a layered system is assumed to be homogeneous. Therefore, the presence of layering, including both the orientation of the layering and the thickness of the layers relative to the wavelength measurement, should be accounted for in the relationships between dielectric constant and water content or saturation .  66  Table 4-1: Properties of components Property  Value  Source  Sand Dielectric constant of solid grain (K dsolid) Porosity (</>) Threshold pressure (P ) Pore-size distribution index(A)  4.5 0.395 3.5 N/m 0.247  Clay Dielectric constant of solid grain (^clay-solid) Porosity (0) Threshold pressure (P ) Pore-size distribution index (A)  11.8 0.482 18.6 N/m 0.0877  Silty Clay Dielectric constant of solid grain (Jc^ty i y. ud)  7.6  san  t  t  c a  so  t  Silt Loam Dielectric constant of solid grain (x  sUt  Porosity (<j>) Threshold pressure (P ) Pore-size distribution index (A) t  2  0.492 17.4 N/m 0.0962  Porosity (0) Threshold pressure (P ) Pore-size distribution index (A) i -solid)  5  oam  W  2  2  3  0.485 56.6 N/m 0.189  Water Dielectric constant (K )  from 01hoeft(1979) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978)  2  from 01hoeft(1979) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978)  Calculated from Olhoeft (1979) and Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978)  Calculated from Olhoeft (1979) and Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978) from Clapp and Hornberger (1978)  80.36  from Roth et al. (1990)  1.00  from Roth et al. (1990)  Air Dielectric constant (K ) a  67  0.2  0.4  0.6  0.8  Fine Fraction  Figure 4-1  Total porosity versus fine fraction for a homogeneous mixture of sand and fines.  68  EMT Xf\>  Figure 4-2  Ray Theory  10  X/t<l  Illustration of layer thickness to wavelength ratio for the effective medium theory and ray theory regimes. The layer thickness is much less than the wavelength in the EMT regime and much more than the wavelength in the ray theory regime.  69  Figure 4-3  Dielectric constant as a function of fine fraction and global water saturation calculated using the EMT parallel relationship. The dielectric constants of the individual layers were calculated using the TP model. This plot is for a sand-clay system. Similar plots can be generated for other relationships and other soil systems.  70  35  T  30 +  4  A Li A A X  25 + c CO  A •  A  in c 20 + o u  •.  A Perpendicular EMT • Perpendicular Ray  .ft •  X  • Parallel EMT  15 +  o TP Model XTopp Equation  5  10 +  <x<>% *  0  x  x^ X  +  0.00  +  0.10  1  0.20  1  0.30  1  0.40  0.50  Water Content  a)  35  T  4  30  S ra +  A •  2 5  in  g U o  A  20 +  A Perpendicular EMT A A # A •  A § ^  H 6  2  I 5  X X X X  B  0.2  Figure 4-4  21  •  x x  +  0.4  x  • Perpendicular Ray  • X  5o  10 + A  b)  • •  U 0> V  5  _  X  • Parallel EMT o TP Model XTopp Equation  X  r— 0.6  0.8  Saturation  A comparison of dielectric constant calculated for the sand-clay system using EMT for layering perpendicular to the propagation direction ( • ) , ray theory for layering perpendicular to the propagation direction (•), EMT for layering parallel to the propagation direction (•), the TP model (•), and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50.  71  30 T  • 25 +  c 20 to  in c o o  A  • Perpendicular Ray  15 + o& •  o 0)  5  A Perpendicular EMT  *X  10  o<* W  5 +  v/X  • Parallel EMT  •  x  O TP Model  X  XTopp Equation  V • •  . O S ? * * * '  0.1  —I  h-  0.2  —I—  0.3  —I  0.4  0.5  Water Content  30  x  25 +  A  c 20 + (A C  5  A A • A • A • gj  15 +  u a;  •  A  CD  5  A •  fi ^ 5 +  n  *  10 +  x  8  X  v  H  o  x x  • o  •  X O  A Perpendicular EMT • Perpendicular Ray • Parallel EMT O TP Model XTopp Equation  x X  x x x 0.2  +•  0.4  +•  0.6  0.8  Saturation  b) Figure 4-5  A comparison of dielectric constant calculated for the sand-silty clay system using EMT perpendicular to the propagation direction ( A ) , the ray theory perpendicular to the propagation direction (•), the EMT parallel to the propagation direction (•), the TP model (•), and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50.  72  x  30  X  x i 25 +  x4 x#  20 +  c  AXi  to  AX»  iii  c o o  15 +  A*  ot *  o a;  4)  o  o°|x  x  A Perpendicular EMT  1  • Perpendicular Ray  A  • Parallel EMT  •  O TP Model  10 +  XTopp Equation  U^  5  40.1  1—  0.2  0.3  —I—  —I  0.4  0.5  Water Content _aj_ 30  T  25  4 £ 20 ID  C o o  A Perpendicular EMT 15 +  • Perpendicular Ray • Parallel EMT  o cu  5  O TP Model  10 +  5 +  x  i b) Figure 4-6  XTopp Equation  89  6 °  x  x x  x  x x x 0.2  +  1—  0.4  0.6  0.8  Saturation  A comparison of dielectric constant calculated for the sand-silt loam system using EMT perpendicular to the propagation direction ( A ) , the ray theory perpendicular to the propagation direction (•), the EMT parallel to the propagation direction (•), the TP model ( • ) , and the Topp equation (x). a) The upper plot shows dielectric constant versus water content at a fine fraction of 0.50. b) The lower plot shows dielectric constant versus saturation at a fine fraction of 0.50.  73  0.12  T  0.10 + c c o o  0.08 +  A ' ©  CD +*  0.06 + A  01  tS 10 0.04 ± c  0.02 +  u c  i_  0.00  5  -0.02 +  CD  1 IT 4A  CD  A A  -0.04 •  •  • TP Sand-Clay  A  • TP Sand-Silty Clay  ^•••HAA  4A  CD  A  A • o A 0.5  A o o  —I— 0.2  0.1  • • •  —h0.3  0.4  TP Sand-Silt Loam Topp Sand-Clay Topp Sand-Silty Clay Topp Sand-Silt Loam  •  -0.06  a)  Global Water Content  0.20 0.10 0.00 -0.10  0.2  A  0^4  A  g  0.8  (0  co -0.20 (0  c  -0.30  CD  -0.40  CD  -0.50  u c  f  A  •  a •  ©  A O  •  -0.60  •  A  A  A O  O O  9 TP Sand-Silty Clay A TP Sand-Silt Loam • Topp Sand-Clay  • A  A • A •  A  -0.70  o Topp Sand-Silty Clay A Topp Sand-Silt Loam  •  -0.80  Figure 4-7  A  •  CD  b)  • TP Sand-Clay  n  Global Water Saturation  The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) and global water saturation (bottom). These plots are for thin layers parallel to the EM wave propagation direction at a fine fraction of 0.50. The solid symbols show the error produced when the TP model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares (•) are for the sand-clay system; the circles (•) are for the sand-silty clay system; and the triangles ( A ) are for the sand-silt loam system.  74  0 0 3  I # •  0.02 • c 0) c o  o  c  •  * 4-  0.00  (0  A ^  •  0.01 - -  d) - 0 . 0 1 c  •  A  +  0.1 A A A A  •  ©  a  ©  e  ©  m  m  F|  +  0.2 A  A  H  A  ^  -A—r-  0.3  °?4  A  A A A A A A  o  • 0 . 0 3 4-  0.5  n  A TP Sand-Silt Loam  • D  °  °  °  O  • Topp Sand-Clay  „ o O ° O o O O o  o Topp Sand-Silty Clay  n  A Topp Sand-Silt Loam  -0.04 +  •  •0.05 +  6 a •  -0.06 - -0.07 - -  a)  • TP Sand-Clay © TP Sand-Silty Clay  •  • 0 . 0 2 -J- A  01 <1>  B  A  •  °  D  •  m  n  • • • • •  •  °  Global Water Content  c o CO 3 CO CO  • TP Sand-Clay  cu co  © T P Sand-Silty Clay A TP Sand-Silt Loam  _c  O Topp Sand-Clay  0) o c cu  o Topp Sand-Silty Clay A Topp Sand-Silt Loam  w  *:  n  •  ° • _  O A O A  O  I |  . • A A  .  § A  • • • • b)  Figure 4-8  Global Water Saturation  The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) a n d global water saturation (bottom).  These  plots are for thin layers perpendicular to the EM wave propagation direction at a fine fraction of 0.50. T h e solid symbols show the error produced w h e n the T P model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares ( • ) are for the sand-clay system; the circles ( • ) are for the sand-silty clay system; and the triangles ( A ) are for the sand-silt loam system.  75  0.03  i A  0.02 + A  0 01 c a> c o 0 00 O ate  i_  O  A A  0.2  -0 01  0.3 o ~ O°  o o o o o o o  c -0 02  °  o  u c cu -0 03 cu £ Q -0 04  •  D  • ••••  • •  D  0.5  •  •  Ol  0.4  •  •  •  m TP Sand-Clay © TP Sand-Silty Clay ATP Sand-Silt Loam • Topp Sand-Clay O Topp Sand-Silty Clay A Topp Sand-Silt Loam  -0.05 + -0.06  -L  a)  Global Water Content  0.20 0.10 Jl 0.00  • • ,  1 0.4  -0.10 co 4)  13  -0.20 -0.30  _C -0.40 CU  o -0.50 c cu -0.60 cu 5 -0.70  A  O A A O O  •  •  •  a A  •  •  •  •  0.6  0.8  1 B TP Sand-Clay o TP Sand-Silty Clay A TP Sand-Silt Loam  a  O A • o A o A • O A • O  •  A  •  °  A A  • •  n  • •  -0.90  Figure 4-9  A  O A  -0.80  b)  • Topp Sand-Clay o Topp Sand-Silty Clay A Topp Sand-Silt Loam  Global Water Saturation  The difference between actual water content/saturation of a layered system and those determined from interpretation schemes which assume a homogeneous system as a function of global water content (top) and global water saturation (bottom). These plots are for thick layers perpendicular to the EM wave propagation direction at a fine fraction of 0.50. The solid symbols show the error produced when the TP model is used to extract information while the hollow symbols show the error produced when the Topp equation is used to extract information. The squares (•) are for the sand-clay system; the circles (•) are for the sand-silty clay system; and the triangles (•) are for the sand-silt loam system.  76  Chapter 5 Accounting for Saturation Heterogeneity in Obtaining Estimates of Water Content From Dielectric Data*  INTRODUCTION A critical parameter in modeling contaminant transfer in the vadose zone is water content or saturation. Geophysical methods can be very useful in determining water content or saturation because of their non-invasive remote sensing capabilities. However, in order to interpret geophysical field data correctly, one must use the correct relationship between the subsurface geological properties and the geophysical parameters. Dielectric measurements are used to detect the presence of water in the system because the dielectric constant of water is so high (-80) and dominates the dielectric response of any other materials in the system such as minerals and air (-1-6). In order to get a better understanding of the measured dielectric constant of a sedimentary system, the dielectric properties of a sedimentary system can be modeled. The proportions and dielectric properties of the components of the system (i.e. air, water, and minerals) all contribute to the overall dielectric constant of the sedimentary system. In chapter four, I show that the arrangement of the components affects dielectric response. Currently, it is common to assume that a sedimentary system is a homogeneous mixture of its components. Under this assumption, the dielectric constant-water saturation relationship can be described by the time propagation (TP) model:  * A modified form of this chapter is published as Chan, C . Y . , and Knight, R.J., 1999, Accounting for Saturation Heterogeneity in Obtaining Estimates of Water Content from Dielectric Data: Proceedings of The Symposium on the Application of Geophysics to Engineering and Environmental Problems, 435-444. Used with permission. Copyright by the Environmental and Engineering Geophysical Society.  77  where  K  avg  is the average dielectric constant,  d  t  is the volumetric fraction of V component, th  and K- is the dielectric constant of / component. t n  However, many sedimentary systems can be modeled as a system of layers. As seen in chapter four, the presence of the layers must also be taken into account when modeling the dielectric constant of the sedimentary system. In a layered system both the relative thickness of the layers and the orientation of the layers becomes important in establishing the dielectric constant-water content and dielectric constant-water saturation relationships. I characterize the relative thickness of the layers by X/t, the ratio between the sampling wavelength X and the average thickness of the layering t. I also consider layers that are either perpendicular to the propagation direction of the electromagnetic (EM) wave or parallel to the propagation direction (see Figure 5-1). If thick layers are present and perpendicular to the E M wave propagation direction, X/t is small and ray theory describes the system (Brown, 1956):  where 0 is the volumetric fraction of each layer, and K is the dielectric constant of the t  t  layer. If thin layers are present and perpendicular to the E M wave propagation direction, X/t is large and perpendicular effective medium theory (1EMT) describes the system (Brown, 1956): «v  K  g  =  X  •  (5-3)  If thick layers are present and parallel to the E M wave propagation direction, X/t is small and the wave only samples a single clay or sand layer. If thin layers are present and parallel to the E M wave propagation direction, X/t is large and the parallel effective medium theory (IIEMT) describes the system (Brown, 1956):  If a layered system is incorrectly interpreted as a homogeneous system, significant error can occur in the determination of water content and especially of water saturation. Likewise if one type of layered system is incorrectly interpreted as a different type of 78  layered system, significant error can also occur. As seen in chapter four, if a layered sandclay system is assumed to be homogeneous, errors of up to 0.104 for water content and over 1 for water saturation can occur. I model this chapter after Klein et al. (1997) where the idea of electrical conductivity anisotropy is introduced. Klein et al. (1997) acknowledge that measurements of electrical conductivity both parallel and perpendicular to layering will give different values. By measuring electrical conductivity in both directions, they can extract more information about the environment they are sampling. Likewise, in this chapter, I take advantage of the fact that measurements of dielectric constant both parallel and perpendicular to layering will give different values. Thus, by measuring dielectric constant in both directions, I can extract more information about the environment, in particular more information about water content and water saturation.  DETERMINING W A T E R CONTENT A N D W A T E R SATURATION F R O M DIELECTRIC FIELD M E A S U R E M E N T S I now propose a method for determining water content and saturation from field measurements. Common field methods for measuring dielectric constant include ground penetrating radar (GPR) and time domain reflectometry (TDR). Under average field conditions, GPR measurements have a wavelength of approximately 1.0 m, and TDR measurements have a wavelength of approximately 0.15 m. Layers in sedimentary environments can range from millimeters to many meters. Therefore, actual field measurements can be in either the E M T or ray theory regime. The procedure in this study only attempts to distinguish between the simplest possible sedimentary systems: a homogeneous mixture of sand and clay, alternating thin layers of sand and clay, and alternating thick layers of sand and clay. For simplicity, I use sand and clay as the two soils because they are mineralogically distinct. The homogeneous system consists of either a sand matrix with clay minerals filling the pores between the sand grains or a clay matrix  79  with suspended sand grains. The layers are assumed to fall in either the ray theory regime or the E M T regime: I do not address the case where X/t falls in the transition zone between the two regimes. I assume also that if layers are present they are parallel to the surface. I use dielectric measurements taken perpendicular to the surface of the earth and measurements parallel to the surface to determine whether the sedimentary system is homogeneous or layered. I then use the dielectric measurements and lithologic information determined from direct sampling to calculate the water content and saturation of both the global system and the individual layers if they are present. The procedure is summarized in the flow chart in Figure 5-2. Step 1: Taking the Measurements To begin determining water content and water saturation from dielectric measurements, I need the perpendicular (Kj_)and parallel (Xj|) measurements of the dielectric constant. Perpendicular measurements can be taken with surface GPR or TDR probes inserted in the surface. Parallel measurements can be taken with borehole GPR or TDR probes inserted in the face of a trench or cliff. It is important to ensure that the same frequency measurement is used for both the perpendicular and parallel directions. I also need the proportions of clay and sand in the system. Using direct samples from the system (i.e. well-borings) and sieve analysis, I separate the sediments into clay and sand fractions. The clay and sand are then repacked into separate fractions. From these fractions, I determine the volumetric proportions of the porous clay (t9 ) and sand (0 ). These c  S  volumetric proportions include both the solid grains and their associated porosity (0 C and <j>  s  for clay and sand, respectively). Using grain densities or porosimetry, the porosities of the pure clay and pure sand is also determined. Finally, I need to know the dielectric constant of the individual components: air (K ), water (K ), air  water  sand grain (K ). sand  80  solid clay grain (K i ), and solid c ay  Step 2: Determining the Geometry  My next step is to determine whether the system consists of a homogeneous mixture, thin layers, or thick layers. If K±=K\\, then the system is a homogeneous mixture of air, water, sand grains, and clay grains. The system will either be a sand matrix with porous clay in the space between the sand grains or a clay matrix with suspended sand grains. If K j * K j | , then the system is layered. The layers will either be thick or thin with respect to the wavelength of the dielectric measurement. Homogeneous Systems Once I know that the system is a homogeneous mixture, I still need to determine the geometry of the homogeneous mixture and the volume fractions of the solid clay (9 i ) c ay  grains and solid sand (6 ) grains. The system can consist either of a sand matrix with sand  dispersed clay particles or of a clay matrix with suspended sand grains. If t9 >^, the c  system consists of solid sand grains suspended in a porous clay matrix, and (5-5)  = °A  0w = l-0  c  0*,=0cO-* )e  (5-6) (5-7)  If 0 <(j) , the porous clay fill in the pore space within the solid sand matrix, and c  s  4>* = 4>,-ofi-4>c)  0«*r = l - f c  e  clay  = e {i-t)c  (5-8)  (5-9)  (5-io)  Layered Systems If Ki<Kj|, then the system has thick layers and can be described using the ray theory; in this case Kj| is sampling a thick high dielectric constant clay layer. Clay layers have high dielectric constants because clay minerals have relatively high dielectric constants and because clay layers tend to hold more water which also increases the dielectric constant. If Kj>Kj|, the system is layered, but there are two possibilities: Kj| is sampling either the average dielectric constant of a system of thin layers (i.e. E M T is valid) or a  81  single thick low dielectric constant sand layer (i.e. ray theory is valid). I can determine for most systems which case truly exists with further testing. By analyzing different soil combinations, I have empirically derived a test to determine whether E M T or ray theory is valid. However, because this is an empirically derived test, this test does not always determine the geometry correctly. In this procedure I assume the system falls under E M T unless the test indicates that the system actually falls under ray theory. This method will predict more measurements to be thin (i.e. fall under EMT) than is actually true. When  KJ>KJI,  I define two test parameters in order to determine  whether the system falls under E M T or ray theory: testl = ^±-  (5-11)  (  test2 =  V  sand  K  s c  +1  (5-12)  If test\>testl, then the system consists of thick layers and can be described using ray theory: in this case Kj| is sampling a thick low dielectric constant sand layer. If test\<testl, then I assume that the system consists of thin layers and can be described using EMT. However, as mentioned above, test\<testl is not a guarantee that the system consists of thin layers. In a small number of cases, when test\<test2 the layers will actually be thick and fall under ray theory. Step 3: Determining Water Content and Water Saturation Once the geometry of the system has been determined, I can then determine the . water content of each system. For the homogeneous systems, I determine the water content and water saturation of the whole system. For layered systems, I determine the water content and water saturation for each type of layer and then determine the global water content and water saturation for the whole system. Homogeneous Systems For a homogeneous system, a number of different relationships between dielectric constant and water content or water saturation have been determined. However, many of  82  these relationships such as the Topp equation are empirically derived and are not robust for all soils. Therefore, in this procedure I use the TP model presented in Equation 5-1. The TP model has been shown to be relatively robust for sand-clay mixtures (Knoll et al., 1999; Knoll and Knight, 1999). For homogenous systems, I can substitute all the known values into Equation 5-1, and solve for water content of the entire system (6 ). W  V^il' ~'Qmix^ air  ~ ^clay ^  K  Q w  _ ,  K  /ST"  hi—-  =  clay ~ ^sand ^ sand  K  "  (  }  y* water V * air  Alternatively, if an empirical relationship is available for a specific site, that relationship can be used to determine water content. To calculate water saturation, I normalize water content by the porosity of the homogeneous system. 5 =|^. W  (5-14)  Tmix  Thinly Layered Systems For thinly layered systems, I solve Equations 5-2 and 5-4 for the dielectric constant of the clay (K ) and sand (K ) layers: C  S  -b + Jb  -40JK«K,  1  K =  ±  C  -^-±  (5-15)  c  20  c  ^ Sr^  -  =5  where  (5  16)  b = 6 K -0**,, - K .  (5-17)  2  S  N  L  From Equations 5-16 and 5-17,1 calculate the water content of the clay (0 ) and sand Wc  (d ) layers using Equation 5-1: w  hi—- nr  <  w  =  "V  water "V  4 water  ^  K  K  (  }  air  air  _  (5"19)  Global water content and saturation for the entire system is now given by 0 =0 0 w  c  +0 0 S  w  c  (5-20) Wj  V  0 S=  *  w  83  .  (5-21)  As mentioned above, the test to determine whether the system falls under E M T or ray theory does not correctly identify all the systems that fall under ray theory. However, if 6 >(j) or 6 <0, then the assumption above that the system consists of thin layers was w  c  W  incorrect, and the system actually consists of thick layers and can be described by ray theory below. However, even with these additional checks, there may still be a few systems that are incorrectly identified as falling under E M T when they actually fall under ray theory. Thickly Layered Systems For systems with thick layers, if  Kj<Kj|,  I know that the layer sampled in the  parallel measurement is that with a higher dielectric constant, in this example clay. Setting K = Kj|, I can solve for K using Equation 5-3 C  S  (5-22) Likewise, if Kj>Kj|, I know that the layer sampled in the parallel measurement is the material with the lower dielectric constant, in this case sand. Setting K = KJJ, I can solve S  for  K,  K..  (5-23)  =  v  c  e  Now that I know the values for K and K , I can calculate the water contents and water C  S  saturations of the sand and clay layers and of the whole system using Equations 5-18 to 521. The method outlined above takes advantage of the fact that measurements of dielectric constant perpendicular and parallel to layering will give different values. In this procedure I can not only determine the global water content and saturation of the whole system, but I can also determine the water contents and saturations of the sand and clay layers if they are present. With this method, I can determine which systems consist of homogeneous sand-clay mixtures and which are layered. If the system is layered, I assume that the layers are thin with regard to the wavelength of the measurement. Testing, 84  however, identifies some of the systems as having layers that are thick with regard to the wavelength of the measurement. Unfortunately, the testing does not identify all the thicklayered systems, and some thick-layered systems will occasionally be incorrectly assumed to be thinly layered.  EXAMPLE I now present two simple examples which show the amount of error that can be incurred if this method is not followed. These examples are modeled on the examples in Klein et al. (1997). In my examples, I have a binary system of randomly layered clay and sand. The clay has a porosity of 0.485, a threshold pressure of 18.6, and a pore-size distribution index of 0.0877 (Clapp and Hornberger, 1978). The sand has a porosity of 0.395, a threshold pressure of 3.5, and a pore-size distribution index of 0.247 (Clapp and Hornberger, 1978). The dielectric constant of air, water, solid clay grain, and solid sand grain are 1.0, 80.36, 11.8, and 4.5, respectively (Gueguen and Palciauskas, 1994). Generating the Example I generate two random series of sand and clay layers with a total thickness of 600 feet (see Tables J - l and J-4 in Appendix J for the details of the sand and clay layers). In one series all the layers are thin (1 foot or 0.3 m) (see Figure 5-3a and Figure 5-5a) and in the other series all the layers are thick (6 feet or 1.8 m) (see Figure 5-4a and Figure 5-6a). The water table is fixed at 550 feet, the saturation of the vadose zone above the water table is calculated using a hydrostatic capillary pressure gradient (see Figures 5-3b and 5-4b). The water content of each layer is also calculated (see Figures 5-5b and 5-6b). The saturation Sw for each layer is calculated using the simple relationship given by Clapp and Hornberger (1978): ^  = (£f  85  (5-24)  where P is the threshold pressure, P is the capillary pressure, and A is the pore-size t  c  distribution index. These water saturation values are converted to water content 6 using W  6 = <i>S . w  w  (5-25)  Notice that at a given depth or capillary pressure, the sand layers will contain less water than the clay layers. Next the layers are blocked into 30 foot intervals. For each interval, the average water saturation is calculated (see Figures 5-3c and 5-4c), and the average water content is calculated (see Figures 5-5c and 5-6c). I also record the amount of porous sand and porous clay in each interval. The dielectric constant of each layer is calculated using the TP model (Equation 51). I then calculate the average dielectric constant for each 30 foot interval. For the thinlayered system (Figure 5-3), I calculate the average dielectric constant in both the perpendicular and parallel directions for each interval using Equations 5-2 and 5-4. For the thick-layered system (Figure 5-4), I use Equation 5-3 to calculate the average dielectric constant in the perpendicular direction for each interval. For the parallel direction, I set the dielectric constant equal to the value of the center layer within each interval. Determining Water Content and Water Saturation In order to test the method presented above, I estimate the average water saturation and water content of each interval from the average dielectric constant. I compare the water saturation and water content estimates assuming that each interval is homogeneous with the water saturation and water content estimates using the flowchart method. (Details for all the calculations and estimates are in Appendix J). First, assuming each interval is homogeneous and the average dielectric constant for the interval is equal to the perpendicular average, I follow the left stream of the flow chart to determine the supposed proportions of sand solid, clay solid, and pore space in the interval (Equations 5-5 to 5-10). Continuing along this stream, I use Equation 5-14 to estimate water content from the average dielectric constant (see Figures 5-5d and 5-6d). I then use Equation 5-15 to estimate water saturation (see Figures 5-3d and 5-4d). The  86  differences between the true average water saturation for each interval and the estimated water saturation assuming homogeneity are shown in Figures 5-3f and 5-4f. The differences between the true average water content for each interval and the estimated water content assuming homogeneity are shown in Figures 5-5f and 5-6f. Next I estimate the average saturation of each interval assuming that layering can be present. First I determine the type of layering: I assume the system has thin layers unless JCj_<K|| or test\>test2. For the thin layers, I use Equations 5-16 and 5-17 to determine the dielectric constants of the clay and sand layers. I then proceed to calculate the water content of the individual layers using Equations 5-18 and 5-19. If 6 >^> or 6 <0, then the system W  C  W  actually consists of thick layers and not thin layers. For the thick layers, I determine the dielectric constant of the sand and clay layers using Equations 5-22 and 5-23. Using Equations 5-18 and 5-19,1 calculate the water contents of the sand and clay layers. Finally, I estimate the global water content using Equation 5-20. The differences between the actual global water contents and these computed global water contents are also calculated (see Figures 5-5g and 5-6g). I then calculate the global water saturation for all the layered systems using Equations 5-21. The differences between the actual global saturations and these computed global saturations are also calculated (see Figures 5-3g and 5-4g). Discussion As can be seen from Figures 5-3 and 5-4, saturation is overestimated (i.e. negative errors) at almost every interval when the intervals are assumed to be homogeneous. This error can be quite large: in fact, for many of the intervals the calculated saturations are untenably over one. In the thinly layered system, the difference in the true and estimated water saturation is less than 0.15 only in the top interval. In the thickly layered system,  87  only three intervals have differences between the true and estimated water saturations that are less than 0.1. When using the method presented in the flow chart above, there are no intervals in the thinly layered system which have errors in water saturation estimation that are of the same magnitude as the errors assuming homogeneity (Figure 5-3). The largest error is a difference of 0.0012. This error occurs in the one interval that contains the top of the water table. This interval consists of fully saturated sand and clay layers and partially saturated sand and clay layers. The method presented above is only valid for systems with two different types of layers. Because this interval contains four different types of layers, the method does not work. When using the flow chart, saturation is correctly predicted for all the intervals in the thickly layered system (Figure 5-4). Even though the flow chart method can misinterpret a thickly layered interval as a thinly layered interval, all of the intervals in this example are correctly interpreted as thick layers. Figures 5-5 and 5-6 show the same results but in terms of water content for the thin-layered sequence and the thick-layered sequence, respectively. Because of the relationship among saturation, water content, and porosity, the errors for water content appear to be less significant than the errors in water saturation. However, in some applications, a difference of 0.01 in water content can be very significant. For the thin layered system, assuming homogeneity does not result in large errors in water content. In most intervals, assuming homogeneity results in slightly underestimating the amount of water (i.e. positive errors). Only the interval containing the water table has an error over 0.01. For the thick layered system, assuming homogeneity results in larger errors in water content than in the thin layered system. Only three intervals have errors less than 0.01. In all the other intervals, the true water content is significantly more than the estimated water content. When using the method presented in the flow chart above, there is only one interval in the thinly layered system which has a significant error in water content estimation  88  (Figure 5-5). The largest error is a difference of 0.0006. Once again this error occurs in one interval that contains the top of the water table where the assumption of only having two kinds of layers breaks down. When using the flow chart, no intervals in the thickly layered system show errors in water content of the same order of magnitude as the errors when homogeneity is assumed (Figure 5-6). The flow chart method correctly predicts a thickly layered system for every interval.  CONCLUSIONS As can be seen in the examples, interpreting geophysical data collected in two directions can result in obtaining more accurate estimates of water content. The interpretation method given in this chapter is based on fundamental principles and is easily followed given all the parameters. This method accurately describes the state of a system when homogeneity and thin layers are present and only occasionally incorrectly assumes the existence of thin layers when thick layers are present. However, all the thick layers intervals were correctly identified in the example presented in this study. As compared to the current practice of assuming that the subsurface is homogeneous, this approach gives significantly lower errors in determining water saturation from dielectric measurements. Although in some cases, the error in water content estimation assuming homogeneity is quite low, the approach presented in this paper still produces smaller errors. As a consequence, effort in the field should be made to collect the required data. When heterogeneity in the form of layers in saturation and lithology is not accounted for, higher errors in water content and saturation will result. When these erroneous values are used in hydrological modeling, highly inaccurate models of transport processes in the vadose zone can result.  "As for me, this mystery has been revealed to me, not because I have greater wisdom than other living men, but so that you ... may know the interpretation and that you may understand Daniel 2:30  89  Homogeneous  Ray Theory A./t <1  II EMT  1EMT  M>10  X/t>10  I  •AT I  1  Figure 5-1 Schematic of E M wave propagation through homogeneous and layered systems  90  Step 1: Taking the Measurements  Step 2: Determining the Geometry  System has S y s E m has  _ , System has  clay matrix  r sand matrix  c  <Pmix = Q<Pc smd=  e  1  _  Ipmix = (ps  hick layers '  <Pc)  c  e  8cla>=9 ( l -<p ) c  Step 3: Calculating Water Saturation and Water Content  *x  "St  ft  Yes system has thick layers  A. 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Because the dielectric constant of water is much higher than those of minerals or air, measurements of dielectric constant are often taken in the field using ground penetrating radar or time domain reflectometry to estimate the amount of water in the subsurface. A common assumption when interpreting dielectric constant data is that the subsurface is homogeneous. In this thesis I challenged that assumption and investigated the relationship between dielectric constant and water saturation for heterogeneous conditions. I only addressed the simplest heterogeneous case of layering which is a reasonable description of heterogeneity in many sedimentary environments. In order to estimate water content accurately, the heterogeneity of the earth must be taken into account. In this thesis, I focused on how to interpret dielectric measurements more accurately under the simple heterogeneous case of layering. In the first half of this thesis, I explored waves and scale by investigating how the interplay between the characteristic wavelength of the propagating E M wave and the average thickness of layers influences the average dielectric constant of the system. In chapters two and three I investigated the validity of effective medium theory (EMT) and ray theory. Effective medium theory is valid when thin layers are present; ray theory is valid when thick layers are present. The numerical and experimental data in chapters two and three confirmed that it is the ratio between the characteristic wavelength of an E M wave and the average layer thickness that is the critical parameter in determing the relationship between dielectric constant and water saturation. The thickness of the layers relative to the wavelength, rather than the absolute thickness of the layers, determines whether a layered system falls under the ray theory regime or the effective medium theory regime. For many layered systems, if the wavelength to layer thickness ratio is less than 4, the system falls under the ray theory regime. If the wavelength to layer thickness ratio is  96  more than 6, the system falls under he effective medium theory regimes. The transition zone between the ray theory regime and the effective medium theory regime in general is quite narrow {4<X/t<6). However, the characteristics of the transition zone for any particular system depend on properties of the system. The transition zone is narrower when the dielectric contrast between layers is small and broader when the contrast is large. The transition zone is also narrower when there is more of the material with the higher dielectric constant; the transition zone is broader when there is more of the material with the lower dielectric constant. A wide distribution of layer thicknesses also leads to a broad transition zone. These results can be explained in terms of the wavelength to thickness ratio for each individual layer. However, conductivity does not affect the transition zone. At typical GPR field conditions, the transition zone falls at layer thicknesses of around one meter. At typical TDR field conditions, the transition zone falls at layer thicknesses of around 10 centimeters. Because sedimentary environments can have a large range of layer thicknesses, the systems can fall on either side of the transition zone. Therefore, it is important to know which regime the sedimentary environment falls within and to determine the consequences in water content estimation when the system is incorrectly described. In the second half of this thesis, I investigated sand and water by examining the relationships among lithology, water saturation and dielectric constant for layered and homogeneous systems. Because field measurements of dielectric constant are used to estimate the amount of water in the subsurface, it is important that the correct relationship between dielectric constant and water saturation is established. Currently, it is common to assume homogeneity of the subsurface when estimating water content from dielectric constant. However, in chapters four and five I show the errors in water saturation estimates that can occur if a layered system is incorrectly assumed to be homogeneous.  97  Because many sedimentary environments are layered, the assumption of homogeneity is often not valid. If layers are present, the interplay between the wavelength and the average layer thickness becomes important when estimating water saturation from the measured dielectric constant. The different soils within the system can have different saturations and consequently different dielectric constants. These different dielectric constants and the relative thicknesses of the layers contribute to determining the average dielectric constant measured over a large volume of soil. I examined five different relationships between water saturation, lithology, and dielectric constant. To describe homogeneous systems, I modeled how dielectric constant changed with water saturation and clay content using the TP model and the Topp equation. To describe the layered systems, I examined how dielectric constant changed with water saturation and clay content under three different measurement conditions. For measurements taken perpendicular to thick layers, I used the ray theory model. For measurements taken perpendicular to thin layers, I used the perpendicular E M T model. And for measurements taken parallel to thin layers, I used the parallel E M T model. These five relationships between dielectric constant, water saturation, and lithology have different characteristics. If layers are present and the system is assumed to be homogeneous, errors in estimating water saturation arise if the TP model or the Topp equation is used. These relationships for homogeneous systems tend to overestimate water saturation of layered systems. The errors are the most pronounced when assuming a measurement perpendicular to thick layers is a measurement through a homogeneous mixture. The errors become very large at high water saturations and sometimes even predict water saturations that are over 1. If measurements of dielectric constant are taken in two directions, the accuracy of estimating water content can be improved. The method presented in this thesis first determines whether the system is homogeneous or layered. If layers are present, the method then determines whether the system has thick or thin layers. Once the geometry  98  has been determined, the water content and saturation of the system is calculated. Because part of the method is empirically derived, occasionally a system with thick layers is interpreted as having thin layers. Nevertheless, the method is robust and predicts the water content and saturation in the two examples of layered systems quite well. When I assume homogeneity for these two layered examples, I incur much larger errors in estimating water content and saturation. When heterogeneity in the form of layers in saturation and lithology is not accounted for, high errors in water content and saturation can result. The measured dielectric constant of a sedimentary system contains information about both water content and lithology. The accuracy with which water content can be determined depends upon the relationship used to relate dielectric constant to water content and lithology. Most of the theoretical and laboratory studies to date have addressed these relationships in homogeneous materials. But many sedimentary systems can be better described with a simple layered model. Therefore, in this thesis I address the relationships between dielectric constant and water saturation in layered materials. Both the orientation of the layering and the thickness relative to the wavelength should be accounted for in these relationships. If not, large errors in estimating water content can occur. As accurate estimates of water content are essential in many fields such as agriculture, geotechnical engineering, and environmental engineering, the results of this thesis can have strong implications for applied disciplines today. "By faith we understand that the universe was formed at God's command, so that what is seen was not made out of what was visible." Hebrews 11:3  99  BIBLIOGRAPHY Alharthi, A., Lange, J., 1987, Soil water saturation: dielectric determination: Water Resources, Research, 23, 591-595. Birchak, J.R., Gardner, C.G., Hipp, J.E., Victor, J. M . , 1974, High dielectric constant microwave problems for sensing soil moisture: Proc. IEEE, 62, 93-98. Born, M . , Wolf, E., 1980, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light: Pergamon Press, New York, 808 p. Brown, W.F., 1956, Dielectrics in Encyclopedia of Physics, 17: Springer-Verlag, Berlin, 154 p. Carcione, J.M., Kosloff, D., Behle, A., 1991, Long-wave anisotropy in stratified media: A numerical test: Geophysics, 56, 245-254. Chan, C.Y., Knight, R.J., 1999, Determining Water Saturation from Measured Values of Dielectric Constant: Water Resources Research, 35, 85-93. , Knight, R.J., 1999, Accounting for Saturation Heterogeneity in Obtaining Estimates of Water Content from Dielectric Data, Proceedings of The Symposium on the Application of Geophysics to Engineering and Environmental Problems, 435-444. Christensen, R . M . , 1979, Mechanics of Composite Materials: John Wiley & Sons, New York, 348 p. Clapp, R.B., Hornberger, G . M . , 1978, Empirical equations for some soil hydraulic properties: Water Resources Research, 14, 601-604. Dasberg, S., Hopmans, J.W., 1992, Time domain reflectometry calibration for uniformly and nonuniformly wetted sandy and clayey loam soils: Soil Sci. Soc. Am. J, 56, 1341-1345. Davis, J.L., Annan, A.P., 1977, Electromagnetic detection of soil moisture: Progress report 1: Canadian Journal of Remote Sensing, 3 , 76-86. , Chudobiak, W.J., In situ meter for measuring relative permittivity of soils, Geological Survey of Canada, 1974. Dobson, M . C . , Ulaby, F.T., Hallikainen, M.T., El-Rayes, M A., 1985, Microwave dielectric behavior of wet soil, II, Dielectric mixing models: IEEE Trans. Geosci., Remote Sens., GE-23, 35-46. Greaves, R.J., Lesmes, D.P., Lee, J.M., Toksbz, M . N . , 1996, Velocity variations and water content estimated from multi-offset, ground-penetrating radar: Geophysics, 61, 683-695. Gueguen, Y., Palciauskas, V., 1994, Introduction to the physics of rocks: Princeton University Press, 294 p. Hokett, S.L., Chapman, J.B., Russell, C.E., 1992, Potential use of time domain reflectometry for measuring water content in rock: Journal of Hydrology, 138, 8996. Hubbard, S.S., Rubin, Y., Majer, E., Ground-penetrating-radar-assisted saturation and permeability estimation in bimodal system: Water Resources Research, 33, 971990. Jacobsen, O.H., Schj0nning, P., 1993, Field evaluation of time domain reflectometry for soil water measurements: Journal of Hydrology, 151, 159-172. , Schj0nning, P., 1993, Laboratory calibration of time domain reflectometry for soil water measurement including effects of bulk density and texture: Journal of Hydrology, 151, 147-157. Kennett, B . L . N . , Kerry, N.J., 1979, Seismic waves in a stratified half space: Geophysical Journal of the Royal Astronomical Society, 57, 557-583. Klein, J.D., Martin, P.R., and Allen, D.F., 1997, The Petrophysics of Electrically Anisotropic Reservoirs: The Log Analyst, 38, 25-36. Knight, R.J., Endres, A L., 1990, A new concept in modeling the dielectric response of sandstones: Defining a wetted rock and bulk water system: Geophysics, 55, 586594. 100  Knoll, M.D., Knight, R.J., 1996, Electrical properties of dry sand-clay mixtures in the frequency range 100 kHz to 10 MHz: Geophysics, submitted. , Knight R.J., Brown, E., 1996, Can accurate estimates of hydraulic permeability be obtained from measurements of dielectric properties?: Geophysics, submitted. Lichtenecker, K., Rother, K., 1937, Deduction of the logarithmic mixture law from general principles: Physics Zeitschrift, 32, 255-260. Marion, D., Mukerji, T., Mavko, G., 1994, Scale effects on velocity dispersion: From ray to effective medium theories in stratified media: Geophysics, 59, 1613-1619. Mavko, G., Mukerji, T., Dvorkin, J., 1998, The Rock Physics Handbook: Tools for seismic analysis in porous media: Cambridge University Press, 329 p. Nur, A., Marion, D., Yin, H., 1991, Wave velocities in sediments, in Hovem, J.M., Richardson, M.D., Stoll, R.D., Eds., Shear waves in marine sediments: Kluwer Academic Publishers, Netherlands, 131-140. Olhoeft, G.R., 1979, Tables of room temperature electrical properties for selected rocks and minerals with dielectric permittivity statistics: U . S. Geol. Survey Open File Report 79-993, 24 p. Pedrotti, F.L., Pedrotti, L.S., 1993, Introduction to Optics: Prentice Hall, Englewood Cliffs, NJ, 602 p. Roth, K., Schulin, R., Fluhler, H., Attinger, W., 1990, Calibration of time domain reflectometry for water content measurement using a composite dielectric approach: Water Resources Research, 26, 2267-2273. Selker, J.S., Graff, L., Steenhaus, T., 1993, Noninvasive time domain reflectometry moisture measurement probe: Soil Sci. Soc. of Amer. J., 57, 934-936. Sen, P.N., Scala, C , Cohen, M . H . , 1981 A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads: Geophysics, 46, 781795. Topp, G.C., Davis, J.L., Annan, A.P., 1980, Electromagnetic determination of soil water content: measurements in coaxial transmission lines: Water Resources Research, 16 , 547-582. , Davis, J.L., Annan, A . P., 1982, Electromagnetic determination of soil water content using TDR: I. Applications to wetting fronts and steep gradients: Soil Science Society of America Journal, 46, 672-678. , Yanuka, M . , Zebchuk, W.D., Zegelin, S., 1988, Determination of electrical conductivity using time domain reflectometry: soil and water experiments in coaxial lines: Water Resources Research, 24 , 945-952. Ursin, B., 1983, Review of elastic and electromagnetic wave propagation in horizontally layered media: Geophysics, 48, 1063-1081. Ward, S.H., Hohmann, G.W., 1994, Electromagnetic Theory for Geophysics Applications, in Nabighian, M . N . , Ed., Electromagnetic Methods in Applied Geophysics: Investigations in Geophysics, Society of Exploration Geophysicists, 1, 131-311. Zegelin, S.J., White, I., Jerkins, D. R., 1992, A critique of the time domain reflectometry technique for determining soil-water content in Advance in measurement of soil physical properties: bringing theory into practice, SSSA Special Publication 20: Soil Science Society of America, Madison, Wis, 187-208.  101  Appendix A Dependence of Velocity and Attenuation on Frequency and Conductivity  The contents of Appendix A are available on the enclosed C D - R O M . Table A-1 Table A-2 Table A-3  Effect of Alpha on Velocity and skin depth A-l Conductivities Corresponding to Values of Alpha and Frequency for a Sand with Dielectric Constant of 2.5 A-2 Conductivities Corresponding to Values of Alpha and Frequency for a Clay with Dielectric Constant of 2.15 A-3  102  Appendix B Typical Wavelengths for Selected Materials  The contents of Appendix B are available on the enclosed CD-ROM. Table B - l Table B-2 Table B - 3  Typical Wavelengths for Different Materials when Conductivity is not taken into account B-l Typical Velocities for Different Materials when Conductivity is taken into Account B-l Typical Wavelengths for Different Materials when Conductivity is taken into Account B-l  103  Appendix C Results from Numerical Simulations  The contents of Appendix C are available on the enclosed C D - R O M . Table C-1 Table C-2 Table C-3 Table C-4 Table C-5 Table C-6 Table C-7 Table C-8 Table C-9 Figure C-1 Figure C-2 Figure C-3 Figure C-4 Figure C-5 Figure C-6 Figure C-7 Figure C-8 Figure C-9 Figure C-10 Figure C-11 Figure C-12  Simulated Data for 50% Dielectric Constant 15-50% Dielectric Constant 5 C-l Simulated Data for 50% Dielectric Constant 24-50% Dielectric Constant 2 C-2 Simulated Data for 50% Dielectric Constant 11-50% Dielectric Constant 9 C-2 Simulated Data for 75% Dielectric Constant 15-25% Dielectric Constant 5 C-3 Simulated Data for 25% Dielectric Constant 15-75% Dielectric Constant 5 C-3 Simulated Data for 50% for Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-4 Simulated Data for 50% for Very Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-4 Simulated Data for 50% Dielectric Constant 15 Conductivity 0.7 mS/m50% Dielectric Constant 5 Conductivity 0.2 uS/m C-5 Simulated Data for 50% Dielectric Constant 15 Conductivity 35 mS/m50% Dielectric Constant 5 Conductivity 10 uS/m C-5 Waveforms Constant 5 Waveforms Constant 5 Waveforms Constant 5 Waveforms Constant 2 Waveforms Constant 2 Waveforms Constant 2 Waveforms Constant 9 Waveforms Constant 9 Waveforms Constant 9 Waveforms Constant 5 Waveforms Constant 5 Waveforms Constant5  at 750 M H z for 50% Dielectric Constant 15-50% Dielectric at 200 M H z for 50% Dielectric Constant 15-50% Dielectric at 50 M H z for 50% Dielectric Constant 15-50% Dielectric at 750 M H z for 50% Dielectric Constant 24-50% Dielectric  C-6 C-7 C-8  C-9 at 200 M H z for 50% Dielectric Constant 24-50% Dielectric C-10 at 50 M H z for 50% Dielectric Constant 24-50% Dielectric : C-ll at 750 M H z for 50% Dielectric Constant 11-50% Dielectric C-l2 at 200 MHz for 50% Dielectric Constant 11-50% Dielectric C-l3 at 50 M H z for 50% Dielectric Constant 11-50% Dielectric C-14 at 750 M H z for 75% Dielectric Constant 15-25% Dielectric C-l 5 at 200 M H z for 75% Dielectric Constant 15-25% Dielectric C-l 6 at 50 M H z for 75% Dielectric Constant 15-25% Dielectric C-l 7  104  Figure C-•13 Figure C--14 Figure C--15 Figure C--16 Figure C--17 Figure C--18 Figure C-•19 Figure C--20 Figure C--21 Figure C--22 Figure C--23 Figure C--24 Figure C--25 Figure C--26 Figure C--27  Waveforms at 750 M H z for 25% Dielectric Constant 15-75% Dielectric Constant 5 C-18 Waveforms at 200 M H z for 25% Dielectric Constant 15-75% Dielectric Constant5 C-19 Waveforms at 50 M H z for 25% Dielectric Constant 15-75% Dielectric Constant5 C-20 Waveforms at 750 M H z for 50% for Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-21 Waveforms at 200 M H z for 50% for Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-22 Waveforms at 50 M H z for 50% for Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-23 Waveforms at 750 M H z for 50% for Very Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-24 Waveforms at 200 M H z for 50% for Very Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-25 Waveforms at 50 M H z for 50% for Very Uneven Layer Thicknesses Distribution Dielectric Constant 15-50% Dielectric Constant 5 C-26 Waveforms at 750 M H z for 50% Dielectric Constant 15 Conductivity 0.7 mS/m-50% Dielectric Constant 5 Conductivity 0.2 uS/m C-27 Waveforms at 200 M H z for 50% Dielectric Constant 15 Conductivity 0.7 mS/m-50% Dielectric Constant 5 Conductivity 0.2 uS/m C-28 Waveforms at 50 M H z for 50% Dielectric Constant 15 Conductivity 0.7 mS/m-50% Dielectric Constant 5 Conductivity 0.2 uS/m C-29 Waveforms at 750 M H z for 50% Dielectric Constant 15 Conductivity 35 mS/m-50% Dielectric Constant 5 Conductivity 10 uS/m C-30 Waveforms at 500 M H z for 50% Dielectric Constant 15 Conductivity 35 mS/m-50% Dielectric Constant 5 Conductivity 10 uS/m C-31 Waveforms at 200 M H z for 50% Dielectric Constant 15 Conductivity 35 mS/m-50% Dielectric Constant 5 Conductivity 10 uS/m C-32  105  Appendix D Characteristics of Sands and Clay  The tables for Appendix D are available on the enclosed CD-ROM.  Sources The coarse sand and fine sand used in these experiments is Ottawa sand donated by Wedron Silica Sand Company, PO Box 119, Wedron, IL 60557, U S A (phone: 1-800435-7856, Jody). The coarse sand is grade 390, and the fine sand is grade 705. These are clean washed and dried sands and silts. The clay used in these experiments is kaolinite donated by E C C International, PO Box 471, Sandersville, G A 31082, U S A (phone: 912-553-5208, Bob Pruett). This clay is also available in small amounts from the Source Clay Minerals Repository, Clay Minerals Society, Department of Geology, University of Missouri, Columbia, M O 65211, U S A (phone: 573-882-0786). The grade is KGa-lb and it is a well-crystallized fairly pure clay.  Physical and Chemical Properties Table DI contains physical properties for all three soils as measured in the laboratory. Table D2 contains a typical chemical analysis for both the sand and silt as determined by the supplier and a typical chemical analysis for kaolinite as determined by the Clay Minerals Society. Table D3 contains the grain size distribution for the sand and silt as reported by the supplier. Table D4 contains additional properties for the soils as determined by the Wedron Silica and/or the Clay Mineral Society .  106  Experimental Parameters In these experiments I measure the dielectric constant of the soils with different layer thicknesses. The thicknesses I use are 20 cm, 10 cm, 5 cm, 2 cm, 1 cm, and 0.5 cm. Table D5 shows details of how much soil and water is needed for each layer thickness. Tables D6 to D25 give the exact amount of soil and water used in the laboratory measurements. Table D-1 Table D-2 Table D-3 Table D-4 Table D-5 Table D-6 Table D-7 Table D-8 Table D-9 Table D-10 Table D - l 1 Table D - l 2 Table D-13 Table D-14 Table D-15 Table D-16 Table D-17 Table D-18 Table D-19 Table D-20 Table D-21 Table D-22 Table D-23 Table D-24 Table D-25  Soil Physical Properties Soil Chemical Properties Coarse Sand and Fine Sand Grain Size Distribution Additional Properties of Soils Experimental Parameters for Soil Layers Soil Parameters for 20 cm layers Trial 1 Soil Parameters for 20 cm layers Trial 2 Soil Parameters for 20 cm layers Trial 3 Soil Parameters for 20 cm layers Trial 4 Soil Parameters for 10 cm layers Trial 1 Soil Parameters for 10 cm layers Trial 2 Soil Parameters for 10 cm layers Trial 3 Soil Parameters for 10 cm layers Trial 4 Soil Parameters for 5 cm layers Trial 1 Soil Parameters for 5 cm layers Trial 2 Soil Parameters for 5 cm layers Trial 3 Soil Parameters for 5 cm layers Trial 4 Soil Parameters for 2 cm layers Trial 1 Soil Parameters for 2 cm layers Trial 2 Soil Parameters for 2 cm layers Trial 3 Soil Parameters for 2 cm layers Trial 4 Soil Parameters for 1 cm layers Trials 1 & 2 Soil Parameters for 1 cm layers Trials 3 &4 Soil Parameters for 0.5 cm layers Trials 1 & 2 Soil Parameters for 0.5 cm layers Trials 3 & 4  107  D-1 D-1 D-2 D-2 D-3 D-4 D-4 D-4 D-4 D-4 D-4 D-5 D-5 D-5 D-5 D-6 D-6 D-6 D-7 D-7 D-7 D-8 D-8 D-9 D-10  Appendix E Details for Coaxial Cell Measurements The figures for Appendix E are available on the enclosed CD-ROM. Measurement Technique The data presented in chapter three were collected using a coaxial cell measurement system (see Figures 3-2 and 3-3). This system was originally designed by Christina Chan in 1997 and built by Ray Rodway in the geology machine shop. The measurement apparatus consists of a center rod and a concentric outer shield. The center rod is 1 cm in diameter and the inner radius of the outer shield is 10 cm. Both the center rod and the outer shield are segmented into 10 cm pieces. The center rod segments screw together while the outer shield segments stack on top of each other. The outer shield segments are also scored every half centimeter and are sealed with O-rings. The total height of the apparatus is 1.20 m. However, because the apparatus is segmented, the total height does not need to be used for every measurement. The bottom segment is fixed to a base. In the base is a coaxial cable connection which connects the coaxial cell to the data collection instrument. Also in the base is a mechanical switch which causes a short between the center rod and the outer shield. The base also has a valve to allow fluid flow. To pack the cylinder, a premeasured weight of soil is placed in bottom segment of the cylinder and tamped down until it fills the desired layer thickness. For saturated layers, the calculated amount of water poured onto a mesh which lies on top of the soil and distributes the water evenly throughout the soil. This process is repeated until the desired number of layers is reached. Cylinder segments are added to the apparatus as they are needed.  108  Data Collection The data were collected using a Tektronix 1502C metallic cable tester TDR. This instrument is controlled by a 486 PC using an RS232 interface and the program TDR-Main written by David Redman at the University of Waterloo. This program records the amplitude of the TDR trace between two specified times. The data is converted to a text format and imported to MatLab for data analysis. For each measurement three TDR traces are collected—an open trace, a trace where the cable is shorted at the base of the cylinder using the mechanical switch mentioned above, and a trace where the center rod and the outer sheath are shorted using a metal rod immediately above the top layer of soil. The three traces are plotted together. The times when the shorted traces deviate from the open trace are used to calculate the travel time through the sample and in turn the average dielectric constant. See Figure 3-4 for a sample trace.  Calibration Measurements In order to assure the accuracy of the apparatus, measurements were taken on air and water. Measurements were taken at every layer (i.e. every 10 cm) of the cylinder. The dielectric constants calculated from the measurements of air are plotted versus cylinder height along with the mean and standard deviations in Figure E l . The results are within a few percent of the expected value of 1. The dielectric constants calculated from the measurements of water are plotted versus cylinder height along with the mean and standard deviations in Figure E2. The results are within a few percent of the expected value of 80. In addition, measurements were taken of dry coarse sand, dry fine sand, wet fine sand, and dry clay. These measurements were made both as a reference and as a check on packing technique. Measurements were taken at every cylinder layer, or every 10 cm, while the cylinder was being filled with soil and while the cylinder was being emptied. These data are shown versus cylinder height in Figure E3 to E6. For each soil, the  109  predicted values using the TP model and the volumetric fractions of air, water, and soil are also shown. The error bars are calculated from the uncertainty in picking arrival times, weighing the soil, and measuring the volume of the soil. The predicted and measured dielectric constants for the coarse and dry sands agree within experimental error. For the most part, the measured values fall close to the predicted values. However, the predicted values for the clay calibration fall above all the measured values. This discrepancy may be due to using the incorrectly high value for the dielectric constant of the clay mineral. These plots also confirm that measured dielectric constants taken when the cylinder is less than 30 cm high do not agree as well with the predicted values. Figure E - l Figure E-2 Figure E-3 Figure E-4 Figure E-5 Figure E-6  Measured Dielectric Constant of Air Measured Dielectric Constant of Water Measured and Predicted Dielectric Constant of Dry Coarse Sand Measured and Predicted Dielectric Constant of Dry Fine Sand Measured and Predicted Dielectric Constant of Wet Fine Sand Measured and Predicted Dielectric Constant of Dry Clay Sand  110  E-l E-2 E-3 E-4 E-5 E-6  Appendix F Results from Measured Laboratory Experiments  The contents of Appendix F are available on the enclosed C D - R O M . Figure F-1 Figure F-2 Figure F-3 Figure F-4 Figure F-5 Figure F-6 Figure F-7 Figure F-8 Figure F-9 Figure F-10 Figure F - l 1 Figure F-12 Figure F-13 Figure F-14 Figure F-15 Figure F-16 Figure F-17 Figure F-18 Figure F-19 Figure F-20 Figure F-21 Figure F-22 Figure F-23 Figure F-24 Figure F-25 Figure F-26 Figure F-27 Figure F-28 Figure F-29 Figure F-30 Figure F-31 Figure F-32 Figure F-33 Figure F-34 Figure F-35 Figure F-36 Figure F-37 Figure F-38 Figure F-39 Figure F-40 Figure F-41 Figure F-42 Figure F-43 Figure F-44  Measured Data 20 cm layers, trial 1, height 120 cm Measured Data 20 cm layers, trial 2, height 120 cm Measured Data 20 cm layers, trial 3, height 120 cm Measured Data 10 cm layers, trial 1, height 60 cm Measured Data 10 cm layers, trial 1, height 80 cm Measured Data 10 cm layers, trial 1, height 100 cm Measured Data 10 cm layers, trial 1, height 120 cm Measured Data 10 cm layers, trial 2, height 60 cm Measured Data 10 cm layers, trial 2, height 80 cm Measured Data 10 cm layers, trial 2, height 100 cm Measured Data 10 cm layers, trial 2, height 120 cm Measured Data 10 cm layers, trial 3, height 60 cm Measured Data 10 cm layers, trial 3, height 80 cm Measured Data 10 cm layers, trial 3, height 100 cm Measured Data 10 cm layers, trial 3, height 120 cm Measured Data 10 cm layers, trial 4, height 60 cm Measured Data 10 cm layers, trial 4, height 80 cm Measured Data 10 cm layers, trial 4, height 100 cm Measured Data 10 cm layers, trial 4, height 120 cm Measured Data 5 cm layers, trial 1, height 40 cm Measured Data 5 cm layers, trial 1, height 50 cm Measured Data 5 cm layers, trial 1, height 60 cm Measured Data 5 cm layers, trial 1, height 70 cm Measured Data 5 cm layers, trial 1, height 80 cm Measured Data 5 cm layers, trial 1, height 90 cm Measured Data 5 cm layers, trial 1, height 100 cm Measured Data 5 cm layers, trial 2, height 40 cm Measured Data 5 cm layers, trial 2, height 50 cm Measured Data 5 cm layers, trial 2, height 60 cm Measured Data 5 cm layers, trial 2, height 70 cm Measured Data 5 cm layers, trial 2, height 80 cm Measured Data 5 cm layers, trial 2, height 90 cm Measured Data 5 cm layers, trial 2, height 100 cm Measured Data 5 cm layers, trial 3, height 40 cm Measured Data 5 cm layers, trial 3, height 50 cm Measured Data 5 cm layers, trial 3, height 60 cm Measured Data 5 cm layers, trial 3, height 70 cm Measured Data 5 cm layers, trial 3, height 80 cm Measured Data 5 cm layers, trial 3, height 90 cm Measured Data 5 cm layers, trial 3, height 100 cm Measured Data 5 cm layers, trial 4, height 40 cm Measured Data 5 cm layers, trial 4, height 50 cm Measured Data 5 cm layers, trial 4, height 60 cm Measured Data 5 cm layers, trial 4, height 70 cm 111  F-1 F-1 F-l F-2 F-2 F-2 F-2 F-3 F-3 F-3 F-3 F-4 F-4 F-4 F-4 F-5 F-5 F-5 F-5 F-6 F-6 F-6 F-6 F-7 F-7 F-7 F-7 F-8 F-8 F-8 F-8 F-9 F-9 F-9 F-9 F-10 F-10 F-10 F-10 F-l 1 F-l 1 F-11 F-11 F-12  Figure F-45 Figure F-46 Figure F-47 Figure F-48 Figure F-49 Figure F-50 Figure F-51 Figure F-52 Figure F-5 3 Figure F-54 Figure F-55 Figure F-56 Figure F-57 Figure F-58 Figure F-59 Figure F-60 Figure F-61 Figure F-62 Figure F-63 Figure F-64 Figure F-65 Figure F-66 Figure F-67 Figure F-68 Figure F-69 Figure F-70 Figure F-71  Measured Data 5 cm layers, trial 4, height 80 cm Measured Data 5 cm layers, trial 4, height 90 cm Measured Data 5 cm layers, trial 4, height 100 cm Measured Data 2 cm layers, trial 1, height 30 cm Measured Data 2 cm layers, trial 1, height 40 cm Measured Data 2 cm layers, trial 2, height 30 cm Measured Data 2 cm layers, trial 2, height 40 cm Measured Data 2 cm layers, trial 3, height 30 cm Measured Data 2 cm layers, trial 3, height 40 cm Measured Data 2 cm layers, trial 4, height 30 cm Measured Data 2 cm layers, trial 4, height 40 cm Measured Data 1 cm layers, trial 1, height 30 cm Measured Data 1 cm layers, trial 1, height 40 cm Measured Data 1 cm layers, trial 2, height 30 cm Measured Data 1 cm layers, trial 2, height 40 cm Measured Data 1 cm layers, trial 3, height 30 cm Measured Data 1 cm layers, trial 3, height 40 cm Measured Data 1 cm layers, trial 4, height 30 cm Measured Data 1 cm layers, trial 4, height 40 cm Measured Data 0.5 cm layers, trial 1, height 30 cm Measured Data 0.5 cm layers, trial 1, height 40 cm Measured Data 0.5 cm layers, trial 2, height 30 cm Measured Data 0.5 cm layers, trial 2, height 40 cm Measured Data 0.5 cm layers, trial 3, height 30 cm Measured Data 0.5 cm layers, trial 3, height 40 cm Measured Data 0.5 cm layers, trial 4, height 30 cm Measured Data 0.5 cm layers, trial 4, height 40 cm  F-12 F-12 F-12 F-13 F-13 F-13 F-13 F-14 F-14 F-14 F-14 F-15 F-15 F-15 F-15 F-16 F-16 F-16 F-16 F-17 F-17 F-17 F-17 F-18 F-18 F-18 F-18  Table F - l Table F-2 Table F-3 Table F-4 Table F-5 Table F-6  Measured Data for 20 cm layers Measured Data for 10 cm layers Measured Data for 5 cm layers Measured Data for 4 cm layers Measured Data for 2 cm layers Measured Data for 0.5 cm layers  F-19 F-19 F-19 F-19 F-19 F-19  112  Appendix G Results from Modeled Laboratory Experiments  The contents of Appendix G are available on the enclosed C D - R O M . Figure Figure Figure Figure Figure Figure Table Table Table Table Table Table  G-1 G-2 G-3 G-4 G-5 G-6  G-1 G-2 G-3 G-4 G-5 G-6  Modeled Data for 20 cm Layers Modeled Data for 10 cm layers Modeled Data for 5 cm layers Modeled Data for 2 cm layers Modeled Data for 1 cm layers Modeled Data for 0.5 cm layers  G-1 G-2 G-3 G-4 G-5 G-6  Modeled Data for 20 cm Layers Modeled Data for 10 cm Layers Modeled Data for 5 cm Layers Modeled Data for 2 cm Layers Modeled Data for 1 cm Layers Modeled Data for 0.5 cm Layers  G-7 G-7 G-7 G-7 G-7 G-7  113  Appendix H Details of Calculations for Theoretical Study  The contents of Appendix H are available on the enclosed C D - R O M . Table Table Table Table Table Table Table Table  H-1 H-2 H-3 H-4 H-5 H-6 H-7 H-8  Table H-9 Table H-10 Table H - l 1 Table H-12 Table H - l 3 Table H-14 Table H - l 5 Table H-16 Table H-17 Table H-18 Table H-19 Table H-20 Table H-21 Table H-22 Table H-23 Table H-24 Table H-25 Table H-26 Table H-27 Table H-28 Table H-29 . Table H-30 Table H-31 Table H-32  Bulk Porosity for Homogeneous Sand-Clay Example H-1 Bulk Water Content for Homogeneous Sand-Clay Sample H-l TP Model for Homogeneous Sand-Clay Example H-2 Topp Equation for Homogeneous Sand-Clay Example H-2 Sand Saturation for Sand-Clay Example H-3 Clay Saturation for Sand-Clay Example H-3 Global Water Content for S and-Clay Example H-3 Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-4 Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-4 Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-5 Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-5 Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-6 Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-6 Bulk Porosity for Homogeneous Sand-Silty Clay Example H-7 Bulk Water Content for Homogeneous Sand-Silty Clay Sample H-7 TP Model for Homogeneous Sand-Silty Clay Example H-8 Topp Equation for Homogeneous Sand-Silty Clay Example H-8 Sand Saturation for Sand-Silty Clay Example H-9 Clay Saturation for Sand-Silty Clay Example H-9 Global Water Content for Sand-Silty Clay Example H-9 Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-10 Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-10 Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-11 Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-11 Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-l2 Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-l2 Bulk Porosity for Homogeneous Sand-Silt Loam Example H-l3 Bulk Water Content for Homogeneous Sand-Silt Loam Sample H-13 TP Model for Homogeneous Sand-Silt Loam Example H-14 Topp Equation for Homogeneous Sand-Silt Loam Example H-14 Sand Saturation for Sand-Silt Loam Example H-15 Clay Saturation for Sand-Silt Loam Example H-15 114  Table H-34 Table H-35 Table H-36 Table H-37 Table H-38 Table H-39  Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-16 Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-16 Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Layers H-17 Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-17 Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-18 Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Layers H-18  115  Appendix I Details of Theoretical Error Calculations  The contents of Appendix I are available on the enclosed C D - R O M . Table I-1 Table 1-2 Table 1-3 Table 1-4 Table 1-5 Table 1-6 Table 1-7 Table 1-8 Table 1-9 Table I-10 Table I-11 Table 1-12 Table 1-13 Table 1-14  Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-1 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-1 Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-2 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-2 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-3 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-3 Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-4 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-4 Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-5 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-5 Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-6 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-6 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-7 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-7  116  Table I-15 Table I-16 Table 1-17 Table I-18 Table 1-19 Table 1-20 Table 1-21 Table 1-22 Table 1-23 Table 1-24 Table 1-25 Table 1-26 Table 1-27 Table 1-28 Table 1-29 Table 1-30 Table 1-31 Table 1-32  Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-8 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-8 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-9 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-9 Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-10 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-10 Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-11 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-11 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-12 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-12 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-13 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-13 Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-14 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-14 Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-15 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-15 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-16 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-16 117  Table 1-33 Table 1-34 Table 1-35 Table 1-36 Table 1-37 Table 1-38 Table 1-39 Table 1-40 Table 1-41 Table 1-42 Table 1-43 Table 1-44 Table 1-45 Table 1-46 Table 1-47 Table 1-48 Table 1-49 Table 1-50  Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-17 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-17 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-18 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-18 Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-19 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-19 Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-20 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-20 Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-21 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-21 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-22 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-22 Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-23 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-23 Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-24 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-24 Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-25 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-25 118  Table 1-51 Table 1-52 Table 1-53 Table 1-54 Table 1-55 Table 1-56 Table 1-57 Table 1-58 Table 1-59 Table 1-60 Table 1-61 Table 1-62 Table 1-63 Table 1-64 Table 1-65 Table 1-66 Table 1-67 Table 1-68  Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-26 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-26 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-27 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-27 Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-28 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-28 Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-29 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-29 Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-30 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-30 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-31 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-31 Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-32 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-32 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-33 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-33 Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-34 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-34 119  Table 1-69 Table 1-70 Table 1-71 Table 1-72 Table 1-73 Table 1-74 Table 1-75 Table 1-76 Table 1-77 Table 1-78 Table 1-79 Table 1-80 Table 1-81 Table 1-82 Table 1-83 Table 1-84 Table 1-85 Table 1-86  Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-35 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-35 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-36 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-36 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-37 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-37 Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-38 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-38 Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-39 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-39 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-40 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-40 Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-41 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-41 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-42 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-42 Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-43 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-43 120  Table 1-87 Table 1-88 Table 1-89 Table 1-90 Table 1-91 Table 1-92 Table 1-93 Table 1-94 Table 1-95 Table 1-96 Table 1-97 Table 1-98 Table 1-99 Table I-100 Table I-101 Table 1-102 Table 1-103 Table 1-104  Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-44 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-44 Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-45 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-45 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-46 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-46 Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-47 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-47 Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-48 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silty Clay Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-48 Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-49 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-49 Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-50 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-50 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-51 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-51 Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-52 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-52 121  105 106 107 108 109 10 11 12 13 14 15 16 17 18 19 120 121 122  Predicted Saturation Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-53 Error in Saturation Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-53 Predicted Water Content Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-54 Error in Water Content Using the TP Model from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-54 Predicted Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-55 Error in Saturation Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-55 Predicted Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-56 Error in Water Content Using the Topp Equation from the Parallel E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-56 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-57 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-57 Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-58 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-58 Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-59 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-59 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-60 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-60 Predicted Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-61 Error in Saturation Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-61 122  • 123 -124 -125 •126 •127 -128 -129 -130 -131 -132 -133 -134 • 135 -136 -137 • 138 -139 -140  Predicted Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-62 Error in Water Content Using the TP Model from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-62 Predicted Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-63 Error in Saturation Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-63 Predicted Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-64 Error in Water Content Using the Topp Equation from the Perpendicular E M T Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-64 Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-65 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-65 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-66 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-66 Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-67 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-67 Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-68 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using TP Model to Determine the Dielectric Constant of Individual Errors 1-68 Predicted Saturation Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-69 Error in Saturation Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-69 Predicted Water Content Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-70 Error in Water Content Using the TP Model from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-70 123  Table 1-141 Table I-142 Table 1-143 Table 1-144  Predicted Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-71 Error in Saturation Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-71 Predicted Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-72 Error in Water Content Using the Topp Equation from the Ray Theory Average for Sand-Silt Loam Example Using Topp Equation to Determine the Dielectric Constant of Individual Errors 1-72  124  Appendix J Results from interpretation examples  The contents of Appendix J are available on the enclosed C D - R O M . Table J - l Table J-2 Table J-3 Table J-4 Table J-2 Table J-3  Depth, Lithology, Saturation, Dielectric Constant for Thin Layered Example J-l Average Dielectric Constants and Real and Estimated Average Saturations for Thin Layers J-7 Average Dielectric Constants and Real and Estimated Average Water Contents for Thin Layers J-7 Depth, Lithology, Saturation, Dielectric Constant for Thick Layered Example J-8 Average Dielectric Constants and Real and Estimated Average Saturations for Thick Layers J-9 Average Dielectric Constants and Real and Estimated Average Water Contents for Thick Layers J-9  125  

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