Electric characterization of sands with heterogenous saturation distribution Ainhoa G. Gorriti Electric characterization of sands with heterogenous saturation distribution Propositions belonging to the thesis: Electric characterization of sands with heterogenous saturation distribution 1. More than any other method, the Propagation Matrices method provides physical insight and a simple mathematical treatment of multisection transmission lines. (Chapters 3 and 5 of this thesis) 2. The frequency dependence of the permittivity of a mixture is not only determined by the permittivity and volume fractions of its components, but also by their geometrical distribution. (Chapter 6 of this thesis) 3. The arithmetic mean and the CRIM mixing law characterize the same layered media at two limits, the low and high frequency limits respectively. (Chapter 6 of this thesis) 4. True scientific progress is achieved only when physical experiments and theoretical developments drive each other. 5. Our fear of being wrong inhibits our creativity. 6. Many problems would not exist if people understood that nationalisms, patriotisms and faiths are circumstantial and not causal. 7. Contrary to what the “Real Academia de la Lengua Española” dictionary says, the present day meaning of the word “discusión” is an escalating argument rather than a discussion. This reflects part of the nature of the Spanish. 8. Bureaucracy is a vile steeplechase and most participants feel as if they joined the race with two broken legs! 9. Women liberation was not meant to get us out from behind vacuum cleaners to the cover of Hustler. 10. The meaning of life is to come to peace with its finiteness. These propositions are considered defendable and as such have been approved by the supervisors, Prof.dr.ir. J.T. Fokkema and Dr.ir. E.C. Slob Stellingen behorende bij het proefschijft: Elektrische karakterisering van zanden met heterogene saturatieverdelingen 1. Meer dan welke methode ook geeft de propagatie matrices methode fysisch inzicht en een eenvoudige wiskundige behandeling van aaneensgeschakelde transmissielijnen. (Hoofdstukken 3 en 5 van dit proefschrift) 2. De frequentie-afhankelijkheid van de permittiviteit van een mengsel wordt niet alleen bepaald door de permittiviteitwaardes en de afzonderlijke volume fracties van de componenten, maar ook door hun geometrische verdeling. (Hoofdstuk 6 van dit proefschrift) 3. Het arithmetrische gemiddelde en de CRIM mengregel karakteriseren hetzelfde materiaal voor twee limietgevallen, respectievelijk de laag- en hoog frequentie limiet. (Hoofdstuk 6 van dit proefschrift) 4. Werkelijke wetenschappelijke vooruitgang wordt slechts geboekt wanneer fysische experimenten en theoretische ontwikkelingen elkaar voortstuwen. 5. Onze angst het verkeerd te hebben smoort onze creativiteit in de kiem. 6. Veel problemen zouden niet bestaan wanneer mensen zouden begrijpen dat nationalismen, pattriottismen and geloofsovertuigingen een gevolg zijn van omstandigheden en geen oorzaak. 7. Tegensteld aan wat het woordenboek van de ”Real Academia de la Lengua Española” zegt, betekent het woord “discusión” in dagelijks gebruik eerder ”escalerende redetwist” dan ”discussie”. Dit is een afspiegeling van de Spaanse natuur. 8. Bureaucratie is een laaghartige hordenloop en voor de meeste deelnemers voelt het alsof ze met twee gebroken benen aan de race meedoen! 9. Vrouwenbevrijding was niet bedoeld om ons van achter de stofzuigers weg te krijgen naar de omslag van Hustler. 10. De betekenis van het leven is vrede te vinden in haar eindigheid. Deze stellingen worden verdedigbaar geacht en zijn als zonadig goedgekeurd door de promotors, Prof.dr.ir. J.T. Fokkema en Dr.ir. E.C. Slob Electric characterization of sands with heterogenous saturation distribution Proefschrift ter verkrijging van de graad van doctor aan de Technische Universtiteit Delft, op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College van Promoties, in het openbaar te verdedigen op 2 november 2004 te 15:30 uur door Ainhoa GONZÁLEZ GORRITI Licenciada en Ciencias Fı́sicas, especialización Fı́sica de la Tierra y el Cosmos Universidad Complutense de Madrid, geboren te Madrid, Spanje. Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. J.T. Fokkema Toegevoegd promotor: Dr.ir. E.C. Slob Samenstelling promotiecommissie: Rector Magnificus Prof.dr.ir. J.T. Fokkema Dr.ir. E.C. Slob Prof.dr. K. Holliger Prof.dr. A. Sihvola Prof.dr.ir. L. Ligthart Prof.dr. S.M. Luthi Dr. J. Bruining voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, toegevoegd promotor Swiss Federal Institute of Technology Helsinki University of Technology Technische Universiteit Delft Technische Universiteit Delft Technische Universiteit Delft Hans Bruining heeft in belangrijke mate bijgedragen aan de totstandkoming van dit proefschrift. c Copyright 2004 by A.G. Gorriti Cover design: Jesús G. Gorriti ISBN 90-9018709-X Printed in the Netherlands. A mi familia, en especial a mi abuelo. Y a Jara y a Ron, in memoriam. Financial support The research reported in this thesis is financially supported by the Netherlands Organisation of Scientific Research (NWO) under contract number 809.62.013, which support is gratefully acknowledged. Uno no es lo que es por lo que escribe, sino por lo que ha leı́do∗ . J.L. Borges (1899-1986) ∗ Traduction of the author: One is not what he is because of what he has written, but because of what he has read. CONTENTS ix Contents 1 Introduction 1 1.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scientific framework . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Basic electromagnetic equations 9 2.1 The electromagnetic field equations . . . . . . . . . . . . . . . . . 2.2 The constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 EM Field Equations in the Complex Frequency Domain . . . . . . . 16 3 Experimental design 9 23 3.1 Measuring the permittivity of dielectric materials . . . . . . . . . . 23 3.2 Different representations for a coaxial transmission line . . . . . . . 27 3.3 Γ and Υ: independent measurements . . . . . . . . . . . . . . . . . 34 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Tool for accurate permittivity measurements 37 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Technical design and measurement characteristics . . . . . . . . . . 39 4.3 Forward Model and Calibration . . . . . . . . . . . . . . . . . . . . 43 4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 x Contents 5 Reconstruction methods for permittivity from measured S-parameters 55 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Analytical vs Optimization . . . . . . . . . . . . . . . . . . . . . . 87 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6 Permittivity states of mixed-phase; two and three component sands 91 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Saturation technique and permittivities . . . . . . . . . . . . . . . . 93 6.3 Averaging ε∗12 and ε∗21 over averaging the S-parameters . . . . . . . 111 6.4 2-layer samples 6.5 Anomalies of layered samples . . . . . . . . . . . . . . . . . . . . . 117 6.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7 Conclusions and recommendations 143 A General solution for the wave equation in polar coordinates 147 A.1 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A.2 General solution for the wave equation . . . . . . . . . . . . . . . . 148 B Waveguides 151 B.1 Maxwell’s equations in polar coordinates . . . . . . . . . . . . . . . 151 B.2 Coaxial Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B.3 Coaxial-Circular Waveguide . . . . . . . . . . . . . . . . . . . . . . 160 Contents C Transmission Lines xi 163 C.1 Transmission Line Equations . . . . . . . . . . . . . . . . . . . . . 163 C.2 Local Reflection and Transmission coefficients . . . . . . . . . . . . 166 C.3 Line parameters for an ideal section . . . . . . . . . . . . . . . . . 168 C.4 Global Reflection and Transmission coefficients . . . . . . . . . . . 169 D Scattering Matrix 171 D.1 S-parameters and the total reflection and transmission coefficients of a transmission line . . . . . . . . . . . . . . . . . . . 172 D.2 Unitary property of the Scattering Matrix . . . . . . . . . . . . . . 173 E Permittivity models 175 E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 E.2 Classical approaches to pure materials . . . . . . . . . . . . . . . . 176 E.3 Bounds for the effective permittivity of mixtures . . . . . . . . . . . 178 E.4 Mixing Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 E.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Bibliography 185 Summary 191 Samevatting 195 Acknowledgements 199 About the author 201 xii Contents Twenty years from now you will be more disappointed by the things that you didn’t do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover. Mark Twain (1835-1910) Chapter 1 Introduction A proper understanding of the electrodynamic response of soils will lead to an improvement in the forward and inverse modelling of electromagnetic (EM) geophysical techniques. 1.1 Statement of the problem Electromagnetic waves have been beautifully described by J.C. Maxwell (1873). He related the material response to EM fields through constitutive parameters that are representative for the macroscopic electromagnetic properties of the media. These parameters are the electrical permittivity and conductivity, and the magnetic permeability. Constitutive parameters can be treated from two different points of view. On one hand the macroscopic and empirical one, in which case they are derived from appropriate experiments. On the other the microscopic and theoretical one, where they are derived from a microscopic model in which hypotheses about the interaction between elementary building blocks of matter are made and the constitutive parameters follow from an appropriate spatial averaging, (Hippel, 1954). Comparison between models and experiments should lead to a better understanding of these relations. As geophysicists we are interested in the imaging capabilities of electromagnetic waves. The goal is to disturb the subsurface of the Earth with EM fields and infer an image from the response to these fields. It is then essential to know how the materials that are commonly found in the subsurface of the Earth react to applied EM fields. In the literature, these materials have been studied from both the macroscopic and microscopic point of views, but in any case, models are validated through experimental results. 1 2 Chapter 1. Introduction We think that the best methodology to study rocks and soils is, not to have a preconceived idea about their microscopic properties, but to test them with appropriate experiments. That way we are probing the material at a reasonable macroscopic scale and no assumptions, apart from the constitutive relations defined by Maxwell, are needed. Taking into consideration the complexity of rocks and soils this is an advantage. In order to study the EM properties of rocks and soils, convenient experiments have to be performed. These have to be accurate and reliable. Furthermore, the materials are very complex as they are composed of several constituents and there is coexistence of phases (solid, liquid and gas). Due to this complexity, many different parameters can have an impact on the properties under study and they must be under control. Much of this thesis is, therefore, devoted to the design and calibration of a tool and methodology to study the response of these materials to an applied EM field, and to accurately reconstruct their permittivity values. We have restricted our study to the complex permittivity because most materials of geophysical interest are non-magnetic, while the conductivity is incorporated in the complex permittivity. From reliable and accurate experimental results theoretical models can be derived and existing ones can be validated. In fact, in the design of any experiment an a-priori model of the response of the material is needed (in our case, it would be the constitutive relations postulated by Maxwell) and it has to be verified. This has to be done with an accurate tool or otherwise errors could camouflage its validity. Currently, the computer modelling algorithms used in EM exploration discretise the Earth’s interior in blocks. The permittivity assigned to each block is generally the one that happens to lie beneath each grid point, or an average of the properties of the different materials coexisting in a specific block. But how this average should be taken is not clear in all situations; what happens in case of time-varying fields? How does this affect the averaging? On the other hand, most Mixing Laws compute permittivity from the volume fraction and permittivities of the constituents. Theoretically, they are valid for quasistatic fields and differ for different types of media. Nevertheless, they are generally applied over a much wider spectrum and the limitation of their application beyond the frequency for which they have been derived is not clear. It is also unclear how the different compositions of the same components and volume fractions lead to different results. With this thesis we want to validate the effective medium theories in the particular case of the electrodynamic response of soils. Existing experimental evidence does not favor any model in particular, which supports our questioning. 1.2. Scientific framework 3 We have chosen to investigate the permittivity of partially saturated sands when the same saturation is distributed differently. We also study its frequency dependency. In that way, very simple experiments can be used to test the relevance of the heterogeneities and the validity of considering the response of sands with a single parameter. 1.2 Scientific framework This section contains relevant literature that frames our work into the appropriate scientific context. This thesis deals with theoretical and experimental issues concerning permittivity, therefore, the literature is very varied and extensive. Distinct scientific disciplines study permittivity at different frequency ranges, depending on the scale of their interest. Even different nomenclatures are used; in the low frequency range it is referred to as dielectric constant, in optics as the complex refraction index and in telecommunication as the complex propagation factor. For simplicity, we only include the most relevant publications for our work and we have divided them into categories; theoretical models, experiments and validation of models through experiments. Theoretical models The beginning of the systematic investigation of the dielectric properties can be established with the works of Mossotti (1850) and Clausius (1879), as they attempted to correlate the dielectric constant with the microscopic structure of the materials. They considered the dielectric to be composed of conducting spheres in a non-conducting medium, and succeeded in deriving a relation between the dielectric constant and the volume fraction occupied by the conducting particles . Debye (1929), in the beginning of the 20th century, realized that some molecules had permanent electric dipole moments and that it gave rise to the macroscopic dielectric properties of the materials. He succeeded in extending the ClausiusMossotti theory to take into account the permanent moments of the molecules. His theory, later extended by Onsager (1936) and Kirkwood (1939), linked the dispersion of permittivity to the characteristic time needed for the permanent molecular dipoles to reorient following an alternating EM field. He deduced that the time lag between the average orientation of the moments and the field becomes noticeable when the frequency of the applied field is of the order of the reciprocal relaxation time. Debye’s model is still being used for polar liquids, where the dipoles are relatively far away from each other. However, the dielectric behavior of solids deviates significantly from Debye’s theory of relaxation. Cole & Cole (1941) pioneered the 4 Chapter 1. Introduction first approach to interpret the non-Debye relaxation of materials by means of a superposition of different relaxation times and Jonscher (1983) postulated that the relaxation behavior at molecular level is intrinsically non-Debye due to the cooperative molecular motions, the ”many-body interactions” approach. All these researchers modelled the macroscopic behavior of dielectric materials composed of a single phase and a single type of micro-particles, that is, they did not describe the electromagnetic properties of mixtures. However, most materials are indeed mixtures and researchers have struggled, and still do, to characterize their physical properties with effective parameters. The literature on the effective properties of materials and mixtures is very vast. Some comprehensive reviews are Wang & Schmugge (1980), Shutko & Reutov (1982), Dobson et al. (1985), Chelidze & Guguen (1999a), Sihvola (1999) and Choy (1999). Most of the models for the permittivity of mixtures only consider the volume fraction of the constituents and their permittivities. They are valid for macroscopic homogeneous and isotropic media and quasi-static fields. With the advent of numerical computing many theoretical studies are being published but they lack experimental corroboration. Experiments Each part of the spectrum has a specific physical measuring principle, and each principle has several techniques. In the microwaves the standing waves methods are used, and among these there are transmission line methods, wave guides, open resonators, closed cavities, etc. And every field of interest requires different setups for the specific characteristics of the samples; human tissues, seeds, polymers, wood, ceramics, composites, rocks, etc. The technique chosen depends on the frequency of interest and the sample requirements. Afsar et al. (1986) reviews the existing techniques for the microwave region. The permittivity of porous media is usually measured from very low frequencies up to the giga-Hertz region, where it becomes constant for most natural materials. This is also the range of interest for many field applications; geo-radar and Time Domain Reflectometry (TDR) in the shallow subsurface, and dielectric logging tools for petroleum reservoir characterization. In the low frequency regime, up to hundreds of mega-Hertz, the so-called direct methods are being used. Generally, the material is placed between two parallel plates and the impedance or admittance of this capacitor is measured so that the permittivity is calculated directly from these measured quantities, see for e.g. Shen et al. (1987) and Bona et al. (1998). 1.2. Scientific framework 5 In the high-frequency regime, the relation between the permittivity and the measured quantities is no longer linear for reasonably sized sample holders and more complex set-ups have to be implemented. For broad band measurements, reflection and/or transmission methods are used. Resonant-cavity methods are limited to a few frequency points and for medium- to low-loss samples. The reflection method gives good results for low- and medium-loss samples but not for high-loss samples, contrary to the transmission method that works best for relatively highloss samples. The S-parameter method, combines the reflection and transmission methods and overcomes the disadvantages of both. It has become very popular in the study of dielectric properties. It is common to place the material in a coaxial transmission line (Shen (1985) and Nguyen (1998)), or in a coaxial-circular wave guide (Taherian et al. (1991)) and measure the S-parameters of the set-up with a Network Analyzer. The reconstruction of permittivity from measured S-parameters can be done analytically or via an optimization procedure. Analytical methods compute the electrical permittivity from analytical expressions involving the S-parameters. Until now there were no straight forward methods, the S-parameters had to be compensated, in one way or an other (Rau & Wharton (1982), Kruppa & Sodomsky (1971), Freeman et al. (1979), Shen (1985) and Chew et al. (1991)). Then different reconstruction formulas were used: Nicolson & Ross (1970), Weir (1974) and Stuchly & Matuszewsky (1978). In this thesis we develop a novel method, the Propagation Matrices Method, that reconstructs the permittivity from the measured S-parameters of the tool. It provides a representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler than existing methods. It also enables us to show how all reconstruction formulas correspond to the same method, and, therefore, suffer from the same instabilities. The EM properties of the sample can also be computed via an optimization procedure. Belhadj-Tahar et al. (1990) and Taherian et al. (1991) applied this method to a coaxial-circular waveguide, and we have used it for our measurements. In this thesis we compare all existing reconstruction methods. Validation of models through experiments Different researchers have compared measured permittivities of different soils, but their results are mainly qualitative and certainly not conclusive. In the high frequency band, the following researchers have published studies over a broad variety of soils, different water contents and temperature: Shutko & Reutov (1982), Hallikainen et al. (1985) Dobson et al. (1985) and Chelidze & Gueguen (1999b). To our knowledge, up to this thesis, there has not been any study of heterogeneously saturated samples. 6 Chapter 1. Introduction Permittivity states of soils Introduction 2 Basic electromagnetic equations ' 3 Experimental design Technical design and measurement characteristics Forward model and calibration Reconstruction Methods % ' Analytical ' Optimization % % Permittivity states of mixe-phase; 2 and 3 component mixtures ' ' ' 7 Saturation Techniques 2-layer samples Interpretation of results Conclusions and Recommendations Figure 1.1: Schematic outline of the thesis. Evidence of permittivity states 6 Resonant frequencies Propagation Matrices Compensation Parameters calibrated coax TL + forward and inverse models Tool for accurate permittivity measurements ' ' 5 Representations for coaxial transmission lines The Propagation Matrices representation TOOL = ' ' 4 EM field equations Preliminaries 1 1.3. Outline of the thesis 1.3 7 Outline of the thesis A schematic structure of this thesis is shown in Figure 1.1. This introductory chapter is followed by the basic electromagnetic equations, presented in Chapter 2. It contains Maxwell’s theory of propagation of EM waves and the constitutive relations. The next three chapters deal with the design and calibration of the tool to probe the electrodynamic response of sands and the reconstruction of permittivity. Chapter 3 reviews various experimental designs for measuring the permittivity of soil samples and justifies our choice for a coaxial transmission line. We also introduce a new notation (Propagation Matrices) to describe these set-ups. This notation provides a representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler than existing methods. It also enables us to prove the independence of the transmission and reflection measurements. The characteristics and calibration of our specific tool are presented in Chapter 4. It is a customized coaxial transmission line that allows for fluid flow through the sample, and whose S-parameters can be determined in the frequency range from 300MHz to 3GHz. We show the calibration needed for the forward model to be in very good agreement with the measured data, and its high sensitivity (it can detect relative changes in permittivity in the order of 1%). Chapter 5 contains the comparison of existing reconstruction techniques and the novel one based in the Propagation Matrices. All the reconstruction methods lead to a permittivity value per frequency without a pre-defined model of frequency dependency. We show the advantages and disadvantages of these reconstructions (analytical and optimized) and how we can use the tool as a resonator for low-loss materials. The Propagation Matrices Representation shows again valuable since its analytical inversion identifies all existing inversions equal to either one of the two fundamental and independent solutions. This explains why indistinctively of the expression used, the reconstruction of permittivity from analytical expressions always suffers from the same problems at resonant frequencies and with low-loss materials. We successfully reconstruct the relative permittivity of air within ±1% error. After this thorough study of the tool, in Chapter 6 we investigate the effect of the distribution of constituents with measured and modelled experiments. We perform experiments with sands composed of loose quartz grains; dry, partially and fully saturated. The saturation technique has an effect in the reconstructed permittivity and extra care is needed for proper measurements. We perform gravity drainage experiments so that the same sample can be measured at different saturation levels. 8 Chapter 1. Introduction Samples saturated in that way result in distinctive 2-layer samples, whose reconstructed permittivity exhibit anomalies due to the interface. These samples can be represented by a 2-layer model very accurately, if the width of the layers is found from the phase of the transmission coefficients. Hence, we can use the tool to monitor the movement of the fluid front. With the aid of numerical experiments, we show that the distribution of layers has a strong impact on the permittivity, and that heterogeneities as small as λ/100 are detectable. We also compare our experiments with existing Mixing Laws and find the best mixing formula for partially and fully homogeneously saturated sands to be a Power Law equation. However, depending on the saturation level and whether we are interested in the real or imaginary parts of the permittivity, different exponents are needed. We think that our results on multilayered samples can explain the variety of exponents encountered in this thesis and in the existing literature. Depending on the desired precision of the model, 2-layer samples can be modelled via a Power Law with three constituents (permittivity and volume fraction of solid grains, air, water) or two (permittivity and volume fraction of the two layers). The first is less accurate and requires different exponents and the second one is very accurate and uses a single exponent, plus it has a clear physical sense. In Chapter 6 we also present numerical and experimental examples that contradict the theory derived by del Rio & Whitaker (2000a). Finally, we conclude this thesis with Chapter 7 where we also make recommendations for future research. We also include five appendices that contain relevant equations and derivations. Appendix A shows a general solution of the EM wave equation in cylindrical structures, and Appendix B presents the solution of these waves in coaxial and circular waveguides. It also contains information on how to modify the tool to measure solid cores, in a combination of coaxial-circular waveguide. Appendix C shows the equations of wave propagation through a transmission line; the reformulation of Maxwell’s wave equation in terms of circuit parameters and voltage and current waves. And Appendix D gives a short introduction to the representation of 2-port networks with scattering matrices. The last appendix, (Appendix E) is a short summary of the theoretical models that try to explain the EM behavior of materials and mixtures. From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. Richard P. Feynman (1918-1988) Chapter 2 Basic electromagnetic equations In this chapter the theoretical background of electromagnetic waves propagating through confined regions is presented. We introduce Maxwell’s equations, the constitutive relations and the generalized wave equations. 2.1 The electromagnetic field equations We use the vector notation and to specify position, we employ the coordinates {x, y, z} with respect to a Cartesian reference frame with origin O and three mutually perpendicular base vectors {x̂, ŷ, ẑ} of unit length each. In the indicated order, the base vectors form a right-handed system. The time coordinate is denoted by t and the position is specified by the vector x = xx̂ + yŷ + zẑ. The electromagnetic phenomena under study occur in a determined domain D in IR3 . The filling material of D is characterized by electric parameters, in the case of free space by the electric permittivity of free space ε0 and magnetic permeability µ0 . Together they determine the velocity of light in free space as c0 = √ 1 def = 299792458[m/s]. ε0 µ0 The value of c0 has been defined, while the value of the free space magnetic permeability is fixed and given by µ0 = 4π × 10−7 [H/m]. Hence the exact value for the free space permittivity must be determined from these two. It is given by ε0 = 1 ≈ 8.854 × 10−12 [F/m]. µ0 c20 9 10 Chapter 2. Basic electromagnetic equations When the medium is different from free space but linear, isotropic, locally reacting and time-invariant, the electric permittivity, ε(x, t), the conductivity σ(x, t) and the magnetic permeability µ(x, t) may depend on position due to inhomogeneities of the medium and on the history of the applied fields in the medium (relaxation processes in time) The electromagnetic field is characterized by two field quantities: E H is the electric field strength [V/m], is the magnetic field strength [A/m], and the action of sources is represented by Je Ke 2.1.1 is the source volume density of electric current [A/m2 ], is the source volume density of magnetic current [V/m2 ]. Basic equations for electromagnetic fields We start with the basic equations for electromagnetic fields in vacuum, where they assume their simplest form. Sources of electromagnetic field are essentially composed of matter and are therefore introduced in the volume densities of electric and magnetic currents that describe the electromagnetic action of matter. The induced parts of these volume densities then describe the reaction of a piece of matter to an electromagnetic field. − ∇ × H + ε0 ∂t E = −J mat , ∇ × E + µ0 ∂t H = −K mat (2.1) , (2.2) In a vacuum domain, the material volume densities {J mat , K mat } are zero-valued. In the presence of matter, we distinguish between the active part and the passive part. The active part describes the external source behavior, that generates the field. The volume densities of electric and magnetic currents are denoted J e and K e , respectively. The induced, or passive, part describe the reaction of matter to the presence of an electromagnetic field and are generally field dependent. They are denoted J i and K i , respectively, J mat = J i + J e J i = J + ∂t P K mat = K i + K e , (2.3) K i = µ0 ∂t M , (2.4) 2.1. The electromagnetic field equations 11 where J P M is the volume density of electric current [A/m2 ], is the electric polarization [C/m2 ], is the magnetization [A/m]. Further it is customary to introduce the quantities D = ε0 E + P , (2.5) B = µ0 (H + M ). (2.6) where D B is the electric flux density [C/m2 ], is the magnetic flux density [T]. Upon substituting equations (2.5) and (2.6) into equations (2.1) and (2.2), we arrive at the two Maxwell equations in matter, − ∇ × H + J + ∂t D = −J e , e ∇ × E + ∂t B = −K , (2.7) (2.8) From Maxwell’s equations two other equations can be derived that are not independent of Maxwell’s equations, the compatibility equations. This means that field quantities that satisfy Maxwell’s equations by definition also satisfy the compatibility equations. Compatibility equations are obtained by applying the divergence operator (∇·) to equations (2.7) and (2.8), respectively. This results in, ∇ · (J + ∂t D) = −∇ · J e , e ∂t ∇ · B = −∇ · K . (2.9) (2.10) Historically, the volume density of electric charge is introduced as, ρ = ∇ · D. (2.11) Obviously, from the above equations it is clear that there is a relation between charge and electric current. The relation between charge and current densities is clear if we combine equations (2.9) and (2.11) and assume that there are no sources of electric current present ∇ · J + ∂t ρ = 0, known as the charge conservation law. (2.12) 12 2.2 Chapter 2. Basic electromagnetic equations The constitutive relations The constitutive relations provide information about the environment in which electromagnetic fields occur. It is customary to relate the quantities {J , D, B} to {E, H} through the constitutive parameters, which are representative for the macroscopic electromagnetic properties of the media. For general media, which are linear, locally reacting, time invariant, instantaneously reacting and isotropic, the constitutive parameters are scalars. If the medium is inhomogeneous the coefficients change with position, therefore their dependence on x, but if the medium is homogeneous they become constants. J (x, t) = σ(x)E(x, t), (2.13) P (x, t) = χe (x)E(x, t), (2.14) M (x, t) = χm (x)H(x, t), (2.15) where σ χe χm is the electrical conductivity [S/m], is the electric susceptibility [-], is the magnetic susceptibility [-]. Substituting equations (2.14) and (2.15) into equations (2.5) and (2.6) give the relations between {D, B} and {E, H} D(x, t) = ε(x) E(x, t) = ε0 εr (x)E(x, t) (2.16) B(x, t) = µ(x)H(x, t) = µ0 µr (x)H(x, t), (2.17) where ε εr µ µr is is is is the the the the (absolute) permittivity [F/m], relative permittivity [-], (absolute) permeability [H/m], relative permeability [-], and can be expressed as ε = ε0 (1 + κe ) or εr = 1 + κe , (2.18) µ = µ0 (1 + κm ) or µr = 1 + κm , (2.19) 2.2. The constitutive relations 13 where κe κm is the relative electric susceptibility [-], is the relative magnetic susceptibility [-]. We restrict ourselves to variations in the electric and magnetic parameters, such that they are piecewise constant functions of position. Then substitution of equations (2.13)-(2.17) to obtain Maxwell’s equations in the electric and magnetic field strengths only, gives − ∇ × H + σE + ε∂t E = −J e , e ∇ × E + µ∂t H = −K , (2.20) (2.21) And the compatibility relations read σ∇ · E + ε∂t ∇ · E = −∇ · J e , e µ∂t ∇ · H = −∇ · K . (2.22) (2.23) Only causal solutions of the differential equations (2.20) and (2.21) are acceptable from a physical point of view. Assuming that the sources start to act at the instant t = 0, causality of the time behavior of the electromagnetic field is then ensured by putting the field values of E and H equal to zero prior to t = 0. Constitutive parameters can be treated from two different points of view. On one hand the macroscopic and empirical one; they are then derived from appropriate experiments. And on the other the microscopic and theoretical one; in this case they are derived from a microscopic model in which hypotheses about the interaction between elementary building blocks of matter are made and the constitutive parameters follow from an appropriate spatial averaging. Comparison between models and experiments should lead to a better understanding of these relations. Material with relaxation The constitutive parameters of a material with relaxation have to show the effect of this type of causal behavior. Assuming a linear, time invariant, locally reacting and isotropic media the relations between {J , P , M } and {E, H} are replaced by a time convolution 14 Chapter 2. Basic electromagnetic equations J (x, t) = P (x, t) = Z t t0 =0 Z t t0 =0 M (x, t) = Z t t0 =0 σ(x, t0 )E(x, t − t0 )dt0 , (2.24) χe (x, t0 )E(x, t − t0 )dt0 , (2.25) χm (x, t0 )H(x, t − t0 )dt0 , (2.26) where {t0 ∈ R; t0 > 0} and σ conduction relaxation function [S/ms], χe relaxation function [s−1 ], χm magnetic relaxation function [s−1 ]. Expressing the constitutive relations by a time convolution, it is mathematically taken into account the fact that the values of the fields {E, H} between the instant t − t0 (cause) and the instant t (effect) contribute to the values of {J , P , M }. The present reaction of the material is influenced by the history of the electromagnetic fields. Boltzmann (1876) was the first one to use time convolutions to mathematically describe mechanical relaxation processes in the study of solid deformations. However, the fact that the relaxation process is time invariant requires the medium to return to its original state after some time so there is no permanent deformation of state, and thus, this is only valid for small disturbances, as was already realized by Boltzmann. Since the electric field (E) and volume density of electric current (J ) are related by measurable quantities, J = σE and E = %J , (2.27) the conductivity (σ) by measuring J for an applied E and the resistivity (%) by measuring E for an applied J , both σ and % (σ = 1/%) should be causal time-functions. They cannot be minimum-phase functions, because it would not represent a strictly passive medium, which is required for natural media in thermodynamic equilibrium. Their real and imaginary parts should however obey the Kramer-Kroning causality relations, discussed in Subsection 2.4.1. The same argument holds for χe and χm . The relations between {D, B} and {E, H} read D(x, t) = ε0 E(x, t) + ε0 Z t t0 =0 B(x, t) = µ0 H(x, t) + µ0 Z κe (x, t0 )E(x, t − t0 )dt0 , t t0 =0 κm (x, t0 )H(x, t − t0 )dt0 . (2.28) (2.29) 2.3. Boundary conditions 2.3 15 Boundary conditions Upon crossing the interface of two adjacent media that differ in their constitutive parameters, the electric and magnetic field strengths will in general vary discontinuously. Since all physical quantities have bounded magnitudes, the relevant discontinuities are restricted to finite jump discontinuities. Because of their discontinuous behavior, the electric and magnetic field strengths are no longer continuously differentiable in a domain containing (part of) an interface, and therefore equations (2.7) and (2.8) cease to hold. Since we assume time invariance for the properties of the media, the non-differentiability is restricted to the dependence on the spatial derivatives. The electromagnetic field equations must therefore be supplemented by conditions that interrelate the field values at either side of the interface, the so-called boundary conditions. Let S denote the interface and assume that S has a unique tangent plane everywhere. Let ν denote the unit vector along the normal to S such that upon traversing S in the direction of ν we pass from the domain D(1) to the domain D(2) , D(1) and D(2) being located on either side of S (see Fig. 2.1). Figure 2.1: Interface between two media with different electromagnetic properties. The partial derivatives perpendicular to S meet functions that show a discontinuity across S, which would lead to interface Dirac distributions located on S. These would physically represent impulsive interface sources. In the absence of such sources, the absence of such interface impulses in the partial derivatives across S must be enforced. Looking at Maxwell’s equations we see that the tangential components are differentiated in the direction normal to the interface, so impose the following boundary conditions: ν ×H ν ×E is continuous across S, is continuous across S, (2.30) (2.31) and looking at the compatibility equations we see that the normal components 16 Chapter 2. Basic electromagnetic equations are differentiated in the direction normal to the interface and hence we have the conditions: ν · (J + ∂t D) ν ·B is continuous across S, is continuous across S. (2.32) (2.33) These conditions allow the continuous exchange of electromagnetic energy between the two domains across the interface S. Notice that equations (2.32) and (2.33) are not independent conditions, they are conditions corresponding to the compatibility equations (2.9)-(2.10), which are themselves direct consequences of the Maxwell equations (2.7)-(2.8), for electromagnetic fields. At the surface of an electrically impenetrable object (it cannot sustain in its interior a non-identically vanishing electric field while the boundary condition of the continuity of the tangential part of the electric field strength on its boundary surface ∂D is maintained) a boundary condition of the explicit type can be given lim ν × E(x + hν, t) = 0 h↓0 for any x ∈ ∂D, (2.34) Electrically impenetrable materials arise as limiting cases of materials whose conductivity and/or permittivity go to infinity. 2.4 The electromagnetic field equations in the complex frequency domain To obtain the Maxwell equations in the frequency domain a Laplace transformation with respect to time is carried out. Time Laplace transformation Switching on the sources at the instant t = 0 the time domain T in which the source affects the field is defined as T = {t ∈ R; t > 0}. (2.35) The complement T 0 of the domain T and the boundary ∂T between the two domains are defined according to T 0 = {t ∈ R; t < 0}, ∂T = {t ∈ R; t = 0}. (2.36) (2.37) 2.4. EM Field Equations in the Complex Frequency Domain 17 The characteristic function χT (t) of the set T is introduced as χT (t) = {1, 1/2, 0} when t ∈ {T , ∂T , T 0 } , (2.38) and the Laplace transform of a function f (x, t) in space-time, defined for t ∈ T , is fˆ(x, s) = Z exp(−st)χT f (x, t)dt, (2.39) t∈R where s is the Laplace transformation parameter, and must satisfy the condition Re(s) > 0. The inverse Laplace transformation can be carried out explicitly by evaluating the Bromwich integral in the complex s plane: 1 2πj Z s0 +j∞ exp(st)fˆ(x, s)ds = χT (t)f (x, t). (2.40) s=s0 −j∞ The path of integration is parallel to the imaginary s-axis in the right half of the complex s-plane where fˆ is analytic. Symbolically equation (2.39) is written as fˆ(x, s) = Lf (x, t), (2.41) and the Laplace transform of a partial differentiation of a function, ∂t f (x, t), equals L∂t fˆ(x, s) = sf (x, t), (2.42) assuming zero initial conditions. 2.4.1 Constitutive parameters: Kramers-Kronig causality relations Considering a medium with relaxation we can carry out the Laplace transformation of equations (2.24)-(2.29) and obtain Jˆ(x, s) = σ̂(x, s)Ê k (x, s), (2.43) D̂(x, s) = ε̂(x, s)Ê k (x, s), (2.44) B̂(x, s) = µ̂(x, s)Ĥ j (x, s), (2.45) 18 Chapter 2. Basic electromagnetic equations where σ̂ = Z ∞ exp(−st)σc (x, t0 )dt0 , (2.46) t0 =0 ε̂ = ε0 (1 + κ̂e ) Z κ̂e (x, s) µ̂ = µ0 (1 + κ̂m ) = Z κ̂m (x, s) = ∞ exp(−st)χe (x, t0 )dt0 , (2.47) exp(−st)χm (x, t0 )dt0 . (2.48) t0 =0 ∞ t0 =0 The complex frequency-domain conduction, electric and magnetic relaxation functions are the Laplace transforms of causal functions of time. Therefore their real and imaginary parts for imaginary values of s = jw, with w ∈ R, satisfy the Kramers-Kronig causality relations (see de Hoop, 1995) For a general relaxation function κ̂(x, s) = Z ∞ exp(−st)κ(x, t0 )dt0 , (2.49) t0 =0 for s = jw it can be separated into its real κ̂0 and imaginary κ̂00 parts κ̂(x, jw) = κ̂0 (x, w) − jκ̂00 (x, w) for w ∈ R . (2.50) The Kramers-Kronig causality relations are given by κ̂0 (x, w) = ∞ κ0 (x, w0 ) 0 dw for w ∈ R , 0 w=−∞ w − w 1 ∞ κ00 (x, w0 ) 0 dw for w ∈ R , π w=−∞ w0 − w κ̂00 (x, w) = − 1 π Z Z (2.51) (2.52) and imply that κ0 and κ00 form pairs of Hilbert transforms. Thus, whether the non-dissipative part κ0 is known experimentally or it is determined by some theory, it is possible to construct the dissipative part κ00 by taking the Hilbert transform. Similarly, if the response of κ00 is known as a function of frequency, then the Hilbert transform can infer a constraint on κ0 . Although this property is not used in this thesis, it allows to control complex effective medium models, since they must obey these relations. It also shows the strong coupling of the real and imaginary parts of the permittivity. From the decomposition of the Laplace transformed relaxation function into its real and imaginary parts it follows that κ0 is and even function of w and κ00 is an odd function of w 2.4. EM Field Equations in the Complex Frequency Domain κ0 (x, −w) = κ0 (x, w) 00 00 κ (x, +w) = −κ (x, w) for all w ∈ R , for all w ∈ R . 19 (2.53) (2.54) The inverse of κ, 1/κ, must also be causal and obey the Kramers-Kronig causality relations. They cannot be minimum phase functions when they represent strictly passive or active media for all frequencies. 2.4.2 Maxwell equations in the frequency domain The Maxwell equations in the frequency domain for linear, locally reacting, time invariant and isotropic media are obtained Laplace transforming equations (2.20) and (2.21) e − ∇ × Ĥ + η̂ Ê = −Jˆ , e ∇ × Ê + ζ̂ Ĥ = −K̂ , (2.55) (2.56) and the compatibility equations (2.9) and (2.10) e ∇ · (η̂ Ê) = −∇ · Jˆ , e ∇ · (ζ̂ Ĥ) = −∇ · K̂ , (2.57) (2.58) where η̂ = σ̂ + sε̂ is the transverse admittance per length of the medium, or total conductivity ζ̂ = sµ̂ is the longitudinal impedance per length of the medium When the medium is instantaneously reacting the constitutive coefficients σ̂, ε̂ and µ̂ are independent of s. Assuming a homogeneous medium, applying the curl (∇×) to equation (2.55) and combining it with equation (2.57) we end up with a second order differential equation for the electric field, and similarly for the magnetic field 20 Chapter 2. Basic electromagnetic equations 1 e e e ∇2 Ê − γ̂ 2 Ê = ∇ × K̂ + ζ̂ Jˆ − ∇(∇ · Jˆ ), η̂ (2.59) 1 e e e ∇2 Ĥ − γ̂ 2 Ĥ = −∇ × Jˆ + η̂ K̂ − ∇(∇ · K̂ ), ζ̂ (2.60) γ̂ 2 = η̂ ζ̂. (2.61) where Both equations have the same structure and are known as wave equations. Maxwell’s equations determine then, the propagation and the form of the electromagnetic waves. Of course, this is very general, and to obtain the exact form of the waves we have to impose the source characteristics as well as the boundary conditions. In the rest of the thesis, we restrict ourselves to the special case of propagation along transmission lines. For that purpose it is easier to work in the angular-frequency domain, where the Laplace parameter s = jw and the transverse admittance and the longitudinal impedance of a medium with conductive, and dielectric and magnetic losses are now η̂(jw) = jwε̂∗ , (2.62) ∗ (2.63) ζ̂(jw) = jwµ̂ , where ε̂∗ = ε̂0 − j(ε̂00 + σdc w ) is the complex permittivity, 0 ε̂0 = ε0 ε̂r is the real part of the permittivity, or dielectric constant, 00 ε̂00 = ε0 ε̂r is the imaginary part of the permittivity, or dielectric loss, and the term ε̂00 + σwdc accounts for dielectric and conductive losses, µ̂∗ = µ̂0 − j µ̂00 is the complex permeability, ε̂∗ and µ̂∗ cannot be minimum phase functions if they have to represent strictly passive or active media for all frequencies, therefore ε̂0 > 0 0 µ̂ > 0 and and ε̂00 ≥ 0, 00 µ̂ ≥ 0, (2.64) (2.65) 2.4. EM Field Equations in the Complex Frequency Domain 21 and we choose the positive branch of the square roots, so that <(γ̂ ≥ 0). (2.66) For non-magnetic materials (µ̂∗ = µ0 ) the propagation constant becomes σdc w2 0 00 γ̂ = −w ε̂ µ̂ = − 2 ε̂r − j ε̂r + , wε0 c0 2 2 ∗ ∗ (2.67) and the Maxwell’s equations (2.55) and (2.56), the compatibility equations (2.57) and (2.58), and the wave equations (2.59) and (2.60) remain the same, taking into account the new expressions for the medium parameters. The propagation constant in free space is then γ̂02 = −w2 ε0 µ0 = − w2 , c20 (2.68) It is possible to define a source-free domain when the source field at the boundary of the domain is known. Then the problem reduces to solving Maxwell’s equations in a source-free domain ∇2 Ê − γ̂ 2 Ê = 0, 2 2 ∇ Ĥ − γ̂ Ĥ = 0. (2.69) (2.70) Along the thesis, the hat that indicates that we are in the frequency domain has been removed for simplicity in the notation. 22 Chapter 2. Basic electromagnetic equations Problems worthy of attack prove their worth by fighting back. Paul Erdos (1913-1996) Chapter 3 Experimental design This chapter is introduced with a short discussion on the very broad field of permittivity measurements, followed by a more specific one devoted to porous media and the reasoning for choosing a coaxial transmission line as the tool for the measurements present in this thesis. It also includes an introduction to the notation we have developed to treat this type of configurations and several characteristics of the measurements. 3.1 Measuring the permittivity of dielectric materials Permittivity is the property that determines the behavior of dielectric materials to an applied EM field, as seen in Chapter 2. Any non-metal, and even metals as a limiting case, can be considered dielectrics. The permittivity is, then, the key to understand the behavior and composition of matter from an EM point of view, and therefore, to many applications. The microscopic arousal of this property is not the topic of this thesis and it can be found in any elementary electromagnetic text book. I would specially recommend von Hippel (1954), as he presents most of the existing microscopic phenomena and their theories in a very comprehensive way. Distinct scientific disciplines study this property at different frequency ranges, depending on the scale of their interest. Even different nomenclatures are used, in the low frequency range it is referred as the dielectric constant, in optics as the complex refraction index and in telecommunication as the complex propagation factor. We will stick to the complex permittivity, since that is how it appears in Maxwell equations, and no matter what part of the spectrum you deal with, they are the governing equations of any electromagnetic process. Measuring at different frequency ranges involves the use of different experimental techniques. From direct current to, approximately 1 MHz, bridges and resonant cir23 24 Chapter 3. Experimental design cuits are used. From hundreds of MHz up to 1011 Hz we are in the microwave region. The microwaves, from decimeter to millimeter wavelengths, are the waves of the spectrum with more scientific and engineering applications. These include, among others, medical (tissue characterization for imaging), agriculture (seeds health, root intake), food and pharmaceutical industries, electrical engineering (telecommunication, circuit components, etc), and of course, geophysics (logs, soil saturation, geo-radar). At these frequencies standing waves methods of measurements are used. Afsar et al. (1986) review the existing techniques for the microwave region. From the infrared to the ultraviolet ([1012 − 1016 ] Hz) we are in the geometrical optics field and permittivity is determined by reflection and transmission measurements. Molecular chemists master the upper end of the microwaves and the infrared since their wavelengths are comparable to the size of the molecules. Above 1016 Hz the size of the atoms and the molecules and their separation become comparable to the incident wavelength, we are in the X-ray region and interference techniques are used by solid physicists to study the fundamental material properties, which often are tensors. CT-scans also operate in this range. Further up, we enter the Gamma ray region (above 1019 Hz), where quantum effects are of relevance and particle physicists try to reveal the fundamentals of the atoms in particle accelerators. Each part of the spectrum has a specific physical measuring principle, and each principle has several techniques. For example, in the microwaves the standing waves methods are used and among these there are transmission line methods, waveguides, open resonators, closed cavities, etc, and every field of interest requires different set-ups for the specific characteristics of the samples; human tissues, seeds, polymers, wood, ceramics, composites, rocks, etc. The technique chosen will then depend on the frequency of interest and the sample requirements. Multi-frequency or single frequency measurements will also affect this selection. In the next section we justify our choice. 3.1.1 Measuring the permittivity of porous media The permittivity of porous media is usually measured from very low frequencies up to the giga-Hertz region, where it becomes constant for most natural materials. This is also the range of interest for many field applications; geo-radar and Time Domain Reflectometry (TDR) in the shallow subsurface, and dielectric logging tools for petroleum reservoir characterization. On the low frequency regime, up to hundreds of mega-Hertz, the so-called direct methods are being used. Generally, the material is placed between two parallel plates and the impedance or admittance of this capacitor is measured so that the permittivity is calculated directly from these measured quantities, see for e.g. Shen et al. (1987) and Bona et al. (1998). 3.1 Measuring the permittivity of dielectric materials 25 Most theories modelling the EM behavior of porous media are developed for static fields (see Appendix E). EM fields can be considered quasi-static below 1 MHz. However, most field applications operate on the decimeter spectrum of the microwaves (0.3-3 GHz), and the static fields theories are being used in many applications, since on the high frequency regime, there are yet no theoretical models to explain and predict the behavior of the media. We think that this extrapolation must have its limitations as the particles present in the medium cannot behave in the same way when a static field or a rapidly alternating one is applied. But, this interaction is very complex and it is almost impossible to theoretically model it. We will instead, study the limitations of such an extrapolation from an experimental point of view in Chapter 6. First we need to obtain reliable permittivity measurements, which is already a difficult task and to which this first part of this thesis is devoted. On the high-frequency regime, the relation between the permittivity and the measured quantities is no longer linear for reasonably sized sample holders and more complex set-ups have to be implemented. For broad band measurements, reflection and/or transmission methods are used. Resonant-cavity methods are limited to a few frequency points determination and for medium to low-loss samples. The reflection method gives good results for low and medium-loss samples but not for high-loss samples, contrary to the transmission method that works best for relatively high-loss samples. The S-parameter method, combines the reflection and transmission methods and overcomes the disadvantages of both. It has become very popular in the study of dielectric properties. It is common to place the material in a coaxial transmission line, Shen (1985) and Nguyen et al. (1998), or coaxial-circular wave guide, Taherian et al. (1991), and measure the S-parameters of the set-up with a Network Analyzer. Since we want to study the behavior of the permittivity of sands at different degrees of saturation with different fluids in the microwave part of the spectrum for application purposes, we need broad band measurements and a standing wave device. We chose a coaxial transmission line, where the propagation of the TEM mode only is ensured, to simplify the modelling and for better accuracy. The tool is connected to the two ports of an ANA (Automated Network Analyzer) that makes sweeps in frequency, and measures the response of the tool from 300 KHz to 3 GHz on both ports, determining its full S-parameters matrix. The obtention of the permittivity from S-parameters measurements is discussed extensively in Chapter 5, but here, I would like to introduce the three main theoretical representations for this type of measurements; full wave, transmission line and 2-port networks, plus a new approach, a combination of the transmission line representation with the traditional method of propagation matrices. This new representation has a clear advantage above the other three as we will show in the next section. 26 Chapter 3. Experimental design Figure 3.1: Equivalent representations of a sequence of dielectric layers traversed by TEM waves. 3.2. Different representations for a coaxial transmission line 3.2 27 Different representations for a coaxial transmission line In the literature different authors have approached the problem of propagation of EM waves along coaxial transmission lines differently. Here is a summary of the three main traditional representations plus the new one we developed. Full wave representation This approach is very convenient for waveguides in which the sole propagation of TEM waves cannot occur, either by its shape or by the frequency of operation. It is then necessary to model all the modes propagating along the guide solving Maxwell equations for the particular set-up, and to impose boundary conditions on the interfaces of the different sections. In Appendix B we present their solution for a coaxial waveguide together with the cut-off frequencies for the different modes. The fields are represented by series expansions. Belhadj-Tahar et al. (1990) and Taherian et al. (1991) use this representation as they model the response of a coaxial-circular waveguide. Each section of the line is characterized by its EM properties, the electrical permittivity ε∗ and the magnetic permeability µ∗ , see Figure 3.1a. In contrast, when the propagation of only the TEM mode can be ensured, it is simpler to treat the coaxial waveguide using characteristics of lumped element equivalent circuits. Transmission Line representation In a coaxial waveguide operated below cut-off only the TEM mode propagates and it is then possible to express the waves in terms of voltage and current. Full treatment and equations can be found in Appendix C. Each section of the line is characterized by its impedance Zn and propagation constant γn , and they are determined by the transmission line parameters, see Figure 3.1b. As explained in Appendix C, if such a representation is chosen, recursive expressions for the total reflection and transmission of the line can be computed via equations (C.30) and (C.31). 28 Chapter 3. Experimental design 2-port network: Scattering matrix representation Electrical engineers use the scattering matrix ([S]) representation of 2-port networks to relate the reflected voltage (V − ) to the incident voltage (V + ) on circuit components with two distinct ports or connections, V − = [S]V + . It is a way to describe the reflection/transmission response of electrical components and their interaction, see Appendix D. An N-multisectional transmission line can be treated as N 2-port networks connected in series (Figure 3.1c) and each section is represented by its own scattering matrix [Sn ]. The total response of the line is obtained by the Redheffer’s star product (Redheffer, 1961) of the individual matrices. Shen (1985) used this approach to compute the permittivity of soil samples. We discuss it in more detail in Subsection 5.2.4. These three representations provide the same solution with a different set of equations. In a multisection transmission line, the sample is placed in a single section and it is the properties of the sample that we would like to obtain from those equations. The simpler the equations the smaller the propagation of error through numerical computations. Moreover, the representation in which the interactions of the sections is clear is most desirable for a thorough understanding. However, these three representations provide rather cumbersome expressions, series expansions for the full wave, recursive equations for the transmission lines and tedious Redheffer’s star products for the 2-port network. It is then, relatively easy to loose track of the section interactions and the relevant features. With this in mind, we have developed a new representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler. Propagation Matrices: a combined representation In physics it is common to represent the down-going and up-coming wave fields in a layered medium in matrix notation, for a seismic example see Claerbout (1968). This technique is known as the Propagation Matrices method. It can be applied to any medium composed of layers of different properties where waves are propagating, such as a succession of sections of a coaxial transmission line filled with different dielectrics. This formulation, for the specific case of a transmission line, is presented in the next section. In essence, it consists of rewriting the Transmission Line formulation (Appendix C) in matrix form, and relating the S-parameters to the total reflection and transmission of the line seen as a 2-port network (Appendix D). 3.2 Different representations for a coaxial transmission line 3.2.1 29 Propagation Matrices: a combined representation Operating the line below the cut-off frequency, only the TEM mode will propagate and a Transmission Line formulation can be adopted. The solution to the transmission line equations (C.2) and (C.3) in between two interfaces n − 1 and n, placed at zn−1 and at zn respectively, can be written as Vn (z) = Vn+ e−γn (z−zn−1 ) + Vn− e−γn (zn −z) , 1 [V + e−γn (z−zn−1 − Vn− e−γn (zn −z) ], In (z) = Zn n (3.1) (3.2) where γn is the propagation constant of section n given by equation (C.4) and Zn is the impedance given by equation (C.8). At any n-th interface, the fields must be continuous, and this can be expressed in a propagation matrices notation as e−γn dn 1 −γn dn Zn e 1 − Z1n Vn+ Vn− = 1 1 Zn+1 e−γn+1 dn+1 1 − Zn+1 e−γn+1 dn+1 + Vn+1 + Vn+1 , (3.3) or in its compact form as R ML n V n = M n+1 V n+1 , where Vn= Vn+ Vn− (3.4) , (3.5) and ML n = e−γn dn 1 −γn dn Zn e 1 − Z1n and MR n = 1 1 Zn e−γn dn − Z1n e−γn dn . (3.6) Then, for a multisectional line, as shown in Fig. 3.1, it is easy to relate the fields at the first interface, n = 1 to those at the last one, n = N , via the expression: −1 V 1 = (M L 1) "N −1 Y n=2 # L −1 MR MR n (M n ) NV N. (3.7) Now, we can group its elements into three main sections with four interfaces, see Fig. 3.2. 30 Chapter 3. Experimental design Figure 3.2: Simplified 3 sections model of a multisectional coaxial transmission line The sample holder is in its central part P , anything at its left is included in L and at its right in R, as follows L −1 P = MR p (M p ) , L = where p stands for the sample holder −1 (M L 1) "n=p−1 Y L −1 MR n (M n ) n=2 n=N Y−1 R = n=p+1 −1 R R (M L n) Mn MN, # , (3.8) (3.9) (3.10) and equation (3.7) can now be rewritten as V 1 = LP RV N . (3.11) But, V 1 and V N are related to the S-parameters of the line, and that is what effectively the Network Analyser is measuring. Then, when an electromagnetic wave impinges from Port 1, equation (3.11) transforms into 1 S11 1 S22 = LP R S12 0 , (3.12) = ĹṔ Ŕ S21 0 . (3.13) and when it does from Port 2 3.2 Different representations for a coaxial transmission line 31 Note, that we can keep the same formulation for this case, if we use a mirror 0 image of the transmission line. So L corresponds to the sections between Port 2 0 and the sample holder, represented now by P . And the sections from the sample 0 0 holder to Port 1 are included in R . Notice that P = P , when it is filled with a homogeneous material, because on both cases it is composed of only the sample holder. To avoid writing mirror equations and since the properties derived for the reflection and transmission coefficients of one specific pair are shared by the other pair, from now on, we will use Γ to represent both reflections, S11 or S22 , and Υ for the transmissions, S12 or S21 . The transmission coefficients are always equal to each other: [S12 = S21 ]always , since the path the waves travel is the same, but the reflection coefficients are equal only when the multisectional line is perfectly symmetric and filled with homogenous materials: [S11 = S22 ]symmetric line . Having expressed the full reflection/transmission of TEM waves along a transmission line in a propagation matrices formulation enables us to compute the electric permittivity and magnetic permeability of any sample filling the sample holder. We show this method in Chapter 5, together with other techniques, but first, we illustrate this representation for an ideal case, in the next section. The Propagation matrices representation consists in rewriting the Transmission Line formulation (Appendix C) in matrix form, and relating the S-parameters to the total reflection and transmission of the line seen as a 2-port network (Appendix D). It is, therefore, very difficult to dissociate these terms. However, as a rule, when talking about S-parameters, we will be referring to the reflection/transmission measurements at the reference planes of a transmission line. 3.2.2 Combined representation of the ideal case Let us consider the simplest configuration possible, where the material of study is placed in region 2 of a coaxial transmission line, such as the one in Figure 3.3, and has a length dp . Regions 1 and 3 are filled with air so that their impedance is Z0 . The walls of the line are made of perfect conducting metals and the measurement planes are at the interfaces of the material. In practice, the commercial transmission lines available (perfectly continuous cylinders filled up with air and with appropriate connectors), are the closest set-ups to the ideal case. Their special configuration, however, is responsible for uncertainties of sample length and position, BakerJavis et al. (1990). In any case, in many situations, the sample specimens are not suitable for these type of fixtures, e.g. loose sands, large sample sizes. Moreover, the commercial fixtures don’t allow for any special measurement feature, e.g. fluid flow. In those cases, the transmission line has to be customized and, in fact, many transition sections between the reference measurement planes and the sample are needed. The problem with these measurements lies in the difficulty of removing the effect of these extra sections, and we treat this extensively in the following chapters, 32 Chapter 3. Experimental design but for the moment, we focus on this simple case, as it proves to be very useful in the theoretical understanding of this type of problems. It is this configuration that most of the published methods consider for the determination of permittivity from transmission and reflection measurements, Weir (1974), Stuchly & Matuszewsky (1978) and Ligthart (1983). REGION 1 REGION 2 REGION 3 a e* b e e 0 0 dp Ref. plane 1 Ref. plane 2 Figure 3.3: Simplified coaxial transmission line For such a configuration, we can write the extended form of equation (3.12) as: 1 Γ 1 = 4 1 Z0 1 −Z0 e+γp dp + e−γp dp Zp (e+γp dp − e−γp dp ) 1 +γp dp − e−γp dp ) e+γp dp + e−γp dp Zp (e 1 1 Υ , (3.14) 1 1 0 Z0 − Z0 where Z0 is the impedance of sections 1 and 3, and Zp , γp and dp are, respectively, the impedance, propagation constant and length of section 2. It is clear from equation (3.14) that the input on reference plane 1 is equal to the product of 3 propagation matrices, first to go from port 1 into the sample holder, then the propagation in the sample holder itself, and finally from the sample holder into port 2, and the output at reference plane 2. As the reference planes coincide with the sample holder interfaces, L and R have no exponential terms but only an impedance contrast, as the source and receiver cannot be placed on the actual sample. 3.2 Different representations for a coaxial transmission line 33 Working out the matrices product of equation (3.14) we obtain, 1 Γ e+γp dp = 4 " Z0 2(1 + e−2γp dp ) + ( Z + p Z0 − (Z p 2(1 + e−2γp dp ) Z ( Zp0 − Zp Z0 )(1 Z0 − (Z p Z0 Zp )(1 − Zp −2γp dp ) Z0 )(1 − e e−2γp dp ) − e−2γp dp ) + Zp Z0 )(1 − e−2γp dp ) # Υ 0 , (3.15) some more handling and naming ζ = e−γp dp and r to the local reflection coefficient, given by equation (C.18), we can rewrite equation (3.15) as 1 Γ 1 = ζ(1 − r2 ) 1 − r2 ζ 2 r(ζ 2 − 1) r(1 − ζ 2 ) ζ 2 − r2 Υ 0 . (3.16) And now it is easy to go back to the Transmission Line formulation as it is possible to express the reflection and transmission as Γ = Υ = r(1 − ζ 2 ) , 1 − r2 ζ 2 ζ(1 − r2 ) . 1 − r2 ζ 2 (3.17) (3.18) These are the global reflection and transmission coefficients given by equations (C.30) and (C.31) for this specific configuration. Having expressed the reflection and transmission of a TEM wave along a transmission line as equation (3.16) will prove very useful in the coming sections, where we study the occurrence of resonant frequencies and we prove that Γ and Υ are independent measurements. Resonant frequencies Broad frequency band measurements of standing waves present destructive inter00 ference, at resonant frequencies. For low-loss materials (εr < 10−2 εr 0 ) of constant permittivity over the frequency range, the off-diagonal terms of the matrix that relate the input and output voltages, equation (3.16), are zero, since ζ 2 = 1 at those frequencies, and therefore Γ = 0 and Υ = ±1. The implications that arise from these frequencies are many, even for high-loss materials, but we are too early in this thesis to give an extensive explanation about them. Here, we simply introduce the concept of resonant frequencies for low-loss 34 Chapter 3. Experimental design materials as they are a recurrent topic in the coming chapters. In the next chapter, we see how the measurement accuracy is poorer around the resonant frequencies (Subsection 4.2.2). In Chapter 5 we explain how it is possible to compute the permittivity from these frequencies for special cases, see Subsection 5.2.1, a well known method from interference optics for single frequency measurements. We also discuss the deterioration of the accuracy in the reconstructed permittivity at the vicinity of these frequencies depending on the method used for reconstruction in Section 5.4. 3.3 Γ and Υ: independent measurements To prove that Γ and Υ are indeed independent measurements we follow a similar procedure as used by (Claerbout, 1968). While he used it to prove that for the acoustic equivalent of a shorted transmission line the reflection and transmission response of a lossless line are not independent, we use it to prove that Γ and Υ are independent in absence of a perfect reflection at one side of the transmission line. In essence, in both, the seismic and the electromagnetic case, the product of k layer matrices can be written as, V+ V− 1 1 = k ζ F (ζ) G(ζ) ζ 2k G(1/ζ) ζ 2k F (1/ζ) V+ V− , (3.19) k where V + and V − stand for the down going and up going fields. And the determinant of the propagation matrix is det = [F (ζ)F (1/ζ) − G(ζ)G(1/ζ)] = 1. (3.20) By taking the up and down going wave fields at either side of the last interfaces according to equation (3.11), + + V 1 V Υ = and = . (3.21) V− 1 Γ V− k 0 Equation (3.21) is just another representation of equation (3.11) and the reflection and transmission responses are found as, 1 F (ζ)Υ ζk Γ = ζ k G(1/ζ), 1 = (3.22) (3.23) and in view of equation (3.20) no relation between F (ζ) and G(1/ζ) can be found and therefore, Υ and Γ are independent. 3.4. Conclusions 3.4 35 Conclusions In this chapter, we have justified our experimental choices and theoretical representation. We are interested in the study of the permittivity of sands partially saturated, in the meters to the decimeter range of the spectrum. We also want to achieve those saturations by flowing liquids through the sample. For those reasons we found it most convenient to use a customized coaxial transmission line, whose technical characteristics are presented in the next chapter. Coaxial transmission lines have been treated differently by different authors and we have developed a new combined representation to do it, the Propagation Matrices representation. It consists of rewriting the Transmission Line representation in matrix form, and relating the S-parameters to the total reflection and transmission of the line seen as a 2-port network. It simplifies the understanding and computation of TEM propagation through transmission lines. This combined representation has many advantages as we show in the coming chapters, but as a start, let us point to the fact that it has allowed us to prove the independence of the reflection and transmission coefficients. 36 Chapter 3. Experimental design In theory, there is no difference between theory and practice. But, in practice, there is. Jan L.A. van de Snepscheut (1953-1994) Chapter 4 Tool for accurate permittivity measurements In this chapter we present the tool we designed and used for measuring the permittivity of sands, its measurement characteristics, the calibration procedure and the comparison of the measurements with the calibrated forward model. 4.1 Introduction We are interested in the electromagnetic behavior of sandy environments in a broad frequency range, and as we saw in the previous chapter, the most suitable set-up is a coaxial transmission line. Our tool is, therefore, designed as such a line, for which all four S-parameters can be measured. This is the first time, to our knowledge, that all four S-parameters of a coaxial transmission line are modelled so accurately with transmission line theory. We present the tool and its measurement characteristics in the next section and then, its forward model is introduced in Section 4.3. To obtain such an accurate model we had to make several calibration steps, as we show in Subsection 4.3.2, where we also comment on several model assumptions. In Section 4.4 we show a comparison of the model with measurements, together with a study of the sensitivity of the model, and we predict expected accuracies for the reconstruction of permittivity from measurements. 37 38 Chapter 4. Tool for accurate permittivity measurements Coaxial Transmission Line TUL.1 SH TUL.2 TUR.2 C TUR.1 fluid outlet fluid inlet Transition Unit Sample Holder Transition Unit Ref. plane 2 Ref. plane 1 Figure 4.1: Coaxial Transmission line probe for permittivity measurements of porous media and its sectioning. SH: sample holder, TU: transition unit, L: to the left of SH, R: to the right of SH, C: connector. Fluid distributor unit 1 Carved fluid distributor grey: teflon white: carved fluid path 2 Inlet into carved path 3 Fluid inlets/outlets 4 Metallic outer ring and inner pin TOP VIEW 3 1 3 2 3 4 3 1 4 SIDE VIEW 3 Figure 4.2: Fluid distributors units. 3 4 4.2. Technical design and measurement characteristics 4.2 39 The Tool: technical design and measurement characteristics The technical description of the tool is presented in Subsection 4.2.1. However, before presenting the results that support this design, in Section 4.3, we first discuss the measurement characteristics of the tool, in Subsection 4.2.2, to be able to separate model errors from measurement errors. 4.2.1 Technical Design The tool was designed such that it is possible to determine the full S-parameter matrix of the system (measuring both the transmission and reflection coefficients of a sample is crucial in the determination of the permittivity). The geometry and size of the probe ensure that only the TEM mode propagates along the line, and, therefore, it can be described with Transmission Line Theory. However, by simple modifications the sample holder can be transformed into a circular waveguide, similar to the one described in Taherian et al. (1991) and it is possible to determine the permittivity of rigid cores with a full wave description, see Section B.3. The sample holder was chosen to be of a representative volume (10 cm long, 3 cm in outer diameter and 0.9 cm in inner diameter) and allowing for fluid flow. It is gold plated to ensure low losses of energy in the line conductors, and its several parts can be characterized separately for accurate modelling. The probe (Figure 4.1) consists of three main sections; two transitions units (TU) and the sample holder (SH). Both transition units are again, composed of three sections: a conical part, a cylindrical part and a fluid distributor. They can be dismounted, enabling separate measurements of the two transition units together for high accuracy calibration measurements. The transition units are Teflon filled and long enough to prevent any higher order mode generated from reaching the measurement plane. The conical part of the transition unit eliminates the impedance jump between the cable connection and the sample holder such that the generated higher order modes are negligible. When the line is completely filled with Teflon the impedance throughout is very close to 50 Ω. The fluid distributors can be connected to four inlets and four outlets for fluids. Figure 4.2 describes their features. The fluid enters the sample through the inlets located on the sides of the gold plated outer ring and into the carved teflon distributors, appropriate filters are placed on top. The carved fluid path ensures a reasonably homogeneous flow through the sample. 40 4.2.2 Chapter 4. Tool for accurate permittivity measurements Measurement Characteristics The probe can be connected to the two ports of an S-parameter test set, and the full S-parameter matrix of the network is measured with a Network Analyzer, controlled by a PC. We use an HP 8357A Network Analyzer together with a 85046A S-Parameter set, so that the frequency range at which we can operate goes from 300 KHz to 3 GHz. However, depending on the material, the effective lower limit of this range can be in the order of several hundreds of MHz. Any influence from the cables that connect the tool to the S-parameter test set is compensated for by doing a Full 2-port calibration, so that the measurement plane is moved to the connection of the cables to the tool, see Figure 4.1. For a full 2port calibration, reflection measurements for an open, a load and a short connector on both cables are done, as well as, thru-measurements and isolation (HP, 1993). Once the cables are calibrated, the three main bodies of the probe are assembled. First one transition unit to the sample holder, that is then filled with the material in study and finally the other transition unit. In case of flow experiments, convenient filters are placed at the interfaces between the sample holder and the transition units. These three pieces are tightly screwed together. And the cables are screwed to the end connectors at both transition units. These are very delicate processes and a good connection is essential to obtain accurate measurements. The tool components were design such that the transition units could be compensated by doing a ”user defined” calibration (HP, 1993). The process is equivalent to the calibration of the cables but the standards were specially designed and constructed to match the transition units characteristics. This calibration attains the best results since it removes any undesirable effects from the units. Unfortunately it was impossible to obtain a proper short circuit of the transition units and we could not calibrate them experimentally. Since the experimental calibration of the transition units failed, we then fined tuned the forward model of the tool with multiple measurements of the different components. We refer to it as the calibration of the model. Note, that it is not an experimental compensation of the transition units but a calibration of the model with measurements. 4.2. Technical design and measurement characteristics 41 Precision of the measurements From a statistical analysis on one hundred measurements for different materials, we determined that the measurements are very stable and their precision can be improved by stacking, if needed. For our study, single measurements are accurate enough, as they have three significant figures. This is clear from Figure 4.3 where, as an example, the standard deviation of 10 groups of 2 measurements each, have been plotted for the real and imaginary parts of the reflection (S22 ) and transmission (S21 ) when the sample holder is filled with air. For the whole frequency range the difference between any two transmission measurements is always lower than 3·10−3 , and lower than 10−3 for reflection. This leads to a true dynamic range of 50-60 dB. It is useful to look at the relative precision of these measurements in amplitude and phase. By doing so, we are able to derive important measurement characteristics. In Figure 4.4 we have plotted the relative precision of reflection (S22 ) and transmission (S21 ) when the sample holder is filled with air. From the figure it is clear that the phase is always more accurate than the amplitude but that below certain frequencies, 300 MHz for air, it is not well determined, specially for reflection data. This corresponds to a wavelength ten times larger than the sample holder. The accuracy in the amplitude of the measurements depends on whether the measurement is a reflection or a transmission coefficient. For transmission there is a slow linear increase of inaccuracy with frequency but it is maintained below 5 · 10−3 . On the other hand the accuracy of a reflection coefficient varies significantly and not linearly with the frequency. The inaccuracy increases on both the high and low limit of the frequency spectrum as well as in the vicinity of the resonant frequency, 1.5 GHz for air. This enhanced inaccuracy around resonant frequencies is observed in other materials. In between these peaks the inaccuracy is maintained below 10−3 . The explanation and occurrences of these resonant frequencies is due to the destructive interference introduced in Section 3.2.2. The inaccuracy around resonant frequencies increases because the amplitude of the signal is very small and the Network Analyzer has troubles determining its phase. This can also happen with the transmission coefficient of lossy materials, as the signal damps out above certain frequencies. For amplitudes around 4 · 10−3 the error can be of the order or 30% and increase up to 50% for half those amplitudes. 42 Chapter 4. Tool for accurate permittivity measurements ℜ 0 10 −2 10 10 22 std(S ) −2 −4 −4 10 10 −6 10 −6 0.5 1 1.5 2 2.5 3 0 10 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 0 10 10 −2 −2 10 10 12 std(S ) ℑ 0 10 −4 −4 10 10 −6 10 −6 0.5 1 1.5 2 2.5 Frequency (GHz) 3 10 Figure 4.3: Standard deviation for 10 groups of 2 measurements for the real and imaginary parts of S22 and S21 relative precision of S Phase 0.02 0.008 0.01 0.006 0 0.004 −0.01 0.002 0 0 12 relative precision of S 22 Amplitude 0.01 1 2 3 0.01 −0.02 0 1 2 3 0.02 0.008 0.01 0.006 0 0.004 −0.01 0.002 0 0 1 2 frequency (GHz) 3 −0.02 0 1 2 frequency (GHz) 3 Figure 4.4: Relative precision for a group of 2 measurements. In amplitude and phase of S22 and S21 for an air sample. 4.3. Forward Model and Calibration 43 Number of points The Network Analyzer HP 8357A can perform from single point frequency measurements up to 1601 points. The measurement and computing time increases with increasing number of points. We decided to use the maximum number of points offered by the apparatus to obtain a high density of permittivity values over the spectrum. That way we expect to detect any significant event. If a particular frequency region is of special interest, the frequency range can be narrowed, increasing the frequency points density in that region, in order to obtain a more precise description. 4.3 Forward Model and Calibration To model the S-parameters of the probe we used the standard transmission line theory for a multi-section line presented in Appendix C. And for the calibration we used multiple measurements of the different components that form the tool, to fine tune the forward model. 4.3.1 Forward Model Transmission line theory can be applied when only the TEM mode is propagating. This is ensured by operating the line below the cut-off frequency of the first order mode, T M01 mode (see Appendix B). It will propagate when √ λcutoff ≥ 2 εr (a − b), where a and b are the outer and inner radius of the conductors that form the line. For our sample holder this will occur when it is filled with a material whose relative permittivity is higher than 25 and only for very high frequencies. For water, at 25◦ C, this mode propagates above 1.6 GHz (λ ≈ 2 cm) enhancing the inaccuracy of the model. However, for most sandy samples, whose relative permittivity values are generally smaller than 25, the propagation of only the TEM mode is ensured. In Appendix D, it has been proven that the diagonal scattering parameters are the same when measured at the reference planes of a perfectly symmetric line with a homogenous filling material. They are equal to the global reflection coefficient ΓN including all internal reflection and transmission effects. The off-diagonal elements of the S-parameter matrix are always equal to each other and to the global transmission coefficient ΥN , including all internal reflection and transmission effects. Our probe is not symmetrical (see Figure 4.1) and therefore the diagonal Scattering parameters are not equal to each other, but they are equal to the corresponding global reflection coefficients measured at the corresponding port. Then, 44 Chapter 4. Tool for accurate permittivity measurements S11 S12 S21 S22 = Γ11 Υ21 Υ12 Γ22 , (4.1) where the global reflection and transmission coefficients are given by the recursive expressions (C.30) and (C.31), and the subindices refer to the departing and measuring planes 1 or 2 (see Figure 4.1). From those expressions and from the ones for the propagation constant γn , (C.4), the impedance Zn (C.8) and the transmission line parameters for a coaxial transmission line, equations (C.10) to (C.13), we can see that our model is governed by {an , bn , dn } {µ∗n , ε∗n } w = 2πf the dimensions of the sections that form the line; outer radius, inner radius and the length of the sections. the electromagnetic properties of the materials filling the sections of the line; the complex magnetic permeability and the complex electric permittivity. the radial frequency of operation. Direct comparison of this model with actual measurements reveal significant differences between both. These differences are due to several reasons and are explained in detail in the next subsection. So, although theoretically everything is known except for the permittivity of the material in the sample holder, in practice, it is necessary to perform many different measurements on cables, connectors, transition units and the whole probe to calibrate the tool. In Figure 4.5 we have plotted the real and imaginary parts of the measured [S msr ] reflection [S22 ] and transmission [S21 ] when the sample holder is filled with air, together with the modelled parameters [S mod ] and their difference [S msr − S mod ]. The model is qualitatively valid, but quantitatively it is out of phase and significantly different from the measurements. To reduce this difference, we spent quite some effort in the calibration of each component and the final tuning of the model. We show this calibration together with several model assumptions, and their justification, in the next subsection. 4.3.2 Calibration The calibration of the tool consists in fine tuning the parameters that control the forward model. Comparing the measurements to the model we were able to find the optimum parameters that describe the behavior of our probe. For that purpose, two main test materials were used: air and Teflon. In the comparison of measurements and model we looked at all the reflection and transmission coefficients in their complex form, real-imaginary or amplitude-phase. We assumed that the conical sections of the transition units could be represented by a single cylinder and that the hollow fluid distributors could be modelled as solid teflon disks. We then, followed a step by step procedure, looking at every component and parameter that 4.3. Forward Model and Calibration 45 ℜ 0.5 0.5 0 0 −0.5 −0.5 S 22 1 −1 0.5 1 1.5 2 2.5 3 −1 1 1 0.5 0.5 0 0 −0.5 −0.5 12 S ℑ 1 −1 0.5 1 1.5 2 2.5 Frequency (GHz) 3 −1 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 Figure 4.5: Measured (solid line) and modelled (dashed line) reflection and transmission coefficients for an air sample, and their difference (dotted line) . could improve our model. We started with the simplest model and geometrical data taken from the technical specifications of the probe, changing one parameter at a time. Let us first justify the model assumptions. Assumptions of the calibrated forward model In the forward model we modelled the conical sections of the transition units as a single cylinder and the fluid distributors, although formed by a hollow teflon disk, are represented by a solid teflon disk. Here, we justify these assumptions. 1. Modelling the conical sections of the transition units: The transition units of the probe have a conical section to eliminate the impedance jump between the cable connection and the sample holder such that hardly any energy is converted to higher order modes. But on a circuit model a cone has to be replaced by series of cylinders that will mimic the behavior of the cone. It was proven that only one cylinder of the same length of the cone and a proper radius was needed to model the cone. Adding more cylinders did not improve the accuracy of the model. The improvement caused by the 46 Chapter 4. Tool for accurate permittivity measurements replacement of the cone by 100 cylinders was smaller than the measurement accuracy and therefore there is no need to use a more complicated model, so that the cone is treated as one single cylinder. 2. Fluid distributors: Between the transition units and the sample holder there are fluid distributors (Figure 4.2), which ensure a more homogeneous flow into the sample. These distributors are Teflon made, but they are not solid cylinders and fluid conductors have been carved into them. By performing measurements with the actual fluid distributors filled with tap water and comparing them to the model with solid Teflon cylinders in place of the distributors we could check the validity of this assumption. In Figure 4.6 we have plotted the amplitude and phase of the transmission coefficient S12 of a sample of tap water with hollow fluid distributors and the model (with solid disks) for the same sample. It is clear that the assumption is valid, at least up to the cut-off frequency of the TEM mode for water: 1.6 GHz. Amplitude Phase 1 0 exp: fluid distributors mod: solid disks 0.9 −10 0.8 −20 0.7 −30 S12 0.6 0.5 −40 0.4 −50 0.3 −60 0.2 −70 0.1 0 0 1 2 Frequency (GHz) 3 −80 0 1 2 Frequency (GHz) 3 Figure 4.6: Measured transmission coefficients for a tap water sample. The solid line represents measurements done with the fluid distributors, and the dotted line to the modelled coefficients with the solid disks in the place of the actual fluid distributors. 4.3. Forward Model and Calibration 47 Calibration steps Here we list the calibration steps, and in Figure 4.7 we show the improvement of each step by plotting the absolute difference between the model and the measurement for every step when the probe is filled with air. Since the initial difference between the model and the measured data is bigger for transmission than for reflection, the advantage of this calibration shows more clearly in the transmission data. 1. Fitting geometrical parameters: Most geometrical parameters (length and radii of the sections shown in Figure 4.1) of the probe can be measured independently and others have to be extracted from their technical specifications or from independent measurements. However, these may vary due to construction, or compression when the probe is mounted, or expansionscontractions from temperature changes. Optimizing for these parameters proved to be crucial, and resulted in a general reduction in both reflection and transmission differences. Their values are listed in Table 4.1 Table 4.1: Geometrical parameters for the tool. [mm] dn an bn TU1.1 57.5 8.9 2.7 TU1.2 46 15 4.5 SH 100 15 4.5 TU2.2 46 15 4.5 TU2.1 57.5 8.9 2.7 C 18.3 2.7 0.8 Although the transition units can be modelled as symmetric their geometrical parameters vary some milliliters from their technical specifications. 2. Permittivity of Teflon: Over the working frequency range of the Network Analyzer, the relative permittivity of Teflon is reported in literature to be in between 2 and 2.1. The minimum difference between the model and the measurement was found at 2.05. We have tried using a measured permittivity for Teflon per frequency, but it did not improve the accuracy of the model. The deviations of the measured permittivity for Teflon from the mean value of 2.05 are so small that the model is insensitive to them. 3. Losses: a lossless ideal line has a unitary scattering matrix (equation (D.7)). So, by plotting the sum of the squares of the reflection and transmission coefficient we can check how ideal our probe is. In Figure 4.8 we have plotted this sum (solid line) for the transition units connected together and for the whole tool, with air as sample. It is clear that for both cases there is a linear deviation with frequency. The loss of amplitude due to the skin effect has been plotted as a dashed line. 48 Chapter 4. Tool for accurate permittivity measurements ℜ ℑ 0.5 0 0 S22 0.5 −0.5 0.5 1 1.5 2 2.5 3 −0.5 0.5 0 0 S21 0.5 −0.5 0.5 1 1.5 2 2.5 Frequency (GHz) 3 −0.5 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 Figure 4.7: Difference between the measured and modelled reflection and transmission coefficients for an air sample, at the different steps of the calibration of the tool. Before calibration (solid line), step 1 (dashed line), step 2 + 3 (dotted line). Tool: AIR only Transition Units 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 0.75 0.75 |S22|2 + |S21|2 1 exp with Rc with δ 0.7 0.5 1 1.5 2 2.5 Frequency (GHz) 3 0.7 0.5 1 1.5 2 2.5 Frequency (GHz) 3 Figure 4.8: Unitary property check for the transition units connected together (tool without sample holder), right figure, and for the whole tool when the sample holder is filled with air, left figure. Experimental data (solid line), including conduction losses (dashed line), including a loss δ (dotted line). 4.4. Experimental Results 49 As we can see in the figure, conduction losses alone, cannot account for the deviation from unity. However, its linear trend and its consistency, whether the sample holder is measured or not, suggested that we could model it as a δ loss in the exponential terms of the conduction along the transition units, transforming equation (C.23) with c0 γ̂Transition Units = e−j ω [(εr Teflon )1/2 −jδ ], (4.2) The model with a loss (δ = 1.5 · 10−3 ) has been plotted in Figure 4.8 as a dotted line. For both cases, whole tool or only transition units, it accounts very well for the total loss trend present in the measurements, without considering conduction losses. We have tried to include experimental losses into the model in different ways and this has been the most successful one. 4.4 Experimental Results In the previous sections we have described the probe, its forward model and the calibration steps, needed to reduce the difference between the model and the measurements. In this section we present two experimental results on materials of known permittivity to show the validity of the calibrated model. We also report on the sensitivity of this model. 4.4.1 Results on materials of known permittivity Air Figure 4.9 shows that the difference between the calibrated model and the measurement has been reduced significantly. This is clear, if we compare it to the difference between the non calibrated model and the same measurement, shown in Figure 4.5. This improvement is due to the calibration of the model itself. Even though the maximum difference is bigger than the precision of the measurements by one order of magnitude, the obtained accuracy is to our knowledge still better than reported in the literature. On average, the model differs from the measurements, 2% for reflection and 1% for transmission, for both real and imaginary parts, although at some frequencies it is three times as much. 50 Chapter 4. Tool for accurate permittivity measurements ℜ ℑ 1 0.5 0.5 0 0 −0.5 −0.5 S22 1 −1 0.5 1 1.5 2 2.5 3 −1 1 0.5 0.5 0 0 −0.5 −0.5 S21 1 −1 0.5 1 1.5 2 2.5 Frequency (GHz) 3 −1 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 Figure 4.9: Measured (solid line), reflection and transmission coefficients for an air sample, its calibrated model (dashed line), and their difference (dotted line). Ethanol In Figure 4.10 we have plotted the reflection and transmission coefficients for measured data of ethanol (99.9% pure), the model for these data and their difference. The permittivity of ethanol is well defined from its Debye parameters (Gemert, 1972), but it cannot be used as calibration material since its permittivity is not standardized and small temperature and purity changes affect it. Still, we can compare model and measurements qualitatively. The transmission is very well modelled, although the loss of signal above 1.5 GHz increases the inaccuracy in the phase of the measured data. The measured and modelled reflections are slightly out of phase. Nevertheless, varying the Debye permittivity of ethanol in 1% results in a noticeable increase in the difference between data and model results over the whole frequency range. We will use this tech- 4.4. Experimental Results 51 ℜ ℑ 1 0.5 0.5 0 0 −0.5 −0.5 S22 1 −1 0.5 1 1.5 2 2.5 3 −1 1 0.5 0.5 0 0 −0.5 −0.5 S21 1 −1 0.5 1 1.5 2 2.5 Frequency (GHz) 3 −1 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 Figure 4.10: Measured (solid line) Reflection and Transmission coefficients for an ethanol (99.9% pure) sample, its calibrated model (dashed line), and their difference (dotted line). nique in the coming section to estimate expected errors in the reconstruction of permittivity with the presented forward model. 4.4.2 Sensitivity of the model To study the sensitivity of our model, we performed experiments with numerically modelled and measured data. It is possible to determine the accuracy in the permittivity if the difference between the model and the measurements is known. Conclusions can be drawn about the maximum error of the possible reconstructed permittivity from the accuracy of the measurement itself and the differences between the measurement and model results as a function of permittivity. In Figure 4.11, we show the result of this analysis. The plot shows the maximum error in permittivity, as a function of normalized wavelength, that gives a change in the model result, which is just observable such that it leads to an increased difference 52 Chapter 4. Tool for accurate permittivity measurements between data and model result above an uncertainty threshold. For the uncertainty threshold we use the maximum error in the data, which leads to the worst result scenario. It is plotted as a function of normalized wavelength, because the resonances at large wavelengths are expected to prohibit the accurate permittivity reconstruction ability. These resonances are related to the ratio of wavelength and sample holder length, rather than frequency. relative error in permittivity % 50 % error 40 30 20 10 0 0 10 20 30 40 50 60 70 5 % error 4 3 2 1 0 0 3 6 λ/L 9 12 15 sample holder Figure 4.11: Maximum error in electric permittivity, which does not lead to observable model error as a function of normalized wavelength. With an air filled sample, the normalized wavelength of 60 is equivalent to a frequency of 50 MHz. It is clear that if we would allow a maximum error of 5% in the reconstructed permittivity, we cannot use the set up below 250 MHz for air. Of course the lower bound of the usable frequency goes down proportional with the inverse square root of the permittivity of the material filling the sample holder. So for most soil samples, varying in permittivity from 4 to 25 (dry to wet), the lowest usable frequency would be 125 MHz and 50 MHz, respectively. These limits are confirmed by experimental data in Chapter 5. For lower frequencies, other techniques 4.4. Experimental Results 53 should be used. Robust information on the permittivity from higher frequencies could be used to stretch this lower limit, but it will only be an extrapolation. For frequencies higher than twice the lowest usable frequency the permittivity can be reconstructed with an error below 1%. The smallest error that we are able to achieve is 0.1%, which is due to the limited accuracy of the measurements. Sample length This type of study enables us to forecast sample length requirements for optimum measurements. For example, we can determine the shortest sample holder we can use if we want to ensure a 1% accuracy on a certain material. This is done by plotting the frequency versus the sample holder length for a certain accuracy. 3 600 2.5 500 Frequency limit (MHz) Frequency limit (GHz) In Figure 4.12 we have represented the curve for a minimum accuracy of 1% for an air sample. If we were going to use a 3 cm sample holder the measurements will be 1% accurate for frequencies above 2 GHz, whereas if we use a 30 cm sample holder we could use most of the spectrum, from 200 MHz onwards. 2 1.5 1 0.5 0 0 400 300 200 100 5 10 15 Lsample holder (cm) 20 0 0 50 Lsample holder 100 (cm) Figure 4.12: Frequency limit depending on the sample length for an air sample to ensure a maximun error of 1%. 54 Chapter 4. Tool for accurate permittivity measurements When performing dynamic flow experiments it is important to keep the samples as short as possible, but as we can see from Figure 4.12 for permittivity purposes we would like to have them as long as possible. With this technique we can reach a compromise to satisfy both requirements. 4.5 Conclusions We have constructed a coaxial transmission line for accurate measurements of permittivity from 300MHz to 3GHz. The forward model representing the reflection and transmission along the line is in very good agreement with the measured data after a thorough calibration. It has been a hierarchal tuning and it involved calibrating the length and radii of every section forming the line, finding the most suitable permittivity for Teflon in the proper frequency range, modelling the losses present along the line and corroborating the assumptions made in the model in relation with the conical sections and the fluid distributors. We have shown that relative changes in the permittivity in the order of 1% can be detected over a wide frequency band up to 3 GHz, while the lowest usable frequency depends on the permittivity of the material filling the sample holder. Let us now summarize the most important technical and measurement characteristics of the presented tool: Characteristics of the tool Technical Measurements Coaxial Transmission Line allows Fluid Flow Transmission Line Theory applies True dynamic range of 50-60 dB 4 S-parameters measured Transmission more reliable than Reflection a careful calibration is needed to obtain accurate results Results? Why, man, I have gotten lots of results! If I find 10,000 ways something won’t work, I haven’t failed. I am not discouraged, because every wrong attempt discarded is often a step forward. Thomas Edison (1847-1931) Chapter 5 Reconstruction methods for permittivity from measured S-parameters In this chapter, we present and compare analytical and optimizing reconstruction methods of materials of known EM properties. We also show in detail the occurrence of resonant frequencies, and their implications. 5.1 Introduction The reconstruction of permittivity from measured S-parameters is a broad and complex problem. In the high-frequency regime, the relation between the EM properties and the measured quantities is highly non-linear, making the reconstruction cumbersome and unstable. Many researches have tackled this problem and solved it in different ways. There are two main streams: Analytical and Optimization methods. Analytical methods The electrical permittivity is computed from analytical expressions involving the S-parameters of the sample holder only. Although this chapter is dedicated to the measurement of the electrical permittivity of porous media, the nature of the problem allows to solve also for the magnetic permeability. So we will present the solution of both properties and the inherent problems associated with it. There are three ways to obtain the reflection and transmission of the sample holder only. First, if a commercial fixture is used, then moving the reference planes from 55 56 Chapter 5. Reconstruction Methods the test ports to the sample interface is a matter of removing the phase delay, e.g. Rau & Wharton (1982), since the empty line characteristic impedance matches the impedance of the instrument’s test ports. However, reported accuracy problems arise from the uncertainty of the sample location and not every sample is suitable for these type of lines. Second, it is possible to compute the permittivity and permeability from the measurements at the test ports via analytical expressions with the new technique we have developed, see Subsection 5.2.2. And third, you can remove the contribution of unwanted transition sections by compensation methods. Kruppa & Sodomsky (1971), Freeman et al. (1979), Shen (1985) and Chew (1991) apply basically the same technique. They compensate for the transition sections by a tedious experimental calibration, which is also prone to additional measurement errors. We discuss this method in Subsection 5.2.4. Once the S-parameters of the sample holder only are obtained different reconstruction formulas can be used. Nicolson & Ross (1970) and Weir (1974) combined the equations for the scattering parameters such that the system of equations could decouple. But its solution is divergent for low-loss materials at frequencies corresponding to integer multiples of one-half wavelength in the sample. Stuchly & Matuszewsky (1978) obtained two explicit equations for the permittivity by a slightly different derivation. They are unstable for low-loss materials at frequencies corresponding to integer multiples of one-half wavelength in the sample. Palaith & Chang (1983) analyzed all three equations emphasizing in the inherent errors they produced and narrowing them down by a mapping method. Ligthart (1983) presented a method for shorted line measurements where the scattering equations for the permittivity were solved over a calculated uncertainty region and the results were then averaged, but they suffer from the same problems when low-loss materials are considered. Finally, Baker-Javis et al. (1990) minimized the instability of the Nicolson-Ross-Weir equations by considering only non-magnetic materials and an iterative procedure. All the methods consider only the ideal case presented in Subsection 3.2.2 and assume that the S-parameters have been properly compensated, but the procedure is not trivial. In Subsection 5.2.3 we show how they all correspond to the same method, and therefore suffer from the same instabilities. Optimization methods The EM properties of the sample can also be computed via an optimization procedure, minimizing a cost function involving the measured and modelled S-parameters of the tool where the material is placed. No compensation is needed. Belhadj-Tahar et al. (1990) and Taherian et al. (1991) applied this method to a coaxial-circular waveguide. The first used a gradient technique and the second modified Newton method. Their sample sizes are relatively small and the method results are inaccurate for low-loss samples as well as for low frequencies since the 5.1. Introduction 57 wavelength becomes too large compared to the sample size. Nguyen (1998) used a Nelder-Mead Simplex method on shorted reflection data of a coaxial waveguide. The possibilities are as many as the drawbacks. It will be ideal to use the most suitable method for each specific material, but that is almost unfeasible. Therefore, we compare analytical and optimization reconstructions of different materials to choose the best way to proceed. 5.1.1 Chapter outline First, we present the analytical methods in Section 5.2, where we start off by studying the occurrence of resonant frequencies in our tool (Subsection 5.2.1). Resonant methods are usually set aside when broad band measurements are done. However, it is very favorable to study their occurrence in detail, specially for low-loss materials whose permittivity is constant in our frequency range of interest, and to by-pass some inaccuracies of the reconstruction methods. Then we present the Propagation Matrices Method (PM) in Subsection 5.2.2. We have developed this technique that, in essence, is a method for moving the reference planes from the measurement planes to the interfaces of the sample holder, taking into account impedance discontinuities among sections and losses along the line, and applying an analytical expression to compute the permittivity of the sample. Writing the solution of the propagation matrices for the ideal case (considering only the sample holder), in Subsection 5.2.3, enables us to prove how all the published analytical expressions are in fact, the same method. And in Subsection 5.2.4 we discuss the validity of the application of the compensation methods (CO) for our tool and frequency range. Finishing the analytical methods section with Subsection 5.2.5 where we compare the performance of the propagation matrices and the compensation methods. Then we move forward to optimization methods (OP), in Section 5.3, where we minimize different cost functions with a modified Newton optimization routine. In Section 5.4 we compare all the different solutions and state a preference. Finally, in Section 5.5 we conclude the chapter. 5.1.2 Experimental considerations All the methods presented in this chapter, have been tested with measured and modelled data for materials of known permittivity. The modelled data were calculated with the calibrated forward model from the previous chapter and the corresponding permittivity. We will refer to it as [S mod ], and it comprises all four S-parameters of the tool when the sample holder is filled with a particular material. The model takes the measurement planes at the connections with the cables, just as in a real measurement. 58 Chapter 5. Reconstruction Methods The measured data are actual measurements of the tool when the sample holder is filled with a certain material. We refer to it as [S exp ] since it is obtained experimentally. To acquire the four S-parameters that form [S exp ], the material of study is placed in the sample holder and the tool is assembled together and connected to the Network Analyzer (NA) via the calibrated cables, as explained in Subsection 4.2.2. Then each pair of measurements [Γ, Υ] are recorded by the computer at the corresponding port to form the full matrix. The NA sweeps 1601 points over the specified frequency range. The measurements present in this chapter were done using the whole range available at the HP 8357A NA, from 300 KHz to 3 GHz, to be able to study the performance of the methods in the entire spectrum. As tests materials we differentiated between non-frequency and frequency dependent materials to check the ability of the different methods in reconstructing distinct permittivity profiles. On one hand, materials that maintain a constant permittivity over the broad frequency range used in these experiments, also present a low-loss. To represent these type of materials we chose air and alumina. Air has a well known relative permittivity of 1, that allows for a quantitative comparison of results. Alumina is a composite of aluminum oxide that presents a relatively high permittivity of around 9. On the other hand, frequency dependent materials are also highly dispersive. To represent these we chose ethanol, a polar liquid, whose permittivity is defined by Debye’s formula, given in Appendix E, and its Debye parameters. These parameters depend on temperature and purity. Reported Debye parameters are in many cases incomplete. Landolt-Börnstein (1996) only lists the static parameter for different temperatures. The only complete set that was close in temperature conditions to our experiment (23◦ C) is the one measured by Van Gemert (1972) (24◦ C) but he does not report on the purity of his samples and his parameters have significant errors. Therefore, an absolute comparison with standard values cannot be performed, but we can corroborate if our results are in agreement with the accepted model for polar liquids or not. In order to do so, once the permittivity of the sample was reconstructed per frequency with the appropriate method, we fitted a Debye model to the result. Obtaining the Debye parameters from optimizing the following cost function costDebye = Pnf ∗ ∗ i=1 |(εr )rec − (εr )Debye | , Pnf ∗ i=1 |(εr )Debye | (5.1) where nf is the total number of frequencies, (ε∗r )rec is the reconstructed permittivity and (ε∗r )Debye is the Debye permittivity. In this manner, we fit a frequency dependent model to the permittivity of ethanol computed without any frequency model, and as a reference we refer to the values measured by van Gemert (1972). 5.2. Analytical Methods 59 Along this chapter the structure of the sections, devoted to the PM, CO and OP reconstructions, is very similar. First the method is presented and then it is tested in two extreme situations; the first one with a material of low-loss and low contrast, air, whose permittivity is constant, and the second one with a material of high loss and frequency dependent permittivity, ethanol. To test and to be able to weight the performance of the individual methods presented in this chapter, we compare the reconstructed permittivity with that of the sample. Their accuracy is derived from this comparison. 5.2 Analytical Methods In this section we have included all the existing methods to compute the electrical permittivity from analytical expressions. It is structured as follows: In Subsection 5.2.1 we introduce the method of the Resonant Frequencies that allows us to calculate the real part of the permittivity from the resonances of the line. This is the first time, to our knowledge, that this technique is applied in a transmission line, as it is usually reserved for resonant cavities. In Subsection 5.2.2 we present the method we have developed to compute the permittivity and permeability from the S-parameters measured at the test ports. This is done via an analytical expression. In Subsection 5.2.3 we show how all existing analytical expressions for the computation of permittivity are in fact, the same. The compensation method, presented in Subsection 5.2.4, suffers from instabilities at many frequency points over our spectrum and it is necessary to use a modelled compensation for our tool. Finally, in Subsection 5.2.5 we compare the reconstruction capabilities of the presented methods. 5.2.1 Resonant frequencies: permittivity implications Resonant methods are very accurate in reconstructing the permittivity of a sample at fixed frequency points. For each frequency the dimensions of the resonator have to be changed, making broad band measurements almost impossible. They also require small samples, specially for high-loss materials. 0 00 In Chapter 3 we saw that for low-loss materials (εr < εr /100) of constant permittivity over the frequency range, the reflection coefficient goes to zero at resonant frequencies, since ζ 2 = 1. At those frequencies, the accuracy of the reflection coefficient is worse than at any other part of the spectrum, see Section 4.2.2. This obstructs the reconstruction of permittivity from broad band measurements, but we can extract valuable information from the occurrence of these resonances. 60 Chapter 5. Reconstruction Methods We introduced ζ, in Subsection 3.2.2, as ζ = e−γp dp , while the propagation constant √ of non-magnetic materials, γ, can be written as γ = j cw0 ε∗r , see Section 2.4.2, so that the resonant frequencies occur when 4πfk p ∗ εr dp = 2kπ < c0 for k = 1, 2 . . . (5.2) where dp is the length of the sample. They are, thus, periodic and depend on the length of the sample holder and the permittivity of the material. Since ε∗r is a complex quantity and can depend on the frequency, let us distinguish between different cases: Low-loss non-frequency dependent materials For materials whose relative permittivity is not a function of frequency and it is 0 real; ε∗r = εr , the reflection coefficient will go to zero at periodic intervals and its permittivity over the whole frequency range can be computed from these single frequency points as k c0 εr = 2 fk dp 0 2 with fk = kf for k = 1, 2 . . . (5.3) In Figure 5.1 we have plotted the amplitude of the measured S11 for an air sample and for an alumina sample, placed in our tool (solid line), together with the model (dotted line) for those materials considering only the sample holder (ideal case, see Subsection 3.2.2). From the figure, it is clear, that the resonances of the tool depend only on the material placed in the sample holder. The measured reflection of the tool goes to zero at the same frequencies as the model considering only the sample holder. Note that the measurements at the reference planes (solid line) are not very different from the model for the sample holder only, so that the effect of the necessary transition units is minimized. We think, that these amplitude differences are caused by losses occurring at the connections between the different sections that form the tool. For these samples, resonant frequencies, fk /k and computed permittivities are listed in Table 5.1 5.2. Analytical Methods 61 Air 0.5 Tool measurement Sample Holder model 0.4 |S11| 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 2 2.5 3 Alumina 1 0.8 |S11| 0.6 0.4 0.2 0 0 0.5 1 1.5 frequency (GHz) Figure 5.1: Measured (full tool: solid line) and modelled (ideal case: dotted line) |S11 | for Air (top) and Alumina (bottom) samples. fk (GHz) Table 5.1: Resonant frequencies, period and computed permittivity for an Air and Alumina sample. 1.51 2.95 mean values 0.50 1.01 1.52 2.04 2.54 mean values fk /k (MHz) Air 1510 1475 1492 ± 18 Alumina 500 505 507 510 508 505 ± 5 0 Computed εr 0.98 1.03 1.00 ± 0.03 9.0 8.8 8.7 8.6 8.7 8.8 ± 0.2 62 Chapter 5. Reconstruction Methods The resonances are periodic within 1% error for both air and alumina. This indicates that their permittivities are not dependent on frequency over the frequency range of interest and that the reconstructed permittivities are valid over the whole frequency range. This can also be checked by the comparison between the measured and the modelled reflection coefficient plotted in Figure 5.1. S11 has been modelled with the computed mean permittivity, that present a standard deviation of 1% for air and alumina. The modelled resonant frequencies coincide with the measured ones, validating the computed permittivity. High-loss non-frequency dependent materials We couldn’t find any material with a high loss and a constant permittivity in our frequency range of interest. However, it is interesting to take a look at the modelled reflection of such a material, see Figure 5.2. ε* = 4 − 0.2j 0.35 0.3 0.25 |S11| 0.2 0.15 0.1 0.05 0 0 0.5 1 1.5 frequency (GHz) 2 2.5 3 Figure 5.2: Modeled (ideal case) |S11 | for a sample of constant permittivity ε∗ = 4 − 0.2j. 5.2. Analytical Methods 63 The hypothetical material has a loss of about 5% of the real permittivity value, ε∗ = 4 − 0.2j. But the reflection coefficient still presents periodic minima, and the √ computed < ε∗r from these minima is 3.99. The presence of a loss results in an inaccurate calculation of the real part of permittivity, but if the loss is not very big, √ the value computed for < ε∗r is a good approximation of ε,r . The presence of a constant small loss in the material doesn’t change the location of the minima, it only varies the amplitude of these; decreasing amplitude with increasing frequency. So if a reflection pattern of periodic minima of decreasing amplitude is encountered, we can be certain that the material’s permittivity is constant over the frequency range and that it’s loss is small but not negligible. Frequency dependent materials For these materials, the mentioned method of computing permittivity from resonant frequencies for the whole frequency range is not valid since their occurrence is not periodic. It can still be used to obtain the real part of permittivity at certain frequency points from the broad band measurement, where the reflection coefficient has a minimum. This could be handy since the accuracy of the measurements is not so good in the vicinity of these minima so that the reconstruction of permittivity around them can suffer from high inaccuracy, and via resonant methods we can determine quite accurately the value of the real part of permittivity for those resonant frequencies. To compute the loss of the material from resonant frequencies is not as straight forward as calculating the real part of permittivity for non-lossy materials. Anyhow, there are no materials with high losses that maintain it constant over the frequency range so that we would not gain anything from computing the loss on a single frequency point. We refer the reader to von Hippel (1954) for further insight. 5.2.2 The propagation matrices method We introduced the notation needed for this method in Chapter 3, Subsection 3.2.1. We mentioned there that we represent both reflection coefficients [S11 , S22 ] with the term Γ and both transmission coefficients [S12 , S21 ] by Υ. This enables us to write only one set of equations but we have to keep in mind that there are two sets of measurements, [S11 , S12 ] and [S22 , S21 ], one pair at each port, and therefore, there will be two sets of solutions. Equation (3.12) can now be written as 1 Γ = LP R Υ 0 . (5.4) 64 Chapter 5. Reconstruction Methods The EM properties of the material under study are contained within the P matrix that can be calculated from equations (3.6) and (3.8) P= 1 2 Zp (e+γp dp − e−γp dp ) e+γp dp + e−γp dp e+γp dp + e−γp dp − e−γp dp ) 1 +γp dp Zp (e . (5.5) Now, we can rewrite equation (5.4) into a more convenient form as A B 1 Γ =P C D , C D (5.6) where A B −1 =L and =R Υ 0 . (5.7) Substituting P into equation (5.6) and eliminating the exponential terms, we find a solution for the impedance Zp of the sample under test as, A2 − C 2 . (5.8) B 2 − D2 If, instead, the impedance terms are eliminated, a solution in terms of the exponentials is found Zp2 = e+γp dp + e−γp dp AB + CD = cosh (γp dp ) = . 2 AD + BC We hence find and expression for the propagation factor as, AB+CD ) acosh( AD+BC . γp = ± dp For simplicity, we will further refer to the quotient β= AB + CD . AD + BC AB+CD AD+BC (5.9) (5.10) as β, hence: (5.11) The electric permittivity and magnetic permeability of the sample are related to 0 the impedance and propagation factor via equation (C.21). The conditions εr > 0, 00 0 00 εr ≥ 0, µ > 0 and µ ≥ 0 are physical constraints, cf. equation (2.64), and the sign for γp is determined, cf. equation (2.66), and so is the sign for the acosh function. 5.2. Analytical Methods 65 We find two coupled equations for ε∗r and µ∗r s Zp µ∗r = ∗ εr Z0 p c0 γp ε∗r µ∗r = jω the impedance method, (5.12) the propagation method, (5.13) with Zp and γp defined in equations (5.8) and (5.10), respectively. In Maxwell’s theory ε∗r and µ∗r appear coupled in characteristic combinations and therefore, any method trying to measure the permittivity will also be able to measure the permeability, specially in their coupled forms. Equations (5.12) and (5.13) are a perfect example. The first one represents the impedance of the medium and the second one the inverse of the velocity at which EM waves propagate. The coupling can be a drawback since the accurate determination of both properties is not always possible. Zp and γp are a combination of the measured reflection and transmission coefficients at the measurement planes. The propagation matrices method effectively moves the measurement planes to the interfaces of the sample holder, taking into account phase and impedance changes, as well as losses along the line. If we divide and multiply equations (5.12) and (5.13) we find the solutions for µ∗r and ε∗r µ∗r = ε∗r = c0 Zp γp , jw Z0 c0 Z0 γp . jw Zp (5.14) (5.15) We have to remember that we always obtain a pair of solutions depending on the pair of measurements used in their calculation. Let us reconstruct µ∗r and ε∗r for an air sample. Magnetic permeability and electric permittivity of air To test the validity of the Propagation Matrices method, we first apply it to modelled data. The relative permittivity and permeability of air are real and 1 for the whole frequency range. 66 Chapter 5. Reconstruction Methods * * εr µr 1.02 1.04 mod rec 1.02 0.98 1 ℜ 1 0 1 2 3 0 −3 20 1 2 3 −3 x 10 5 0 10 −5 5 −10 0 −15 ℑ 15 x 10 −5 0 1 2 frequency GHz 3 −20 0 1 2 frequency GHz 3 Figure 5.3: Modelled and reconstructed electrical permittivity and magnetic permeability of Air. In Figure 5.3 we have plotted the modelled and the reconstructed EM properties for an air sample. The method reconstructed the modelled data perfectly well, for both real and imaginary parts, except at the resonant frequencies where the reconstructed data diverges. Let us now see how it works with experimental data. In Figure 5.4 we have plotted the standard and the reconstructed EM properties for an air sample. This time, the reconstructed data diverges significantly from the true value of 1, for both real and imaginary parts, specially at the resonant frequencies. 5.2. Analytical Methods 67 * * εr µr 1.5 2.5 std rec 2 ℜ 1 1.5 0.5 0 0 1 1 2 3 0.5 0.5 0 1 2 3 2 1.5 0 ℑ 1 0.5 −0.5 0 −1 0 1 2 frequency GHz 3 −0.5 0 1 2 frequency GHz 3 Figure 5.4: Standard and reconstructed electrical permittivity and magnetic permeability of Air. The propagation matrices method works perfectly well with modelled data but fails to accurately reconstruct the EM properties of the sample from true data. For an explanation, we have to look at the modelled and experimental Zp and cosh (γp dp ) for air, see Figure 5.5. The modelled Zp and cosh (γp dp ) are very stable and this makes the method work perfectly with modelled data. But the experimental Zp suffers from instability while cosh (γp dp ) does not. This Zp instability affects the reconstruction of the permeability and permittivity. From the expressions (5.8) and (5.9), it is clear, that Zp is more prone to be unstable than cosh (γp dp ). However, cosh (γp dp ) is not exempt of problems. For materials with no or small loss, the imaginary part of cosh (γp dp ) is close to zero which causes the imaginary part of the acosh(β) to be poorly determined. In Figure 5.6 we have plotted the real and imaginary parts of the experimentally determined cosh (γp dp ) for an air 68 Chapter 5. 4 2.5 x 10 Z Reconstruction Methods cosh(γ d ) p p p 1 from exp from mod 0.9 2 0.8 amplitude 0.7 0.6 1.5 0.5 1 0.4 0.3 0.5 0.2 0.1 0 0 1 2 Frequency (GHz) 3 0 0 1 2 Frequency (GHz) 3 Figure 5.5: Experimental and reconstructed Zp and cosh (γp dp ) for an Air sample. sample (left hand side of the figure) together with its acosh, in the center of the figure. It is clear that =[acosh(β)] suffers from computational instabilities, but it can be amended by applying a careful unwrapping. In the right hand side of Figure 5.6 we have plotted the real part of acosh(β) and its linearized imaginary part. Two techniques for linearization can be applied. If =(β) is of the order of measurement accuracy it could be set to zero and then the instabilities disappear, although its unwrapping still has to be amended, according to the sign choice given by equation (2.66). In case =(β) is bigger than measurement accuracy, as it is for certain frequencies for air, we cannot just set a linear imaginary part of acosh(β), but instead we take its absolute value and again amend its unwrapping. 5.2. Analytical Methods 69 β acosh(β) 1 ℜ 0.5 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 −0.5 −1 1 2 3 0.05 ℑ acosh(β) 0.1 0 1 2 3 0 8 8 6 6 4 4 2 2 1 2 3 0 0 −0.05 1 2 3 Frequency (GHz) −2 experimental 1 2 3 Frequency (GHz) 0 −2 linearized 1 2 3 Frequency (GHz) Figure 5.6: Experimental β for an Air sample (left hand side), its acosh in the center, and in the right hand side the same acosh with the imaginary part linearized. Zp is experimentally unstable and β = cosh (γp dp ) can produce numerical instabilities for low loss materials. Therefore, any solution including Zp will suffer from instability, making the method less accurate, and any solution including cosh (γp dp ) should be revised in the case of low loss materials, and amended if necessary. Since the propagation matrices method reconstructs the permittivity and permeability of a sample from both, Zp and cosh (γp dp ), it does not do it accurately. However, we deal with non-magnetic sands and, in the next section, we eliminate Zp from the solution, making it more robust. In that sense, we compute the velocity of propagation from equation (5.13) and assume the magnetic permeability to be that of free space so we can compute the permittivity. 70 Chapter 5. Reconstruction Methods Non-magnetic materials For non-magnetic materials, the relative magnetic permeability is 1, and then, two different solutions for the permittivity can be derived from equations (5.12) and (5.13) Zp Z0 2 ε∗r = ε∗r c0 γp 2 = jω the impedance method, (5.16) the propagation method, (5.17) We have already seen, that the Zp method will be very unstable and we will only use the propagation method. However, it is important to explicitly write these two solutions, as we will show in the coming section, Subsection 5.2.3, that all the analytical solutions published till now, are these same two methods. But first, let us see how it works on actual experimental data Non-frequency dependent materials: AIR In Figure 5.7 we have plotted the standard and the reconstructed electrical permittivity for an air sample with the PM propagation method, for both pairs of measurements, [S11 , S12 ] and [S22 , S21 ] . The reconstructed permittivity presents three significant features: around 500 MHz on both the real and the imaginary parts, and below that frequency, the accuracy of the [S22 , S21 ] pair worsens. Below 250 MHz both solutions start to diverge. And at 1.5 GHz there is a clear jump. These features were already predicted from the measurements characteristics in Section 4.4.2, although the experimental accuracy threshold of 1% is lower (500 MHz) than predicted (650 MHz). From 500 MHz to 3 GHz the error of the real and imaginary parts of both solutions is always between ±1%, as predicted in Chapter 4. The 5% threshold is set below 50 MHz and the 2% around 250 MHz . A result within 1% for air, for such a broad band, is a great achievement. It is also a confirmation of the accurate reconstructions we expect for materials of unknown permittivity, since the air result is a measure for the errors introduced by the tool. 5.2. Analytical Methods 71 ε* r 1.04 1.02 ε , 1 0.98 0.96 0.94 0.5 1 1.5 2 2.5 3 0 −0.01 ε ,, −0.02 −0.03 std PM rec11,12 PM rec −0.04 22,21 −0.05 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 5.7: Standard and PM reconstructed permittivity for an Air sample. Frequency dependent materials: ETHANOL Applying the same method to ethanol (99.9% pure), a polar liquid, whose permittivity varies along our frequency range, we find the reconstructed permittivity (dashed line) plotted in Figure 5.8 for the pair [S11 , S12 ] and Figure 5.9 for [S22 , S21 ]. The fitted Debye model is also shown (dotted line). The reconstructed permittivities are very well captured by the frequency dependent Debye model. They present three significant features: around 300 MHz, the imaginary part of εrec deviates from the Debye fitted model. Below 50 MHz the solution starts to diverge. And at high-frequencies, above 1.5 GHz the influence of noise is clear. These frequency limits correspond to the ones predicted in Section 4.4.2 for ethanol. Above 1.5 GHz the signal is no longer transmitted through the sample and the Network Analyzer has difficulties determining the phase of the transmission coefficient, boosting up the inaccuracies in the reconstructed permittivity, however, the general trend is captured. 72 Chapter 5. Reconstruction Methods * εr 30 PM rec 11,12 Debye fit ε , 20 10 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0 ε ,, −5 −10 −15 Figure 5.8: PM reconstructed permittivity of ethanol (dashed line), for the pair [S11 , S12 ], and the fitted Debye model (dotted line). * εr 30 PM rec 22,21 Debye fit ε , 20 10 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0 ε ,, −5 −10 −15 Figure 5.9: PM reconstructed permittivity of ethanol (dashed line), for the pair [S22 , S21 ], and the fitted Debye model (dotted line). 5.2. Analytical Methods 73 The fitted Debye model and the reconstructed permittivities differ less than 1% between 300 MHz and 2 GHz. If the Debye parameters are changed by 1% this difference increases between 2 and 3 times. The fitted Debye parameters are listed in Table 5.2. EthanolGemert Ethanolfit 11,12 Ethanolfit 22,21 T (◦ C) 24 23 23 εs 25 ± 1 25.2 26 ε∞ 4.6 ± 0.7 4.6 4.6 fr (M Hz) 900 ± 60 866 834 σdc (µS/m) 0.8 ± 0.1∗ 6 27 Table 5.2: Experimental and fitted Debye parameters. Data for ethanol taken from van Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ. The fitted Debye parameters have been optimized to the reconstructed permittivities. Note that the fitted parameters are within the error bounds of the van Gemert values. The conductivity given as a van Gemert parameter was, in fact, measured in the laboratory. The fitted conductivity is orders the magnitude bigger, but this enables the Debye model to capture the low frequency behavior of the imaginary part of the reconstructed permittivity. We have shown that the combination of the propagation matrices, PM and the propagation method reconstructs satisfactorily the permittivity of non-frequency and frequency dependent non-magnetic materials without a-priori information. It shows frequency thresholds predicted from measurement characteristics. It is simple to program and very fast. We can learn more about it in the next Subsection 5.2.3, where we will simplify the problem to the ideal case to prove that all other analytical solutions are equivalent, and therefore, suffer from the same problems. 5.2.3 Analytical solutions for the ideal case As mentioned in the introduction of this chapter, many researchers published different analytical solutions. They all involve the knowledge of the S-parameters at the interfaces of the sample holder. That is, they all were derived for the ideal case presented in Chapter 3. To prove that in fact they are all the same solution let us first write the expressions for the Zp -γp methods. We can find A, B, C and D from equation (5.7) for the ideal case and then rewrite equations (5.16) and (5.17) as 74 Chapter 5. (1 − Γ)2 − Υ2 (1 + Γ)2 − Υ2 2 c20 1 + Υ2 − Γ2 ∗ εr = − 2 2 acosh ω dp 2Υ ε∗r = Reconstruction Methods the Zp method, (5.18) the γp method. (5.19) The impedance and the propagation reconstruction methods for an ideal line, equation (5.18) and equation (5.19) are already equivalent to those derived by Stuchly & Matuszewsky in (Stuchly & Matuszewsky, 1978). They claimed that the propagation equation is ambiguous for certain sample lengths, but we do not see such ambiguity since the acosh of a complex number z is well defined as √ √ acosh(z) = ln z + z + 1 z − 1 . (5.20) and physical constraints determine the sign for the complex logarithm, as expressed for equation (5.9). At resonance, where z = ±1, the logarithm is zero and the permittivity will not be accurately determined. This occurs with low-loss materials when almost no reflection and total transmission occur at resonant frequencies. Palaith and Chang (Palaith & Chang, 1983) comprise a comparison of three methods, their 1/Z 2 method is the first equation present in Stuchly and Matuszewsky (Stuchly & Matuszewsky, 1978) and of course the impedance method. The second is their so-called K 2 method that they claim to be new, while it is equal to the second equation from Stuchly and Matuszewsky (Stuchly & Matuszewsky, 1978) and the same as the propagation method, since they write: −γp dp ζ=e 1 = 2Υ 2 1+Υ −Γ 2 ± h 2 1+Υ −Γ 2 2 2 − 4Υ i1/2 . (5.21) To find γp we would take the natural logarithm of the right hand side which can then be written as a acosh in virtue of equation (5.20) ending up with the same expression for ε∗r as equation (5.19). As a third solution, Palaith & Chang (Palaith & Chang, 1983) analyze the Nicolson-Ross-Weir method, published by Nicolson and Ross in (Nicolson & Ross, 1970) and by Weir (Weir, 1974) for both magnetic permeability and electrical permittivity. We restrict ourselves to the solution for the permittivity. Their equation is as follows: ε∗r c0 1 − r ln(ζ), =j ωdp 1 + r (5.22) where r is the local reflection coefficient and ζ is expressed in terms of Γ and Υ by equation (5.21), and again we obtain the propagation method. Note that the 5.2. Analytical Methods 75 expression of equation (5.22) is somewhat misleading because the local reflection coefficient in the right-hand side contains the unknown complex permittivity of the √ ∗ h 1−r i sample under test: εr = 1+r , so that, in fact, it is not another equation but the same as the square root of equation (5.19). These methods are obviously the same and they naturally all suffer from the same problems. They are not well behaved for low-loss materials, especially at frequencies corresponding to integer multiples of one-half wavelength in the sample, see Stuchly & Matuszewsky (Stuchly & Matuszewsky, 1978), or in other words at resonant frequencies. Ligthart (Ligthart, 1983) and Baker-Javis et al. (Baker-Javis et al. , 1990) have tried to by-pass this ill behavior in two different ways but using the same equations. Ligthart (Ligthart, 1983) presented a method for shorted line measurements where the scattering equations for the permittivity were solved over a calculated uncertainty region and the results were then averaged, but he could not avoid the low-loss problem. Finally, Baker-Javis et al. (Baker-Javis et al. , 1990) minimized the instability by an iterative procedure, without truly solving the problem. 5.2.4 Compensation Parameters The compensation method has been used by several authors. Kruppa & Sodomsky (1971) published the system of equations needed and Freeman et al. (1979), Shen (1985) and Chew (1991) applied it, respectively, to two commercial fixtures and a large customized cell. It is based on a Scattering Matrix representation (Section 3.2) of the coaxial transmission line, dividing it into three major sections each represented by its own scattering matrix. This method accounts for experimental errors introduced by the transition units. The S-parameters of the sample holder only are obtained from the S-parameters measured at the reference planes, by a cumbersome set of operations. Six short circuit measurement and a through measurement of only the transition units connected together are needed to compensate and to characterize the scattering matrix of both transition units. Figure 5.10 is an example of this method applied to modelled data of an air sample. On the left hand side, we have plotted the amplitude and phase of both reflection and transmission of the whole tool. In the middle of the figure, the two plots correspond to the compensated parameters, or in other words, the S-parameters of the sample holder only. The amplitudes remain unchanged while the phases are different. This is easy to understand if we consider that this is a modelled case in which both transition units behave ideally and therefore, the compensation is only a phase move-out. 76 Chapter 5. Smod r sample holder 1 1 0.8 2 0.8 12 |S | and |S | ε, Smod tool S11 S12 0.6 1 0.2 0.2 11 0.4 1 2 1.5 0.6 0.4 0 Reconstruction Methods 3 0 0.5 1 2 3 0 1 2 3 θ(S11) and θ(S12) (°) ,, εr 0 100 100 0 0 −0.02 −0.04 −0.06 −100 −100 1 2 3 Frequency (GHz) −0.08 1 2 3 Frequency (GHz) −0.1 1 2 3 Frequency (GHz) mod Figure 5.10: Modelled S-parameters of the whole tool (Stool ) and of only the sample mod holder (Ssampleholder ) for an air sample and the computed permittivity. In the right hand side of Figure 5.10 we find the reconstructed permittivity from the compensated parameters with the propagation method. The real part of the permittivity is well detemined, while the imaginary part is well determined except for two frequency points. These frequencies are related to resonances. In the compensation equations there is one term obtained as the inverse of a subtraction between two short-circuited measurements. These measurements introduce new resonances from the teflon filling of the tool, that are not present in the data measurements, and propagate along the compensation. For actual measurements this method doesn’t behave that well in our spectrum, precisely due to the propagation of the mentioned resonances. In Figure 5.11 we have plotted the amplitude and phase of the S-parameters of the whole tool at the left hand side, and in the center the S-parameters of only the sample holder exp compensated with actual measurements, Ssampleholder . While, in the right hand exp mix side S has been compensated with modelled data, Ssampleholder . 5.2. Analytical Methods 77 Sexp Sexp tool sample holder 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.8 12 |S | and |S | 1 S11 S12 11 0.6 0 θ(S11) and θ(S12) (°) Smix sample holder 1 2 3 0 1 2 3 0 100 100 100 0 0 0 −100 −100 −100 1 2 3 Frequency (GHz) 1 2 3 frequency (GHz) 1 2 3 1 2 3 frequency (GHz) exp ) and of only the sample Figure 5.11: Experimental S-parameters of the whole tool (Stool exp mix ) compenholder (Ssampleholder ) compensated with experimental data and (Ssampleholder sated with modelled data. From the figure it is clear that the compensated parameters with true measurements suffer from instability, with major deviance at resonant frequencies. Reconstructing permittivity from these parameters will not succeed. However, if S exp is compensated with modelled measurements the instability is reduced to only two frequency points. This technique involves only compensating for the phase move-out from the reference planes of measurement to the sample holder, or in other words, the travel distance along the transition units. Previous authors successfully applied this method to their tools because they were air filled and their spectrum was such that the resonances did not affect the calibration since they fallen out of it. Chew et al. (1991) applied it in a broadband modifying the compensation in the high frequency range by using modelled short circuited reflections. 78 Chapter 5. Reconstruction Methods The compensation is unstable for broad frequency measurements and it has to be modelled to succeed in the reconstruction of the permittivity. As an example, we discuss the reconstructions for air and ethanol. Non-Frequency dependent materials: AIR In Figure 5.12 we have plotted the reconstructed permittivity from the modelled compensated experimental S-parameters (right hand side of Figure 5.11), here indicated as CO. This reconstruction presents the same features observed in the PM reconstruction. The same frequency limits and accuracies hold. And their major differences are the instability points already foreseen in the modelled S-parameters example. It is very similar to that found with the Propagation Matrices method since, in essence, they are very similar methods. * εr 1.04 1.02 ε , 1 0.98 0.96 0.94 0.5 1 1.5 2 2.5 3 0 −0.01 ε ,, −0.02 −0.03 std CO rec11,12 CO rec22,21 −0.04 −0.05 0.5 1 1.5 Frequency (GHz) 2 2.5 Figure 5.12: Standard and CO reconstructed permittivity for an Air sample. 3 5.2. Analytical Methods 79 Frequency dependent materials: ETHANOL Applying the same compensation procedure to the ethanol measured S-parameters, as that employed with air, the reconstructed permittivities (dashed line) plotted in Fig. 5.13 for the pair [S11 , S12 ] and Fig. 5.14 for [S22 , S21 ], show again the same features as the ones reconstructed with the propagation matrices method, in Section 5.2.2, although the difference between the fitted Debye model (dotted line) and the reconstructed ε∗r is worse. The two peaks present in the air permittivity reconstruction due to resonances in the compensation do not show in the reconstruction of the permittivity of ethanol because their effect is of smaller amplitude due to the very small transmission above 1.5 GHz. With this method, however, the fitted Debye model and the reconstructed permittivities differ less than 1% between 800 MHz and 1.5 GHz for the real part and less than 2% for the imaginary part. The fitted Debye parameters are listed in Table 5.3. EthanolGemert Ethanolfit 11,12 Ethanolfit 22,21 T (◦ C) 24 23 23 εs 25 ± 1 25.3 24.9 ε∞ 4.6 ± 0.7 4.6 4.6 fr (M Hz) 900 ± 60 906 926 σdc (µS/m) 0.8 ± 0.1∗ 7600 8400 Table 5.3: Experimental and fitted Debye parameters. Data for ethanol taken from van Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ. The Optimized Debye parameters have been fitted to the CO reconstructed permittivities. Note that the fitted parameters are within the error bounds of the van Gemert values. The conductivity given as a van Gemert parameters was, in fact, measured in the laboratory. The fitted conductivity is orders the magnitude bigger, but this enables the Debye model to capture the low frequency behavior of the imaginary part of the reconstructed permittivity. 80 Chapter 5. Reconstruction Methods * εr 30 CO rec 11,12 Debye fit ε , 20 10 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0 ε ,, −5 −10 −15 Figure 5.13: CO reconstructed permittivity of ethanol (dashed line), for the pair [S11 , S12 ], and the fitted Debye model (dotted line). * εr 30 CO rec 22,21 Debye fit ε , 20 10 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0 ε ,, −5 −10 −15 Figure 5.14: CO reconstructed permittivity of ethanol (dashed line), for the pair [S22 , S21 ], and the fitted Debye model (dotted line). 5.2. Analytical Methods 5.2.5 81 Propagation Matrices vs Compensation Parameters The difference between the PM and the CO reconstructed permittivities, for both air and ethanol and both their real and imaginary parts, above their 2% accuracy thresholds (250 MHz for air and 50 MHz for ethanol) is of the order of measurement accuracy (10−3 ). This reinforces the similarity of the results of these methods for an almost ideal tool as ours. If the tool had not been designed as a match load and big impedance contrasts were to be present amongst the different sections that form the transition units the results from the Propagation Matrices method will differ from the ones computed with the modelled compensation slightly more, since the latter does not correct for them, and its reconstruction will deteriorate. Ideally, the CO method with experimental compensation would, in principle, be the most suitable one as it characterizes experimentally the true electromagnetic behavior of the transition units. However, apart from being a tedious task, since it includes dismantling and connecting the different sections of the tool many times, which is prone to experimental errors, it also limits the frequency range of the measurements. This limitation, as we have seen, comes from the fact that the experimental compensation suffers significantly from resonances and, instead, a modelled compensation must be performed. Therefore, the PM method is more suitable to correct for the effect of the transition units, as it accounts for both impedance and phase changes. In addition to this, the fit of the Debye model to the CO reconstructed permittivities for ethanol is worse than that of the PM. In Section 5.4 we will compare the analytical and optimized reconstructions, but since, the PM method provides a better reconstruction than the CO method, we will take the PM method as representative for analytical solutions. We can already conclude that for broadband permittivity reconstruction from S-parameters measurements via analytical expressions a good and accurate model of the tool in use is mandatory. Any of the existing methods involve certain modelling of the transition units connecting the sample holder to the Network Analyzer, and therefore, the calibration of the model to true measurements is of great importance. The deviations between the tool measurements and the model limit the accuracy of the reconstruction. 82 5.3 Chapter 5. Reconstruction Methods Optimization methods This technique does not use analytical expressions to reconstruct the permittivity, instead, it minimizes a cost function involving the measured and modelled S-parameters of the tool. The attainment of the S-parameters of the sample holder only is not an issue, but the convergence of the method can become one. They are necessary if only reflection or transmission is measured, since the analytical methods require always a combination of the two. The optimization methods can be very varied in cost functions to minimize and the minimizing procedure is limited to non-linear techniques. Belhadj-Tahar et al. (1990) and Taherian et al. (1991) applied this method to a coaxial-circular waveguide. The first applied a gradient technique to an L2 norm, and the second, a modified Newton method to an L1 norm. Nguyen (1998) used a Nelder-Mead Simplex method on shorted reflection data of a coaxial waveguide. We have used a Quasi-Newton algorithm implemented in the optimization toolbox of Matlab, (1996), as fminunc. It is an unconstrained nonlinear optimization routine that saves computing time by approximating the Hessian with an appropriate updating technique. It is limited to real numbers and we by-passed the problem by splitting the output of the cost functions to their real and imaginary components. The tolerance was set to measurement accuracy and computing time varied with the cost functions. The optimization is done per frequency and over the full spectrum. An initial guess of the permittivity of the sample, for the first frequency point, is given and S mod is computed with the forward model. By minimizing a specific cost function, the optimized permittivity is calculated and used as initial guess for the next frequency point. This is repeated until the permittivity of the sample has been reconstructed for the whole frequency range. We have tested different cost functions to study their performance. With modelled data all of the tested functions reproduced the permittivity introduced in the model, but with true measurements they varied in their performance. We tried normalized and non-normalized cost functions. We only show the results obtained with normalized cost functions, which showed a better performance. We have made a summary of these in Table 5.4. Both cost functions involving products of the S-parameters diverge for high-loss materials whose permittivity is a function of frequency. And their performance with low-loss materials with constant permittivity is worse than that of L1 and L2 norm. 5.3. Optimization methods 83 Table 5.4: Optimized cost functions and summary of performance. Methods based on the L1 and L2 norm succeed in reconstructing the permittivity of air but have to be weighted when ethanol is considered. L2 fails for frequencies above 1.5 GHz. This weighting is necessary because of the small transmission present in highly lossy materials above certain frequencies and the sole reliability in reflection is not enough for the optimization to converge. However, a careful weighting has to be done because if the transmissions are emphasized too much the solution does not converge either. The weighting consists in finding the proper reflection/transmission ratio that will produce a convergent optimization. We also tested the performance of these same cost function but using only one pair of reflection and transmission measurements instead of the four of them. We obtained the same behavior, but the accuracy was slightly worse. It is then preferable to use the complete Scattering matrix with L1 norm as the best suitable cost function to optimize, i.e. first row of Table 5.4. Its results are presented in the next section. 84 Chapter 5. Reconstruction Methods ε* r 1.04 1.02 ε , 1 0.98 0.96 0.94 0.5 1 1.5 2 2.5 3 0 −0.01 ε ,, −0.02 −0.03 std OP rec −0.04 −0.05 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 5.15: Standard and OP reconstructed (dotted line) permittivity for an Air sample. ε* r 30 OP rec Debye fit ε , 20 10 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0 ε ,, −5 −10 −15 Figure 5.16: OP reconstructed permittivity of ethanol (dashed line), and the fitted Debye model (dotted line). 5.3. Optimization methods 85 Non-frequency dependent materials: AIR In Figure 5.15 we have plotted the OP reconstructed electrical permittivity for an air sample. As with the analytical methods, the OP reconstruction worsens below 500 MHz on both the real and the imaginary parts and below 250 MHz the solution starts to diverge. This time, the resonant frequency, at 1.5 GHz, is not seen in the OP reconstructed permittivity. From 500 MHz to 3 GHz the accuracy of the solution is mostly of the order of 1% for the real part, but this time, the imaginary part is within measurement accuracy. This result is very similar to that obtained with the analytical methods, slightly worse for the real part and better for the imaginary part. Frequency dependent materials: ETHANOL With ethanol the optimized reconstruction is slightly different to that obtained with the analytical methods. In Figure 5.16 we have plotted the OP reconstructed electrical permittivity for this material. The solution holds up very well in the entire frequency range and only below 10 MHz it starts to diverge. At high-frequencies, above 2 GHz, the influence of noise is clear, and at the very end of the spectrum 0 a few points are off for εr . The fitted Debye model and the reconstructed permittivities differ less than 1% between 300 MHz and 1.5 GHz for the real part and from 500 MHz to 1.5 GHz for the imaginary part. The fitted Debye parameters are listed in Table 5.5. EthanolGemert Ethanolf it T (◦ C) 24 23 εs 25 ± 1 24.7 ε∞ 4.6 ± 0.7 4.4 fr (M Hz) 900 ± 60 922 σdc (µS/m) 0.8 ± 0.1∗ 700 Table 5.5: Experimental and fitted Debye parameters. Data for ethanol taken from van Gemert (1972). ∗ The conductivity has been measured with a conductivity meter in-situ. The fitted Debye parameters have been optimized to the OP reconstructed permittivities. Again, the fitted parameters are within the error bounds of the van Gemert values, and the low frequency behavior of the imaginary part of the reconstructed permittivity is captured by the conductivity. 86 Chapter 5. Reconstruction Methods ε* r 1.04 1.02 ε , 1 0.98 0.96 0.5 1 1.5 2 2.5 3 0 −0.01 ε ,, −0.02 −0.03 std PM OP −0.04 −0.05 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 5.17: PM and OP reconstructed permittivity for an Air sample. |ε* −ε* |/|ε* | rec fit rec 0.05 PM OP 0.04 ℜ 0.03 0.02 0.01 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 Frequency (GHz) 2 2.5 3 0.05 0.04 ℑ 0.03 0.02 0.01 0 Figure 5.18: Differences between the PM and OP reconstructed permittivities for Ethanol and their respective fitted Debye models. 5.4. Analytical vs Optimization 5.4 87 Analytical vs Optimization Along this chapter we have presented the different methods and their independent reconstructions. It is now time to compare the reconstruction of permittivity via analytical and optimization methods. There are three ways to analytically reconstruct the permittivity of a sample from measured S-parameters. We presented them together with experimental results in Section 5.2. The determination of permittivity from resonant frequencies is limited to low-loss materials that present a constant permittivity over the frequency range and therefore we will not compare it to any other reconstruction method. We propose it as an alternative for the reconstruction of low-loss materials. In Subsection 5.2.5 we already stated our preference for the Propagation Matrices method above the Compensation Parameters. Although, we showed that the results difference was of the order of measurement accuracy, but for ethanol, the fit of the Debye model performed better on the PM reconstructed permittivities. The analytical reconstructions solve for two permittivities, depending on which pair of measurements is used. For simplicity, in this section we only show the results of a single pair, as the results do not vary significantly if the other pair is chosen. The optimization methods are diverse in cost functions and in Section 5.3 we ruled out several of them and chose the L1 norm as the best possible. In Figure 5.17 and Figure 5.18 we have plotted the reconstructed permittivities for air and ethanol, with the Propagation Matrices method and that from optimization. For both materials, the optimization reconstruction (OP rec: dashed line) is slightly better than the Propagation Matrices reconstruction (PM rec: dotted line). The two main accuracy thresholds, 2% and 5%, vary for reconstruction methods, materials and real and imaginary parts of permittivity. The air reconstruction is peculiar, in its real part, because of the jump present at 1.5 GHz where the PM reconstruction goes from overestimation at smaller frequencies to underestimation, while the OP reconstruction stays overestimating its value. However, both reconstructions lie within ±1%, and the presence of the jump confirms the resonance of air at that frequency. For the imaginary parts, the optimization is definitely superior to the propagation matrices method. It is clear that both methods exceed 5% accuracy in the real part below 50 MHz and around 250 MHz they exceed the 2% threshold. Ethanol follows a less complicated trend, and both solutions are well captured by the Debye model. It is not conclusive whether the propagation matrices reconstruction is better than the optimization one or viceversa. 88 Chapter 5. Reconstruction Methods We can then conclude that any method is valid for the reconstruction of permittivities, whether they are low-loss and constant over the frequency range or highly lossy and frequency dependent. We could, maybe, argue that the optimization reconstruction is slightly better in its accuracy and that its solution is more stable for low frequencies. It would be, then, most adequate to use both methods to obtain results in a broader spectrum. However, for the rest of this thesis we only use the propagation matrices method because the optimization is very time consuming and it is necessary to tune the cost function to each particular material, as the weighting needed for its convergence depends in the ratio of the amplitudes of reflection and transmission. On the contrary, the propagation matrices method is very fast and easy to program and it always gives the same solution given the same data. 5.5 Conclusions Along the chapter we have presented and compared the different reconstruction methods for permittivity from measured S-parameters. We already evaluated their individual performance with different materials and compared the three main methods presented. This section is intended as the closing summary of the chapter, although it is not in order of appearance of the sections but in an order more in accordance to the natural progression of the research. As we have seen, there are two ways to obtain the permittivity of a sample from its measured S-parameters: analytically or via an optimization. Both methods determine the permittivity at each frequency independent of a predefined model of the frequency dependence of the permittivity. Traditionally, obtaining the frequency dependent permittivity of a material via analytical expressions was done in two steps. First, the measured S-parameters of the transmission line were moved from the measurement planes to the interfaces of the sample holder. Secondly, they were introduced into analytical expressions to compute the permittivity. The nature of these expressions allow to reconstruct one permittivity from each pair of reflection/transmission coefficients, and in the case, the full scattering matrix is measured, two solutions can be computed. There are two ways to move the measured S-parameters of the transmission line from the measurement planes to the interfaces of the sample holder. If it is an ideal tool a simple phase correction is sufficient, see Rau & Wharton (1982), but in practice this technique does not take into account errors present in true experimental set-ups. Compensation techniques were developed to experimentally account for them, it is cumbersome and very time consuming, and as we have shown in Subsection 5.2.4 it cannot be applied for broad band measurements. In fact, it must 5.5. Conclusions 89 be replaced by a modelled compensation in order to obtain reasonable results. Complementary, in this chapter, we have developed a new method, the propagation matrices (5.2.2). While it moves the measured S-parameters of the transmission line from the measurement planes to the interfaces of the sample holder taking into account impedance jumps between sections and phase changes, the nature of the solution technique allows to directly find analytical expressions to compute both permittivity and magnetic permeability. It is based on a simple concept and it is easily programmable. The analytical expressions, equations (5.12) and (5.13), found from the propagation matrices approach, depend on Zp and cosh (γp dp ), two different combinations of the corrected S-parameters. The instability of Zp is responsible for the considerable errors encountered in the reconstruction of permittivity and permeability of air. Therefore, since we are interested in non-magnetic soils, we restrict ourselves to the reconstruction of permittivity from the propagation method (equation (5.17)). It is a successful approach and we are able to reconstruct, both real and imaginary parts, of the relative permittivity of air within ±1% error and obtain for ethanol, a permittivity that fits the Debye model. For low-loss materials, =(acosh(β)) needs special attention. This approach also enabled us to show that all the existing analytical expressions are the same (Subsection 5.2). For simplicity we only considered the solutions proposed for permittivity reconstruction, but it is also valid for permeability. Writing Zp and cosh (γp dp ) for the ideal case, so that no correction is needed and they are a direct combination of the reflection and transmission coefficients at the interfaces of the sample holder (see equations (5.18) and (5.19)), showed that they were equivalent to the equations proposed by Stuchly & Matuszewsky (1978), and later by Palaith & Chang (1983). By further mathematical considerations we also showed their equivalence to the Nicolson-Ross-Weir equation. So, indistinctively of the expression used, the reconstruction of permittivity from analytical expressions always suffers from the same problems at resonant frequencies and with low-loss materials, because they are essentially the same methods. It is only now, that we have showed it. We also showed the superiority of the methods based in the cosh (γp dp ) parameter above the unstable ones based on Zp (see for example Stuchly & Matuszewsky (1978)). The results obtained with the propagation matrices and with the modelled compensation parameters are comparable because of our almost ideal tool (see Subsection 5.2.5), but their different corrections and the better fit of the Debye model to the PM reconstructed permittivity of ethanol (PM accounts for impedance and phase changes and CO only for the latter) make the PM method superior. The accuracy thresholds predicted in Chapter 4 are confirmed experimentally by the PM and CO solutions. For air, below 50 MHz the difference between the 90 Chapter 5. Reconstruction Methods true permittivity of air and the reconstructed one is bigger than 5%. And the 2% threshold is located around 250 MHz. From 500 MHz to 3 GHz the solution is always between ±1% of the true data. Around the resonant frequency of 1.5 GHz there is a clear deterioration of the solution. For ethanol, the comparison cannot be quantitative but must be qualitative, and we have shown that a Debye model can be fitted to the reconstructed permittivity. The obtained parameter lie within the accuracy interval given by van Gemert (1972). The fact that these methods present inaccuracies around resonant frequencies for low-loss materials, encouraged us to introduce the resonant frequencies method (5.2.1), within the analytical methods. To our knowledge this is the first time that it is applied for a transmission line and not for a resonator. For materials of constant ε∗r and with a low loss, it is possible to compute its permittivity for the whole frequency range from a single resonant frequency or a set of them. Their periodicity give a measure of the constancy of the property, and the amplitude of the reflection coefficient at those points indicate the presence or absence of loss. This method, then, overcomes the problems present in other analytical solutions with low-loss materials. When the reconstruction of the permittivity of a sample is not done from analytical expressions, it must be reconstructed via an optimization procedure. In Section 5.3 we proved that the normalized L1 norm is the best cost-function to minimize (of the ones we considered) and that the results obtained with it are slightly better than those computed with the PM method, specially in the lower limit of the spectrum. It would be, then, most adequate to use both methods to obtain results in a broader spectrum. However, for the rest of this thesis we only use the propagation matrices method because the optimization is very time consuming. Based on the results obtained with the propagation matrices method for air, we expect to reconstruct permittivities within ±1% error from 3λ/L up to 3 GHz, while the lower limit can be relaxed by using optimization. However, for lossy materials, this upper limit is conditioned by its loss, as the transmission gets smaller the accuracy of the phase of the measurements deteriorates taking its toll in the reconstructed permittivity, that clearly presents noisy behavior. However, it stills captures its trend, and it can be amended with noise reduction techniques. All our science, measured against reality, is primitive and childlike — and yet it is the most precious thing we have. Albert Einstein (1879-1955) Chapter 6 Permittivity states of mixed-phase; two and three component sands This chapter is dedicated to the study of the permittivity of dry, partially and fully saturated sands, and how the distribution of saturation affects the EM properties. 6.1 Introduction In Chapter 4 we saw that the probe used to measure the permittivity of our samples allows for fluid flow through them, and in Chapter 5 we studied different reconstruction techniques for the permittivity of calibration samples. It is time to put our tool to work and investigate its performance in the measurement of samples at different saturations. After all, the main interest for field applications, is to find out the fluid content of the soil from permittivity measurements. To do this, we compare our measurements with existing Mixing Laws, see Appendix E. Furthermore, since our tool allows for fluid flow, it also enables us to study different saturation distributions and their influence. The sample can be saturated in different ways, fluid flowing perpendicular or parallel to the wave’s direction of propagation and at any angle in between, or, as most researchers do, it can be saturated outside the tool and then filling the sample holder with it. All these saturation techniques generate different saturation distributions in the sample even when the average saturation is the same, but how this influences the permittivity measurements has not been studied yet. 91 92 Chapter 6. Permittivity states of mixed-phase In this chapter we compare the different saturation techniques and we show how a simple 2-layer model can account for the case in which the flow is perpendicular to the direction of propagation of the waves, and we can find the fluid front position from the measured S-parameters. It is now important to clarify two troublesome terms, homogeneous and heterogeneous. They come from ancient Greek and literally mean of the same kind and other kind. These nouns and their adjectival and adverbial forms are widely used in this chapter in particular, and in science in general, with a slightly altered meaning, or should I say scale? Science is renown for its relativeness and these terms are not exceptions to the rule. A certain volume of sand with fluid in its pores is certainly heterogenous at the microscale but we can refer to it as homogenous in the mesoscale if the fluid is homogeneously distributed in that volume. The sample as a whole turns into an entity with a homogenous average saturation, porosity, permeability and even permittivity. What happens when the fluid is not evenly distributed, for example in the case of layering? In this chapter, we will then, be referring to two different heterogeneities, one at the microscale, pore to pore, and the other one at the mesoscale, differently saturated sections of the sample. The correct use of the terms is intrinsically linked to the scale of interest and to the physical property under study. A single porosity might be representative of a certain volume of sand but the distribution of saturation may be so that an average saturation does not correctly describe the sample. Therefore, it is important to place the terms in the proper context to correctly understand its meaning. In the specific case of referring to the permittivity of a sample it is important to remember that the forward model, used along this thesis, considers the sample holder as a whole and the Propagation Matrices method reconstructs a single effective permittivity for each pair of reflection/transmission parameters, even when the sample holder is filled with two or more materials. As we will see in the coming sections the reconstructed permittivities are sensitive to certain heterogeneities. 6.1.1 Chapter Outline In Subsection 6.2.2 we give a detailed description of the samples, the experimental procedures for the sample preparation and saturation and the S-parameters measurements. And in Subsection 6.2.3 we present the permittivities of three samples saturated in different ways. The permittivities are computed from the measured S-parameters with the Propagation Matrices method presented in Chapter 5. In Section 6.3 we show the advantages of averaging the two permittivities obtained with the mentioned method over averaging the S-parameters and in Section 6.4 we 6.2. Saturation technique and permittivities 93 model the case in which the flow is perpendicular to the direction of propagation of the waves with a simple 2-layer model. In Section 6.5 we study the anomalies present in the reconstructed permittivities of the 2-layer samples, whereas in Section 6.6 we compare our measurements with existing Mixing Laws, and finally, we conclude this chapter with Section 6.7. 6.2 Saturation technique and permittivities To study the effect of fluid flow on the permittivity of the samples under investigation, we performed three sets of experiments for comparison. One in which the flow was assumed to be perpendicular to the propagation of the TEM waves, another in which it was parallel, and a third one in which the saturation was homogenously distributed over the sample, for control purposes. For each case we measured 6 steps, from a fully dry sample to a fully saturated one. These are explained in detail in Subsection 6.2.2, while in Subsection 6.2.3 we present the reconstructed permittivities and their comparison in Subsection 6.2.4. Let us first give a detailed description of the samples. 6.2.1 Samples All the samples were from loose quartz grains with grain sizes in the range [112 − 280]µm. Their grain size distribution is shown in Figure 6.1. Two thirds of the grains have radii of 160 or 180 µm. This narrow distribution of radii ensures homogeneous packing and a fairly close range of porosities among different samples. Grain Size Distribution of Quartz 0.5 0.45 0.4 Mass Ratio 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 100 120 140 160 180 200 Grain Size (µm) 220 240 260 280 Figure 6.1: Grain size distribution for the quartz grains that form the samples. 94 Chapter 6. Permittivity states of mixed-phase The quartz grains were dried in an oven for two days at 70◦ C to ensure that no water was present at the beginning of each experiment. The sample holder was filled with these grains in different ways, as explained in the next section, and the fluid used to flow into the sample was distilled water. The temperature in the lab remained in the range [21 − 23]◦ C throughout all the experiments. The porosities of the samples were computed from mass-volume relationships and their inherent error is of the order of 0.5%. 6.2.2 Saturation technique In the three flow experiments, the filling of the sample holder with the dry grains differed. Since the resulting porosities were not equal for all the samples, we not only looked at water saturation (Sw ) in volume ratio, but also at the mass ratio water/quartz (mw /mq ) to be able to compare results. The measurements are very sensitive to sample preparation. No-Flow: saturation homogeneously distributed To obtain a homogeneously distributed saturation, we added the desired amount of water to a certain weight of quartz grains and stirred it well until no visible lumps were present. Then, the sample holder was filled up, from above, with the mixtures and the final porosity and saturation of the samples were computed from mass measurements. Figure 6.2 represents these samples. Figure 6.2: No-Flow: saturation homogeneously distributed. 6.2. Saturation technique and permittivities 95 Filling the sample holder with already made partially saturated samples resulted in samples with different porosities, and although we tried to ensure a homogenous distribution of water throughout the sample the filling of the sample holder could be responsible for some layering and/or trapped air. One could argue that with this type of set-up and due to gravity, the distribution of water in the sample varies with time. However, the sample is only 10 cm long and the time it takes from the moment of sample preparation to the measurement of the S-parameters is in the order of 10 minutes. Whereas gravity will take over the capillary forces in matter of several hours. Flow perpendicular to the direction of propagation: Gravity drainage All throughout this experiment, the sample holder is maintained perpendicular to the ground. Figure 6.3 shows a schematic representation of this procedure. The sample holder was first filled with water and then dry quartz grains were carefully added so that the first measurement was done on a fully saturated sample (SW ). During the whole filling process the tool is under vibration to help the compaction of the grains and to remove possible trapped air. This technique resulted in the lowest porosity for all three experimental sets, 40%. Later we let the water flow out with the help of gravity (top to bottom), in steps. When no more water flows out of the sample, we assume the sample to have a residual water saturation (Sr ). Note that this is just a way of naming the saturation of this step and that it is not the true residual water saturation. It refers to the obtained saturation at the last step of a gravity drainage. To dry up the sample (S0 ) we connected a vacuum pump to the fluid outlets of the sample holder and let it pump for 24 hours. The amount of water that came out was monitored by a balance and because of the geometry of the set-up, the tubes connecting the sample holder to the water balance were always full, so that saturation determination was very simple and precise. At every step we monitored the changes in the S-parameters and the measurement was taken when there was no noticeable change in them, sometimes having to wait for a few minutes. At the end, we had determined the S-parameters of the sample in several stages of saturation, from fully dry to fully saturated. This gravity drainage is possible with enough height between the sample and the balance. In our case, this was of 1.25 meters. This is a very porous sample with most grains being of the same size, and although a sharp interface between the residual and fully saturated regions is a simplistic assumption, we can think of the fluid front as being perpendicular to the direction of propagation of the TEM waves. 96 Chapter 6. Permittivity states of mixed-phase Figure 6.3: Flow perpendicular to the direction of propagation. We also tried to perform imbibition experiments, but those resulted in misleading results. During the imbibition, the grains compacted and a fine layer of water formed on top of the sample. Flow parallel to the direction of propagation First, the sample holder is put in an upright position and filled with the dry grains. By this procedure we obtained a porosity of 43%. Then, the sample holder is turned and maintained parallel to the ground and the fluid is injected step by step via a controlled syringe until it overflows. Figure 6.4 shows this procedure. The amount of water that is injected into the sample is measured with the controlled syringe, but there is uncertainty to it because of trapped air in the tubes connecting the sample holder with the in-flow system. It was not possible to maintain a constant filling of the tubes and therefore the amount of water in the sample holder was not accurately known. In this type of saturation it is hard to know the distribution of the fluid front, and with the flow inwards a dry sample there is risk of grain replacement and further compaction. 6.2. Saturation technique and permittivities Figure 6.4: Flow parallel to the direction of propagation. 97 With these three different techniques we saturated very similar samples. In the next subsection, we show results for all cases and compare them. We should keep in mind that the permittivity reconstruction with the Propagation Matrices method yielded two permittivities, one computed with the pair [S11 , S12 ] and another with the pair [S22 , S21 ] (Subsection 5.2.2). From now on, we will refer to these parameters as ε∗12 and ε∗21 so that it is clear from which S-parameters they have been calculated. 6.2.3 Permittivities for the differently saturated samples For each technique, six different saturations have been reached, from fully dry to fully saturated. For the no-flow case, the saturation is assumed to be homogeneously distributed, but for the other two cases, it is not and therefore, the value given corresponds to an effective saturation. Furthermore, as mentioned earlier, the porosities are not equal for all samples due to the experimental procedure, and in order to be able to compare the results it is more convenient to refer to the gravimetric moisture content, mw /mq (water/quartz mass ratio). Unfortunately it was not possible to obtain samples with equivalent ratios and the comparison is more qualitative than quantitative. Each subsection corresponds to a saturation technique, and in each of them, we list the measurement steps together with the porosity of the sample and their effective water saturation, in volume and mass ratio. Errors are stated when significant. We also show ε∗12 and ε∗21 for each step. 98 Chapter 6. Permittivity states of mixed-phase Homogeneously mixed 30 6 ε , 20 5 10 4 3 2 1 0 0.5 1 1.5 2 2.5 0 1 2 3 −1 4 −2 5 ε ,, 3 −3 6 −4 −5 0.5 1 1.5 2 Frequency (GHz) 2.5 3 Figure 6.5: Complex permittivity for six samples homogeneously saturated. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 6.2. Saturation technique and permittivities 99 No-Flow: saturation homogeneously distributed Having to saturate the sample outside of the sample holder, results in samples with slightly different porosities. Table 6.1: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±0.5) and water-to-sand mass ratio (∆(mw /mq ) = ±0.2) for the six samples mixed outside the sample holder. All quantities are given in %. Step φ Sw mw /mq 1 2 3 4 5 6 43 45 45 46 45 42 0 16 32 47 66 100 0 5 10 15 21 28 Figure 6.6 presents the real and imaginary component of both ε∗12 and ε∗21 . A blown up version of this figure (Figure 6.8) can be found in pages 104-105. The real parts are more or less constant for the whole frequency range and their values increase with increasing water content. The imaginary parts show a linear increase with frequency and the loss enlarges with water content. Their trend is less smooth than their real counterparts, and they clearly diverge for low frequencies due to the low frequency limit of the measurement technique. In Chapter 5 we saw that when the samples were fully homogeneous (air and ethanol) ε∗12 and ε∗21 differed within measurement accuracy, and, as we will see, their similarity can be used to estimate the homogeneity of a sample. In this case, we have tried to homogeneously distribute the water saturation throughout the sample, and from the results, it is clear that the dry sand is very close to this assumption and could be considered as a homogenous sample with an effective permittivity of around 2.5 − 0.03j. As the water content increases, the difference between the ε∗12 and ε∗21 also increases, being maximum for the last two steps. However, this difference is still acceptable and always smaller than 2% above 1 GHz. The permittivity of step 5 shows two clear deviations from the constant trend of 0 00 both εr and εr around 500 MHz. As we will see in the coming sections, the reconstructed permittivities present synchronized anomalies when the samples are composed of 2 or more layers, and we think that these deviations are indicators of layering and due to the sample preparation. This effect is more pronounced in the 00 anomalies of the otherwise smooth εr . In a smaller scale this is also present in step 4 around 1 GHz. 100 Chapter 6. Permittivity states of mixed-phase Flow perpendicular to the direction of propagation 30 6 20 ε , 5 4 10 3 2 1 0.5 1 1.5 2 2.5 3 0 1 2 −1 3 −2 4 5 ε ,, 0 6 −3 −4 −5 0.5 1 1.5 2 Frequency (GHz) 2.5 3 Figure 6.6: Complex permittivity for the six steps of a flowing experiment when the flow is perpendicular to the direction of propagation of the TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 6.2. Saturation technique and permittivities 101 Gravity Drainage: Flow perpendicular to the direction of propagation In this case, the sample is the same all throughout the experiment, and therefore, the porosity remains constant. It is also, the lowest porosity obtained with the three different methods. Table 6.2: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±0.5) and water to sand mass ratio (∆(mw /mq ) = ±0.2) for the six flowing steps, when the flow is perpendicular to the direction of propagation of the waves. All quantities are given in %. Step φ Sw mw /mq 1 2 3 4 5 6 40 40 40 40 40 40 0 27 53 67 83 100 0 7 13 17 21 25 Figure ?? presents the real and imaginary components of both ε∗12 and ε∗21 . A blown up version of this figure (Figure 6.9) can be found in pages 105-106. The reconstructed permittivities, are certainly not as smooth as the ones computed when the sample was homogenously saturated, except, of course for the dry, residual water and fully saturated steps (1, 2 and 6). For all the other steps, the real parts are more or less constant for frequencies above 1 GHz and their value increases with increasing water content, but below 1 GHz they present anomalies. The imaginary parts show a linear increase with frequency and the loss increases with water content but their trend is certainly more erratic than their real counterparts. The anomalies on the real part are also present in the imaginary part. Again, as the water content increases, the difference between the two computed permittivities also increases, being maximum for the fifth step. The fully saturated sand behaves as the homogenously mixed sample from the previous experiment. Above 1 GHz the difference in the real part of permittivity is still acceptable and smaller than 2%, but the imaginary parts differ considerably. 102 Chapter 6. Permittivity states of mixed-phase Flow parallel to the direction of propagation 30 6 5 4 10 3 2 ε, 20 1 0.5 1 1.5 2 2.5 3 0 1 −2 2 −4 5 63 45 4 ε ,, 0 −6 −8 −10 0.5 1 1.5 2 Frequency (GHz) 2.5 3 Figure 6.7: Complex permittivity for the six steps of a flowing experiment when the flow is parallel to the direction of propagation of the TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 6.2. Saturation technique and permittivities 103 Flow parallel to the direction of propagation As in the previous flowing experiment, the sample is the same all throughout the experiment, and therefore, the porosity remains constant. Table 6.3: Porosity (∆φ = ±0.5), average water saturation (∆Sw = ±7) and water to sand mass ratio (∆(mw /mq ) = ±2) for the six flowing steps, when the flow is parallel to the direction of propagation of the waves. All quantities are given in %. Step φ Sw mw /mq 1 2 3 4 5 6 43 43 43 43 43 43 0 7 36 54 72 100 0 2 11 16 21 29 The reconstructed permittivities ε∗12 and ε∗21 for this experiment are presented in Figure 6.7. A blown up version of this figure (Figure 6.10) can be found in pages 108-109. They behave differently than those computed in the two previous experiments, but still, they increase with increasing water content. The real parts are no longer constant but present a growing tendency with frequency, except for the fully dry and saturated samples. The permittivity of the fully saturated sample is consistently higher than that of its equivalent counterparts and the difference in porosity cannot account for it. From experimental experience, we think that this is due to the saturation technique used. As we already mentioned, flowing into a dry sample allows for grain repositioning and compaction that could result in a small empty chamber at the top of the sample holder where more water could be stored explaining the high value of the permittivity. The anomalies show up during a broader part of the spectrum and the imaginary parts are not consistent, since the apparent loss in partially saturated samples is greater than the fully saturated sample. This time, the difference in the real part of permittivity is only smaller than 4% for frequencies above 2 GHz, but the imaginary parts differ considerably. As we will see in the coming sections, the anomalies present in the previous flow experiment can be explained with a layered model. We think that the anomalies present in this experiment is also due to layering. However, this layering cannot be modelled with homogeneous cross sections and therefore cannot be included in our model. Since, we cannot make a thorough study of these samples we will not use it for comparison with the other two techniques. ε, Homogeneously mixed 15 5 10 4 3 1 Chapter 6. Permittivity states of mixed-phase 2 5 3 2.5 2 1.5 1 0.5 0 104 25 6 20 1 2 −0.5 3 −1 4 −1.5 ε ,, −2 5 −2.5 −3 6 −3.5 6.2. Saturation technique and permittivities 0 −4 −4.5 −5 0.5 1 1.5 Frequency (GHz) 2 2.5 3 105 Figure 6.8: Complex permittivity for six samples homogeneously saturated. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 106 Flow perpendicular to the direction of propagation 25 6 20 5 15 ε, 10 3 5 2 1 0 0.5 1 1.5 2 2.5 3 Chapter 6. Permittivity states of mixed-phase 4 1 2 ε ,, −0.5 −1 3 −1.5 4 −2 5 −2.5 6 −3 −3.5 6.2. Saturation technique and permittivities 0 −4 −4.5 −5 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.9: Complex permittivity for the six steps of a flowing experiment when the flow is perpendicular to the direction of propagation of the TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 107 108 Flow parallel to the direction of propagation 30 6 25 5 4 15 3 10 2 5 1 0 0.5 1 1.5 2 2.5 3 Chapter 6. Permittivity states of mixed-phase ε, 20 1 −1 2 −2 3 −3 6 4 ε,, −4 5 −5 −6 −7 6.2. Saturation technique and permittivities 0 −8 −9 −10 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.10: Complex permittivity for the six steps of a flowing experiment when the flow is parallel to the direction of propagation of the TEM mode. The dashed line corresponds to ε∗12 and the dotted line to ε∗21 . 109 110 6.2.4 Chapter 6. Permittivity states of mixed-phase Comparison and interpretation of results Here, we compare only the results of the first two sets of experiments, because we ruled out the last one since the saturation technique provided samples that violated the model assumptions (homogeneity in the cross-section). All the samples consisted of quartz grains with the same grain size distribution, and although saturated in different ways, their permittivities are comparable. The first step corresponds to the fully dry state. The porosities of the two samples differ by 2% and their permittivities by 5%. The sample with lowest porosity has a higher permittivity, since the quartz to air ratio is higher. ε∗12 and ε∗21 for both samples are within experimental error and the dry sands can be treated as homogenous. 0 As water is added the results are consistent, higher values of εr and greater losses for higher water content. From steps 2 to 4 the water to quartz ratio is always bigger for the sample subject to gravitational drainage than for those homogenously mixed outside of the sample holder, and so are their permittivities. Step 5 has the same mw /mq for both experiments, however, the permittivity of the sample with 40% porosity (gravity drainage) is 2% bigger than the one with 45% (homogeneously mixed). This is related to the different ratios of quartz/air/water they posses. When the samples are fully saturated (step 6), their permittivities are again comparable, and they are higher for the one with higher water to quartz ratio. In principle, when all possible water has been drained with the aid of gravity, the remaining residual water will be evenly distributed along the sample and it can be treated as homogeneous (step 2 of the experiment in which the flow is perpendicular to the direction of propagation of the TEM waves). Only steps 3 to 5 of the sample are subject to gravitational drainage and present remarkable anomalies, whereas steps 1, 2 and 6 do not present these anomalies. Furthermore, ε∗12 and ε∗21 are within experimental errors for steps 1 and 2 and differ by less than 2% for the sixth step. It is thus clear that the heterogeneous samples can be identified from the reconstructed permittivity. The consistency of the anomalies in all heterogeneous samples, and the distortion of ε∗ for the case in which the fluid front is parallel to the direction of propagation, points to heterogeneity. Furthermore, the fact that we are able to reproduce the anomalies by modelling the sample as a two layer sample (see Section 6.4) encourages this explanation. In Section 6.5 we study their occurrence and distribution. The interpretation of the results would not be complete without the attempt to model these results with known mixing models, but we have preferred to do so at the end of the chapter, when all the needed information is available. 6.3. Averaging ε∗12 and ε∗21 over averaging the S-parameters 6.3 111 Averaging ε∗12 and ε∗21 over averaging the S-parameters Along the previous subsections we have presented results on both ε∗12 and ε∗21 . Their advantage is clear when dealing with heterogeneous samples, as the more inhomogeneous a sample is the more different they are. However, two permittivities complicate the analysis and comparison, so for simplicity we have decided to average them out per frequency. Even more so, when it is possible to do it without loosing heterogeneity information. Obtaining a single effective ε∗r per frequency can be done in two different ways, averaging the permittivities into a single value per frequency or reducing the 4 S-parameters into 2, averaging both reflections and transmissions into single parameters, also per frequency, and invert to obtain a single permittivity. Reducing the two pairs of reflection and transmission data into one single pair and then invert, has to be done on a perfectly symmetric tool, otherwise it would not be valid, because the reflections would be different due to the asymmetry of the tool. For our tool, we have to compensate for the connector that enables the connection with the cable from Port 2 (see Chapter 4), a simple phase correction takes it into account. In an ideal case of a perfectly homogeneous sample and with the compensated symmetric tool, both reflections and transmissions should be equal. For our tool this is true within experimental errors when the sample holder is filled with a homogeneous material. Then, averaging the S-parameters per frequency yields the same result as if two permittivities are computed from the 4 S-parameters and then averaged. However, the samples we are interested in, always have a certain degree of heterogeneity, and it is logical that the average of the S-parameters will involve a loss of information about it. To corroborate it, we have performed these two averaging approaches on the No-Flow experiment where the samples were assumed to be homogeneous. With the obtained permittivity we have reconstructed the data and looked at the similarities between the measured and reconstructed data. We show results only for step 4 of the experiment, but in all cases the results were similar. Averaging the measured S-parameters and then, reconstructing the permittivity (ε∗s ) always resulted in permittivity values between 2% and 4% bigger than those computed from the averaged ε∗12 and ε∗21 (ε∗m ), so that both the real part and the loss were greater, see Figure 6.11. The phase of the reconstructed S-parameters was equally well modelled but the amplitude of reflection and transmission was underestimated by ε∗s , see Figure 6.12. 112 Chapter 6. Permittivity states of mixed-phase 11 ε , 10 9 8 0.5 1 1.5 2 2.5 3 0 εm ε 12 ε 21 ε ε,, −0.5 −1 s −1.5 −2 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.11: Averaged complex permittivities for step 4 of the no-flow experiment. ε∗12 and ε∗21 are represented by dotted lines, their mean ε∗m by the solid line, and the dashed line corresponds to the permittivity computed from the averaged S-parameters. e es Figure 6.12: Amplitude of the measured and reconstructed S11 and S12 , for step 4 of the no-flow experiment. The solid line corresponds to the measured parameters, the dotted line to the reconstructed parameters from ε∗m and the dashed line to the reconstructed parameters from ε∗s 6.4. 2-layer samples 113 From now on, we will show a single permittivity per sample/water content, that is, the average of ε∗12 and ε∗21 , i.e. ε∗m . As we have seen, it is a better representation than ε∗s , plus the anomalies due to heterogeneities do not disappear and a single effective parameter represents the sample. In the next section, we model the response of the sample in which the flow is perpendicular to the direction of propagation of the TEM as a 2-layer sample, obtaining the thickness of the layers in two different ways. 6.4 2-layer samples For the case in which the flow is perpendicular to the direction of propagation of the TEM mode (Figure 6.3) we can picture the sample from steps 3 to 5, as consisting of two layers, one fully saturated and the other one with residual water saturation and assume each layer to be homogeneous. If we knew the height of these two layers we could model the S-parameters corresponding to such a configuration and test the validity of the 2-layer assumption. Figure 6.13 shows this model. Although, it is a very simple approach, the model yields a good approximation. Figure 6.13: Model for a sample in which the flow is perpendicular to the direction of propagation of the TEM mode. SW is the saturation of the fully saturated layer and equal to 100% (step 6 in Table 6.2) and Sr is the saturation of the residual water saturated layer and equal to 27% (step 2 in Table 6.2). H is the total height of the sample, 10 cm, and h is the height of the fully saturated layer. To find out h, there are many different methods, but we will limit ourselves to two. The first one is based on an effective saturation assumption and the second one is a phase approximation. 114 Chapter 6. Permittivity states of mixed-phase 6.4.1 hSw from saturation If we assume an effective water saturation (Sw ) for the whole sample and a homogeneously distributed porosity, then, equation (6.1) must hold, and since all the other parameters are known hSw can be obtained. (6.1) Sw · H = Sr · (H − hSw ) + SW · hSw 6.4.2 hθw from phase In this case, we do not make any assumptions about the saturation of the sample, but instead, we take the phase of the transmission coefficients to be linear and suppose that the contribution of multiples can be neglected. Now, θw is the measured phase of Υ (θ(Υ)) for any step from 2 to 6 of Table 6.2, θW is θ(Υ) of the fully saturated sand (step 6) and θr is θ(Υ) of the residual saturated sand (step 2). θw · H = θr · (H − hθw ) + θW · hθw 6.4.3 (6.2) Results Solving equations (6.1) and (6.2) we found the different heights listed in Table 6.4. The heights are given in cm, and the ones computed from the saturation assumption, have an error of ±0.1. For those computed from the phase approximation it is not possible to estimate the error a-priori. Table 6.4: Heights [cm] of the fully saturated sand layer for the five flowing steps, from Sr at 2 to SW at 6, when the flow is perpendicular to the direction of propagation of the waves. Step h Sw hθw 2 3 4 5 6 0 3.5 5.5 7.7 10 0 3.9 6.0 7.9 10 With the heights from Table 6.4, and the reconstructed permittivities for the fully and the residual water saturated sands we modelled the sample as two layers and generated computed S-parameters for such a model. Comparing the measured S-parameters and the synthesized ones we found that hθw is a better approximation to the true height of the saturated layer than hSw . 6.4. 2-layer samples 115 In Figure 6.14 we have plotted the real and imaginary parts of S11 and S12 for step 4. The solid line corresponds to the measured data, the dotted line to the modelled data taking hSw , and the dashed line to the modelled data taking hθw . From the figure it is clear that using hθw results in a closer fit to the measured data. Specially, for the transmission, where the modelled data with hSw is clearly out of phase. These results are consistent in steps 3 and 5. Then, the assumption that a sample, where a two-phase flow (air-water) is occurring and a fluid front present, can be treated with an effective saturation underestimates the position of the fluid front. In contrast, the linear approximation to the phase of the transmission coefficient yields a good estimation. However, if the error in the saturation was of the order of 5% (10 times bigger than the computed error) the estimates of the layer height would be comparable. The error difference is considerable, but the experimental conditions were not optimum, and, we can only positively affirm that the phase approximation is more reliable. ℜ ℑ 1 0.5 0.5 0 0 −0.5 −0.5 S11 1 −1 1 0.5 1 1.5 2 2.5 3 meas mod h(Sw) mod h(θ ) −1 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 1 w 0.5 0 0 −0.5 −0.5 S12 0.5 −1 0.5 1 1.5 2 2.5 Frequency (GHz) 3 −1 Figure 6.14: Real and imaginary parts of S11 and S12 for step 4 of the experiment in which the flow is perpendicular to the direction of propagation of the TEM. The solid line corresponds to the measured data, the dotted line to the modelled data taking hSw and the dashed line to the modelled data taking hθw 116 Chapter 6. Permittivity states of mixed-phase From the modelled S-parameters with hSθw we can reconstruct a modelled effective permittivity (dashed line) and compare it to the one reconstructed from the measured data (solid line), corroborating the good approximation of the height of the saturated layer from the phase of the transmission coefficient (Figure 6.15). Note, that the reconstructed modelled permittivity also has the anomalies present in the permittivity reconstructed from measured data. In Figure 6.15 we have plotted the reconstructed permittivities for step 5 of the gravitational drainage experiment from the measured S-parameters [S exp ] (solid line) together with the reconstructed permittivities from the synthesized S-parameters ]. with a two layer model in the sample holder, [Shmod ] and [Shmod S θ w w Gravity Drainage Experiment 25 ε, 20 15 10 0.5 1 1.5 2 2.5 3 0 ε,, −1 −2 meas mod h(Sw) mod h(θw) −3 −4 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.15: Averaged complex permittivities for step 5 of the experiment in which the flow is perpendicular to the direction of propagation of the TEM. The solid line corresponds to the measured data, the dotted line to the modelled data taking hSw and the dashed line to the modelled data taking hθw . 6.5. Anomalies of layered samples 117 It is clear that the best solution is given from taking hθw instead of hSw . The real parts coincide except for the regions around the anomalies present in ε∗hθ . These w anomalies match those of ε∗rec below 500 MHz, but are more numerous. In fact, there seems to be a relation between the height of the fully saturated layer and the number and amplitude of anomalies present. We think that the sharpness of the interface introduced in the model could be responsible for the clearer appearance of anomalies, while in the experimental case the interface between fully and residual water saturated sand is more gradual. In the next section we will test this hypothesis. The imaginary parts are not so well captured by the 2-layer model because the more frequent anomalies mask the data. The results obtained for steps 3 and 4 are comparable. 6.5 Study of the anomalies present in reconstructed permittivities of layered samples In Subsection 6.2.3 we saw how the reconstructed permittivities of layered samples present anomalies in their real and imaginary parts, steps 3 to 5 of Figure 6.9. However, the ε∗rec of the same sample when the saturation was homogeneously distributed, steps 1, 2 and 6 of Figure 6.9 were smooth. This pointed out the fact that those anomalies could be originated from the inhomogeneity of the sample. Moreover, in Section 6.4 by modelling the sample with 2 layers, one fully saturated and another with residual water saturation, we were able to reconstruct the measured S-parameters and their associated permittivities, anomalies included. In this section we want to investigate further these anomalies and the parameters that influence them. To study these anomalies we have worked with modelled data and we have created 3 sets of numerical experiments with different conditions. In all three sets we have used an approximation to the permittivities for the residual (ε∗r ) and fully water (ε∗W ) saturated quartz sand (steps 2 and 6 of Figure 6.9). Over the frequency range of operation, the real parts can be considered constant and the imaginary parts can be approximated with a linear dependence on frequency (f ). Their values are ε∗r = 5 − j · 0.83 · 10−10 · f ε∗W = 21.5 − j · 9.17 · 10−10 · f In the first place, we study the effect of the length of the sample holder, reconstructing the permittivities of samples of 10, 20 and 30 cm long, for four different volume fractions of the 2 different materials. All the samples studied are shown in Figure 6.16, the grey layers correspond to ε∗W and the white ones to ε∗r , the 118 Chapter 6. Permittivity states of mixed-phase numbers on the left reflect the volume fraction of material with permittivity ε∗W to that of ε∗r . Figure 6.17 shows the reconstructed real permittivity for these samples. 0 Note that the scale of the εr axis is very big. Figure 6.16: Modelled samples for different sizes of sample holder. 2/10 100 H = 10 cm H = 20 cm H = 30 cm 50 4/10 0 50 6/10 0 50 8/10 0 50 0 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.17: Real permittivity for four different volume fractions in three sample holders of different length. 6.5. Anomalies of layered samples 119 For constant volume fraction, the anomalies are ordered, they occur at lower frequencies for longer sample holders, and move to higher frequencies and they expand their size for broader frequency spans for shorter sample holders. As the volume fraction of the wet quartz increases so do the amplitude of the anomalies. This indicates that the contrast in the permittivity of the layers has a role in the amplitude of the anomalies, and although, the sample holder length influences the frequency span and location, making the sample holder longer doesn’t make the anomalies disappear. Secondly, we investigate the role of the position of the layer. Figure 6.18 shows the modelled samples we used, they consist of a 2 cm long wet layer at different locations in a 10 cm sample holder. The rest of the sample holder is filled with residual water saturated sand. This time, we show the amplitude and phase of the synthesized S-parameters for such samples in Figure 6.19. Figure 6.18: Modelled samples for different positions of the wet layer. The numbers correspond to the position of the layer in cm from the left hand side of the sample holder. The phase of the transmission coefficients are equal, and so are their amplitudes for equivalent samples, 0 and 8, and 2 and 6. Whereas the amplitudes and phases of the reflection coefficient are certainly different. Note that for the first 100-200 MHz all coefficients are equal. Of course, when reconstructing the permittivity from these S-parameters, the anomalies are distinct for distinct positions of the wet layer, but, in the low frequency limit they all have the same permittivity and as frequency increases they oscillate around a constant value of permittivity lower than that of the low frequency limit. We discuss this further at the end of this section. The position of the wet layer certainly matters and a single effective permittivity cannot capture the EM response of these samples, as it would produce the same S-parameters for the six samples. 120 Chapter 6. Permittivity states of mixed-phase Amplitude Phase 1 4 0.8 2 S11 0.6 0 0.4 −2 0.2 0 0.5 1 1.5 2 2.5 3 1 −4 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 4 0.8 2 S12 0.6 0.4 0.2 0 0 0 2 4 6 8 0.5 1 1.5 2 2.5 Frequency (GHz) −2 3 −4 Figure 6.19: Modelled S-parameters for samples with the same volume fraction wet/residual, but with the wet layer at different positions (in cm from the left of the left hand side of the sample holder). The last numerical experiment focuses in the width of the layers, and consists in maintaining the volume fraction of wet/residual constant for a 10 cm sample, and evenly distributing the initial 2 cm wet layer in smaller and smaller layers throughout the sample holder. Figure 6.20 shows the samples studied. The synthesized S-parameters, Figure 6.21, are different for the three first samples and equal for the last three, implying that the width and distribution of the layers play and important role in whether the samples can be considered equivalent from an EM point of view, or not. Again, for the first 100-200 MHz the S-parameters are equal for all samples, but this time, the phases of the transmission coefficients are slightly different for the first three samples. 6.5. Anomalies of layered samples 121 Figure 6.20: Modelled samples for different distributions of wet layer. Amplitude Phase 1 4 0.8 2 S11 0.6 0 0.4 −2 0.2 0 0.5 1 1.5 2 2.5 3 1 S12 0.4 0.2 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 Frequency (GHz) 3 4 0.8 0.6 −4 2 1 3 7 15 31 63 0.5 1 1.5 2 2.5 Frequency (GHz) 0 −2 3 −4 Figure 6.21: Modelled S-parameters for samples with the same volume fraction wet/residual, but with diminishing wet layer width and increasing number of layers (in the legend). 122 Chapter 6. Permittivity states of mixed-phase Let us now take a look at the reconstructed permittivities of the last two experiments. The volume fraction of wet sand to that of residual water saturated sand is constant and equal to 0.2. Figure 6.22 shows ε∗ for all six samples in which the width of the wet layer is maintained constant but its position changes. 10 0 2 4 6 8 ε, 9 8 arim crim 7 6 0.5 1 1.5 2 2.5 3 0 ε,, −0.5 crim arim −1 −1.5 −2 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.22: Reconstructed permittivities from modelled S-parameters for samples with the same volume fraction wet/residual, but with the wet layer at different positions (in cm from the left of the left hand side of the sample holder). They are all different but it is interesting to note that at the lowest frequency they all have the same value, corresponding to the static prediction of the arithmetic mean, εarim , equation (E.3). Above 1.5 GHz, the real parts of the permittivity all tend to the value predicted by the CRIM (Complex Refractive Index Model) model, equation (E.19) with c = 0.5. The arithmetic mean and the CRIM model are explained in detail in Appendix E, but for convenience, we list their expressions for this particular example. 6.5. Anomalies of layered samples 123 ε∗arim = fr ε∗r + fW ε∗W , (ε∗crim )0.5 = fr (ε∗r )0.5 + (6.3) fW (ε∗W )0.5 , (6.4) where fr is the volume fraction of material with permittivity ε∗r (residual water saturated) and fW that of material ε∗W (fully saturated). Above 1.5 GHz, the real parts of the permittivity all tend to the value predicted ε∗crim , with a maximum standard deviation is 3%. The imaginary parts are unfortunately more erratic. At 1.5 GHz, the wavelength corresponding to the permittivity of the mixture given by the CRIM model (λcrim ) is approximately four times the width of the wet layer. Figure 6.23 shows ε∗ for all six samples in which the volume fraction is maintained constant, but the width of the wet layer is reduced, and the thinner but more numerous wet layers are distributed along the sample. 10 ε, 9 arim 8 crim 7 6 0.5 1 1.5 2 2.5 3 0 ε ,, −0.5 −1 −1.5 −2 1: 3: 7: 15 31 63 crim arim 20.0 6.67 2.86 : 1.33 : 0.64 : 0.32 0.5 1 1.5 Frequency (GHz) 2 2.5 3 Figure 6.23: Reconstructed permittivities from modelled S-parameters for samples with the same volume fraction wet/residual, but with diminishing wet layer width and increasing number of layers (odd sequence) [number of wet layers: width of wet layers]. 124 Chapter 6. Permittivity states of mixed-phase This time, as the width of the inclusions decreases, both real and imaginary parts of the permittivity, tend to ε∗arim . Only the first two samples at high frequencies tend to ε∗CRIM . When the layer width is 2.86 mm the standard deviation of the real part of ε∗ is about 2% for the first half of the spectrum and increases up to 6% at the high frequency end. When they are 1.33 mm it is already smaller than 0.5% for the first half and around 1% for the second half. The imaginary parts deviate slightly more than their real counterparts but we can consider the sample with 15 equally spaced layers of 1.33 mm wide to be homogeneous and with a permittivity given by ε∗arim . The fact that this experiment considers an odd sequence of layers (the change in width ratios is not constant) makes it difficult to find the relation between layer width and the frequency at which ε∗arim changes to ε∗crim . In figure 6.24 we have plotted the permittivities of a series of samples equivalent to those of figure 6.21 but with an even sequence of layers, such that their width ratio is always a factor of two. Effective permittivities for layered systems 12 1:20.0 2:10.0 4:5.00 8:2.50 16:1.25 32:0.62 11 10 ε, r 9 8 7 6 5 8 10 10 9 10 10 10 11 frequency [Hz] Figure 6.24: Reconstructed permittivities from modelled S-parameters for samples with the same volume fraction wet/residual, diminishing wet layer width and increasing number of layers (even sequence). In the legend [number of wet layers: width of wet layers]. 6.5. Anomalies of layered samples 125 For this specific case, the frequency at which ε∗arim changes to ε∗crim , ft corresponds to approximately wet ft = fc · Nlayers +1 with fc = 1.1GHz, (6.5) wet ) is related to the width of these layers, and since the number of wet layers (Nlayers 2 as dlayer = N wet [cm], the transition frequency can be rewritten as layers ft = fc · 2 dlayer +1 . (6.6) The transition frequency, ft depends on the width of the layers and on fc that is determined by the volume fraction and permittivities of the two components. For very narrow layers this frequency is very high but finite. Note that for frequencies smaller than ft the values of permittivity values correspond precisely to ε∗arim while for frequencies bigger than ft they oscillate around ε∗crim . Local electrodynamic equilibrium Rio & Whitaker (2000a) constrained the validity of using one-equation models for the dynamic electric and magnetic fields in a two component mixture with equations (E.11), (E.12) and (E.13). They derived them using the method of volume averaging. In essence, and according to del Rio and Whitaker, when a mixture violates these constrains it cannot be represented by single effective electromagnetic properties, and a Two-Equation Model must be used (Rio & Whitaker, 2000b). Moreover, if the mixture can be represented by a One-Equation Model, then its effective permittivity is given by the arithmetic mean, equation (E.3). fh fi (εh − εi )2 1, (fh εh + fi εi )(εh + εi ) fh fi (µh − µi )2 1, (fh µh + fi µi )(µh + µi ) fh fi (σh − σi )(εh − εi ) 1. (fh σh + fi σi )(εh + εi ) (6.7) (6.8) (6.9) The samples used in this modelled experiments do not obey the Local Electrodynamic Equilibrium. Since their condition for permittivity, equation (E.11), is in this particular case 0.2, certainly not very small compared to 1, and yet a single permittivity is a very good representation of the sample. The other two conditions are met for non-magnetic (µh = µi = µ0 ) and low-loss materials (σh ≈ σi ≈ 0). 126 Chapter 6. Permittivity states of mixed-phase In the case of thin layers evenly distributed in the sample holder the permittivity is given by the arithmetic mean, as del Rio & Whitaker (2000a) claim, but for high enough frequencies and a 2 cm wet layer located at different positions within a water residual saturated layer it is given by the CRIM model, equation (E.19), contradicting their conclusions. From these modelled experiments we can conclude that the anomalies present in the reconstructed permittivities of layered samples are dependent on the permittivity contrast between the different layers, the distribution of these layers and the length of the sample holder. However, this dependency is not trivial. It is important to note that the distribution of the layers has a strong impact in the permittivity, and, while at low frequencies they all tend to the arithmetic mean, at higher frequencies it depends on the electrical length of the layers. At this point, we are ready to interpret the reconstructed permittivities, of the two valid saturation techniques, from an effective medium point of view. 6.6 Interpretation Mixing Laws relate the effective permittivity of a mixture in terms of the permittivities of each constituent and their volume fraction. The literature on the effective properties of materials and mixtures is very vast (Appendix E) and their applications range from the design of new composites for electronic components to the estimation of water content of porous materials (biological tissues, wood, bedrock . . . ). Estimating the fluid content of a rock core or soil sample from measured permittivities is yet a rather imprecise task. The permittivity depends on the soil structure, texture, composition, porosity, fluid nature, content and distribution, temperature, and the mixing models only consider the permittivity of the components of the mixture and their volume fractions. Moreover, the most widely used Mixing Laws have been developed under the consideration of static or quasi-static fields but are applied in the high frequency range of the spectrum. Different researchers have tried to find the best mixing law for different types of soils and rocks. Topp (1980) found an empirical relation for frequencies between 1 MHz and 1 GHz, and its has been extensively used. Shutko & Reutov (1982) found the Complex Refractive Index Method to give a good approximation to the permittivity of different soils in the range [1-10] GHz and Bruggeman-Hanai at 10 GHz as a good prediction for soils with low moisture content. Hallikainen et al. (1985) pointed out that although soil texture may not affect the permittivity of dry soils, it 6.6. Interpretation 127 certainly does so for wetted soils, specially at the lower end of the frequency range. Dobson et al. (1985) found that both the Four Component Model and a Power Law with a coefficient c = 0.65, are capable of describing the complex dielectric permittivity above 1.45 GHz for different soils (sand, silt and clay). According to Chelidze & Gueguen (1999b) the Bruggeman-Hanai model with L = 1/3 works fine for the Megahertz region of the spectrum and different rocks. Finally, Seleznev et al. (2004) studied the permittivity of partially saturated carbonate rocks, from 300 KHz to 3 GHz, finding that the best fit was that of the CRIM model. In this section, we compare our results for the homogeneously mixed and gravity drainage experiments with existing Mixing Laws. We also show that even if the Local Electrodynamic Equilibrium is not satisfied, again, a single effective permittivity can represent the sample and not necessarily by the arithmetic mean. Our samples have 2 components and two phases for steps 1 and 6, dry and fully saturated sands, and 3 components and three phases in the intermediate saturations, quartz grains, air and water. For convenience, we have listed the parameters used in this section. ε∗q ε∗a ε∗w φ Sw Sr SW 6.6.1 permittivity of quartz permittivity of air permittivity of water porosity of the sample saturation of the sample residual water saturation full saturation 4.65 (Parkhomenko, 1967) 1 Debye model for water (Section E.2.2) Tables 6.1 and 6.2 Tables 6.1 and 6.2 27% 100% Dry sand The dry samples were obtained in different ways and possess slightly different porosities and permittivities. Their frequency dependence is negligible and we can represent them with their mean. To model their permittivities we have used the relation proposed by Dobson et al. (1985) and given by equation (E.22), and the reinterpretation of the Claussius-Mossotti model, equation (E.23). According to Dobson et al. (1985) the permittivity of dry soils (εs ) can be computed from their density (ρs ), given in [gr/cm3 ], via εs = (1 + 0.44ρs )2 − 0.062. (6.10) 128 Chapter 6. Permittivity states of mixed-phase It is surprising to relate a unitless property to the square of a density, but these are empirical fittings to experimental data on different dry soils. We think it is more appropriate to compute the permittivity from the following reinterpretation of the Clausius-Mossotti relation, equation (E.1), given in Section E.2.1, and rewritten here for simplicity ε= 1 + 23 N α ε0 . 1 − 13 N α (6.11) Since this expression has been successfully used to compute the permittivity of different gases from known polarizabilities, we wanted to test it for mixtures. For non-interactive materials we can compute α of each material from its permittivity and equation (6.11), and by superposition we can find the permittivity of the dry sand mixture as 3 + 2[Nq αq + Na αa ] , (6.12) ε∗cm = 3 − [Nq αq + Na αa ] where the subscript a refers to the air (αa = 0), and the subscript q refers to the quartz. According to Parkhomenko (1967) the permittivity of quartz over the frequency range of interest is 4.65. The permittivities computed with the different models are listed in Table 6.5 Table 6.5: Relevant data for two dry sands samples, including reconstructed and modelled permittivities. φ ρs ε∗rec ε∗Dobson ε∗cm ε∗mod Homogeneously Mixed Gravity Drainage 43 % 1.47 gr/cm3 2.49 - 0.02j 2.68 2.36 2.47 - 0.005j 40 % 1.60 gr/cm3 2.64 - 0.03j 2.88 2.47 2.57 - 0.005j The empirical Dobson model overestimates by around 8% the reconstructed values of permittivity for these two samples. Dobson’s equation is in fact a fit to different dry soil types and dependent on experimental conditions. The other empirical relation, by Shutko, equation (E.21) gives even higher predictions, and we have not included it here. The Claussius-Mossotti model, ε∗cm , underestimates the permittivity a 5%. However, if the ”dry” sand contained a very small quantity of water (Sw = 1%), double the accuracy of the water saturation, the underestimation of the Claussius-Mossotti model could be explained. Using the CRIM equation to model the dry sand with such a small moisture content the result is ε∗mod , and not only is the error in the estimation of the permittivity considerably smaller, but also the loss is present in 6.6. Interpretation 129 the result, although it is underestimated. For this particular case ε∗mod = ε∗cm (1 − φSw ) + ε∗w φSw . (6.13) Overall, we consider the Claussius-Mossotti reinterpretation to give the most adequate estimate for the permittivity of dry sands. If these sands were not perfectly dry, as it could have absorb moisture from the environment when filling the sample holder in the case of the homogeneously mixed samples, or in case of the gravity drainage the sample was dried with a vacuum pump during 24 hours and total absence of moisture cannot be ensured, their permittivity is very well captured by the Claussius-Mossotti reinterpretation together with the CRIM model, assuming a very small quantity of water. 6.6.2 Fully saturated sand The 6th step of the Homogeneously Mixed and Gravity Drainage experiments correspond to fully saturated samples. That is, they are 2 component mixtures: quartz grains and water, occupying the total porosity of each sample. In the modelling of the fully saturated sands we have tried the CRIM model, (ε∗crim )0.5 = (ε∗q )0.5 (1 − φ) + (ε∗w )0.5 φ (6.14) the Power Law model with c = 0.65 proposed by Dobson et al. (1985), (ε∗Dobson )0.65 = (ε∗q )0.65 (1 − φ) + (ε∗w )0.65 φ (6.15) the Maxwell-Garnett formula given by equation E.15, and the symmetric Bruggeman formula, equation E.17. The results on both samples were similar. The best fit is given by the CRIM model, however, it overestimates the permittivities by 10%. All the others overestimate or underestimate its value by even greater margins. A smaller exponent in the Power Law model would result in a more accurate result. This can be observed in Figures (6.25) to (6.28) in the next section. In view of these results, in the next subsection we will compare the reconstructed permittivities at different saturations with the Power Law model and different exponents. 130 Chapter 6. Permittivity states of mixed-phase Local electrodynamic equilibrium In Section 6.4 we showed how our modelled data did not obey the Local Electrodynamic Equilibrium proposed by del Rio & Whitaker (2000a) and nevertheless could be represented by a single permittivity. Do these fully saturated sands also contradict del Rio & Whitaker theory? They are two component and their volume ratios result in a Local Electrodynamic Equilibrium of around 0.5, which is not very small compared to 1, and therefore, this mixture does not obey the condition on permittivity given by del Rio & Whitaker (2000a). Moreover, and in contradiction to their theoretical results, the reconstruction of its permittivity gave a solid value throughout the whole frequency range not equal to the arithmetic mean, which is 60% higher. This experimental result together with the synthesized example of Section 6.4 suggest that the derived conditions by del Rio & Whitaker (2000a) for mixtures are not confirmed by our numerical or experimental results. Our numerical results are confirmed by the experimental results; one-equation models can be used up to high frequencies for two component mixtures and the high frequency limit for the effective relative permittivity is the CRIM model. 6.6.3 Partially saturated sands For both experiments, the saturation varies from fully dry to fully saturated in 6 steps. We have modelled the results with the Power Law model for 3 components (c3 ) and different exponents, c = [0.3, 0.4, 0.5, 0.6] (equation (6.16)). They read (ε∗c3 )c = (ε∗q )c (1 − φ) + (ε∗q )c Sw φ + (ε∗a )c (1 − Sw )φ, (6.16) Note that the Power Law model changes name with changing exponent, c = 0.5 corresponds to the CRIM model, c = 0.65 to the coefficient found by Dobson et al. (1985) and c = 1/3 to the Looyenga equation. For the Gravity Drainage Experiment, we have also modelled the partially saturated sands with the CRIM model for 2 components, taking as components the CRIM model for each layer. Then, the permittivity of the two layer samples is modelled as 0.5 (ε∗ )2layers = (ε∗r )0.5 (H − h) + (ε∗W )0.5 h, (6.17) where ε∗W is the permittivity of the fully saturated layer, step 6 in Table 6.2, h is its height (Table 6.4), ε∗r is the permittivity of the residual water saturated layer, step 2 in Table 6.2, and H is the total height of the sample (10 cm). The permittivities 6.6. Interpretation 131 of both layers have been also modelled with the CRIM model. For step 2 (the residual water saturated sand) it is given by equation (6.16) taking Sw = 1, and for step 6 (the fully saturated sand) Sw = 0.27. In this way, we introduce geometry in the CRIM model for steps 3 to 5 of the Gravity Drainage experiment, as the two layer model presented in Section 6.4. Figures 6.25 to 6.28 show the real and imaginary parts of the reconstructed permittivity versus saturation, at 500 MHz, 1, 2 and 3 GHz for the homogenously mixed experiment. They also include the Power Law model with different coefficients, equation (6.16). For all frequencies, the best model compared to the data depends 0 00 on the saturation and on ε or ε . 00 0 The 3-component modelling of ε requires higher coefficients than ε . In both real and imaginary parts of the permittivity, there is a clear jump around saturations of 50%, where the best fitting model changes exponent, from a lower value to a higher one. 0 ε below 50% saturation is very well captured by the Power Law model with a coefficient equal to 0.3. Step 4 of the Homogeneously Mixed Experiment (Sw = 47%) is in a transition zone between the Power Models with coefficients 0.3 and 0.4. For higher saturations, the model with exponent equal to 0.4 gives the best fit. This 00 behavior is also found in ε , but now, the jump is from a coefficient equal to 0.5 to 0.6. The real part of the permittivity is always better captured than its imaginary counterpart. However, as frequency increases the imaginary part of permittivity is better captured by the model. Although it is not shown, when comparing the reconstructed permittivities of the Gravity Drainage experiment with the different Power Law models, the same behavior as that of the Homogeneously Mixed samples is found. However, the fitting to the permittivity of the Homogenously Mixed samples is better than to those subject 0 to Gravity Drainage. In the first case, the error of the model for ε is smaller than 1% for all partial saturations, except for the transition saturation that it is 3%. For 00 ε the maximum difference is of 0.1. The accuracy regarding the Gravity Drainage samples is twice as big, although sufficiently small. It is possible to obtain a better fit to the Gravity Drainage samples if the 2 layer model (equation (6.17)) is chosen. Figures 6.29 to 6.32 show the real and imaginary parts of the reconstructed permittivity versus saturation, at 500 MHz, 1, 2 and 3 GHz for the gravity drainage experiment and this model. 0 The error of the fit for ε is smaller than 1% and the fit to the imaginary part results improves significantly. Note that with this model, the exponent used for the real and imaginary parts is the same. From our results, we can coclude, that the CRIM model including geometry, gives the best fit to the permittivities of the Gravity Drainage Experiment samples. 132 Chapter 6. Permittivity states of mixed-phase 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw −3 0 1 GHz ε0.5 rec 0.3 (εc3) (εc3)0.4 (ε )0.5 c3 0.6 (εc3) 0.5 Sw 1 Figure 6.25: Reconstructed permittivities at 500 MHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with coefficients 0.3, 0.4, 0.5, and 0.6. 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw 1 −3 0 GHz ε0.5 rec (εc3)0.3 (ε )0.4 c3 (ε )0.5 c3 (ε )0.6 c3 0.5 Sw 1 Figure 6.26: Reconstructed permittivities at 1 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with coefficients 0.3, 0.4, 0.5, and 0.6. 6.6. Interpretation 133 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw −3 0 1 ε0.5 GHz rec (ε )0.3 c3 (ε )0.4 c3 (ε )0.5 c3 0.6 (εc3) 0.5 Sw 1 Figure 6.27: Reconstructed permittivities at 2 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with coefficients 0.3, 0.4, 0.5, and 0.6. 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw 1 −3 0 GHz ε0.5 rec (εc3)0.3 (ε )0.4 c3 (ε )0.5 c3 (ε )0.6 c3 0.5 Sw 1 Figure 6.28: Reconstructed permittivities at 3 GHz for the Homogenously Mixed experiment versus water saturation (Sw ) and the Power Law model (equation (6.16)) with coefficients 0.3, 0.4, 0.5, and 0.6. 134 Chapter 6. Permittivity states of mixed-phase 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 −2.5 ε0.5 GHz rec )0.5 (ε* crim 2 layers 0 0 0.5 Sw −3 0 1 0.5 Sw 1 Figure 6.29: Reconstructed permittivities at 500 MHz for the Gravity Drainage experiment versus water saturation (Sw ) and the CRIM model of equation (6.17) 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw 1 −3 0 ε1 GHz rec * 0.5 (εcrim)2 layers 0.5 Sw 1 Figure 6.30: Reconstructed permittivities at 1 GHz for the Gravity Drainage experiment versus water saturation (Sw ) and the CRIM model of equation (6.17) 6.6. Interpretation 135 0 25 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw −3 0 1 ε2 GHz rec * 0.5 (εcrim)2 layers 0.5 Sw 1 Figure 6.31: Reconstructed permittivities at 2 GHz for the Gravity Drainage experiment versus water saturation (Sw ) and the CRIM model of equation (6.17) 25 0 −0.5 20 −1 ε, ε,, 15 −1.5 10 −2 5 0 0 −2.5 0.5 Sw 1 −3 0 ε3 GHz rec * 0.5 (εcrim)2 layers 0.5 Sw 1 Figure 6.32: Reconstructed permittivities at 3 GHz for the Gravity Drainage experiment versus water saturation (Sw ) and the CRIM model of equation (6.17) 136 Chapter 6. Permittivity states of mixed-phase Other researchers have found also the Power Law model to be the best model describing the electric properties of different soils. Dobson et al. (Dobson et al. , 1985) found a coefficient c = 0.65 for different soils and frequencies above 1.45 GHz, and Seleznev et al. (2004) found the CRIM model to give the the best fit to the permittivity of partially homogeneously saturated carbonate rocks, in our same frequency range. 6.7 Conclusions In this chapter, we have studied the effect of fluid flow through a sand sample in relation to its permittivity. Here we summarize the most relevant conclusions of our study. Despite the simplicity of the experiments used in this study, and the lack of thorough fluid flow study, the experiments properly illustrate the influence of the fluid distribution in the permittivity. And although the assumptions about this fluid distribution have been very simplistic, the reconstructed permittivities were properly modelled with these assumptions. We showed that the saturation technique has a clear effect on the reconstructed permittivity and that extra care is needed for proper measurements. If the sample is initially dry and then saturated with the flow parallel to the direction of propagation of the TEM waves, the control on the saturation is very poor and there is a risk of compaction of the grains, leading to a higher content of liquid. This extra saturation together with the fact that we cannot model these types of samples, because they have non homogeneous cross sections, made us rule out this saturation technique. The other two techniques are superior. The liquid content is easily controlled and accurately determined, and the reconstructed permittivities are consistent. Their difference rests on the saturation distribution and sample properties. While the technique that mixes the sand outside the sample holder with a certain amount of liquid ensures a homogeneous distribution (HM), except for minor disturbances, the method in which the liquid is let to drain with the aid of gravity (GD) results in distinctive 2-layer samples, one with residual saturation and the other one fully saturated. However, all the samples of the HM were different and possessed different porosities. With this methodology, it is very difficult to obtain samples of the same porosity, plus it is very time consuming. Whereas the sample subject to GD was always the same, with constant porosity, it is a considerably faster experiment and it is easier to determine the permittivity at more saturation levels. The reconstructed permittivities (ε∗12 and ε∗21 ) for perfectly homogenous materials (air and ethanol) are equal within measurement accuracy, while for heterogenous materials they are different. How different depends on the scale and distribution of the heterogeneity and the permittivity contrast, the higher the contrast and the scale, the higher the difference. Of course, there is also a frequency dependence. 6.7. Conclusions 137 The permittivities of the samples of the HM experiment, are equal within 2% above 1 GHz, and therefore we can consider them homogeneous. The real parts of permittivities of the GD experiment are also equal within 2% above 1 GHz, but their imaginary parts difference is slightly higher. The frequencies below 1 Ghz are in an interesting region, because that is the region where the anomalies due to heterogeneities are dominant. 0 00 The ε of partially saturated sands does not depend strongly on frequency, and ε shows an increase with increasing frequency. While the steps corresponding to the dry, residual saturation and fully saturated sands of the GD experiment (steps 1, 2 and 6) have equivalent ε∗12 and ε∗21 , also below 1 GHz, the steps corresponding to 2-layer samples (steps 3, 4 and 5) present anomalies in that frequency region. These anomalies are present only in the samples in which a 2-layer assumption of the saturation distribution is reasonable, and to a smaller extension in steps 4 and 5 of the HM experiment, but not in the rest of the samples with a homogeneous distribution. Their further study in Section 6.5 corroborated the fact that these anomalies are indeed result of heterogeneities in the sample. I think of the similarity between ε∗12 and ε∗21 as a measure for heterogeneity in the small scale, and the anomalies present in both of them as a measure for a larger scale of heterogeneity. For partially saturated sands, the small scale heterogeneities have a small impact on their permittivities (resemblance within 2%) and therefore can be represented by a single parameter. Moreover, if the permittivities have anomalies, due to layer heterogeneities, and they are substituted by a single ε∗ , they do not disappear. We can, then, represent a sample with a single permittivity loosing only redundant information on the small scale heterogeneities that we already know of, since we deal with a 3 component random mixture. To do so, it is best to average ε∗12 and ε∗21 per frequency, and not to average the S-parameters and then compute a single permittivity (Section 6.3). Anyhow, we recommend the reconstruction of both ε∗12 and ε∗21 in order to check these assumptions, and then, if the small scale heterogeneities does not have a big impact, average them out and work with a single parameter. The sample subject to gravity drainage can be represented by a 2-layer model very accurately, if the width of the layers is found from the phase of the transmission coefficients, equation (6.2). The error made if the width is computed from effective saturation assumptions, equation (6.1), is considerable. In addition, in Section 6.4 we showed how the reconstructed permittivities from the 2-layer model (modelled samples with layer widths computed from the phase of the transmission coefficients, and permittivities of the two layers from the reconstructed permittivity of the fully and residual water saturated samples) were equivalent to the measured permittivities of the real experiment. Nevertheless, the anomalies of the modelled samples were more pronounced and extended for a broader frequency range. We think that this is because the interface of the modelled samples represents a sudden change 138 Chapter 6. Permittivity states of mixed-phase of properties, while, the change in the real samples is more gradual, leading to smoother anomalies. Being able to accurately determine the width of the fully saturated layer implies that we can use the tool to monitor the movement of the fluid front, opening new doors in the investigation of fluid flow in porous media by means of electromagnetic measurements. Nguyen et al. (1996) published an equivalent method, at a bigger scale, to find the capillary transition zone from ground penetrating radar data. In Section 6.5 we studied the anomalies present in the reconstructed permittivities of 2-layer samples, with modelled experiments. We showed how the anomalies depend on the permittivity contrast between the different layers, the width of these layers and the length of the sample holder. In addition, the distribution of the layers has a strong impact on the permittivity, and, whereas at low frequencies all sample permittivities tend to the arithmetic mean, no matter their distribution, at higher frequencies it depends on the electrical length of the layers. For a single thick fully saturated layer, the permittivity corresponds to that of the CRIM model above 1.5 GHz, but if that layer is divided into sufficiently thin layers, the volume fraction kept constant, and the layers are evenly distributed, then the permittivity corresponds to the arithmetic mean in our frequency range. There is, therefore, a transition frequency that depends on the width of the layers, the volume fraction and the permittivities of the two components. For very thin layers this frequency is very high but finite. The amount of parameters that influence the anomalies (layer width, permittivity contrast and layer distribution) make it very difficult to draw an absolute conclusion and more investigation is necessary. These results can contribute to the improvement of direct and inverse modelling. Most modern modelling problems are so complicated that they can be solved only using numerical techniques. These can be either local or global techniques. In both cases, the medium needs to be discretized and the optimal discretization of a piecewise continuous medium is an unsolved problem for all wave problems. The optimal average for the partial equations that describe these fields, is unknown. Different researches have applied different techniques. In seismics, Muir et al. (1992) modelled elastic fields across irregular interfaces with the harmonic average, while Moczo et al. (2002), used both harmonic and arithmetic averaging of the elastic parameters in a finite difference modelling scheme. Our results suggest that the arithmetic averaging should be used for low frequency fields and the CRIM model for high frequencies. The most important class of problems for modelling is, of course, when the strong heterogeneities call for grid refinements, which usually introduce spurious scattering at the connection of the fine grid to the coarser grid (Falk, 1998). He reduced the spurious scattering using harmonic averages of the shear modulus and arithmetic averages of the mass density, for velocities he used the harmonic average. 6.7. Conclusions 139 Examples of these heterogeneities are rough surfaces either at the Earth surface or in the subsurface with strong electric parameter contrasts. Schneeberger (2003) successfully accounted for small scale heterogeneities (due to a rough surface) in a single effective parameter, using the Complex Refractive Index Model for modelling surface emission in radiometry measurements. For local modelling methods, apart from averaging the medium parameters across boundaries of discontinuities in the medium parameters, the minimization of the computational domain is of great importance to minimize computational cost, especially for three-dimensional modelling. Since the advent of very efficient boundary conditions, known as Perfectly Matched Layers (PML’s, Berenger (1994)), many other attempts have been reported to optimize these PML’s and the optimal type of layers would be the one represented by a sequence that allows transmission and is reflection free (Hoop et al. , 2002).The study reported in this thesis can increase our understanding of how to construct such PML’s. In the interpretation of the reconstructed permittivities for the HM and GD experiments in terms of Mixing Laws (Section 6.6), we saw that a slight difference in porosity has an effect in the value of the permittivity of dry sands (higher for lower porosities). And that the empirical relations between the permittivities and the sands densities, given by Dobson et al. (1985) and Shutko & Reutov (1982), overestimate their value while the reinterpretation of the Clausius-Mossotti model (equation (E.23)) underestimates it. While, a very small quantity of water could account for this underestimation no logical explanation could be found for the overestimation of the empirical models, besides their self-empirical nature. Due to this, we find that the dry sands should be modelled with the reinterpretation of the Clausius-Mossotti model in opposition to the empirical models. The best mixing formula for partially and fully homogenously saturated sands (HM experiment and steps 2 and 6 of the GD experiment) has proved to be a Power Law equation. Different exponents are needed depending on the saturation level 0 and if we are interested in the real or imaginary parts of the permittivity. The ε of these samples, is in very good agreement (error of 1%) with the Power Law model [c = 0.3 for Sw < 50% and c = 0.4 for Sw > 50%]. This is not in contradiction with other researchers results, where variety is dominant (see Section E.5). In fact, the fit of the Power Law model to the reconstructed permittivities is better than most published data, but our samples were equal in composition. A broader range of sands would certainly lead to lower accuracies, since the dispersion of coefficients will be bigger. For the 2-layer samples (steps 3 to 5 of the GD experiment) the error made by the Power Law model is larger. However, we could still model them in that manner if we allowed for a lower precision (error of 2%). In that sense, the partially saturated sands, whose saturation is clearly distributed in 2-layers, could be considered homogenous. The validity of this approximation decreases with increasing 140 Chapter 6. Permittivity states of mixed-phase permittivity contrast. Instead, we can use the CRIM model to model the 2 layer samples (equation 6.17) and improve considerably the agreement between the measurements and the model results. This approach includes geometry in its modelling since first the permittivities of each layer is computed with the CRIM model and then the permittivity of the combined 2 layer sample is calculated with the same model and the volume fractions are given by the heights of the layers. Del Rio & Whitaker (2000a) derived constrain conditions for the validity of using effective parameters (one-equation models) for the dynamic electromagnetic properties of mixtures. They define the Local Electrodynamic Equilibrium, given by equations (E.11), (E.12) and (E.13), and they claim that if the constrains are met the effective permittivity will be that corresponding to the arithmetic mean, which is true in the low frequency approximation. In this chapter, we have given two examples that contradict their statements: the modelled samples of Section 6.5 (fSW = 0.2 and fSr = 0.8) and the real fully saturated samples, (fSW = 0.4 and fSr = 0.6). Both samples do not meet the Local Electrodynamic Equilibrium constraint and yet, can be represented by an effective permittivity. In the case of the modelled samples, this depends on the frequency and the distribution of the fully saturated layer. Moreover, the saturation distribution affects the value of this ε∗ , when the wet layers are thin enough and evenly distributed, the effective permittivity corresponds to the arithmetic mean, but if it is a single thick layer, then its value is given by the CRIM model for high enough frequencies, whereas, if the sample is fully saturated, then it is better modelled by the Power Law 0.4 . These experimental and modelled results contradict the derived conditions by del Rio & Whitaker (2000a) for the validity of using effective parameters (one-equation models) for the electromagnetic properties of mixtures. Along this thesis we have supported our choice of tool and reconstruction method, and throughout this chapter we have presented the characteristics of different saturation techniques and their corresponding permittivities. It is now time to explain the best experimental methodology according to our studies. 6.7.1 Recommended experimental methodology In the determination of the effective permittivity of dry, partially or fully saturated soils we recommend the use of a Gravity Drainage experiment. The experimental advantages of the GD experiment over the HM mixed are evident: • The same sample can be measured at many saturation degrees. 6.7. Conclusions 141 • It allows for two phase flow, and although in this thesis we have presented results with air and water as the two flowing phases, in principle, they could be any other liquids or gases (limitations given by fluid viscosity and soil permeability) • Extra information about the movement of the effective fluid front along the sample can be easily found very accurately. • Depending on the permittivity contrast between the 2-layers and the desired accuracy, these samples can be modelled as homogenous and the results are analogous to true homogeneous samples. So that, they can be used in the study of the effective properties of soils at different saturations. • A simple 2-layer model together with the CRIM model account extremely well for the permittivity of these samples. A Gravity Drainage experiment is, in principle, always possible but it is only appropriate when the permittivity contrast is small so that the loss of accuracy when treating the sample as homogenous is not large. If the sample cannot be considered to have an effective permittivity, then, this experiment is suitable for the study of 2-layer samples, alone. In the study of effective properties of samples with large permittivity contrasts, the samples must be saturated outside the sample holder. In view of our experience, the sand(SW )/sand(Sr ) contrast (21/5) is in the safe area while a 2 layer sample consisting of one layer of water and another of air is not (80/1). Yet, most sands have a small contrast between the fully and the residually saturated soils and we expect that this technique can be applied for most cases leading to a better understanding of the relations between fluid saturation and permittivity. 142 Chapter 6. Permittivity states of mixed-phase We never stop investigating. We are never satisfied that we know enough to get by. Every question we answer leads on to another question. This has become the greatest survival trick of our species. Desmond Morris (1928- ) Chapter 7 Conclusions and recommendations In this thesis we have shown the successful design of an accurate tool to investigate the electrodynamic response of soils. We have also shown that their response is very much dependent on the distribution of components of the sands, and that heterogeneities as small as λ/100 can be detected. This challenges existing mixing laws, that only consider the components volume fractions and permittivities, and when EM fields can be considered quasi-static. We have also proposed a new method to treat the interfaces present in the soils. This will enhance existing models of the subsurface. The tool is a customized coaxial transmission line that allows for fluid flow through the sample, and whose S-parameters can be determined in the frequency range from 300MHz to 3GHz. This design is most suited to measure loose sand samples. The same tool can be modified to measure solid cores. It would be appropriate to expand this study to lower frequencies and use the same tool as a capacitor. We also introduced a new combined representation of the measurements, the Propagation Matrices Representation. It allowed us to prove the independence of the reflection and transmission coefficients, as well as, to develop a novel analytical inversion of the measurements. Moreover, it provides a representation in which the sections interact by matrix multiplication making its mathematical treatment and physical understanding simpler than existing methods. The forward model representing the reflection and transmission along the line is in very good agreement with the measured data after a profound calibration. We have shown that relative changes in the permittivity in the order of 1% can be detected over a wide frequency band up to 3 GHz, while the lowest usable frequency depends on the permittivity of the material filling the sample holder. 143 144 Chapter 7. Conclusions and recommendations We compared the different inversion techniques to obtain the complex permittivity (ε∗ ) from the measured S-parameters. All techniques determine ε∗ per frequency point without a pre-defined model of the frequency dependency of permittivity. The Compensation techniques have several disadvantages. They can be applied only in a narrow frequency band. They are very time consuming since they require many measurements, which increases the experimental errors. On the other hand, the optimization method gives very good results. However, an appropriate weighting is needed, which is unknown a-priori. We also presented our new method, the propagation matrices, it allows to find explicit expressions to compute both complex electric permittivity and magnetic permeability. It is based on a simple concept and it is easily programmed. The analytical expressions found from the propagation matrices approach, depend on two different combinations of the corrected S-parameters. The one related to the sample’s impedance is unstable and since we are interested in non-magnetic sands, we restrict ourselves to the reconstruction of permittivity from the other method, which is related to the propagation factor of the sample. We successfully reconstructed the relative permittivity of air within ±1% error. And we could fit a Debye model to the reconstructed permittivity of ethanol, within the same error bound. The formulation of the measurements with the propagation matrices method allowed us to prove that all existing analytical expressions are, in fact, the same. This explains why indistinctively of the expression used, the reconstruction of permittivity from analytical expressions always suffers from the same problems at resonant frequencies and with low-loss materials. Our tool, a customized coaxial transmission line together with the propagation matrices inversion, is able to reconstruct permittivities within ±1% error from 3λ/L up to 3 GHz, while the lower limit can be relaxed by using optimization. However, for lossy materials, this upper limit is conditioned by its loss. As the amplitude of the transmission gets smaller, the accuracy of the phase of the measurements deteriorates taking its toll on the reconstructed permittivity that clearly presents noisy behavior. Nevertheless, it stills captures its trend, and it can be amended with noise reduction techniques. To improve the knowledge over the measurements it is desirable to include error propagation in the reconstruction of permittivity. We also studied the effect of fluid flow through a sand sample in relation to its permittivity. We showed how the saturation technique has a clear effect in the reconstructed permittivity and that extra care is needed for proper measurements. Depending on the goal of the measurement it is best to homogeneously saturate the sample outside the sample holder or to perform a gravity drainage to change the saturation. The second technique has the advantage that the same sample can be measured at different saturation levels. However, the saturation distribution results in distinctive 2-layer samples, whose reconstructed permittivity exhibit anomalies 145 due to the interface. These samples can be represented by a 2-layer model very accurately, if the width of the layers is found from the phase of the transmission coefficients. Since we are able to accurately determine the width of the layers present in a sample subject to gravity drainage, we can use the tool to monitor the movement of the fluid front, opening new doors in the investigation of fluid flow in porous media by means of electromagnetic measurements. With the aid of synthetic experiments, we showed how the anomalies depend on the permittivity contrast between the different layers, the distribution of these layers and the length of the sample holder. In addition to this, the distribution of the layers has a strong impact on the permittivity. The amount of parameters that influence the anomalies (layer width, permittivity contrast and layer distribution) make it very difficult to draw a quantitative conclusion and more investigation is necessary. This should be done together with an analytical study of layered systems, which, thanks to the propagation matrices representation can be easily done. When the size of the heterogeneities is smaller that λ/100 (quasi-static field approximation is valid), the permittivity of all samples composed of two effective materials corresponds to the arithmetic mean, while for higher frequencies the distribution of the materials has a direct effect on the value of the permittivity. In case they form a 2-layer sample, the permittivity of the sample will be given by the harmonic mean of the permittivities (velocities) of the components. As mentioned in Section 6.7 these results can contribute to the improvement of direct and inverse modelling and increase the understanding of how to construct PML’s. We found that dry sands should be modelled with the reinterpretation of the Clausius-Mossotti model, in opposition to the empirical models. Their real permittivity is constant over the frequency range as well as their imaginary part that is very small. The real part of the permittivity of partially saturated sands does not depend strongly on frequency, while its imaginary part shows an increase with increasing frequency. The best mixing formula for partially and fully homogenously saturated sands has proved to be a Power Law equation. Different exponents are needed depending on the saturation level and whether we are interested in the real or imaginary parts of the permittivity. We think that our results on multilayered samples can explain the variety of exponents encountered in this thesis and in the existing literature. We saw that the permittivity of a layered sample was given by the arithmetic mean for frequencies smaller than a transition frequency, and above it, it oscillated around that of the CRIM model. The Power Law is also able to model the permittivity of 2-layer samples, but the error it makes is larger than when it is used for modelling homogenous samples. If a lower precision is allowed (error of 2%), the partially saturated sands, whose 146 Chapter 7. Conclusions and recommendations saturation is clearly distributed in 2-layers, can be considered homogenous. The validity of this approximation decreases with increasing permittivity contrast. If we are interested in their heterogeneity, geometry can be included in the modelling using the CRIM model. The permittivities of both layers are also modelled with the CRIM model, and their volume fractions are determined by the height of the layers. It gives the best fit to the permittivities of the Gravity Drainage Experiment samples. We presented numeric and experimental examples that contradict the theory derived by del Rio & Whitaker (2000a). Our experiments did not meet the constraints for the Local Electrodynamic Equilibrium, and yet the samples were adequately represented by an effective permittivity. Furthermore, εef f is not necessarily the arithmetic mean of the permittivities of the components, as stated by del Rio & Whitaker. Based on our results we can conclude that sands can be represented by a single value of permittivity. However, the effective parameter depends very strongly on the constituents, volume fractions and distribution within the sand, and these must be taken into account. It should also be considered that the permittivity of sands can vary with frequency, not because of the frequency dependence of the permittivities of its constituents, but because of their geometrical distribution. Appendix A General solution for the wave equation in polar coordinates This appendix presents the general solution for the Maxwell wave equations in a cylindrical confined region, the symmetry of this structure recommends the use of polar coordinates. A.1 Polar coordinates In polar coordinates a point P in space is represented by three coordinates r, φ, and z, according to Fig. (A.1) x1 = r cos φ, (A.1) x2 = r sin φ, (A.2) x3 = z, (A.3) and the three mutually perpendicular base vectors being {r̂, φ̂, ẑ} of unit length each. A vector G = {Gr , Gφ , Gz } can be written as G = Gr r̂ + Gφ φ̂ + Gz ẑ. 147 (A.4) 148 Appendix A. General solution for the wave equation in polar coordinates Figure A.1: The polar coordinate system Expressing the curl (∇×) and divergence (∇·) of a vector G and the Laplacian of a scalar ψ in polar coordinates will help us later on. 1 1 ∂r (rGr ) + ∂φ Gφ + ∂z Gz , r r 1 ∇×G = ∂φ Gz − ∂z Gφ r̂ + [∂z Gφ − ∂r Gz ] φ̂ r 1 + [∂r (rGφ ) − ∂φ Gr ] ẑ, r 1 1 ∇2 ψ = ∂r (r∂r ψ) + 2 ∂φ ∂φ ψ + ∂z ∂z ψ. r r ∇·G = A.2 (A.5) (A.6) (A.7) General solution for the wave equation The source free wave equations for the electric and the magnetic field, equations (2.69) and (2.70), have the same shape and may be written in terms of Ĝ, standing either for the electric field, Ê, or the magnetic field, Ĥ. ∇2 Ĝ − γ̂ 2 Ĝ = 0, (A.8) it can be solved for a general component F̂i , where the subscript i refers to the r-radial, φ-angular or z-component, by separation of variables. Let Ĝi be written as the product of three independent functions, each depending only on one variable, i.e. R(r) for the radial, Φ(φ) for the angular and Z(z) for the z-component. Ĝi (r, φ, z) = R(r)Φ(φ)Z(z). (A.9) A.2. General solution for the wave equation 149 We solve the homogeneous equation introducing this expression into equation (A.8) and doing the appropriate separation (the sign of the separation constants is chosen so that the solution has physical meaning), lead to the following separate differential equations and solutions, for each variable ∂z ∂z Z − γ̂z2 Z = 0 −→ Z = c1 exp(γ̂z z) + c2 exp(−γ̂z z), (A.10) ∂φ ∂φ Φ + ν 2 Φ = 0 −→ Φ = c3 exp(jνφ) + c4 exp(−jνφ), (A.11) r2 ∂r ∂r R + r∂r R + (k̂c2 r2 − ν 2 )R = 0 −→ R = c5 Jν (k̂c r) + c6 Nν (k̂c r), (A.12) where γˆz and ν 2 are the separation constants, k̂c2 = γˆz 2 − γ̂ 2 , Jν is the Bessel function of the first kind of order ν, Nν is the Bessel function of the second kind of order ν. The separation constants and the coefficients of the linear combination of solutions can be determined imposing initial and boundary conditions. Properties concerning the Bessel functions can be found in (Abramowitz & Stegun, 1972). For k̂c2 = 0 the wave will propagate with the propagation constant of the medium γ̂ (i.e. γˆz 2 = γ̂ 2 ). But for this particular value of k̂c the differential equation on r (A.12) is degenerate, so that the solution given is not valid and it has to be solved separately. This case gives rise to the TEM mode and will be solved explicitly in the Subsection B.2.1. For imaginary values of γ̂, Z(z) is the sum of two waves, one travelling in the positive z-direction, exp(−γˆz z), and the other one in the negative z-direction, exp(γˆz z). At time t = 0, when the source is switched, only one wave travelling in the positive z-direction is present, but when it encounters a discontinuity it will then be reflected, therefore, there will be waves propagating in both z-directions. The coefficients c1 and c2 depend, then, on the amplitude of the exciting source and the reflections that take place within the guide. For the general solution we will take Z(z) = exp(γˆz z), where γˆz is determined from γˆz 2 = k̂c2 + γ̂ 2 , (A.13) and k̂c will come from imposing boundary conditions to the radial solution, and γ̂ depends on the medium properties and is given by equation (2.61). Imposing the continuity of the field and its derivate with respect to the angular coordinate Φ(0) − Φ(2π) = 0 and (∂φ Φ)(0) − (∂φ Φ)(2π) = 0 150 Appendix A. General solution for the wave equation in polar coordinates it is found that the separation constant ν must be an integer, and taking the convention of the right-hand (i.e. clockwise rotation implies positive z-direction propagation) we will take positive integers ν = m = 0, 1, 2, . . . The boundary condition to be imposed on the radial component of the solution is that the tangential component of the electric field is zero along a perfect conducting wall. This depends on the geometry and characteristics of the guide, so we will leave them for later. The solutions (for the non-degenerate case) can then be written as the product of Z(z), Φ(φ) and R(r) ĝim (r, φ, z) = Bm [c1 Jm (k̂c r) + c2 Nm (k̂c r)] exp(jmφ) exp(−γˆz z), (A.14) where Bm are the coefficients of the expansion that can be determined from orthogonality as Z Z [ĝim ĝi∗m ]dS = 1, (A.15) with the integration expanded over the entire cross section of the waveguide. The total field Ĝi is then the sum over m of ĝim Ĝi (r, φ, z) = ∞ X ĝim (r, φ, z) m=0 where ĝim is given by the equation (A.14). To find the specific solutions of propagating waves along any cylindrical structure we have to impose boundary conditions to Maxwell’s equations, but now written in polar coordinates. Appendix B Waveguides The solution of Maxwell equations for a coaxial waveguide can be found in this Appendix, together with cut-off frequencies for the TE and TM modes Electromagnetic waves propagate along waveguides in specific patterns. To find these, Maxwell’s equations have to be solved, subject to the proper boundary conditions, and in a convenient coordinate system. In this appendix we solved them for the specific case of a coaxial waveguide. We also propose a variation of the tool, presented in this thesis, for the permittivity determination of solid rock cores, see Section B.3. B.1 Maxwell’s equations in polar coordinates Maxwell’s equations (2.55) and (2.56) can be written splitted into their polar components, see appendix A. 1 − ∂φ Ĥz − ∂z Ĥφ + η̂ Êr = −Jˆre , r i h − ∂z Ĥr − ∂r Ĥz + η̂ Êφ = −Jˆφe , i 1h − ∂r (rĤφ ) − ∂φ Ĥr , + η̂ Êz = −Jˆze r 151 (B.1) (B.2) (B.3) 152 Appendix B. 1 ∂φ Êz − ∂z Êφ + ζ̂Hr = −K̂re , r i h ∂z Êr − ∂r Êz + ζ̂ Ĥφ = −K̂φe , i 1h ∂r (rÊφ ) − ∂φ Êr . + ζ̂ Ĥz = −K̂ze r Waveguides (B.4) (B.5) (B.6) We will now consider the case of TM and TE modes in each configuration (circular and coaxial waveguides). On each mode the z-components of either the magnetic (for TM mode) or the electric (for TE mode) field is zero and all the other components can be written in terms of the non-zero z-component field. The TEM mode will be derived in the Coaxial Waveguide section, as it can only propagate in that configuration of the two studied here. As the only coefficients and separation constant to be determined are those of the radial component. We can simplify the problem writing the fields as Ê(r, φ, z) = ê(r) exp(jmφ) exp(−γˆz z), (B.7) Ĥ(r, φ, z) = ĥ(r) exp(jmφ) exp(−γˆz z), (B.8) where for each mode we have TM TE êT M = {êr , êφ , êz } ĥT M = {ĥr , ĥφ , 0} êT E = {êr , êφ , 0} ĥT E = {ĥr , ĥφ , ĥz } where the z-components of each mode correspond to a Fourier expansion of the form ζ̂(r) = ∞ X ĝzm (r) with ĝzm (r) = Bm [c1 Jm (k̂c r) + c2 Nm (k̂c r)], (B.9) m=0 again the coefficients of the expansion Bm can be determined from orthogonality, equation (A.15). Introducing these expressions for the electric and the magnetic fields into Maxwell’s equations (B.1)-(B.6) new systems of equations are obtained for the TE and TM modes. B.1. Maxwell’s equations in polar coordinates B.1.1 153 Transverse magnetic: TM Taking into account that the longitudinal component of the magnetic field ĥz = 0, we will write the differential equation for the longitudinal component of the electric field êz and all the other electromagnetic field components can be related to it. An electric source current in the z-direction, Jˆze , will give rise to such a field. However the equations are presented and solved in a source-free domain for simplicity. The system of equations that give rise to this mode reads 1 k̂ 2 1 ∂r ∂r êz + ∂r êz + ∂φ ∂φ êz + k̂c2 êz = − c Jˆze , r r η̂ êr =− ĥφ =− η̂ k̂c2 γˆz k̂c2 ∂r êz , ∂r êz , η̂ jm ∂φ êz , k̂c2 r γˆz jm ∂r êz . ĥr = − k̂c2 r êφ = (B.10) (B.11) (B.12) The electromagnetic field components of the transverse magnetic mode will be determined imposing the appropriate boundary conditions to equation (B.9). B.1.2 Transverse electric: TE For this mode the longitudinal component of the electric field êz = 0 and this time a magnetic source current in the z-direction, K̂ze , gives rise to this mode. Similarly to the previous case 1 1 k̂ 2 ∂r ∂r ĥz + ∂r ĥz + ∂φ ∂φ ĥz + k̂c2 ĥz = − c K̂ze , r r ζ̂ êφ = ĥr =− ζ̂ k̂c2 γˆz k̂c2 ∂r ĥz , ∂r ĥz , ζ̂ jm ∂φ ĥz , k̂c2 r γˆz jm ∂φ ĥz . ĥφ = − k̂c2 r êr = − (B.13) (B.14) (B.15) The solution to equation (B.13) is given for a general case by equation (B.9) and imposing the appropriate boundary conditions, the electromagnetic field components of the transverse electric mode will be determined. 154 Appendix B. Waveguides Figure B.1: Electric and Magnetic fields for a TEM wave propagating along a coaxial waveguide. B.2 Coaxial Waveguide Along this type of configuration (see Fig. (B.1)) the principal mode TEM, together with the TM and TE, can propagate. Let us first consider the TEM mode B.2.1 Transverse electromagnetic: TEM This mode has only two non-zero components, Êr and Ĥφ , and it will propagate when the waveguide is exited with a longitudinal source volume density of electric current, Jˆze . We have already seen that for k̂c2 = 0 the differential equation on r (A.12) is degenerate, so that it has to be solved separately. However the other differential equation on z (A.10) is still valid and γˆz 2 = γ̂ 2 . So that the fields can be written as Êr (r, φ, z) = êr (r, φ) exp(−γ̂z), (B.16) Ĥφ (r, φ, z) = ĥφ (r, φ) exp(−γ̂z). (B.17) Maxwell’s equations, in this particular case, read −γ̂ ĥφ + η̂êr = 0, 1 + ∂r Ĥφ = Jˆze , r −γ̂êr + ζ̂ ĥφ = 0, ∂φ Êr = 0. (B.18) (B.19) (B.20) (B.21) B.2. Coaxial Waveguide 155 Equations (B.19) and (B.21) show the non dependence of the fields with the φcoordinate, and then êr = êr (r) and ĥφ = ĥφ (r), and from equation (B.19) and taking into consideration that the current Iˆ is related to Jˆe according to Iˆ = Z e Jˆ · dl, ĥφ can be expressed in terms of Iˆ as ĥφ (r) = Iˆ , 2πr and êr can be derived from equation (B.18). Finally the expression for the electric and magnetic fields that compose the TEM mode propagating along a coaxial waveguide read Êr " #1/2 Iˆ ζ̂ exp(−γˆz z), = η̂ 2πr Ĥφ = Iˆ exp(−γˆz z), 2πr (B.22) (B.23) h i1/2 Iˆ where we recognize η̂ζ̂ 2π as the initial source potential V0 , which is a customary notation. And now the fields can be written as V̂0 exp(−γ̂z), r 1/2 η̂ V̂0 = exp(−γ̂z). r ζ̂ Êr = (B.24) Ĥφ (B.25) So that the electric and magnetic fields are perpendicular to each other and to the direction of propagation z, see Fig. (B.1). Their propagation factor is γ̂, that can be complex (lossy materials) and therefore the mode can be attenuated and dispersed, but they have no cutoff frequency as they can propagate for any frequency. For a perfect mode (no longitudinal component) to exist the walls of the waveguide should be perfect conductors, but in reality this is not the case and the metals have 156 Appendix B. Waveguides a high, but finite, conductivity, so that the electric field has a slight longitudinal component (êm z 6= 0), and the signal is slightly attenuated as it propagates. B.2.2 TM For this mode the non-zero longitudinal component of the electric field is êzmn (r, φ) = [c1 Jm (k̂cn r) + c2 Nm (k̂cn r)] exp(jmφ). (B.26) The second kind Bessel function Nm goes to infinity when r = 0, but in a coaxial waveguide it is within the inner conductor, so that there are no problems of finiteness and Nn is an admissible solution (c2 6= 0). Since êzmn must vanish at r = a and r = b, − Nm (k̂cn b) Nm (k̂cn a) c1 = , = c2 Jm (k̂cn a) Jm (k̂cn b) (B.27) where k̂cn is given from the non-vanishing roots of Nm (k̂cn a)Jm (k̂cn b) − Nm (k̂cn b)Jm (k̂cn a) = 0. (B.28) Making use of the properties of the Bessel functions of first and second kind (see Abramowitz & Stegun (1972)) and assuming asymptotic behavior, the approximate solution for k̂cn is found to be k̂mn = πn a−b with n = 1, 2, . . . As the system of equations (B.27) is undetermined we can choose c1 and c2 to be c1 = Nm (k̂cn a) and c2 = −Jm (k̂cn a). (B.29) We can now write the expressions for the fields that form this mode in terms of a 0 function ψmn (r) and its differential ψmn (r) ψmn (r) = Nm (k̂cn a)Jm (k̂cn r) − Jm (k̂cn a)Nm (k̂cn r), 0 ψmn (r) = 0 Nm (k̂cn a)Jm (k̂cn r) − 0 Jm (k̂cn a)Nm (k̂cn r), (B.30) (B.31) where 0 Nm (k̂cn r) = " 0 Jm (k̂cn r) = " m k̂cn r m k̂cn r # Nm (k̂cn r) − Nm+1 (k̂cn r) , # Jm (k̂cn r) − Jm+1 (k̂cn r) , (B.32) (B.33) B.2. Coaxial Waveguide 157 so that the field components of this mode read êzmn = ψmn (r) exp(jmφ) Êz = ∞ X ∞ X TM Xmn êzmn exp (−γ̂zn z), ∞ ∞ X X TM Xmn êrmn exp (−γ̂zn z), ∞ X ∞ X TM Xmn êφmn exp (−γ̂zn z), ∞ X ∞ X TM Xmn ĥrmn exp (−γ̂zn z), ∞ X ∞ X TM Xmn ĥφmn exp (−γ̂zn z), m=0 n=1 êrmn = êφmn =− − γ̂zn k̂cn 0 ψmn (r) exp(jmφ) Êr = m=0 n=1 jm γ̂zn ψmn (r) exp(jmφ) r k̂c2 Êφ = m=0 n=1 n ĥrmn = ĥφmn = jm η̂ ψmn (r) exp(jmφ) r k̂cn − η̂ k̂cn 0 ψmn (r) exp(jmφ) Ĥr = m=0 n=1 Ĥφ = m=0 n=1 (B.34) (B.35) (B.36) (B.37) (B.38) TM where the coefficients of the expansion Xmn are determined from equation (A.15) and read TM Xmn =√ 1 1 (m) π k̂c χm where (m) χm = a2 [Jm (k̂cn a)Nm+1 (k̂cn a) − Nm (k̂cn a)Jm+1 (k̂cn a)]2 − b2 [Jm (k̂cn a)Nm+1 (k̂cn b) − Nm (k̂cn a)Jm+1 (k̂cn b)]2 . (B.39) (B.40) Propagation Properties From equation (A.13) we can see that the propagation of this mode depends on the material it propagates through and the frequency at which it does. If γˆz is purely imaginary there will be no attenuation, but in case it has a real part (αz ) it will get attenuated in the z-direction by the factor exp(−αz z), the frequency at which this change occurs is known as cutoff frequency, because below this frequency the mode is evanescent. We need to distinguish between media with and without loss. 158 Appendix B. Waveguides • Lossless medium: the propagation factor becomes a real number γ̂ 2 = −w2 εµ = −w2 /c2 and a cutoff frequency wn can be found for each n-th root k̂cn = πn w2 πn = (γ̂zn )2 + 2 → wcn = c. a−b c a−b (B.41) • Medium with losses: the propagation factor γ̂ has already a real and an imaginary part, so that γˆz also has a real part, no matter what value k̂c takes. For these media the modes behave as attenuated waves for all frequencies. B.2.3 TE For this mode the non-zero longitudinal component of the magnetic field is ĥzmn (r, φ) = [c1 Jm (k̂cn r) + c2 Nm (k̂cn r)] exp(jmφ). (B.42) Since êφmn must vanish at r = a and r = b, and êφmn ∝ ∂r ĥzmn we have − c1 N 0 (k̂c b) N 0 (k̂c a) = m n = m n , 0 (k̂ a) 0 (k̂ b) c2 Jm Jm cn cn (B.43) where k̂cn is given from the non-vanishing roots of 0 0 0 0 Nm (k̂cn a)Jm (k̂cn b) − Nm (k̂cn b)Jm (k̂cn a) = 0. (B.44) The approximate solution for k̂cn is found to be k̂c,m1 = k̂c,mn = 2 for n = 1, a+b π(n − 1) for n = 2, 3, . . . a−b Now c1 and c2 are chosen 0 c1 = N m (k̂cn a) and 0 c2 = −Jm (k̂cn a) (B.45) So that the fields that form this mode are written in terms of the function ϕmn (r) and its differential ϕ0mn (r) 0 0 ϕmn (r) = k̂cn [Nm (k̂cn a)Jm (k̂cn r) − Jm (k̂cn a)Nm (k̂cn r)] ϕ0mn (r) = 0 k̂c2n [Nm (k̂cn a)Jm (k̂cn r) − 0 Jm (k̂cn a)Nm (k̂cn r)] (B.46) (B.47) B.2. Coaxial Waveguide 159 0 (k̂ a) and J 0 (k̂ a) are equal to equations (B.32) and (B.33) when where Nm cn m cn r = a. so that the field components of this mode read ĥzmn = ϕmn (r) exp(jmφ) Ĥz = ∞ X ∞ X TE Xmn ĥzmn exp (−γ̂zn z), ∞ X ∞ X TE ĥrmn exp (−γ̂zn z), Xmn ∞ X ∞ X TE Xmn ĥφmn exp (−γ̂zn z), ∞ X ∞ X TE Xmn êrmn exp (−γ̂zn z), ∞ X ∞ X TE Xmn êφmn exp (−γ̂zn z), m=0 n=1 ĥrmn = ĥφmn =− − ζ̂ 2 γ̂zn k̂c2n ϕ0mn (r) exp(jmφ) Ĥr = m=0 n=1 ζ̂ 2 jm ϕmn (r) exp(jmφ) r γ̂zn k̂c2 Ĥφ = m=0 n=1 n êrmn = − jm ζ̂ 2 ϕmn (r) exp(jmφ) r k̂c2 m=0 n=1 n ζ̂ 2 êφmn = Êr = k̂c2n ϕ0mn (r) exp(jmφ) Êφ = m=0 n=1 (B.48) (B.49) (B.50) (B.51) (B.52) TE where the coefficients of the expansion Xmn are determined from equation (A.15) and read 1 1 TE Xmn = √ (e) π χm (e) χm = a2 − − b2 − m2 ! k̂c2n ! m2 k̂c2n where (B.53) [Jm (k̂cn a)Nm+1 (k̂cn a) − Nm (k̂cn a)Jm+1 (k̂cn a)]2 − [Jm (k̂cn a)Nm+1 (k̂cn b) − Nm (k̂cn a)Jm+1 (k̂cn b)]2 . Propagation Properties Following the same reasoning as in the case of TM modes we find a cutoff frequency when they propagate along a coaxial waveguide filled with a lossless medium, and it reads 160 Appendix B. wc,m1 = wc,mn = B.3 2 c for n = 1, a+b π(n − 1) c for n = 2, 3, . . . a−b Waveguides (B.54) (B.55) Coaxial-Circular Waveguide Belhadj-Tahar et al. (1990) and Taherian et al. (1991) proposed a coaxial-circular waveguide for permittivity measurements of solid rock cores. It consists of coaxial transition units and a circular sample holder to host the core. This has an advantage over performing these types of measurements with a coaxial sample holder, as the core doesn’t need drilling and the flow through the sample will not have a preferential path. Moreover air gaps between the central pin and the core are avoided. The tool presented in this thesis, see Chapter 4, can be modified to work as a coaxial-circular waveguide. It is relatively easy and consists in removing the central conductor of the sample holder and adjusting the ones of the transition units. Figure B.2 shows a diagram for this modified tool. Figure B.2: Modified tool: coaxial-circular waveguide B.3. Coaxial-Circular Waveguide 161 To reconstruct the permittivity of a rock core from measured S-parameters in this type of configuration, a full-wave description of the waveguide is needed. The fields in the coaxial sections will propagate as the TEM, TE and TM modes presented previously in this Appendix, and in the circular section similar equations can be obtained for the TM mode, the only one generated when going from a coaxial into a circular section. Then the total fields at the reference planes can be computed numerically and via an optimization routine the permittivity can be reconstructed. 162 Appendix B. Waveguides Appendix C Transmission Lines This appendix contains the equations of wave propagation through a transmission line, and more explicitly through a coaxial configuration TEM transmission lines are a specific group of waveguides. They are composed of two or more conductors whose cross section perpendicular to the direction of propagation is constant (i.e. two parallel wires, coaxial cable, micro-strip line). In such a configuration, as we showed in Appendix B, TEM waves (transverse electromagnetic; both electric and magnetic fields lie entirely in the plane perpendicular to the direction of propagation) are able to propagate. Other modes, TE (transverse electric; the electric field lies in the plane perpendicular to the direction of propagation) and TM (transverse magnetic; the magnetic field lies entirely in the plane perpendicular to the direction of propagation) are also excited, but the line is usually operated below the cutoff frequency of these modes so they do not propagate. In this situation, it is possible to reformulate Maxwell’s waves equations in terms of circuit parameters and voltage and current waves, see Section C.1. C.1 Transmission Line Equations In Appendix B we show the existence of a purely transversal electromagnetic mode that can propagate along a coaxial waveguide. It can be proven, in a similar way (solving Maxwell’s equations subject to the boundary condition that the tangential component of the electric field vanishes on the surface of the conductors), that this type of waves may propagate in uniform lines, consisting of any combination of conductors and dielectrics where the cross-section of the line is constant (parallel wires, coaxial cables, etc). 163 164 Appendix C. Figure C.1: paths Transmission Lines Integration But Maxwell’s equations applied to the specific case of transmission lines can be ˆ as they can be written expressed in terms of voltage V̂ and current I, Z Z ˆ Ĥ · dl, (C.1) V̂ = Ê · dl and I= Ct C0 where Ct is an integration path transverse to z and C0 a closed contour, see Fig. (C.1). The equivalent to the wave equations for a TEM mode propagating along a coaxial waveguide, equations (2.69) and (2.70), in terms of voltage and current are ∂z ∂z V̂ − γ̂ 2 V̂ = 0, ∂z ∂z Iˆ − γ̂ 2 Iˆ = 0, (C.2) (C.3) R when the line has been exited by an initial voltage, V̂0 = Ct Ê(r, 0) · dl, and γ̂ is again the propagation factor, but has to be written in terms of the transmission line parameters. C.1.1 Propagation factor (γ̂) of a transmission line The propagation factor of general media, with conductivity σ̂, permittivity ε̂, permeability µ̂ and taking s = jw reads q γ̂ = η̂ ζ̂ where η̂ = σ̂ + jwε̂ and ζ̂ = jwµ̂, and its transmission counterpart can be written as γ̂ = p (G + jwC)(R + jwL), (C.4) C.1. Transmission Line Equations 165 where G is the conductance and equivalent to σ̂ [S/m] C is the capacitance and equivalent to ε̂ [F/m] R is the resistance due to the non perfect conductors [Ω/m] L is the inductance and equivalent to µ̂ [H/m] The resistance R is determined by the finite conductivity σc of the non perfect conductors and the skin depth δ. A resistivity Rs can be defined as r 1 2 Rs = where δ= , δσc wµσc and thus R = Rs /(circumference of the line). These parameters are all defined per unit length and depend only on the geometrical aspects of the line and the filling material. Explicit formulas for their calculation can be found in the literature (see, for example, Karmel et al. (1998)). The solutions to the transmission line equations (C.2) and (C.3) are V̂ (z) = V̂0+ e−γ̂z + V̂0− e+γ̂z , ˆ I(z) = Iˆ0+ e−γ̂z + Iˆ0− e+γ̂z . (C.5) (C.6) The voltage and the current waves consist of two waves, one travelling to the right and the other one to the left. The complex numbers V̂0+ , Iˆ0+ , V̂0− and Iˆ0− are not independent of each other as they must obey the transmission line equations, (C.2) and (C.3) and thus V̂0− V̂0+ = − = Z0 , Iˆ+ Iˆ− 0 (C.7) 0 where Zc is the characteristic impedance (Ω) of the transmission line. And it is a function of the material electromagnetic properties, µ̂, ε̂, σ̂ and σc , and the cross-sectional geometry of the line Zc = s R + jwL . G + jwC (C.8) 166 Appendix C. Transmission Lines The impedance at any position is defined as the ratio between the voltage and current on the line Z(z) = V̂ (z) , ˆ I(z) (C.9) and the admittance is the inverse of the impedance Y (z) = 1/Z(z). Transmission Line Parameters for a coaxial transmission line For a coaxial transmission line with inner radius b and outer radius a these parameters become L = C = G = R = C.2 a µ̂ ln , 2π b 2π ε̂ , ln(a/b) 2πσ̂ , ln(a/b) 1 1 1 Rs . + 2π a b (C.10) (C.11) (C.12) (C.13) Local Reflection and Transmission coefficients Consider an interface n located at zn , the medium on the left has impedance Zn and on the right Zn+1 . Part of an incident wave on the interface will be reflected and part will be transmitted, this is represented in Fig. (C.2). Figure C.2: Incident, reflected and transmitted waves on the nth interface C.2. Local Reflection and Transmission coefficients 167 A local reflection coefficient and a local transmission coefficient can be defined as the ratios of the reflected and transmitted components of the voltage wave to the incident component. rn± = t± n = V̂n∓ , V̂n± #±1 " ± Vn+1 Vn± (C.14) . (C.15) In order to find these ratios we need to impose boundary conditions to the voltage and current wave expressions. For zn = 0, these expressions are z<0 z>0 The voltage and current waves on each side of the interface are given by V̂n (z) = V̂n+ e−γ̂n z + V̂n− e+γ̂n z + V̂n+1 (z) = V̂n+1 e−γ̂n+1 z Iˆn (z) = Iˆn+ e−γ̂n z + Iˆn− e+γ̂n z + Iˆn+1 (z) = Iˆn+1 e−γ̂n+1 z a local impedance can be defined as Zn = V̂n+ Iˆn+ − = − V̂ˆn− Zn+1 = In + V̂n+1 + Iˆ n+1 V̂ − n+1 = − ˆ− In+1 and then the waves can be rewritten as + V̂n+1 (z) = V̂n+1 e−γ̂n+1 z V̂n (z) = V̂n+ e−γ̂n z + V̂n− e+γ̂n z Iˆn (z) = 1 Zn h V̂n+ e−γ̂n z − V̂n− e+γ̂n z i Iˆn+1 (z) = + 1 −γ̂n+1 z Zn+1 V̂n+1 e These expressions under the boundary conditions, that the fields must be continuous at the interface lim V̂n+1 = lim V̂n , (C.16) lim Iˆn+1 = lim Iˆn , (C.17) z↓0 z↓0 z↑0 z↑0 168 Appendix C. Transmission Lines lead to the equations for the local reflection and transmission coefficients C.2.1 rn+ = Zn+1 − Zn , Zn+1 + Zn (C.18) t+ n = 2Zn+1 . Zn+1 + Zn (C.19) The terminated transmission line Assume a transmission line of constant impedance Zc which is terminated at a distance z = L from the origin by a load impedance ZL . Its reflection coefficient at the interface L, where the line is terminated, depends thus on the impedance of the load and of the line, according to rL = ZL − Zc , ZL + Zc (C.20) so for different types of load we have: • Short-circuit load ZL = 0 and RL = −1. • Open-circuit load ZL = ∞ and RL = 1. • Matched load ZL = Zc and RL = 0. C.3 Line parameters for an ideal section For ideal lines (resistance due to the conductors that form the line is negligible) the propagation constant and the impedance of each section can be expressed in terms of the electromagnetic properties of the materials filling it γ̂n2 = −ω 2 ε̂∗n µ̂∗n and Zn = g s µ̂∗n , ε̂∗n (C.21) where g is a geometry factor that depends on the geometry of the line. For a coaxial transmission line g = ln(a/b) 2π . C.4. Global Reflection and Transmission coefficients 169 The local reflection coefficient can be rewritten as r rn = r µ̂∗n+1 ε̂∗n+1 µ̂∗n+1 ε̂∗n+1 − + q q µ̂∗n ε̂∗n . (C.22) µ̂∗n ε̂∗n If the materials are non magnetic, µ̂∗ = µ0 these equations simplify to ω2 ∗ ε , c20 r,n Z p 0∗ where εr,n q p ∗ εr,n − ε∗r,n+1 q . p ∗ εr,n + ε∗r,n+1 γ̂n2 = − Zn = rn = C.4 (C.23) Z0 = g r µ0 , ε0 (C.24) (C.25) Reflection and Transmission coefficients for a multi-section transmission line Let us now consider a multi-section transmission line consisting of N sections, see Fig. (C.3). The sections are determined by abrupt changes in impedance at the interfaces between them. At z = zs < 0 the line is driven by a source and at z = zN = L it is terminated by a load impedance ZL . Figure C.3: Multi-section transmission line 170 Appendix C. Transmission Lines If Γn is the local reflection coefficient at the nth interface the voltage and current waves on the left hand side (zn−1 ≤ z ≤ zn ) of the nth interface can be written as i h V̂n (z) = V̂n+ e−γ̂n z + Γn e+γ̂n (z−2zn ) , i 1 + h −γ̂n z Iˆn (z) = V̂n e − Γn e+γ̂n (z−2zn ) , Zn (C.26) (C.27) and on the right hand side (zn ≥ z ≥ zn+1 ) h i + e−γ̂n+1 z + Γn+1 e+γ̂n+1 (z−2zn+1 ) , V̂n+1 (z) = V̂n+1 h i 1 Iˆn+1 (z) = V̂n+ −γ̂n+1 z − Γn+1 e+γ̂n+1 (z−2zn+1 ) . Zn+1 (C.28) (C.29) It is possible to derive a recursive expression for the multiple reflection coefficient Γn and for the multiple transmission coefficient Υn if the continuity of the fields across the nth interface, equations (C.16) and (C.17) are taken into account Γn = rn + Γn+1 e−2γn+1 dn+1 , 1 + rn Γn+1 e−2γn+1 dn+1 (C.30) Υn = n Y (C.31) k=1 P n 1 + Γk e− k=1 γk dk , −2γ d k+1 k+1 1 + Γk+1 e where rn is the local reflection coefficient given by equation (C.18) and dn+1 = zn+1 − zn is the length of the n + 1 section. Appendix D Scattering Matrix Here, we show the explicit relations between the total reflection and transmission coefficients, found in the previous Appendix, with the S-parameters. We also show the unitary property of the Scattering Matrix for a lossless line 2-port networks are described by electrical engineers via three different matrices, Impedance [Z], Admittance [Y ] and the Scattering Matrix [S]. The impedance matrix relates reflected voltage to incident current and viceversa in the case of admittance representation. The scattering matrix relates reflected voltage to incident voltage on circuit components with two distinct ports or connections, V − = SV + , see Figure (D.1). And the entries of the matrix, the S-parameters are related to the reflection/transmission response of the electrical components forming the network and their interaction. And therefore, they are the favorite representation for reflection/transmission problems. This voltage relation can be written as − V01 − V02 = S11 S12 S21 S22 Figure D.1: 2-Port Network. 171 + V01 + V02 . (D.1) 172 Appendix D. Scattering Matrix So that, the diagonal elements of [S], Sii are given by Sii = V0i− , V0i+ (D.2) it is then clear that Sii is the reflection coefficient at the ith port, when every other port is terminated in a matched load. The off diagonal terms, Sij , i 6= j are given by Sij = V0i− V0j+ j 6= i, (D.3) and Sij , i 6= j is the transmission coefficient from the jth port to the ith port, when every other port is terminated in a matched load. D.1 S-parameters and the total reflection and transmission coefficients of a transmission line Assuming a perfectly symmetric and homogeneous line, [S] is symmetric and their components are related to the total reflection, Γn , and total transmission, Υn , coefficients of a transmission line, given by equations (C.30) and (C.31), as follows S11 = S22 = V1− = Γ1 e2γ1 zr , V1+ (D.4) where zr is the position where the voltage ratio is measured. If this reference plane is taken at the origin of the line (zr = 0) then S11 = S22 = Γ1 . (D.5) For the off-diagonal terms and taking the reference plane of measurement at the origin or end of the line, we have S12 = S21 = ΥN . (D.6) D.2. Unitary property of the Scattering Matrix D.2 173 Unitary property of the Scattering Matrix For a lossless ideal line it can be proven that [S] is unitary (Karmel et al. , 1998), and the S-parameters obey the following properties 2 X i=1 2 X |Sik |2 = 1, ∗ Sik Sir = 0, i=1 k = 1, 2 k, r = 1, 2 (D.7) k 6= r. (D.8) For lossless material filling the line, these properties can be used to determine how close a line is to ideal. 174 Appendix D. Scattering Matrix Appendix E Permittivity models This appendix gives a short introduction into the theoretical models that try to explain the EM behavior of materials and mixtures. It is not extensive, for further insight and full derivations we refer to the cited literature. E.1 Introduction The electromagnetic behavior of materials has been, is and will continue to be a very broad research topic, for obvious reasons. Along history, many different authors have been involved in the production of theoretical and empirical models that could explain the experimental evidence. The first capacitor was constructed by Cunaeus and Musschenbroek as far back as 1745, and it became very popular for a variety of experimental purposes under the name of Leyden jar. However, it was Faraday in 1837, who published the first numerical measurements on dielectrics, a term introduced by him (Böttcher, 1952). The beginning of the systematic investigation of the dielectric properties can be established with the works of Mossotti (1850) and Clausius (1879), as they attempted to correlate the dielectric constant with the microscopic structure of the materials. They considered the dielectric to be composed of conducting spheres in a non-conducting medium, and succeeded in deriving a relation between ε and the volume fraction occupied by the conducting particles (Section E.2.1). Debye (1929), in the beginning of the 20th century, realized that some molecules had permanent electric dipole moments and that it gave rise to the macroscopic dielectric properties of the materials. He succeeded in extending the ClausiusMossotti theory to take into account the permanent moments of the molecules. His theory, later extended by Onsager (1936) and Kirkwood (1939), linked the 175 176 Appendix E. Permittivity models dispersion of ε∗ to the characteristic time needed for the permanent molecular dipoles to reorient following an alternating EM field. He deduced that the time lag between the average orientation of the moments and the field becomes noticeable when the frequency of the applied field is of the order of the reciprocal relaxation time. Debye’s model is still being used for polar liquids, where the dipoles are relatively ”far away” from each other. However, the dielectric behavior of solids deviates significantly from Debye’s theory of relaxation. Cole & Cole (1941) pioneered the first approach to interpret the non-Debye relaxation of materials by means of a superposition of different relaxation times and Jonscher (1983) rebates that the relaxation behavior at molecular level is intrinsically non-Debye due to the cooperative molecular motions, the ”many-body interactions” approach. All these researchers modelled the macroscopic behavior of dielectric materials composed of a single phase and a single type of micro-particles, that is, they did not describe the electromagnetic properties of mixtures. However, most materials are indeed mixtures and researchers have struggled, and still do, to characterize their physical properties with effective parameters. A permittivity mixture model expresses ε∗ of the mixture in terms of the permittivities of each constituent and their volume fraction. The literature on the effective properties of materials and mixtures is very vast and we have decided to include in this appendix just a few of the most relevant ones. Nevertheless, you can find comprehensive reviews by Wang & Schmugge (1980), Shutko & Reutov (1982), Dobson et al. (1985), Chelidze & Guguen (1999a), Sihvola (1999) and Choy (1999), among others. This appendix is divided into four sections, the first three contain different models grouped according to their characteristics. Section E.2 presents two classical models to compute the permittivity of materials from known parameters, Section E.3 deals with existing bounds for the permittivity of mixtures and Section E.4 contains the most widely used mixing formulas. The last section summarizes some of the most important experimental results. E.2 Classical approaches to pure materials This section contains the classical models for gases, due to Clausius-Mossoti, and Debye’s model for polar liquids. E.2. Classical approaches to pure materials E.2.1 177 Clausius-Mossotti formula This famous formula is derived from the idealization of a spherical particle with polarizability α embedded in a homogeneous background of permittivity ε0 . The permittivity of N of these particles is given by: ε= 1 + 32 N α ε0 . 1 − 13 N α (E.1) It is also known as the Lorenz-Lorentz equation, since it was independently formulated for the optical range by Lorentz in The Netherlands and Lorenz in Denmark (Hippel, 1954). Note that the permittivity has a singularity at N α = 3, the Clausius-Mossotti catastrophe and it is due to the cavity model they used. E.2.2 Debye’s model As we have mentioned, Debye (1929) developed a model for the complex permittivity of a spherical polar molecule relaxing in a viscous medium, based on the assumption of a single relaxation time for all molecules, and it is described by the following equation ε∗r = ε∞ + σdc εs − ε∞ −j , 1 + jωτr ωε0 (E.2) where j is the imaginary unity, ω is the angular frequency [1/s], ε0 is the permittivity of free space, ε∞ is the permittivity in the limit ω → ∞ and εs when ω → 0, τr = 1/(2πfr ) is the characteristic relaxation time given in [s], and σdc is the direct current conductivity of the material [S/m]. Although not explicitly mentioned in the formula, these parameters are temperature dependent. In Chapter 5 we showed how the reconstructed permittivities per frequency for ethanol, a polar liquid, could be fitted with a Debye model very accurately. Here, we show the Debye parameters for distilled water WaterHasted Waterfit T (◦ C) 20.6 22 εs 79.9 77.4 ε∞ 4.22 4.81 fr (GHz) 17 19.4 σdc (µS/m) 0.6 ± 0.1∗ 0.06 Table E.1: Debye parameters for distilled water from literature (Hasted, 1973) and the fit to our measurements. ∗ The conductivity shown for WaterHasted was measured with a conductivity meter in-situ. 178 Appendix E. E.3 Permittivity models Bounds for the effective permittivity of mixtures Several authors limit the effective permittivity value to lie between bounds, based on general assumptions, rather than finding an explicit solution for the effective permittivity. We present equations for two-component mixtures where εi and εh are the permittivities of the inclusion and host, respectively, and fi and fh are their corresponding volume fractions. E.3.1 Geometric bounds Figure E.1: Geometrical limits of a mixture of 2 materials. The geometries in which 2 materials can be mixed is unlimited, but the most extreme cases are when the materials are distributed as shown in Figure E.1. Then the effective permittivity of any mixture cannot exceed the upper or lower bounds given by the arithmetic and harmonic means εmax ef f = fi εi + (1 − fi )εh , εi εh . εmin ef f = fi εh + (1 − fi )εi (E.3) (E.4) In electromagnetics, these bounds are also known as the Wiener bounds. For N components they read εmax ef f = 1 εmin ef f i=N X = fi εi i=1 i=N X i=1 fi . εi (E.5) (E.6) E.3. Bounds for the effective permittivity of mixtures E.3.2 179 Hashin-Shtrikman bounds For macroscopically homogeneous and isotropic composites, Hashin & Shtrikman (1962), derived bounds for the magnetic permeability of an N-component mixture based on energy considerations. Accordingly, for the permittivity εmin ef f = ε1 + A1 A1 1 − 3ε 1 where A1 = εmax ef f = ε2 + A2 A2 1 − 3ε 2 where A2 = N X fi + 1 3ε1 fi + 1 3ε2 1 i=2 εi −ε1 N −1 X 1 i=1 εi −ε2 , (E.7) , (E.8) where ε1 corresponds to the minimum permittivity found in the N components, and ε2 to the maximum. For a two component mixture, these bounds reduce to the Maxwell-Garnett model, where ε1ef f corresponds to the component i acting as inclusion into the host component h and ε2ef f interchanges the roles of inclusion/host. Ri→h = εh 1 + 3fi 1 − fi Ri→h Rh→i = εh 1 + 3fi 1 + (1 − fi )Rh→i ε1ef f ε2ef f E.3.3 εi − εh , (E.9) 2εh + εi εh − εi . (E.10) = 2εi + εh where Ri→h = where Rh→i Local Electrodynamic Equilibrium constrains Rio & Whitaker (2000a) constrained the validity of using one-equation models for the dynamic electric and magnetic fields in a two component mixture with equations (E.11), (E.12) and (E.13). They derived these constrains using the method of volume averaging. fh fi (εh − εi )2 1, (fh εh + fi εi )(εh + εi ) fh fi (µh − µi )2 1, (fh µh + fi µi )(µh + µi ) fh fi (σh − σi )(εh − εi ) 1. (fh σh + fi σi )(εh + εi ) (E.11) (E.12) (E.13) 180 Appendix E. Permittivity models In essence, and according to del Rio and Whitaker, when a mixture violates the constrains given by equations (E.11), (E.12) and (E.13) it cannot be represented by single effective electromagnetic properties, and a Two-Equation Model must be used (Rio & Whitaker, 2000b). Moreover, if the mixture can be represented by a One-Equation Model, then its effective permittivity is given by the arithmetic mean, equation (E.3). In Chapter 6 we presented numeric and experimental examples where these bounds are not met and yet the samples were adequately represented by an effective permittivity that is not necessarily the arithmetic mean of the permittivities of the components. E.4 Mixing Laws This section gathers the most commonly used mixing laws. Traditionally they are given in implicit form, but here we also present their explicit solution and we only state the solution for spherical inclusions. Most mixing formulae originate from solving the Laplace equation excluding the wave-propagation properties of EM fields. They are a static approach and, according to Sihvola (1999), can be considered quasi-static when the wavelength of the field divided by 2π is bigger than the size of the inclusion. But this is only an approximation. There are extensions of the static mixing formulae for time-varying fields but they are hardly used in geophysical applications, while the use of the static formulae is very wide spread. E.4.1 Maxwell-Garnett The Maxwell-Garnett mixing rule consist in rewriting the microscopic terms of the Clausius-Mossotti equation (E.1) in terms of the permittivities of the host and the inclusion. It reads m X εi − εh εef f − εh fi = , εef f + 2εh εi + 2εh (E.14) i=1 and it is also known as the Rayleigh mixing formula. In its explicit form for two components εef f = εh 1 + 3fi Ri→h , 1 − fi Ri→h (E.15) E.4. Mixing Laws 181 −εh is the Rayleigh reflection coefficient of a small sphere in the where Ri→h = εεii+2ε h quasistatic limit. It is anti-symmetric, if the roles of host and inclusion are interchanged the result is different. This approach of treating one of the components as host medium, and the inclusion as a perturbation against the background has been widely used and there are many Maxwell-Garnett type of mixing formulae (Sihvola, 1999). Equation (E.15) is equal to equation (E.9) and if the host and inclusion components are interchanged it is equal to equation (E.10). The Hashin-Shtrikman bounds for a 2-component mixture correspond to the Maxwell-Garnett model. E.4.2 Bruggeman Bruggeman (1935) derived several mixing rules depending on the geometry of the mixtures. His most famous contribution is a symmetrical formula where, in contrast to the Maxwell-Garnett type of solutions, the effective medium itself is considered as the background against which polarizations are measured. It assumes that the components are homogeneous and randomly distributed, and that the size of the components is very small compared to the size of the mixture. It is also known as the Böttcher equation (Böttcher, 1952). For m components with permittivities εi it reads m X i=1 fi εi − εef f = 0, εi + 2εef f (E.16) and its explicit solution for two components is εef f r εh εi = F + F2 + 2 where F = 1 [(3fi − 1)εi + (3fh − 1)εh ] . 4 (E.17) For dilute mixtures (fi 1), Maxwell Garnett and Bruggeman, predict the same results, since, up to the first order in fi their formulas are the same. E.4.3 Bruggeman-Hanai Bruggeman also derived a non-symmetrical formula later modified by Hanai (1936) and generalized by Sen et al. (1981) 182 Appendix E. Permittivity models εef f − εi εh L = 1 − fi , εh − εi εef f (E.18) where L is a depolarization factor to be fitted with core measurements. In the original formula by Bruggeman L = 1/3. E.4.4 Power Law This series of formulae relate a certain power of the permittivity of the mixture to the same power of the permittivities of the components weighted by their volume fractions. εcef f = m X fi εci , (E.19) i=1 when c = ±1 equation (E.19) describes the geometric bounds given in Subsection E.3.1. Birchak et al. (1974) used c = 1/2, and then equation (E.19) is commonly known as the Complex Refractive Index Method (CRIM), which is a harmonic mean of velocities, or the arithmetic mean of slownesses. By contrast, Looyenga (1965) used c = 1/3. E.4.5 Four-Component Dielectric mixing Model This model was first proposed by Dobson et al. (1985) based on a previous model given by de Loor (1963). It is based on plate-shaped inclusions dispersed in a host medium and reads εef f = 3εs + 2ff w (εf w − εs ) + 2fbw (εbw − εs ) + 2fa (εa − εs ) , s − 1) + fa ( εεas − 1) 3 + ff w ( εεfsw − 1) + fbw ( εεbw (E.20) where the subscripts bw, f w, a and s refer to bound water, free water, air and soil, respectively. They found that εbw = 35 − j · 15 yielded a good fit to the measured data. This is an interesting approach because it treats the bound water separately from the free water. However, it is very difficult to determine the amount of bound water present in the system and its permittivity. E.5. Results E.4.6 183 Dry Sand Dry sand is a two-component mixture and the influence of texture and grain size seem negligible. According to Shutko & Reutov (1982) the permittivity of dry soils (εs ) can be computed from their density (ρs ), given in [gr/cm3 ], via εs = (1 + ρs 2 ) , 2 (E.21) whereas Dobson et al. (1985) found a slightly different relation εs = (1 + 0.44ρs )2 − 0.062. (E.22) It is surprising to relate a unitless property to the square of a density, but these are empirical fittings to experimental data on different dry soils. We think it is more appropiate to compute the permittivity from the following reinterpretation of the Clausius-Mossotti relation Clausius-Mossotti reinterpretation Since equation (E.1) has been successfully used to compute the permittivity of different gases from known polarizabilities, we wanted to test it for mixtures. For noninteractive materials we can compute α of each material from its permittivity and equation (E.1). By superposition we can find the permittivity of the m-materials mixture as P Nj αj 3+2 m Pmj=1 εef f = (E.23) 3 − j=1 Nj αj The results presented in Subsection 6.6.1 support this model. E.5 Results Different researchers have compared measured permittivities of different soils at different water contents and temperatures to existing mixing models. In this section, we summarize existing results but it is a complex problem and, in general, they are only qualitative results. Shutko & Reutov (1982) found the Refractive Index Method to give a good approximation to the permittivity of different soils in the range [1-10] GHz and BruggemanHanai at 10 GHz is a good prediction for soils with low moisture content. Hallikainen et al. (1985) pointed that although soil texture may not affect the permittivity of 184 Appendix E. Permittivity models dry soils, it certainly does for wetted soils, specially at the lower end of the frequency range. Dobson et al. (1985) found that, both, the Four Component Model and a Power Law with a coefficient c = 0.65, were capable of describing the complex dielectric permittivity above 1.45 GHz for different soils. And, according to Chelidze & Gueguen (1999b) the Bruggeman-Hanai model with L = 1/3 works fine for the Megahertz region of the spectrum and different rocks. Finally, Seleznev et al. 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Proceedings of the IEEE, 62(1), 33–36. 190 Bibliography Summary As geophysicists we are interested in the imaging capabilities of electromagnetic (EM) waves. Our goal is to disturb the subsurface of the Earth with EM fields and infer an image from the response to these fields. It is then essential to know how the materials that are commonly found in the subsurface of the Earth react to applied EM fields. Sands are one of the most common materials found in the subsurface. They can be dry, partially or fully saturated. Their electromagnetic response depends mainly on, porosity, saturation, and the electromagnetic properties of the sand grains, the saturating fluid and air. It is possible to obtain the saturation of the material from its EM properties through the Mixing Laws. However, there are many of these laws and the experimental evidence does not favor any one in particular. They only consider volume fractions and electromagnetic properties of the constituents of the sand, while the geometrical characteristics of the saturation distribution and the heterogeneities present in the sand are not taken into account. This thesis tests whether the geometry of the saturation distribution within the sand and the possible heterogeneities present in it have an impact in its electromagnetic properties. In order to do so, we have designed and constructed a customized tool in which fluid flow is allowed through the sample. Much of this thesis deals with the design and calibration of the tool and the reconstruction of the EM properties of the sample. We review various experimental designs for measuring the electromagnetic properties of soil samples, and construct a customized coaxial transmission line whose reflection and transmission coefficients can be determined in the frequency range from 300MHz to 3GHz. The electromagnetic properties of the material are obtained from the inversion of these coefficients, the so-called reconstruction of the EM properties. The forward model that describes the propagation of EM waves along the line, needs to be calibrated to be in very good agreement with the measured data. 191 192 Summary We introduce a new notation (Propagation Matrices) to describe the transmission lines. It provides a representation that simplifies the understanding of the interactions between the different sections that configure these tools. And enables us to prove the independence of the transmission and reflection measurements, and to develop a novel analytical inversion of the measurements. The electromagnetic properties that determine the propagation of EM waves are the magnetic permeability and the electric permittivity. Inverting for both properties results in relatively big errors, and since most sands are non-magnetic materials we restrict our investigation to the study of the permittivity alone. We analyze and compare existing reconstruction techniques with the novel one based in the Propagation Matrices and find it the most convenient. We successfully reconstruct the relative permittivity of air within ±1% error. And we could fit a Debye model to the reconstructed permittivity of ethanol, within the same error bound. The inversion obtained with the Propagation Matrices representation identifies all existing inversions equal to either one of the two fundamental and independent solutions. This explains the inaccurate reconstructions of the permittivity of low-loss materials, indistinctively of the expression used. Near the end of the thesis, we investigate the effect of the distribution of constituents with measured and modelled experiments. With the aid of numerical experiments, we show that a distribution of layers has a strong impact on the permittivity, and that heterogeneities as small as λ/100 are detectable. We also studied the effect of fluid flow through a sand sample in relation to its permittivity. We showed how the saturation technique has a clear effect in the reconstructed permittivity and that extra care is needed for proper measurements. Depending on the goal of the measurement it is best to homogeneously saturate the sample outside the sample holder or to perform a gravity drainage to change the saturation. The second technique has the advantage that the same sample can be measured at different saturation levels. However, the saturation distribution results in distinctive 2-layer samples, whose reconstructed permittivity exhibit anomalies due to the interface. These samples can be represented by a 2-layer model very accurately, if the width of the layers is found from the phase of the transmission coefficients and then, the tool can be used to monitor the movement of the fluid front. And finally, we compare our experiments with existing Mixing Laws and find the best mixing formula for partially and fully homogeneously saturated sands to be a Power Law equation. However, depending on the saturation level and whether we are interested in the real or imaginary parts of the permittivity, different exponents are needed. The Power Law is also able to model the permittivity of 2-layer samples, but the error it makes is larger than when it is used for modelling homogenous samples. If a lower precision is allowed, the partially saturated sands, whose saturation is clearly distributed in 2-layers, can be considered homogenous. The validity of this approximation decreases with increasing permittivity contrast. For the first 193 time in literature, we present a technique to include the geometrical distribution of saturation in a Power Law equation. With this technique the modelling of the permittivity of 2-layer samples is very accurate. Based on our results we can conclude that sands can be represented by a single value of permittivity. However, the effective parameter depends very strongly on the constituents, volume fractions and distribution within the sand, and these must be taken into account. It should also be considered that the permittivity of sands can vary with frequency, not because of the frequency dependence of the permittivities of its constituents, but because of their geometrical distribution. Ainhoa G. Gorriti 194 Summary Samenvatting Elektrische karakterisering van zanden met heterogene saturatieverdelingen We zijn als geofysici genteresseerd in de afbeeldingmogelijkheden van elektromagnetische (EM) golven. Ons doel is een beeld te maken van de ondergrond uit de responsie van de Aarde op de in de ondergrond aangebrachte EM verstoringen. Het is dan van essentieel belang de reactie van de materialen, die vaak in de aardse ondergrond worden aangetroffen, op de gebruikte EM velden te weten. Zand is een van de meest voortkomende ondergrondse materialen; het kan droog zijn of gedeeltelijk of geheel met water verzadigd. De elektromagnetische responsie wordt voornamelijk bepaald door de porositeit en de elektromagnetische eigenschappen van de zandkorrels, de vloeistoffen en lucht die de holle ruimte vullen.Het is mogelijk de verzadigingsgraad van het materiaal te bepalen uit de EM eigenschappen via Mengregels. Er zijn echter veel van deze mengregels en experimentele resulaten lijken geen enkele regel te onderbouwen. De mengregels beschouwen slechts de volume fracties and de elektromagnetische eigenschappen van de verschillende componenten van het zand/vloeistof/lucht mengsel, terwijl de geometrische karakteristiek van de verzadigingsverdeling en de in het zand aanwezige heterogeniteit niet worden meegenomen. Dit proefschrift bekijkt in hoeverre de geometrie van de verzadigingsverdeling in het zand en de mogelijk aanwezige heterogeniteiten van invloed zijn op de elektromagnetische eigenschappen van het geheel. Hiervoor hebben wij een speciaal instrument ontworpen en gebouwd waarin vloeistofstroming door het monster mogelijk is. Een belangrijk deel van dit proefschrift behandeld het ontwerp en de kalibrering van het instrument en een rekenwijze om de elektromagnetische eigenschappen van het monster te reconstrueren uit de metingen. Een aantal experimentele ontwerpen, om elektromagnetische eigenschappen van grondmonster te meten, wordt onder de loep genomen om uiteindelijk te kiezen voor de coaxiale transmissielijn, waarvan de reflectie- en transmissiecofficinten kunnen worden bepaald in de frequentieband van 300MHz tot 3GHz. De elektromagnetische eigenschappen van het materiaal worden verkregen door invertering voor deze cofficinten, de zogenaamde reconstructie van de EM eigenschappen. Het voorwaartse model dat de propagatie van 195 196 Sumenvatting de EM golven langs de lijn beschrijft, moet worden gekalibreerd om tot een zeer nauwkeurige beschrijving van de meetgegevens te komen. We introduceren een nieuwe notatie (Propagatie Matrices) om de transmissielijnen te beschrijven. Deze representatie vergemakkelijkt het begrip van de interacties tussen de verschillende delen van het totale instrument. Het stelt ons in staat te bewijzen dat de reflectie- en transmissiemetingen onafhankelijk van elkaar zijn, alsook om een nieuwe analytische inversie van de meetgegevens te ontwikkelen. De elektrische permittiviteit en de magnetische permeabiliteit zijn de twee elektromagnetische eigenschappen die de propagatie van EM golven bepalen. Inversie voor beide eigenschappen resulteert in relatief grote fouten, en omdat de meeste zanden niet-magnetische materialen bevatten hebben we onze onderzoekingen tot alleen de permittiviteit beperkt. Bestaande reconstructietechnieken worden geanalyseerd en vergeleken met de nieuw ontwikkelde techniek gebaseerd op de Propagatie Matrices, welke de meest geschikte blijkt. De relatieve permittiviteit van lucht wordt binnen een foutmarge van 1 Tegen het eind van het proefschrift onderzoeken we het effect van de verdeling van de bestanddelen aan de hand van fysische en gemodelleerde experimenten. Met behulp van de numerieke experimenten tonen we aan dat de verdeling van laagjes een grote invloed hebben op de permittiviteit en dat heterogeniteiten tot een grootte van een honderdste golflengte detecteerbaar zijn. We bestuderen ook het effect van vloeistofstroming door het zandmonster op de permittiviteit. De manier van satureren heeft een effect en hieraan dient aandacht te worden gegeven bij het uitvoeren van dergelijke experimenten. Afhankelijk van het doel van het experiment zijn de volgende twee procedures het meest aan te bevelen: ofwel wordt het zand met een vooraf bepaalde hoeveelheid vloeistof gemengd tot een homogeen mengsel, ofwel wordt volledig met vloeistof verzadigd zand in de monsterhouder geplaatst waarna met behulp van zwaartekracht de verzadigingsgraad wordt verlaagd onder gecontroleerde omstandigheden. De tweede manier heeft als voordeel dat de porositeit van het monster ongewijzigd blijft bij alle verzadiginggraden. Het nadeel is echter dat de verzadigingsverdeling duidelijk in twee-laags monsters resulteert, waarvan de gereconstrueerde permittiviteit uitschieters vertoont vanwege het grensvlak. Dit soort monsters kunnen zeer nauwkeurig door twee-laags modellen worden beschreven indien de dikte van één van de twee lagen uit de fase van de transmissiemeting wordt bepaald, waarna het instrument gebruikt kan worden om de verplaatsing van het vloeistof grensvlak kan worden gevolgd. Uiteindelijk vinden we op basis van onze experimenten dat de machtsregel, van alle bestaande regels, de beste mengregel is voor, gedeeltelijk en geheel, homogeen verzadigde zanden. Helaas zijn verschillende exponenten nodig bij verschillende verzadigingsgraden en worden zowel het rele als het imaginaire deel met dezelfde macht voldoende nauwkeurig beschreven. De machtsregel kan ook worden gebruikt bij het homogeniseren van twee lagen, maar dan is de nauwkeurigheid minder dan 197 bij homogeen gemengde monster met dezelfde verzadigingsgraad. Dit homogeniseren van gelaagde monsters is mogelijk indien deze verminderde nauwkeurigheid nog voldoende is. Deze verminderde nauwkeurigheid neemt nog verder af met toenemend contrast tussen de twee lagen. Dit is de eerste maal dat in de literatuur de geometrische verdeling van de verzadiging is meegenomen in een machtsregel. Met deze techniek kan de permittiviteit van twee-laags monsters zeer nauwkeurig worden gemodelleerd. We concluderen op basis van onze resultaten dat zanden door een enkele permittivteitwaarde kunnen worden gerepresenteerd. De waarde van deze effectieve parameter hangt echter sterk af van de bestanddelen, volume fracties en de verdelingen in het zand en deze moeten worden meegenomen. Men dient zich te realiseren dat de permittiviteit van zand frequentieafhankelijk kan zijn, niet vanwege de frequentieafhankelijkheid van de bestanddelen maar vanwege de geometrische verdeling ervan. Ainhoa G. Gorriti 198 Sumenvatting No son todos los que están, ni están todos los que son Spanish saying Acknowledgements Finally we get to the part of the writing that most enjoy most, the acknowledgements. If you got here after reading all the previous chapters, congratulations! If, on the contrary, this is the first page you looked for, well . . . congratulations too, there is very little book left. Writing a book takes a lot of effort and can be anything but an individual enterprise. Much of the effort came from myself but this thesis would not be what it is without the help of so many others. First of all, I would like to thank Hans Bruining and Evert Slob for giving me the opportunity of doing this research. All of the committee members for helping me improving this text considerably. Prof. Fokkema for his constant motivation and support, and for his contagious enthusiasm. And my supervisor, Evert Slob, for giving me freedom and confidence in my research. Evert was there whenever I needed him, he knew when to push and when to relax, working with him cannot be dull. I am sure that wherever I go I will miss him. Hartelijke bedank, jefe! The experimental set-up and the experiments present in this thesis were done in the Dietz Laboratory. They would not have been possible without all the technical team. Impossible is not in their dictionary. Andre Hoving, Peter de Vrede, Leo Vogt and Karel Heller I want to thank you for your invaluable help and specially for having those smiles and laughs ready at any moment of the day. I have spent 5 years in the lab and running back and forth to the Mijnbouw. Many people have come and go. I shared great and not so great moments with many. Thank you all! Specially Nikolai for being a wonderful office mate and WillemJan for helping me out with the fluid flow through porous media. Marco, Quoc, Mohammed, Roald and Jeroen for your different point of views. And Pacelli for those evening talks. Coffee breaks played a very important role as depressurises and problem solvers. I am grateful for the martian surrealism, the political activism, the endless resources of Google, and the vending machine. Antonio, Cas, Tanguy, Aletta and Sevgi see you all in Mars and neighboring planets! 199 200 Acknowledgements I thought that at this point I would not need to thank my friends and family. It would be understandable not to have any friends or family left after the last year when I turned off the switch of the real world and crawled into my cocoon. But hey! they supported me a great deal and deserve to be here. You are too many, so I will keep it simple. I thank you all for the scientific discussions, the great days out in the mountains, dinners and nights out ”de baretos”. The losers, the numerous spanish mafia and its Leiden branch. And those back in el patio de vecinas. I am truly grateful to my second family, I+D and los Filipiris, because without them life would be so much boring. And my first family, abuelos, tios y primos, por su gratificante alegria y saber vivir. This list would not be complete without my parents. They introduced me to skepticism and wonder and that is one of the most important things in my life, the roots of my scientific character and a great source of fun. Thanks for believing in me, for your unconditional support and for taking care of me when I needed it. To my brother, for being ready for a caustic chat at any time and for his technical support from the other side of the globe. And at last to Antonio, thank you for being there when Murphy was ruling my world. Muchı́simas gracias a todos, About the author Ainhoa González Gorriti was born in Madrid, Spain, on the 6th of June 1974. During her childhood her family moved around the country and overseas, until they settled down back in Madrid. There, she finished highschool in 1992 and joined the faculty of Physics at the Universidad Complutense. Thanks to a scholarship, the year of ’95 she studied at the University of Montana, and in the summer of ’97 she enjoyed an internship at Norsk Hydro. Finally, in 1999 she graduated. That year, she moved to Delft where she joined the Dietz Laboratory, Faculty of Civil Engineering and Geosciences (CITG), department of Geotechnology. She investigates the electromagnetic characterization of geophysical materials. The work carried out during the last years forms the basis of this thesis. Currently, she works as a postdoctoral researcher in the same field. Apart from geophysics and science in general, she likes to travel all over the world, specially if it involves going up and down mountain ranges. 201 202 About the author Publications Peer-reviewed articles Gorriti, A.G. & Slob, E. C. 2004. A new Tool for accurate S-parameters measurements and permittivity reconstruction, Submitted for publication: Trans. of Geoscience and Remote Sensing Gorriti, A.G. & Slob, E. C. 2004. Synthesis of all known analytical permittivity reconstruction techniques of non-magnetic materials from reflection and transmission measurements, Submitted for publication: Geoscience and Remote Sensing Letters Gorriti, A.G. & Slob, E. C. 2004. Comparison of the different reconstruction techniques of permittivity form S-parameters, Submitted for publication: Trans. of Geoscience and Remote Sensing Proceedings and symposium abstracts Gorriti, A.G. & Slob, E. C. 2004. Permittivity Measurements and Effective Medium Considerations, Proc. 10th International Conference on Ground Penetrating Radar 21-24 June 2004, Delft, pages 747-750. Gorriti, A.G. & Slob, E. C. 2003. Analytical Determination of Permittivity from Measured S-parameters, Proc. 65th EAGE Conference & Exhibition. 2-5 June 2003, Stavanger, paper F27. Gorriti, A.G., Slob, E. C. & Bruining, J. 2002. A Coaxial Transmission Line for Accurate Permittivity measurements from 300KHz to 3GHz, Proc. 8th EEGS-ES Conference, 8-12 September 2002, Aveiro, pages 547-550. Gorriti, A.G., Slob, E. C. & Bruining, J. 2002. Accurate Reconstruction of Permittivity from Coaxial Transmission Line Measurements, Proc. 8th EEGS-ES Conference, 8-12 September 2002, Aveiro, pages 551-555. ISBN 0-387-90593-6

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