Published online 4 August 2005
Published in Soil Sci Soc Am J 69:1372-1378 (2005)
DOI: 10.2136/sssaj2004.0366
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Soil Physics
A Physically Derived Water Content/Permittivity Calibration Model for Coarse-Textured, Layered Soils
D. A. Robinsona,*,
S. B. Jonesa,
J. M. Blonquist, Jr.a and
S. P. Friedmanb
a Dep. of Plants, Soils and Biometeorology, Utah State Univ., Logan, UT 84322-4820
b The Institute of Soil, Water and Environmental Sciences, (ARO) The Volcani Center, Bet Dagan 50250, Israel
* Corresponding author (darearthscience{at}yahoo.com)
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ABSTRACT
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Many empirical formulas relating TDR-measured permittivity (Ka) to volumetric water content (
) have been presented owing to the lack of a robust and accurate physically derived model describing this relationship across a range of soils. Soil-specific calibrations are often infeasible due to the time-consuming gravimetric sampling required for adequate calibration. In this work we propose a sample scale model for the Ka– relationship in coarse-grained media using physically based pore-scale, or calibrated two-point, anchoring. Materials tested include mono-size glass spheres and quartz sand grains in addition to two sandy soils. The performance of the model was comparable with the empirical models of Topp et al. (1980) for the different media. The model accounts for particle shape and bulk density using a two-phase pore-scale mixing model and refers to a wetting or draining profile with a sharp wetting or drying front. Our measurements indicate the absence of dielectric hysteresis for the narrow size distribution materials studied. An alternate calibration approach only requires the measured soil effective permittivities for dry (
dry) and saturated (sat) conditions (i.e., two-phase mixtures) and knowledge of the bulk density. We recommend a general value of 2.8 for dry in soils with predominantly quartz mineralogy; the model then requiring only the sat to develop a soil specific calibration. The results provide insight into the appropriate ‘refractive index’ modeling of layered (wetting/drying) soil profiles with the grain-scale modeled two-phase permittivity providing bounds for the sample-scale three-phase porous medium.
Abbreviations: TDR, time domain reflectometry
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INTRODUCTION
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TOPP ET AL. (1980) proposed a sample-scale empirical calibration between apparent permittivity (Ka) measured using time domain reflectometry (TDR) and soil volumetric water content ():
 | [1] |
This so-called ‘universal calibration’ equation obtained from a fit to four mineral soils is widely used in coarse and medium-textured soils and has proved very successful. However, in some soils containing clays (Bridge et al., 1996) and organic matter (Schaap et al., 1996), or in aggregated (Miyamoto et al., 2003) or anisotropic porous media (Jones and Friedman, 2000), this equation doesn't describe the lower permittivity values often measured. Many other empirical equations have been presented that seek to incorporate the effects of soil texture and density to overcome this (Roth et al., 1992; Malicki et al., 1996; Yu et al., 1997). Complex learning algorithms such as neural networks have been used (Persson et al., 2002) to try and ascertain the relative importance of different soil properties for calibration. This approach has good utility for finding simple and practical equations that can be used to estimate . However, the limitation of such an approach is set by the required input parameters describing the soil attributes used in the analysis.
An alternative approach to understand the permittivity response of unsaturated porous media is the ‘bottom-up’ approach. This approach begins at the grain-scale and endeavors to reconstruct the sample-scale response based on the grain-scale properties. Maxwell (1881) developed a mixing model approach relevant for describing the electrical conductivity of heterogeneous sands. Since then many dielectric-mixing models have been proposed for different applications and geometries (Looyenga, 1965; de Loor, 1968; Dobson et al., 1985; Hilfer, 1991; Heimovaara et al., 1994; Friedman, 1997; Friedman, 1998; Hilhorst et al., 2000). This approach has been used extensively in the geophysics literature where saturated rocks present the simpler case of a two-phase medium (Sen et al., 1981; Kenyon, 1984). The extension of these methods to the problem of three-phase, unsaturated soils presents a difficult conceptual and mathematical problem. Friedman (1998) applied an effective medium approach using two types of configurations for the air, water, and solid phases with some success. In that work, the modeled three-phase Ka– relationship is very sensitive to the solid/water/air geometrical configuration.
We analyze a less complex situation where the wetting/draining of the material leads to distinct, almost saturated and unsaturated layers as is commonly encountered in soils. The aim of the work is to determine if a modeling approach requiring easily measurable parameters can be used to successfully describe the sample-scale dielectric response of the coarse grained layered material. The modeling approach capitalizes on previously acquired insight into the grain-scale dielectric phenomena for the prediction of the sample-scale response, and can be applied as a new field calibration, especially in conditions where a TDR probe is inserted vertically (i.e., perpendicular to the wetting–drying front in a coarse-textured soil). It also serves for exploring how we can better understand sample-scale dielectric measurements in soils. The limitations of the model are that it is developed for coarse-grained media and does not account for the presence of bound water.
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MATERIALS AND METHODS
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Granular Materials and Soils
In our first experiment three coarse-grained media were used to fill TDR cells, two types of 500-µm glass beads (Mo-Sci Corp., Rolla, MO, USA, and Trime-IMKO, Germany) and 500-µm quartz grains (Yerucham Crater, Negev desert, Israel). The coarse-grained media were prepared in the laboratory by washing in deionized water. The quartz grains were also acid-washed to remove oxide coatings. In the second experiment, two sandy soils were used, Kidman and Hyrum, to test the principal of the proposed approach. Particle-size analysis categorized the Kidman soil as sand (Coarse-loamy, mixed, mesic Calcic Haploxeroll) and the Hyrum soil as loamy sand (Loamy-skeletal, mixed, mesic Pachic Argixeroll).
The shape of the grains in the quartz sand and sandy soils was assessed using the method described in Friedman and Robinson (2002). The method provides an estimate of grain shape based on the slope formed by pouring a granular pile under water. The maximum angle of the poured slope is related to the depolarization factors in the dielectric-mixing model described in the theory section.
The solid phase permittivity of quartz sand and glass spheres was measured using the dielectric immersion method (Robinson and Friedman, 2003; Robinson, 2004). Each material was immersed in five dielectric fluids and the point at which the permittivity of the immersion fluid matched the measured effective permittivity gave the permittivity of the solid. The quartz permittivity was measured to be 4.7, in close agreement with measurements for crystals (Fontanella et al., 1974), and the glass permittivity was measured to be 7.6. Both soils are predominantly quartz and were thus assumed to have a solid phase permittivity of 4.7.
Time Domain Reflectometry Measurements
Permittivity measurements were made using a TDR cable tester (Tektronix Inc., Beaverton, OR; model 1502B Metallic Cable Tester) connected to the measurement cell using a 1-m RG-58 coaxial cable. The TDR cable tester was connected to a PC, which was used to collect and analyze waveforms using software developed by Heimovaara and de Water (1993) and by Or et al. (2003). The probes were calibrated for effective length using deionized water and air (Heimovaara, 1993; Robinson et al., 2003a). Soil was repacked into the measurement cell and drained while the permittivity was measured with the temperature being kept constant at 25°C ± 0.5.
Cell Design, Construction, and Packing
The Ka– measurements were conducted in custom made, rectangular (8 x 8 cm) or cylindrical (9-cm diam.), 17-cm tall plexiglass containers (Fig. 1a). The electrodes were made of two parallel, stainless steel plates measuring 2.5 cm wide and 15 cm long and spaced 2 cm apart. The choice of parallel plates as transmission lines for measurements offer certain advantages for laboratory TDR measurement (Robinson and Friedman, 2000; Robinson et al., 2003b). A uniformly oriented electrical field is developed and a more even energy distribution (i.e., than rods) is obtained in the sample. The relative energy storage density (E2) (where E is defined as the electrical field strength) of the parallel plates was calculated using the Arbitrary Transmission Line Calculator (ATLC; Kirkby, 1996; http://atlc.sourceforge.net, verified 12 Apr. 2005) (Fig. 1b). A voltage of +1 V was assumed for the left-hand plate and –1 V for the right. The far field voltage was set to 0 V. The concentration of energy between the plates is clearly observed (Fig. 1b).
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Fig. 1. (A) Dual plate probe used for vertically draining and wetting TDR measurements in partially saturated porous media. The sintered plastic porous membrane at bottom allows water suctions of up to about 70 cm. (B) Energy density (E2) (darker shade = higher density) is demonstrated on the right as a two-dimensional horizontal cross-section.
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Two ceramic cups were mounted in the rectangular cell at the base against the plexiglass walls. In the cylindrical cell design, a sintered plastic membrane (X-7744 T3 sheet, Porex Porous Products Group, Fairburn, GA) having a 7-µm pore size and 0.635-mm thickness was glued into a recessed ring in the base. The sintered plastic membrane was soaked in a mixture of 20% (v/v) Tween 20 (Sigma-Aldrich Co., St. Louis, MO) and 80% Isopropyl Alcohol for 20 min and allowed to dry. Tubes leading from under the porous membrane allowed the cell to be drained and wetted from the base upward.
Oven-dry materials were packed into the water-filled cells to prevent air entrapment in the sample. The containers were tapped on a solid bench surface to achieve tight packing. The water content was measured directly knowing the mass of water in the cell. Porosity (
) was independently verified using the particle density (
s) (measured using the immersion method, Flint and Flint, 2002) and bulk density (b) and found to correspond to better than 0.01 (m3 m–3). For the rectangular cells, water was drained from the cell through the ceramic cups and subsequently rewetted using the same cups. The cell was sequentially drained under tension for water content adjustment over a period of several weeks and then rewetted. Eventually, the cell was placed in the oven at 80°C to remove the last 2% of the water not drained, and measurements were made after the cell had cooled for several days. It was then rewetted under tension for the second time, allowing the full range of water content to be measured.
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THEORY AND MODELING
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Grain-scale Modeling of the Effective Permittivity of a Two-Phase Medium
Perhaps the most widely used two-phase model for describing dielectric spheres in a uniform dielectric background is the Maxwell-Garnett (1904) mixing equation (Sihvola, 1999), given as:
 | [2] |
where f is the volumetric fraction of the solid inclusions (= 1 – , where is the porosity), eff is the effective permittivity of the two-phase mixture, 0 is the permittivity of the background (water or air) and s is the permittivity of the granular inclusions. The model assumes that the inclusions and their respective fields are noninteracting, which is an invalid assumption for densely packed granular materials. This interaction effect has been studied theoretically for regular cubic lattices of spheres (Sangani and Acrivos, 1982; Cheng and Torquato, 1997) and demonstrated experimentally for cubic lattices (D.A. Robinson and S.P. Friedman, unpublished data, 2004). Recognizing this effect, Sihvola and Kong (1988) proposed a virtual, apparent permittivity (a) describing the dielectric constant seen by an individual inclusion (particle or scatterer) in dense packing. Equal to 0 in the dilute case, a in the dense packing is a function of the background and inclusion permittivities and is expected to lie somewhere between the value of the bathing fluid (i.e., water or air) and eff:
 | [3] |
The heuristic parameter, 
, can be considered physically as representing a polarization resulting from the close packing of the particles. The value of is confined between 0 and 1. A value of = 0 results in the Maxwell-Garnett (1904) formula (Eq. [2]) and = 2/3 and 1 lead to the well known symmetric effective medium approximation (Polder and van Santen, 1946) and the coherent potential formulas (Tsang et al., 1985), respectively. A value of = 0.2 was proposed and demonstrated by Friedman and Robinson (2002) to properly describe the neighboring particle effects based on measurements in coarse, densely packed random materials. However, it appears that the value changes depending on the ordering of the packing, with ranging between 0.2 for random packing of mono-size spheres and 0.323 for the simple cubic periodic lattice (D.A. Robinson and S.P. Friedman, unpublished data, 2004).
A further assumption in the Maxwell-Garnett model is that of spherical particles, which holds for glass spheres but not for natural granular materials whose shape is expected to cause added polarization. This means that the Maxwell-Garnett model is expected to be an upper bound where the background has a higher permittivity than the inclusion (water-saturated media), and a lower bound when this is reversed (air-dry media). The Sihvola and Kong (1988) two-phase dielectric-mixing model described by:
 | [4] |
also incorporates particle geometry and provides an implicit equation for calculating the effective permittivity (eff) for non zero values of . In Eq. [4] a, b, and c are the axial directions of the particles. When = 0, the Maxwell–Garnett formula (Eq. [2]) results. The effect of particle geometry is modeled using depolarization factors (Ni) that describe the extent to which the inclusion polarization is reduced according to its shape and orientation with respect to the applied electrical field. The depolarization factors for an ellipsoid of revolution (spheroid, a 
b = c) with an aspect ratio of (a/b) can be approximated by the empirical function presented in Jones and Friedman (2000):
 | [5] |
The depolarization factors for a sphere where a, b, and c are all equal in length are Na,b,c = 1/3, 1/3, 1/3; for thin disks Na,b,c = 1, 0, 0; and for long needles Na,b,c = 0, 0.5, 0.5.
The shape factor for the quartz sand and for the two sandy soils was determined from the maximum angle of the slope of the poured granular material, according to the method described in Friedman and Robinson (2002). The measured angles were 33.0 for the sand, 35.1 for Hyrum loamy sand and 33.0 for Kidman sand. These were converted to the ratio of (a/b) for an equivalent oblate particle according to (Friedman and Robinson, 2002):
 | [6] |
where ß is a constant equal to 0.1698 and 23.1 is the maximum slope angle for a pile of spheres. Equation [6] results in (a/b) values (Eq. [5]) of 0.466 for the quartz sand, 0.411 for Hyrum sand, and 0.467 for Kidman sand.
Sample-Scale Permittivity Response
The drainage of the coarse-grained material gives rise to a sharply contrasting front between the wet material and the drained material. As such it is appropriate to model the sample-scale as two dielectric layers (Birchak et al., 1974). This approach is termed the refractive index mixture theory. The use of refractive index mixture theory to describe a layered material has a theoretical basis. Refractive index mixture theory has also been used to describe mixed multiphase materials; however, it has no rigorous theoretical basis for this application and should be considered a power law approximation (Sihvola, 1999). The TDR measured apparent permittivity (Ka) of a layered material can be derived from the travel time (t) of the TDR signal through layers L1, L2 with combined length of L = L1 + L2:
 | [7] |
where vp is the TDR signal propagation velocity and the subscript x denotes an arbitrary dielectric. Thus, based on refractive index mixture theory, the measured permittivity is:
 | [8] |
This should hold for TDR measurements with relatively thick layers but not for multiple thin layers (Schaap et al., 2003). This expression has been used previously to model layered soils (Birchak et al., 1974; Topp et al., 1982). Making an assumption that the medium described is composed of one wet and one dry layer, we propose that an equation of the following form should describe a bilayered wetting or drying soil profile:
 | [9] |
where is the mean volumetric water content and is the porosity. The two, two-phase mixture permittivities of saturated (sat) and dry materials (dry) can be measured or derived from Eq. [4] and used in Eq. [9].
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RESULTS AND DISCUSSION
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Quartz Sand and Glass Spheres
All the data collected from the experiments for both wetting (imbibition) and drying (drainage and evaporation) of the glass spheres and quartz sand are presented in Fig. 2. Plotted on the same graph are the empirical polynomial equations presented by Topp et al. (1980) for Rubicon sandy loam soil, given as:
 | [10] |
and 450-µm glass beads, described by:
 | [11] |
Topp et al.'s (1980) original data were collected with a horizontally oriented cell; our data however, were collected with a vertically oriented cell but show a very similar result. Comparison of Topp et al.'s (1980) calibration curves for glass beads and Rubicon sandy loam with the layer model, Eq. [9] is provided in Fig. 3. This close correspondence would indicate that Eq. [9] is still valid, to some extent, even when the geometry is not that of a sharp wetting or drying front perpendicular to the direction of wave propagation. Two points are important here: (1) The refractive index averaging constitutes a lower bound as compared with the situation where the saturated and dry soil volumes are not arranged in series to the wave propagation direction, and to situations of smaller layer thickness/wavelength ratios, leading to an arithmetic averaging regime (Schaap et al., 2003); and (2) at a given saturation degree (S = /), the positive curvature of the eff () relationship at the grain scale, releasing air toward S = 1, increases the eff up to sat more than the reverse situation of releasing the same volume of water toward S = 0, decreasing eff down to dry. Thus (1) and (2) partially compensate each other, improving the applicability of Eq. [9] for unsaturated coarse-grained soils.
Application of the Mixing Model
The data presented in Fig. 4 are for the desaturating glass spheres and quartz sand grains and the attempts to model the data using the approach described in the theory section. Values of sat and dry were calculated from Eq. [2] according to the Maxwell–Garnett mixing formula. As expected with the assumptions of noninteracting particle fields and spherical shape, the prediction forms an upper bound to the saturated measurements and a lower bound to the dry measurements (solid symbols, Fig. 4).
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Fig. 4. Permittivity for desaturating glass spheres and sand grains. The filled symbols give the two-phase predictions from the Maxwell-Ganett (1904) model for spheres (Eq. [2]). The lines give the predictions using Eq. [4] to calculate the saturated and dry two-phase permittivity values, which were input into Eq. [9] to calculate the effective permittivity.
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The Sihvola and Kong (1988) formula (Eq. [4]) was then used to predict the same points using the parameters in Table 1. The value of , to account for the depolarization due to close packing was set to 0.2 and the shape parameter (a/b) was set to 1 for spheres and 0.466 for sand grains; having been predicted from the measured maximum slope angle (Friedman and Robinson, 2002). The measured and predicted sat and dry values are presented (Table 1) and show very close agreement. The predicted sat and dry values were inserted into Eq. [9] and the model predictions are presented over the data in Fig. 4 (solid and broken lines). This modeling approach indicates the importance of the grain-scale geometry when endeavoring to describe the permittivity response of the granular material at the sample-scale. A comparison of measured and predicted permittivity values is obtained using the RMSE (root mean squared error) in Table 2.
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Table 1. Parameters describing particle shape used in Eq. [4] due to close packing () and particle shape, Na/(b=c), used in Eq. [4]. Measured and modeled permittivities for the two-phase mixtures (solid-water and solid-air). Glass spheres and sand grains are assumed randomly oriented and the calculation is taken as the summation of the three principle axes in Eq. [4].
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As a further example, the model (Eq. [4] and [9]) was tested on two sandy soils, Hyrum and Kidman (Fig. 5). Again, the model presented with no empirical fitting parameters describes the data very well. Deviations from the model are to be expected for materials with a broad particle-size distribution. Broader particle size tends to reduce the permittivity toward high saturation degrees (Robinson and Friedman, 2001). The modeling provides one physical derivation for the empirical Topp et al. (1980) calibration equation. This general approach should allow this work to be extended to examine the effects of other physical properties such as the particle-size distribution (Robinson and Friedman, 2001) and distributed wetting fronts. Another step to develop and refine the model further will be to incorporate the effects of temperature on the calibration.
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Fig. 5. Permittivity–water content relationships for two soils, Hyrum and Kidman. The solid line is the model prediction based on measured parameters used in Eq. [4] and [9].
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The general bilayer refractive index approach adopted in this work should not be confused with the three-phase refractive index model proposed by Whalley (1993). The equation proposed by Whalley (1993) has proved reasonably successful in an empirical sense. However, strictly speaking it can only be physically derived for a three-layer system perpendicular to the TDR probe (Schaap et al., 2003). It is important to point out some of the limitations of the bilayer modeling approach. First, it addresses only a layered system where one layer is practically water-saturated and the other close to air-dry. The model is designed for a steep wetting/drying front (likely to be encountered in coarse-textured media) and does not describe a distributed water profile. Second, the model does not describe soils with bound water. To extend the modeling to a distributed water profile, two additional extension steps are required: (i) the extension of the refractive index averaging toward an effective medium theory, depending on the steepness of the wetting front (correlation length of the water content) and the dominant wavelength; and (ii) derivation of a proper pore-scale eff() model following, for example, the approach presented in Friedman (1998) which would allow different configurations of the air–water phases.
Wetting and Drying/1 and 2 Point Calibration Model
Results for wetting (imbibition), followed by drying (draining and evaporating), and followed again by rewetting are presented (Fig. 6) for the glass beads (MoSci) and quartz sand. Use of Eq. [9] with measured values of Ksat and Kdry provides a two-point calibration model that describes the data accurately. The lines shown were fitted using the permittivity measured at saturation (Ksat) and oven dry (Kdry); this empirical approach relies on:
 | [12] |
where Ka is instrument measured permittivity and the other terms are as previously described. This form of model fits the data exceptionally well. The data confirms that for two thick enough dielectric layers, the refractive index is the appropriate averaging to use (Schaap et al., 2003). However, it is important to point out that this may not be the case for low frequency (f < 100 MHz) dielectric sensors. In that case, the correct averaging would probably have to be determined through experiment and will lie somewhere between refractive index and arithmetic averaging described as
 | [13] |
because effects of layering are frequency-dependent (Schaap et al., 2003). The very slightly sigmoidal pattern (higher permittivities than expected at lower saturation degree, S and lower permittivities at higher S) observed in the measured data may stem partially from a transition from arithmetic to refractive index averaging as the permittivities/water contents increase and wavelengths decrease.
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Fig. 6. Drying and wetting cycles for glass spheres (left) and quartz sand grains (right) modeled using a simple two-layer model (Eq. [9]) fitted according to the measured saturated and dry permittivity values. Right-hand diagram is for quartz sand, modeled again according to Eq. [9].
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It is also interesting to note that from this data there appears to be no dielectric hysteresis effects in the measurements, as might be expected for a uniform material. This is also due to the sharply defined wetting and drying fronts. After the material was drained, reaching water contents of about 0.02 and 0.04 for the glass and sand, respectively, it was rewetted. This rewetting caused some air entrapment and the wetted water contents were about 2% lower than originally measured. However, after oven drying at 80°C the sand fully rewetted, but the glass spheres still trapped about 2% of the air, never fully reaching a equivalent to the porosity.
We propose that Eq. [12] can provide a very simple single point field calibration if the permittivity of the dry soil is assumed to have a value of 2.8. This value appears justified from both our measurements and those of Topp et al. (1980) and others (Roth et al., 1992; Malicki et al., 1996; Friedman, 1998). Given a measurement of the water-saturated permittivity of the soil, a soil-dependent calibration can be fitted according to Eq. [12], which can be rearranged to determine from TDR measurements:
 | [14] |
It is important to emphasize that this is only appropriate for quartz dominated coarse-textured soils unless tested further.
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CONCLUSIONS
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Results are presented to demonstrate the importance of grain-scale geometry on the sample-scale dielectric response of porous media. The modeling gives insight into the importance of accounting for depolarization caused by close particle packing and particle shape in a two-phase dielectric mixture. Predictions from the grain-scale dielectric modeling are used to anchor a simple refractive index averaging model describing a layered wetting–desaturating porous media. For the frequency at which TDR operates we demonstrate that refractive index averaging is appropriate, however, this may not be the case for sensors operating at frequencies under 100 MHz. We would suggest calibration to determine if arithmetic or refractive index averaging is appropriate at these lower frequencies. The modeling approach describes the data with no requirement of unknown fitting parameters. In addition, it provides a simple single point calibration model that requires permittivity at water saturation to be known to generate a soil specific calibration, assuming the permittivity of the dry soil has a value of about 3.
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ACKNOWLEDGMENTS
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This research was supported in part by research grant No. IS-2839-97, from BARD. The United States-Israel Binational Agricultural Research and Development Fund, and in part by the National Research Initiative Competitive Grant no. 2002-35107-12507 from the USDA Cooperative State Research, Education, and Extension Service.
Received for publication November 24, 2004.
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ABSTRACT
INTRODUCTION
