Soil Science Society of America Journal 65:13-18 (2001)
© 2001 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Numerical Modeling of a Capacitance Probe Response
G. de Rosnya,
A. Chanzyb,
M. Pardéb,
J-C. Gaudub,
J-P. Frangia and
J-P. Laurentc
a Laboratoire Environnement et Développement, Université Paris 7, 2 place Jussieu, F-75251 Paris Cedex 05, France
b INRA, Unité de Science du Sol, Domaine St Paul, F-84914 Avignon Cedex 09, France
c Laboratoire d'étude des Transferts en Hydrologie et Environnement, BP53, F-38041 Grenoble Cedex 09, France
Corresponding author (derosny{at}ccr.jussieu.fr)
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ABSTRACT
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Capacitance sensors are one of the common means of characterizing the soil dielectric constant. Their design depends on their expected applications. In particular, the electrode geometry has a critical influence on the extension of the probed region. Moreover, the soil may not always be viewed as a medium of statistically uniform dielectric constant (because of packing effects, air sheath in the vicinity of the electrodes, stones). Numerical modeling for the behavior of a particular probe was developed. It is based on solutions of the Maxwell's equations in the quasi-static approximation, by a finite element method. This modeling was compared with laboratory measurements in various media (air, ethanol) where heterogeneity was inserted in the vicinity of the electrode. The numerical model reproduces very well the probe response when millimetric scale perturbations were introduced. The numerical model appears to be a promising tool to investigate more deeply the capacitance probe measurements, for instance the extension of the measurement volume or the significance of measurements in highly structured soils.
Abbreviations: FEM, finite element method • PDE, partial differential equation • PVC, polyvinyl chloride • TDR, time domain reflectometry
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INTRODUCTION
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METHODS THAT CHARACTERIZE the soil dielectric constant are used widely to measure soil moisture. For in situ measurements, time domain reflectometry (TDR) and capacitance sensors are the most common techniques to measure the soil dielectric constant (
r). Correlation of the dielectric constant measured by the probes with the soil moisture measurement is not straightforward and involves two steps. First, there is the relation between the soil dielectric constant and soil moisture. This relation has been investigated theoretically and empirically (de Loor, 1968; Wang and Schmugge, 1980; Topp et al., 1980; Dobson et al., 1985; Roth et al., 1990). It can be correctly handled when soil properties such as texture, dry bulk density, temperature, and salinity are known. The second step is to establish the link between the effective dielectric constant measured by the device and the average dielectric properties in the soil volume of interest. In general, this step is implicitly included within the field calibration relationship (Chanzy et al., 1998). The r heterogeneity of the soil in the vicinity of the probe (which refers to the transmission line with the TDR and to the electrodes with the capacitance device) strongly influences the measurements. For instance, Rothe et al. (1997) demonstrated that the soil compaction due to the probe installation strongly affects the TDR measurements. Gaudu et al. (1993) found that the sensitivity of a capacitance probe to the soil moisture is significantly reduced when the soil–probe contact is not well obtained. The volume of influence of the probe is practically limited to a few centimeters around it. However, a precise operational definition of the influence of the medium on the probe response has to be worked out to define unambiguously this volume.
The aim of this study was to provide a tool to represent the measurements made by a capacitance device, which accounts for electrode geometry and r heterogeneity within the volume of influence. This tool will be useful to evaluate the impact of installation artifacts (air gap, compaction), to optimize electrode shape and determine the associated volume of influence, or to simply understand some unexpected measurements. Experimental approaches give some indications on the spatial weighting functions of how the soil contributes to the measurement (Baker and Lascano, 1989; Gaudu et al., 1993). Experiments are limited to a few cases such as the planar discontinuity between two media or a set of water-filled glass tubes. Modeling approaches are probably more relevant to our goal. Existing models representing the TDR or the capacitance probe are all based on quasi-static approximation. Analytical formulations have been proposed for some particular geometries (Annan, 1977; Zegelin et al., 1989; Knight, 1992). With numerical approaches, we can investigate any case without a restriction on the geometry (electrodes, soil heterogeneity). Moreover, improvements in computer performance and the availability of commercial finite element software make numerical methods much easier to use than in the past. Knight et al. (1997) carried out numerical simulations in the case of TDR and found good agreement with results from analytical models. Using numerical simulations, Ferré et al. (1998) defined the sampling area in the plane perpendicular to wave guide of the TDR measurements. They showed how numerical modeling can be used for the geometric design of a TDR wave guide. Straub (1994) has used the boundary element method to compute the admittance of capacitance electrodes. However, in the studies based on numerical modeling we did not find comparisons of simulations against measurements.
This study was devised to check the validity of quasi-static approximation of electromagnetism to represent the capacitance probe response. In order to account for the electrode geometry and the heterogeneity of the studied dielectric medium, we used a finite element approach to compute the electric field. We explain how we implement the finite element method and define the quantities that were taken in the comparisons of experiments and simulations. The simulations were then validated with different media presenting known heterogeneities. To keep a good control of the media, we made the measurements using fluids whose dielectric constants fall within the range of these of soils. Air was used for the dry end, whereas ethanol was taken to represent moderately wetted soil. To validate the capability of the numerical modeling to represent the capacitance probe measurement in heterogeneous media, we compared the impact of perturbations located in the vicinity of the electrodes measured by an actual probe against numerical simulations.
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THEORY
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To enable mathematical treatment of probe behavior, the latter was simplified into a two-conductor system: a central electrode on which a 38-MHz excitation is applied and an annular grounded electrode. Between them, various dielectric materials can be found. It is assumed that the soil influences the probe only through its dielectric constant. Moreover, the wavelength of the electromagnetic field is much larger than the probe dimensions. It is then assumed that the quasi-static approximation of electromagnetism applies.
The probe response is presumed to be a univocal function of the central electrode capacitance, so that the aim of the simulation is to compute this capacitance for various spatial distributions of the dielectric constants. This capacitance constitutes the output of the simulation. The probe response is converted to a dielectric constant scale by calibration in uniform media of known dielectric constants. When studying a nonuniform dielectric medium, the response will be expressed in terms of an effective dielectric constant, the one of a uniform medium that would give the same response. Similarly, the probe capacitance may be converted to an effective dielectric constant, namely the dielectric constant of the uniform medium leading to the computed capacitance. The validation of the model is based on the comparison between measured and computed effective dielectric constants.
The probe capacitance may be computed from the solution of the Laplace's equation satisfied by the electrical potential in the dielectrics connecting the electrodes
 | (1) |
is the gradient operator, (x,y,z) the complex electric permittivity, and V(x,y,z) the complex electrostatic potential. It is assumed that the charge density in the dielectric medium may be neglected. The coordinates x, y, and z specify a position in space. In the present study, the axial symmetry of the probe was used to reduce the dimensionality of the problem: in cylindrical coordinates, a point in space is characterized by its distance r to the axis of symmetry, its position z along this axis, and its azimuth 
. The investigations will be limited to soil dielectric constant distributions that do not depend on the azimuth. With this assumption, the Laplace's equation may be written as
 | (2) |
Exact analytical solutions can be obtained only for particular geometrical configurations and dielectric constant distributions; otherwise, one has to rely on numerical methods.
The probe capacitance can be computed from the potential distribution V(r,z) in the dielectric with the boundary conditions 
on the central electrode and 
on the grounded annular electrode. In this situation, the probe capacitance C is equal to the charge Q of the central electrode, which is related to the potential by the flux relation
 | (3) |
where 
is any surface in the dielectric enclosing the central electrode, 
is the electrostatic displace ment vector, and d
is an outward orientated elementary surface element.
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MATERIALS AND METHODS
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Two-dimensional numerical simulation requires axial symmetry of the probe as well as of disturbances in the medium. To fulfill this requirement, we used the HMS9000 capacitance probe (SDEC, Reignac sur Indre, France), which measures the real part of the dielectric constant. It operates at 38 MHz and is formed by two stainless steel electrodes located along the same axis. One electrode is annular with a length of 10 mm and a diameter of 21 mm. The other is a rod (20 mm long, 2-mm diam.) located just below the annular electrodes. The electrode characteristics correspond with their visible side. In fact, the electrodes are partly hidden by the polyvinyl chloride (PVC) structure, as shown in Fig. 1
. The visible and the hidden side of the electrodes were both taken into account in the numerical simulations. The capacitance probe delivers analog and digital signals, which were scaled to r by an air–ethanol calibration. It is performed by making a measurement in air and by dipping the electrode in a cylindrical glass vessel of radius 85 cm, filled with ethanol. This procedure was applied according to builder's instruction and using a linear relation between the probe output and r.
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Fig. 1. Geometrical layout of the elements used in the simulations. For the polyvinyl chloride material,  is used. db and Yb are the radial and horizontal bottom coordinates, respectively, of the boundaries. The figures between brackets correspond with coordinate nominal values that have possibly been modified in particular simulation conditions
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The ethanol dielectric tangent loss angle is of the order of 0.1 in the megahertz frequency range (Von Hippel, 1995), which generates a signal in quadrature of phase with respect to the capacitance response. The probe has been designed to be insensitive to this signal. The imaginary part of the ethanol dielectric constant may also influence the capacitance signal when other dielectric media, air and PVC in this study, are also present. It has been verified by computation that this influence is completely negligible for the situations encountered in the present study. In consequence, the imaginary part of the dielectric constants will be neglected hereafter.
Measurements
Experiments were conducted in a glass vessel (25-cm height with an 8.5-cm radius) filled with pure ethanol. The capacitance probe was attached to a graduated support (Fig. 2)
. Using this support, the probe can be moved up or down the vertical axis with an accuracy of 
1 mm. The following experiments were carried out.
- The capacitance probe was dipped progressively into the ethanol to assess the influence of the horizontal air–ethanol boundary on the measurement.
- The rod was covered by a variable number of thermo-retractable sheaths to simulate an air gap around the electrode. Sheath thickness ranged from 0.4 to 2 mm, and its dielectric constant was assumed to be 3.
- Polyvinyl chloride tori were located in the vicinity of the electrodes. The torus element was chosen to simulate a disturbance in the probe volume of influence that respects the axial symmetry. All tori had a square section of 3 by 3 mm. The external diameters of the tori were 20, 26, and 32 mm. The tori were held in the center of the ethanol by three nylon threads stretched by ballasts located on the external side of a plastic container as shown in Fig. 2 (10-cm height with an 11-cm diam.). It was observed that the plastic container had little effect on the capacitance probe measurements. The probe was centered and shifted along the torus axis. We assumed that the PVC dielectric constant was three (Weast, 1986) as for the PVC parts of the capacitance probe.
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Fig. 2. Experimental setup used to calibrate the probe in liquids of various dielectric constants and to investigate the effects of various perturbations on the measurements
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Numerical Approximations
In the present study, solutions of the Laplace's equation were computed using the Finite Element Method (FEM), which allows an approximate resolution of partial differential equations with boundary conditions on complex geometrical surfaces and arbitrary dielectric constant distributions. For two-dimensional problems, MathWorks Inc. (Natick, MA) has PC software, the Matlab Partial Differential Equation (PDE) Toolbox, which will be used hereafter.
In order to obtain a single solution to the Laplace's equation in a given region, boundary conditions must be provided for the whole surface enclosing the region. In the present configuration, the region extends to infinity both in the soil and in the air, a situation that the Matlab PDE software does not handle. It is therefore usual to close the region at finite distances by sufficiently remote surfaces to provide solutions expected to be close to the actual ones. For the sake of simplicity, the selected closing surface is a cylinder limited by two horizontal planes. It is represented by a vertical and two horizontal straight lines in the (r,z) plane. The boundary conditions on the closing surface may be either given values of the electrostatic potential (Dirichlet conditions) or given values of the normal component of the electric field (Neumann conditions). The best choice of boundary conditions will emerge later from a comparison with measurements. A Neumann boundary condition was applied to the left boundary to represent the axis of symmetry.
The FEM is based on decomposing the volume in question into a set of triangular elements, the apexes of which constitute a mesh of nodes, and a discretization of the PDE on this mesh. The FEM result is expected to converge to the exact solution when the size of each element drops to zero. To check the quality of the approximation, the stability of the solution with respect to mesh refinement and the position of the boundary was studied. Figure 1 summarizes the various geometrical elements used for the simulation.
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RESULTS
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Uniform Soil Dielectric Constant: Calibrating the Probe Response
The probe capacitance was computed assuming uniform dielectric constants in the range 1 to 34, corresponding with air and soils of increasing water content. The capacitance was computed for increasing mesh density and various boundary positions and conditions (Dirichlet or Neumann). It is found to vary by <1% when the bottom boundary Yb moves from -200 to -300 mm and/or the lateral boundary db moves from 100 to 200 mm (see Fig. 1).
The effect of mesh sizes and boundary conditions may be observed in Table 1, which shows the computed capacitance for 
. From an initial mesh, successive mesh refinements were made both in the whole region and in a region close to the electrodes, where the potential variation is expected to be large.
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Table 1. Values of the probe capacitances computed assuming a uniform dielectric constant r = 20, with Dirichlet and Neumann boundary conditions and increasing number of mesh elements
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To save computation time, the mesh resulting from one global and two local refinements was chosen, although an asymptotic regime is still not encountered when the element number is increased.
The relation between the computed capacitance C and the dielectric constant r depends on the boundary condition (Dirichlet or Neumann), but is in all cases approximately linear: the discrepancy between the computed values and the best linear fit is at most 2% for r in the range 4 to 34, encountered with actual soils. The observed quasi linearity is in agreement with calibration measurements made in different dielectric fluids ranging from 1 (air) to 37.5 (ethylene glycol). The simulated relationships were then used to obtain an effective dielectric constant, which was then used for the simulation–measurement comparisons.
Four cases of boundary conditions were tested. On the one hand, we considered that the whole medium to be homogeneous. On the other hand, we tried to match the experimental conditions by using 
outside of the vessel. We combined both of these descriptions with Neumann and Dirichlet boundary conditions. The four cases led to different results. We further selected the case that best represented the probe response when it was dipped in the ethanol (see below).
Dipping of the Probe in Ethanol
Figure 3
presents the measured and simulated effective dielectric constants as a function of the dipping depth. The ethanol dielectric constant used in the computation is 
. The Dirichlet boundary conditions led to a better agreement than Neumann boundary conditions. Surprisingly, the best agreement was obtained when the effect of being confined in the vessel was not taken into account in the simulation, that is, when the ethanol is not restricted to a radius of 85 mm. In the following, the observations are compared with the simulation assuming Dirichlet boundary conditions and a horizontally homogeneous medium.
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Fig. 3. Effective dielectric constant as a function of dipping depth in ethanol, the depth origin corresponds with the probe tip at the ethanol surface. Black and white circles correspond with two sets of measurements. For Neuman boundary condition, the curves corresponding with  and 1, respectively, outside of the vessel are practically superimposed
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To minimize the effects of the temperature dependence of the ethanol dielectric constant on the comparison, the probe was calibrated in ethanol just before each set of experiments, with the ethanol dielectric constant set to 25 whatever the operating temperature.
Low Dielectric Constant Sheath Around the Probe Electrode
Figure 4
presents the results of measurements and a simulation when the probe tip was coated with an increasing thickness of low dielectric constant material (thermo-retractable polymer). The simulation was performed assuming in the sheath. A dramatic decrease in the effective dielectric constant was observed even for very small thicknesses, and the simulation followed the measurements reasonably well. These observations are of great practical importance as air gaps may be produced when the probe is inserted into the soil.
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Fig. 4. Effective dielectric constant in ethanol, as a function of sheath thickness of thermo-retractable material around the central electrode. Circles correspond with measurements. The line corresponds with a simulation assuming in the sheath
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Toroidal Dielectric Constant Perturbation
Figure 5
presents the effect of the inclusion of tori 
in the ethanol, depending on the distance of the tip of the central probe electrode from the upper part of the torus. The variations in the effective dielectric constant are fairly well described by the model. Moreover, as the measured values should be limited to , corresponding with the absence of any tori, one may infer that the measurements presented in Fig. 5a and 5b should be shifted downward by 0.1. As a matter of fact, a shift of measured dielectric constant of the order of 0.1 may be observed when the probe cables are displaced, which may have happened between the calibration and the measurements. In the situation shown in Fig. 5c, the torus has a diameter greater than the one of the annular grounded electrode. For this case, the effect of the torus decreases as it reaches regions of low electric field around the annular electrode.
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Fig. 5. Effective dielectric constant in ethanol where a torus is dangling, as a function of the distance between the central electrode tip and the top surface of the torus. Negative values correspond with the tip above the torus. The torus section is a square of 9 mm2. Torus outside diameters of (a) 20, (b) 26, and (c) 32 mm, respectively. For the 32-mm torus diameter, the probe annular electrode may be inserted inside the torus. Circles correspond with measured values. Lines correspond with simulations assuming for the torus material (polyvinyl chloride)
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CONCLUSIONS
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A good overall agreement between the measurements and the simulation was observed. The range of the probe response with various kinds of disturbance within the dielectric medium was reproduced very well by the numerical model. We used fluid media as a surrogate soil to work on well-controlled conditions. We show that the numerical model represented accurately the probe response induced by perturbations at a millimetric scale. This opens the possibility of interpreting the impact of the soil structure on the capacitance probe measurements. This is very encouraging for further studies on the volume of influence of capacitance probes in relation to the soil heterogeneity and the probe geometry. Conversely, the influence of perturbations at smaller scale (textural porosity for instance) cannot be analyzed by the proposed numerical model. When implementing the model in soil media, it will allow a study of the influence of the imaginary part of the dielectric constant, related to the soil dielectric relaxation and electric conduction, on the measurements.
The study was performed with commercial software. We would like to highlight that the use of such software is quite simple and does not require specific skill in the field of numerical methods. We believe that a wide community of soil scientists will be able to use this approach for their own research.
A pending question concerns the reason for the better agreement between measurements in ethanol and simulations when the medium is considered as homogeneous in the whole domain instead of accounting for the glass vessel actual dimension and the surrounding air media. The nature of introduced boundary conditions is probably one explanation. Further investigations should be carried out to better handle this step of the numerical model.
Extending the method to a three-dimensional geometry still has to be performed to investigate noncylindrically symmetrical disturbances such as the effect of stones.
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NOTES
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This work was supported by the French "Programme National de Recherche en Hydrologie" (Contract 1997-PNRH24).
Received for publication November 30, 1999.
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REFERENCES
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ABSTRACT
INTRODUCTION
