WATER RESOURCES RESEARCH, VOL. 39, NO. 9, 1230, doi: /2002wr001774, 2003

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1 WATER RESOURCES RESEARCH, VOL. 39, NO. 9, 1230, doi: /2002wr001774, 2003 Differential effective medium schemes for investigating the relationship between high-frequency relative dielectric permittivity and water content of soils Philippe Cosenza, Christian Camerlynck, and Alain Tabbagh UMR 7619, Sisyphe, Université Pierre et Marie Curie, CNRS, Paris, France Received 15 October 2002; revised 27 January 2003; accepted 4 April 2003; published 3 September [1] Differential effective medium (DEM) theory is presented and used to calculate the effective HF dielectric permittivity k of unsaturated soils considered as mixtures of solid and two fluid phases. In the case of coarse-grained soils for which dielectric losses are negligible, a good agreement with the empirical equation of Topp et al. [1980] is obtained using only a spherical grain shape. Moreover, results of DEM schemes have shown that ohmic losses induced by salinity modify significantly the relationship between the apparent dielectric permittivity at 120 MHz and the volumetric water content q, for NaCl concentration greater than mol/l. In the case of clayey soils, simulations based on the same theoretical approach show that the available data can not be modeled by considering only the geometrical effect associated with the platy units and the ohmic losses associated with the surface conductivity. INDEX TERMS: 0699 Electromagnetics: General or miscellaneous; 1866 Hydrology: Soil moisture; 5112 Physical Properties of Rocks: Microstructure; KEYWORDS: relative dielectric permittivity, volumetric water content, differential effective medium theory Citation: Cosenza, P., C. Camerlynck, and A. Tabbagh, Differential effective medium schemes for investigating the relationship between high-frequency relative dielectric permittivity and water content of soils, Water Resour. Res., 39(9), 1230, doi: /2002wr001774, Introduction [2] It is well known that the electromagnetic properties of soils vary as a function of water content. Since the relative permittivity of water is significantly higher than that of other materials commonly found in soils, there has been much interest in using the relative permittivity of soils as an indirect measurement of water saturation. In particular, capacitance sensors [e.g., Tran Ngoc et al., 1972], time domain reflectometry (TDR) probes [e.g., Davies et al., 1977] and ground-penetrating radar (GPR) [e.g., Chanzy et al., 1996] are often used to determine the moisture content of soils. These different types of device have been developed in few dozen of MHz to 1 GHz frequency range and allow to obtain a high-frequency (HF) relative permittivity measurement. [3] In order to transform this measurement into a moisture content value, various empirical equations have been suggested for the relation of the HF relative permittivity k and the volumetric water content q [Birchak et al., 1974; Topp et al., 1980; Wang and Schmugge, 1980; Ruault and Tabbagh, 1980; Hallikainen et al., 1985; Alharti and Lange, 1987; Jacobsen and Schjonning, 1993; Curtis, 2001]. The most commonly used equation is that of Topp et al. [1980]: q ¼ 4:310 6 k 3 5:510 4 k 2 þ 2: k 5:310 2 Copyright 2003 by the American Geophysical Union /03/2002WR ð1þ SBH 1-1 [4] This equation is valid for a wide range of mineral soils and independent of soil bulk density, ambient temperature, and salt content. This led many authors to the use of the term universal for this equation with appropriate caveat that in organic soils or heavy clay soils problems arise which may require site-specific calibration [Zegelin et al., 1992]. [5] In our opinion, in order to improve the inversion of relative permittivity data for the widest range of soils and physicochemical conditions, a better understanding of the effect of the water distribution and the soil texture on the relative permittivity is required [e.g., Tabbagh et al., 2000; Robinson and Friedman, 2001]. This understanding should be based on a quantitative physical description of soils considered as a three-phase mixture at a microscopic scale. [6] A great deal of theoretical work has been devoted to the estimation of relative permittivity of heterogeneous media in terms of electromagnetic properties of the constituents and their geometrical arrangements. There are three distinct approaches: (1) statistical network models (2) numerical models and (3) effective medium approximations. [7] In the first group, the porous media is considered as a three-dimensional network of capacitors [Ansoult et al., 1984; Friedman, 1997]. The probable path of an electric charge in the network is a function of the solid, water, and air components, distributed in their volume fractions. In this statistical approach, the capacitors represent the three components and are randomly distributed in the network. Nevertheless, this approach has three limitations: (1) it includes calibration parameters with no clear physical meaning [e.g., Ansoult et al., 1984], (2) it does not take into account effect of texture and fluid distribution on the relation between relative permittivity and water content, (3) although an electrical analogy is established, it is not based on the

2 SBH 1-2 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS fundamentals of electromagnetic theory (i.e., Maxwell s equations) governing the physical processes involved at the microscopic scale. [8] The second group of approaches is based on a numerical solution of Maxwell s equations written at a microscopic scale. This approach has been used to model dc electrical conductivity [e.g., Adler et al., 1992] and recently relative permittivity by using the moment method [Tabbagh et al., 2000]. These numerical models are efficient tools to study textural effects but may be time-consuming to obtain accurate solutions and often complicated to implement and to validate. This is why effective medium approximations are often preferred. [9] In the effective medium theories, the relative permittivity is calculated by using inclusion-based models that view the porous medium as a matrix with randomly embedded spherical or ellipsoidal inclusions representing individual pores or solid grains [e.g., De Loor, 1983]. The most commonly used assumptions are the dilute approximation, called non-self consistent approximation, and the selfconsistent approximation [e.g., Guéguen and Palciauskas, 1994]. In the dilute approximation, the electromagnetic interactions between the elementary particles are neglected. In the self-consistent approximation, these interactions are overestimated: the elementary inclusions are embedded into the effective medium itself that includes all the others particles. In spite of this approximation, the latter approach has been applied by Friedman [1998] who considered a composite spheres model. This model gave a good agreement with experimental data but Friedman s model does not allow us to study the effects of microscopic liquid distributions on the dielectric properties. [10] The approach presented hereafter is based on differential schemes for the effective relative permittivity, also called iterated dilute limit approximations [e.g., Sen et al., 1981; Norris, 1985]. In a two-phase composite medium, one phase is taken as a matrix and the other is added incrementally in a such way that the newly added material is always in dilute concentration with respect to the current effective medium. This differential effective medium (DEM) theory offers three advantages: (1) it integrates the electromagnetic interactions in a more realistic way than the dilute approximation and the self-consistent model, (2) particle shape effects [e.g., Jones and Friedman, 2001] on effective relative permittivity can also be studied, (3) numerically, it is easy to implement. Moreover, this approach was widely and successfully used in rocks physics in order to model the following properties of sedimentary rocks: (1) elastic properties related to seismic velocities [Le Ravalec, 1995; Le Ravalec and Guéguen, 1996; Berryman et al., 2002]; (2) dc electrical conductivity [Sen et al., 1981; Bussian, 1983; Revil et al., 1998]; (3) dielectric properties in a wide range of frequencies [Feng and Sen, 1985; Chelidze and Guéguen, 1999]. Note that a review and a comparison between the different effective medium theories used for modeling elastic properties [Le Ravalec, 1995; Le Ravalec and Guéguen, 1996] have shown that the DEM schemes provided the best agreements with experimental data. [11] The DEM approach is briefly presented in the first part of the paper. In the second and the third part, it is used to investigate the effects of fluid distribution and ohmic losses on the relative permittivity considering sandy soils and clayey soils, respectively. 2. Differential Effective Medium (DEM) Theory [12] In the DEM theory, the effective relative permittivity of the mixture is built explicitly from an initial material through a series of incremental additions. Let us consider three-phase mixture: a, b, and g. The phases a, b, and g can be alternatively solid, water and air. [13] The procedure begins with the initial material g corresponding to a relative permittivity k g in a volume V 0. Small volumes of phases a and b, dv a and dv b,are imbedded in phase g in such a way that the volume remains fixed at V 0. The effective relative permittivity of the mixture is calculated considering a dilute suspension of inclusions of a and b in a medium g. Now, in the volume V 0, the mixture has a homogenized effective constant dielectric k 2 and constitutes the initial medium of the next step. A schematic of the iterative process is shown in Figure 1. The construction process continues such that (1) at each stage i the embedded inclusions of a and b are in dilute concentration and (2) the required volume fractions f a, and f b are satisfied. [14] Considering two spherical inclusions of phase a and b with small increments of volumes, dv a and dv b, homogeneous relative permittivity k i at the step (i) is given by [e.g., Maxwell-Garnett, 1904]: k i k i 1 k i þ 2k i 1 ¼ dv a V 0 k a k i 1 k a þ 2k i 1 þ dv b V 0 k b k i 1 k b þ 2k i 1 where k i-1 is the relative permittivity at the step (i-1). Since volumes, dv a and dv b are small, we can write at the first order: k i ¼ k i 1 þ dk [15] By substituting equation (3) in (2): dk 3k i 1 ¼ dv a V 0 k a k i 1 k a þ 2k i 1 þ dv b V 0 k b k i 1 k b þ 2k i 1 [16] In equation (4), parameter dk is the increment of relative permittivity of the mixture due the ith addition of such a small volume. In the construction process, the volume of mixture is kept constant: at each replacement, the removed material must have the same volume fractions of materials a, b, and g as the total volume V 0. This incompressibility assumption leads to the following conditions [Norris, 1985]: V 0 df a ¼ð1 f a ÞdV a f a dv b V 0 df b ¼ð1 f b ÞdV b f b dv a ð2þ ð3þ ð4þ ð5aþ ð5bþ where f a, and f b are the volume fractions of a and b, respectively, df a, and df b are the small increments of volume fractions of a and b, respectively. The incompressibility assumptions allow to minimize overlapping of a and b: indeed, the (i+1)th set of inclusions

3 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS SBH 1-3 Figure 1. The iterative process of the DEM theory. a and b will essentially replace material g, but also a part of the previously added inclusions. By using equations (5a) and (5b), parameters dv a and dv b can be replaced by df a, and df b. [17] Consequently, a construction process is uniquely defined by a path in the (f a, f b ) plane (Figure 2). This path, which begins at origin, is called homogenization path or integration path since equation (4) can be considered as a differential equation linking dk, df a, and df b. [18] In Figure 2, it should be noted that an infinity of construction process (i.e., homogenization path) exists. The simplest paths are those parallel to f a and f b axis: the homogenization paths I and II are obtained by adding successively a (resp. b) and then b (resp. a). These particular paths will be called two-phase homogenization path (2PHP) since only two phases are considered during the construction process. In order to illustrate the previous equations and this particular path, let us consider the 2PHP path I for which at the first step, only phase a is added in a uniform phase g (see Figure 2). In equation (4), df b = 0 and it can be integrated from and k = k g and V a /V 0 =0tok i 1 = k and V a /V 0 = f a (required volume fraction of phase a) and Bruggeman s equation is obtained [Bruggeman, 1935]: k k a k a 1=3¼ 1 fa ð6þ k g k a k

4 SBH 1-4 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS Figure 2. The different homogenization paths in the (f a, f b ) plane to construct the material M with the volume fractions. [19] The effective property k is the relative permittivity of a mixture with the required f g and f a. In order to calculate the final effective relative permittivity k f with respect to the required f b, inclusions b are added in the previous mixture. The following equation (7) is derived from equation (6) by substituting parameters k, k a and k g by parameters k f, k b and k, respectively: k f k b k k b k b 1=3¼ 1 fb k f ð7þ where the effective relative permittivity k is calculated from equation (6). [20] In Figure 2, path III is the more general case: phases a and b are added simultaneously. In the following, it will be called three-phase homogenization path (3PHP). 3. Coarse-Grained Soils [21] In this part, we focus on coarse-grained soils for which the following assumptions are satisfied. (1) Solid particles can be considered of spherical shape. (2) Their specific surface area is low and consequently the bounded water content is not significant. (3) We use a water permittivity of 80, a solid permittivity of 4.3 by considering quartz grains [Weast, 1983]. (4) The pore water has a low ionic concentration. Hence, ohmic losses are negligible. [22] Moreover, the empirical equation of Topp et al. [1980] (equation (1)) which holds approximate for coarsegrained soils will be considered as a reference. In the following sections, 2PHP and 3PHP schemes are performed to model the k(q) relationship and compared to equation (1) PHP Paths [23] 2PHP paths have been previously used by Wobschall [1977] and Alharthi and Lange [1987]. However, the authors have compared their results with a low number of data associated with a low saturation degree (less than 60%) and a low frequency (less than 30 MHz). Moreover, Wobschall s approach introduced adjustable parameters, which are difficult to estimate: the dispersed water fraction and the crevice water fraction. Here, we do not consider directly 2PHP paths but a linear combination of 2PHP which will be written as follows: k ¼ ð1 SÞk ls þ Sk hs ð8þ where S is the saturation degree. The parameters k ls and k hs are the relative permittivity at low saturation degree and the relative permittivity at high saturation, respectively. Hence, the factors S and 1-S can be seen as weight functions: close to saturation (resp. dry soil), S is close to 1 (resp. 0) and the property k is given by the relative permittivity at high (resp. low) saturation degree. [24] The characteristic property at low saturation degree, k ls, is obtained from the two following steps (see Figure 3). (1) In a first step, an intermediate and homogenized property k inter is constructed from the liquid phase and the solid phase (dv a = 0) by a 2PHP path (2PHP(1) in Figure 3). The effective property k inter is computed from an equation similar formerly to (6). This first step is intuitively justified by the high closeness between the solid phase and the liquid phase when saturation degree is low. (2) In a second step, the homogenized property k ls is calculated from the previous property, k* inter, and the relative air permittivity by adding the required air phase volume following a 2PHP path (2PHP(2) in Figure 3). [25] A quite similar approach was used to compute the high saturation degree property k hs with the following exceptions (see Figure 4): (1) in a first step, only the liquid phase and the air phase are considered to compute an intermediate effective property k inter, (2) the solid phase is added in the second step to obtain the required property k hs. Figure 3. Diagram of the 2PHP approaches used to model the low saturation degree permittivity e ls.

5 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS SBH 1-5 Figure 4. Diagram of the 2PHP approaches used to model the high saturation degree permittivity e hs. [26] The property k calculated from the latter 2PHP schemes and equation (8) is presented in Figure 5 and compared to the empirical equation of Topp et al. s [1980] equation. A good agreement is obtained PHP Path [27] In this section, a 3PHP scheme is considered with the following additional Feng and Sen s [1985] assumption: with dv a ¼ V a i!1 dv s V i!1 ¼ C s C ¼¼ n q 1 n ð9þ ð10þ where n is porosity. This assumption means that during the construction process, at each step, the ratio dv a /dv s is kept constant. It allows to integrate analytically equation (4) since dv a and dv s are related through condition (9). At step 0, the volume V is filled with water. The same results are obtained as in the work by Feng and Sen s [1985], which involved oil and gas mixtures in sedimentary porous rocks: values used in section 3.1. In comparison with 2PHP schemes (Figure 5), a better agreement is obtained with the empirical equation of Topp et al. [29] In conclusion, by considering the good agreements we obtain in sections 3.1 and 3.2, the DEM approach seems to capture the main physical processes involved in the k(q) relationship in simple three-phase media (i.e., coarsegrained soils and medium-textured mineral soils). This conclusion leads us to use the same approach for investigating more complex porous media: the clayey soils. 4. Clayey Soils 4.1. Phenomenological Background and Assumptions [30] In a wide range of frequency, the permittivity of clayey soils depends on texture (fine-grained materials) and electrochemical/interfacial effects. Both aspects lead to the following features, which do not exist in coarse-grained q ¼ Sn ¼ k w 1=3 ak 2 1=2 þ bk þ c k ak 2 w þ bk w þ c p 2ak þ b ffiffiffiffi D p 2ak þ b þ ffiffiffiffi 2ak p w þ b þ ffiffiffiffi m D p D 2ak w þ b ffiffiffiffi ð11þ D with and m ¼ 1 ½ 2 k a þ 2k s þ Cð2k a þ k s Þ Šp 1 ffiffiffiffi D 8 a ¼ 2ðCþ1Þ >< D¼b 2 4ac b ¼ Cð2k a þ k s Þþð2k s k a Þ >: c ¼ð1þCÞk a k s ð12þ ð13þ [28] Equation (11) for which the effective property k is the unknown, has been solved numerically with a Newton- Raphson scheme [e.g., Press et al., 1992]. The results we obtained for different volumetric water contents and for different porosities are compared with the equation (1) in Figure 6. Model predictions were computed with the same Figure 5. The effective relative permittivity as a function of volumetric water content for two porosities (n = 0.4 and n = 0.6), calculated by a combination of 2PHP schemes (dashed lines). Also shown is the function of Topp et al. [1980] (solid line).

6 SBH 1-6 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS frequency range 100 MHz-1 GHz. Both features, the high imaginary part of k* and the dispersion phenomenon, are illustrated in Figures 7 and 8, from Saarenketo s data. [33] Moreover, the clayey soils show a high content of bound water due to their high specific surface area and thus the dielectric properties of such water k bw may have an influence on the dielectric properties of the soil itself. Indeed, experimental evidence and theoretical arguments have shown that dielectric permittivity of bound water associated with two to three molecular layers of water is between that of ice (about 3.2) and of free water (about 80 at 20 C) [Dobson et al., 1985; Dirksen and Dasberg, 1993; Friedman, 1998; Or and Wraith, 1999]. The effect of bound water on the soil dielectric properties has been modeled in different ways. On the basis of the theoretical works of De Loor [1968], Dobson et al. [1985] have treated the bound water as a separate phase: in their mixing model, the free and bound water are randomly and independently dispersed in a four phases mixture and thus without any spatial correlation. This is a major limitation of such model since Figure 6. The effective relative permittivity as a function of volumetric water content for two porosities (n = 0.4 and n = 0.6), calculated by a 3PHP scheme (dashed lines). Also shown is the function of Topp et al. [1980] (solid line). soils with low pore water salinities: (1) a significantly high imaginary part of the permittivity; (2) a frequency dependent dielectric constant; (3) a high content of bound water with a low permittivity value. [31] The high imaginary part values are due to energy losses associated with polarization and electrical conductivity processes [e.g., Campbell, 1990]. Consequently, the permittivity has to be considered as a complex quantity with a real and imaginary component. The complex permittivity of clayey soils will be designated k* further and is written as follows: k* ¼ k 0 þ ik 00 ð14þ where k 0 and k 00 are the real part and the imaginary part of k*, respectively; i 2 = 1. Note that in equation (14) both dielectric losses and ohmic losses are included in the imaginary part k 00. This unusual notation has been used in order to obtain a clear presentation of both k 0 (q) and k 00 (q) relationships. [32] The frequency dependence of the relative permittivity in natural clayey soils has been experimentally studied by Hoesktra and Delaney [1974], Hipp [1974], Campbell [1990], Wensik [1993], Saarenketo [1998]. The results of these investigations have revealed that the real component of the complex permittivity increases with decreasing frequency, the increase becoming more pronounced below about 25 MHz. For frequencies greater than 1 GHz, another dispersion phenomenon starts to occur, associated with the dielectric relaxation of free water. Consequently, in order to minimize the effect of these dielectric dispersions, the data, which we used in this study, have been measured in the Figure 7. The measured real part k 0 and the measured imaginary part k 00 of the relative permittivity as a function of the volumetric water content in Houston Black clay [from Saarenketo, 1998). Also shown is the function of Topp et al. [1980] (solid line).

7 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS SBH 1-7 l =10 8 cm 1. The average thickness of the water layer d w is approximated by: d w ¼ q r b S sa ð16þ Figure 8. The measured real part k 0 and the measured imaginary part k 00 of the relative permittivity as a function of the volumetric water content in Beaumont clay [from Saarenketo, 1998]. Also shown is the function of Topp et al. [1980] (solid line). where r b is the bulk density and S sa is the specific surface area. Illustrations of the dielectric constant k w calculated according to equation (15) are given in Figure 9. This figure shows that the parameter k w is significantly lower than the bulk value of 80 for high values of S sa, i.e., for very finetextured soils. We are aware that the Friedman s formalism is not fully validated: (1) there are not enough values of the adsorbed water permittivity in the literature to constrain the parameter l, (2) the relative permittivity of bound water is likely a complex quantity for which the impact of the imaginary part on the complex permittivity of clayey soils is unknown. Nevertheless, in spite of these drawbacks, Friedman s equation (15) will be used further since it is simple and allows quantitative estimates. [36] In the following sections, we will focus on two aspects, which may play a significant role in the relation between the complex permittivity of unsaturated clayey soils and volumetric water content: (1) the geometrical effects related to elementary units in clays and (2) the ohmic losses related to water conductivity and surface conductivity. [37] The effects of both aspects, for which the clay literature had paid little attention, will be studied with DEM theory since previous DEM schemes have given satisfactory results for describing the k*(q) relationship of coarse-grained soils. We feel that studying these particular effects separately to the others (bound water, electrochemical phenomenon) would be of great benefit in understanding the role of each phenomenon in the k*(q) relationship of unsaturated clayey soils Geometrical Effects of Clayey Particles [38] Microstructural observations carried out by scanning (SEM) and transmission (TEM) electron microscopy have there is no textural evidence to spatially uncouple these two states of water. Or and Wraith [1999] proposed to compute k bw from either the harmonic average or the arithmetic average of the data of Bockris et al. [1963] who studied the first three molecular layers near metal surfaces. [34] On the basis of Thorp s [1959] work on silica gel, Friedman [1998] suggested an exponential form for the mathematical function that relates the dielectric constant of the water and the distance from the solid phase, its harmonic average for a water layer thickness of d w being: k w ¼ d w þ 1 l d w k h max i ð15þ kmax ðkmax kminþe ldw ln k min [35] This function is controlled by (1) a minimal value k min (taken to be 5.5 by Friedman), (2) a maximal value k max, i.e., permittivity of free water (taken to be 80), and (3) a parameter l, which is in the range [ cm 1 ]. The best fit with Thorp s data is obtained for a value near Figure 9. The average relative permittivity of the water phase as a function of the volumetric water content calculated according to Friedman s approach for two values of specific surface area (40 and 800 m 2 /g) and for two values of porosity (0.4 and 0.6).

8 SBH 1-8 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS Figure 10. Schematic representation of (a) the microstructure of synthetic smectites prepared with diluted solutions, at low suctions (modified from Tessier [1991]) and (b) that of kaolinite system. (c) Also shown is the modeling approach. shown that, the texture of clays is defined by basic units, which are dependent on the clay mineralogy. Considering two extreme cases, the kaolinite clays and the smectite clays, the basic units are for kaolinites, single crystals or for smectites, an assemblage of elementary silicate sheets (layers) referred to as quasi-crystals or tactoids [e.g., Tessier, 1991]. [39] For smectites at low salt concentration, a significant part of water is located between layers inside tactoids, i.e., in intradomain pores (Figure 10). Within tactoids or quasi-crystals, the corresponding interlayer spacing is either 18.6 angstroms for Ca-smectites, or 35 to 100 angstroms for Na-smectites [Tessier, 1991]. Under similar conditions, kaolinite systems are based on elementary crystallites or aggregates of these crystallites. The pores inside aggregates define also a microporosity. [40] Since this intradomain water in microporosity governs partly numerous macroscopic physical properties (electrical conductivity, swelling-shrinkage) [e.g., Mitchell, 1976; Tessier, 1991], one may wonder if this textural arrangement between the silicate layers and water has an influence on the k*(q) relationship. In order to answer to this question, DEM schemes considering ellipsoidal inclusions have been used. In our approach, the microstructure of the microporosity is modeled as an inhomogeneous and oblate ellipsoid, consisting of two layers of different permittivities (Figure 10). An oblate ellipsoid has two equal axes (a = c) and the third one, the axis of revolution b is smaller (Figure 11). The core is solid (phase b in Figure 11) and the shell is water (phase a in Figure 11). [41] The homogenized permittivity of layered ellipsoidal inclusions was derived by Sihvola and Lindell [1990]. A synthetic presentation of their approach is given by Jones and Friedman [2000]. Consider two-layer confocal oblate ellipsoids, lying in a background medium of relative permittivity k i-1. These ellipsoids are randomly oriented in the three phases mixture. This assumption is questionable at low water content and when compaction is high [e.g., Saarenketo, 1998]: clear oriented structures are observed at a microscopic scale [e.g., Tessier, 1991]. However, the related anisotropic effect is difficult to model quantitatively for at least two reasons: (1) no extensive dataset related to clayey soils is available and (2) at low water content, a major part of water in clay systems is bound water for which its pemittivity value is still a subject of considerable debate (see discussion in section 4.1). Hence, for sake of simplicity, Figure 11. approach. The two-layer oblate ellipsoid of the modeling

9 the anisotropic effect related to clayey soils will not be taken into account. [42] Consequently, if no preferred direction is considered, the scalar homogenized permittivity, referred as k i, is [Sihvola and Lindell, 1990]: 1 3 k i ¼ k i 1 þ P i¼x;y;z P i¼x;y;z f a i p ð17þ N1 i f a i p k i 1 where the polarizability components in parentheses are given by: f a i h p ¼ðj k a þ j b Þ ðk a k i 1Þ½k a þ Nb i ðk b k a ÞŠ i 1 þ j b ðk b k a Þ k a þ Na i j a þ j ðk i i 1 k a Þ h b = k i 1 þ Na i ðk a k i 1 Þ ½ka þ Nb i ðk b k a ÞŠ þ COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS SBH 1-9 j b j a þj b N i a ð1 N i a Þðk a k i 1 Þðk b k a Þi ð18þ where f is the number of ellipsoidal inclusions per unit volume, a P i is the polarizability for ellipsoids in the i-direction (i = x, y, z), k a and k b are the complex relative permittivity of phases a and b, respectively, j a and j b are the volumetric fractions of phases a and b, respectively, N a i and N b i are the depolarization factors of the outer and inner ellipsoids, respectively. In case of an oblate ellipsoid, given the axial ratio b/a, the depolarization factors are [Sihvola and Lindell, 1990]: N x j ¼ N y j 1 e2 e 3 ðe tan 1 eþð j ¼ a; bþ ð19þ ¼ Nj z ¼ 1 N j x ð j ¼ a; bþ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffi a e ¼ 2 b 2 1 ð20þ ð21þ In the extreme case, that we will consider further, a j b j (thin discs) and therefore N x j 1 and N y j = N z j 0(j=a,b). Following the latter assumption a j b j, equation (17) can be rewritten as: 8 h i < k i ðk a k iþk bþ j b j a þj ðk b b k aþk i dk ¼ðj a þ j b Þ h i : j b þ where 3k ak b ðj a þj b Þ ðk a k iþk bþj a þj ðk b b k aþk i h 2k ak b ðk a k iþþ j b j a þj ðk b b k aþ h 3k ak b ðj a þj b Þ ðk a k iþk bþ j b j a þj ðk b b k aþk i i dk ¼ k i k i 1 9 = i ; ð22þ ð23þ [43] Considering the construction process in the DEM theory, the parameter dk is seen as the increment of relative permittivity of the mixture due to the ith addition of a small volume associated to the two-layer ellipsoidal inclusion. Hence equation (22) is equivalent to equation (4) in the construction process with spherical inclusions. In the DEM Figure 12. The effective relative permittivity as a function of volumetric water content for two specific surface areas (S sa =40m 2 /g and S sa = 800 m 2 /g) and different porosities (n = 40% and 60%) calculated by DEM schemes with twolayer disk-shaped inclusions. Also shown is the function of Topp et al. [1980] (solid line). schemes that we performed, three additional assumptions were used. [44] 1. In the construction process, at the first stage (stage 0), when there is no ellipsoidal inclusion: the effective permittivity is equal to that of air (k * i=1 = k 0 = 1). Water and solid included in a two-layer ellipsoidal inclusion is added during the next stages with the following condition: with dv w ¼ V w i!1 dv s V i!1 ¼ C 0 C 0 ¼ s q 1 n ð24þ ð25þ Hence, at each stage i of the construction process, the ratio dv w /dv s is kept constant and the current volumetric fractions j a and j b are calculated by combining equations (5a), (5b), (24), and (25). [45] 2. The dielectric constant of water obeys to Friedman s equation (i.e., equation 15) with l =10 8 cm 1 and k min = 5.5. The dielectric constant of the solid is equal to 4.3. [46] 3. The imaginary parts of water and solid are equal to zero. [47] Considering different specific surface areas (S sa = 40 m 2 /g and S sa =800m 2 /g), the results are given in Figure 12. The following results can be pointed out: (1) as

10 SBH 1-10 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS expected, the specific surface area and hence the bound water content have a significant influence on the k*(q) relationship, (2) there is a poor agreement between these theoretical results and the Saarenketo s data given in Figures 7 and 8. Nevertheless, considering the latter, it should be emphasized that it is difficult to compare both sets of results for at least two reasons. [48] Firstly, at high water content, a major part of water may be also located in a macroporosity, which corresponds to inter-domain pores or inter-aggregate pores; and microstructural investigations have shown that this macroporosity can not be modeled simply as a layered structure [e.g., Tessier, 1991]. Indeed, by comparison, DEM schemes with spherical inclusions (i.e., 3PHP scheme) give better results (Figure 13). [49] Secondly, in Saarenketo s experiments, there is no dielectric data corresponding to the water content values associated with bound water, probably lower than 10% (see estimations in Table 1), and it is clear that bound water organization at microscopic scale could be better modeled by this approach. [50] Moreover, Figure 12 shows an unexpected effect associated with porosity at high water content: the higher the porosity, the higher the effective permittivity value. This effect has two origins: (1) the geometrical arrangements of the different phases, (2) the method we used to calculate the permittivity of the water phase. The geometrical arrangement of the three phases at high water content would be Figure 13. The effective relative permittivity as a function of volumetric water content for two specific surface areas (S sa =40m 2 /g and S sa = 800 m 2 /g) and two porosities (n = 40% and n = 60%) calculated by DEM schemes with spherical inclusions (i.e., 3PHP scheme). The water relative permittivity is calculated with equation (15). Also shown is the function of Topp et al. [1980] (solid line). Table 1. Physical and Chemical Properties of the Clayey Soils a Houston Black Clay Beaumont Clay Solid density (10 3 kg/cm 3 ) Clay content (percent by weight) Specific surface area (m 2 /kg) CEC (meq/100g) Estimated bound water content (n = 40%) (percent of total, bound and free, water) Estimated bound water content (n = 60%) (percent of total, bound and free, water) a From Saarenketo [1998]. Bound water content is estimated with equation (16) by considering three layers of water molecules ( 9 angstroms). comparable to the air-water-soil (AWS) configuration in the composite spheres model proposed by Friedman [1998]. Indeed, note the similarity between Figure 12 (with S sa = 40 m 2 /g) and Figure 2e of Friedman [1998]. In the AWS configuration, the air phase is encapsulated within the water phase and the solid phase that constitutes the external shell. Hence, following our assumption, there would be an equivalence between (1) a mixture composed of a high number of randomly oriented discs of water and solid in an air phase and (2) a AWS composite spheres model for which the effective permittivity increases with porosity. [51] Considering the second origin (2), the permittivity of the water phase is calculated by using equations (15) and (16). In equation (16), the bulk density is a decreasing function of porosity n following the product (1-n)r s where r s is the solid density (constant and equal to 2.7 g cm 3 in our calculations). Hence, for fixed q and S sa values, since the water layer d w in equation (16) is an increasing function of porosity, the permittivity of the water phase which is an increasing function of d w (equation (16)), will increase also with porosity. If we consider the [0,40%] range of q and S sa =40m 2 /g (800 m 2 /g resp.), the permittivity of water increases of 4,5% (20,7% resp.) in average when porosity increases from 0,4 up to 0,6. This effect is quite small but it explains the reversal n-dependence observed in Figure 13 for high water content values and S sa = 800 m 2 /g. [52] Therefore, on the basis of the whole set of simulations given in Figure 12, it can be concluded that the geometrical effect associated to the platy shape of an individual basic unit is not the only mechanism that contributes to the k*(q) relationship of swelling clayey soils, at least for the range (15 40%) of volumetric water content which corresponds to the available experimental data. Others phenomena associated with surface effects (surface conduction, relaxation of bound water etc.) or a combination of both as it suggested by Endres and Knight [1993] are involved Impacts of the Ohmic Losses Related to Water Conductivity and to Surface Conductivity [53] As mentioned, when clayey materials are considered, their dielectric properties are complex quantities with a real part and an imaginary part, which is associated with the energy losses in the material. [54] In clayey materials, energy losses arise from two kinds of mechanism: (1) rotation of dipoles or bound charges, spatial polarization (i.e., polarization loss) and (2) motion of charge carriers (i.e., ohmic loss) [e.g., Guéguen

11 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS SBH 1-11 and Palciauskas, 1994]. The latter occurs through the interstitial water, which contains dissolved electrolytes and via the exchangeable cations that reside near the surface of charged particles. The conductivity induced by the exchangeable cations defined the so-called surface conductivity, which plays an important role in clayey soils at low salinities. [55] Regarding the effective medium theory, the water electrical conductivity and the surface conductivity lead to introduce complex quantities for the components in the mixture. Since it is known that the imaginary parts of each components in the mixture may influence significantly the complex effective permittivity [e.g., Sihvola, 2002], one may wonder if the surface conductivity and the water electrical conductivity have a significant effect on the complex k*(q) relationship of clayey soils. [56] In order to answer to this question, a sensitivity analysis based on DEM schemes has been performed with the following assumptions. (1) DEM schemes with spherical inclusions (i.e., 3PHP scheme) have been used since they have given better results than those with ellipsoidal inclusions in the previous section. (2) The real part of water permittivity obeys to Friedman s equation (i.e., equation 15) with l = 10 8 cm 1. In order to compare the results to the Saarenketo s data, the specific surface S sa is equal to 40 m 2 /g and n = 0.5. (3) The imaginary parts of water and solid are not equal to zero and correspond to their ohmic losses, which are defined by the following relationship: k 00 a ¼ s a we 0 ð26þ with w is the pulsation, e 0 is the free space permittivity and is equal to F/m, s a is the dc conductivity of phase a. The subscripts s and w are used for solid and water, respectively. Considering the theoretical works of O Konski [1960], Bussian [1983] and Revil et al. [1998], the conductivity s s is seen as the surface conductivity. Since the purpose of this sensitivity analysis was to study the impact of the ohmic losses of the different constituents, polarization losses are not taken into account in equation (26). Moreover, in order to maximize the effect of ohmic losses and to compare the simulations to Saarenketo s data, the lower frequency (i.e., f = 0.12 GHz) has been chosen in the calculations. This frequency value is also interesting since it corresponds approximately to a mean value of the frequency range [10 MHz, 1 GHz] used in TDR methodology. [57] To study the relative contribution of s w and s s to the k*(q) relationship, two extreme cases were introduced. In the first case, case a, the surface conductivity s s is kept to 0 and the water conductivity is taken in the range [0.1, 6 S/m]. This range is similar to that chosen by Rhoades et al. [1976] who have studied the effect of water conductivity on bulk soil electrical conductivity. This range corresponds to typical values associated with pore water salinity in soils. Obviously, all the results related to case a will not be only relevant to clayey soils but can be generalized to low specific surface soils. [58] In the second case, case b, the water conductivity s w is equal to 0 and the surface conductivity s s is taken in the range [10 2, 0.2 S/m], which corresponds to a set of values obtained by Bussian [1983] in his modeling approach. Figure 14. The calculated real part k 0 and the calculated imaginary part k 00 of the relative permittivity as a function of the volumetric water content for water conductivity values in the range [0.1, 6 S/m] (case a: s s = 0). Also shown is the function of Topp et al. [1980] (thick solid line). [59] The results corresponding to case a (i.e., s s =0)are given in Figure 14. They show that the ohmic losses play an important role in the imaginary part k 00 but not in the real part k 0. The latter is independent on the imaginary component of the liquid phase permittivity. This point suggests that the k 0 (q) relationship could be modeled independently of the conductivity losses associated with fluid salinity. Moreover, our calculations have shown that k 0 (q) relationship does not change with frequency. This means that the dispersion phenomenon associated with k 0, observed in Figures 7 and 8, is not related to the single liquid phase but would result from a complex interaction between solid and liquid. [60] The apparent relative permittivity k a, e.g., that measured with TDR, can be calculated as follows: k a ¼ k0 k002 1 þð1þ Þ1=2 2 k02 When there is no loss; (i.e., k 00 = 0), then k a ¼ k 0 ð27þ ð28þ

12 SBH 1-12 COSENZA ET AL.: DIELECTRIC CONSTANT AND WATER CONTENT OF SOILS The parameter k a governs the velocity of electromagnetic wave in soils and hence the travel time of TDR measurement [e.g., Topp et al., 1980]. As discussed previously, a high fluid salinity value will induce a high value of k 00 w and hence k 00 and k a will increase following the equation (27). Equation of Topp et al. has been determined by using low-loss mineral soils and, in a practical point of view, it is interesting to know what fluid salinity value will modify in a significant way this empirical equation. When the water conductivity value is below 0.5 S/m, theoretical results fall satisfactorily around the empirical equation over the range [0 50%] water content range (Figure 14) with a relative difference lower than 10% in average. This upper bound, 0.5 S/m, corresponds to a NaCl concentration value of about 2630 ppm (or mol/l) [e.g., Tiab and Donaldson, 1996] and the latter is in agreement with the value of 2000 ppm for which Topp et al. [1980] did not observed measurable effect on the k a (q) relationship. [61] Concerning case b (i.e., s w = 0), there is a poor agreement between the theoretical calculations given in Figure 15 and the published data in Figures 7 and 8. Indeed, since Houston Black clay and Beaumont clay are known to be swelling clays, it is expected that they exhibit a significant surface conductivity and associated ohmic losses. The theoretical curves show two features that are not observed in Saarenketo s data. On the one hand, the calculated k 0 increases significantly with the surface conductivity s s. On the other hand, the theoretical jk 00 (q)j curves are concave down. These discrepancies may have two origins. Firstly, although their CEC are quite high (see Table 1), the surface conductivities of theses clays, which are unknown, may be low and hence induce low ohmic losses compared to the dielectric losses. It should be emphasized that surface conductivity is not a measurable property but is in general, indirectly estimated by a modeling approach. The quantitative determination of this property is still a subject of considerable debate. Secondly, the way we chose to model the surface conductivity may not be relevant. The surface conductivity has been associated with the whole solid phase (i.e., the spherical solid inclusion) and has not been treated as a specific property of an interfacial region, i.e., a conductive clayey phase that separates the quartz grains and the pore water. This interfacial region could be included in our approach by considering a concentric and conductive shell surrounded a non-conductive spherical solid inclusion. This will be the purpose of a future work. Figure 15. The calculated real part k 0 and the calculated imaginary part k 00 of the relative permittivity as a function of the volumetric water content for surface conductivity values in the range [10 2, 0.2 S/m] (case b: s w = 0). Also shown is the function of Topp et al. [1980] (bold solid line). 5. Conclusions [62] This study confirms that the DEM theory is an efficient tool to investigate in a simple way the physical macroscopic properties of heterogeneous media. In the case of coarse-grained soils with low fluid salinity, DEM schemes give results, which are in satisfactory agreement with the empirical equation of Topp et al. [1980] established in the ( MHz) frequency range associated with the TDR probes. [63] Moreover, this approach has been used to investigate the k(q) relationship of more complex porous media: clayey soils. The results we obtain show that the geometrical effect associated to the platy units that exist in clayey materials at a microscopic scale, is not the only mechanism that contributes to the k(q) relationship at least for the range of volumetric water content (15 40%) which corresponds to the available experimental data. In order to go further in this analysis, dielectric data for lower volumetric water content are required. This point underlines the need to continue improving experimental techniques in order to obtain accurate data for low volumetric water content values. [64] Considering clayey materials, the study of the impacts of ohmic losses suggests that the complex k(q) relationship cannot be understood by considering separately the different constituents. Surface conductivity and dispersion phenomenon are likely related to interfacial regions, which have to be included in a future modeling approach. However, once again, this modeling approach has to be supported by accurate and extensive experimental data sets that do not exist yet on clayey soils for a wide range of water contents. [65] Acknowledgments. We thank S. Friedman and an anonymous reviewer for their thoughtful comments that have improved significantly the initial manuscript. References Adler, P. M., C. G. Jacquin, and J.-F. Thovert, The formation factor of reconstructed porous media, Water Resour. Res., 28(6), , 1992.

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