Published in Soil Sci. Soc. Am. J. 68:440-446 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—SOIL PHYSICS
Three-Dimensional Numerical Modeling of a Capacitance Probe
Application to Measurement Interpretation
Hervé Bolvin*,a,
André Chambarela and
André Chanzyb
a Complex Hydrodynamics Lab., 33 rue Louis Pasteur, F-84000 Avignon, France
b INRA– Climate, Sol, Environment, Domaine Saint Paul, F-84914 Avignon Cedex, France
* Corresponding author (herve.bolvin{at}univ-avignon.fr).
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ABSTRACT
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In this study, a three-dimensional model represents a capacitance probe measurement. It is based on Laplace's equation, which is solved using the Finite Element Method. A three-dimensional model can accurately represent both the probe geometry and nonaxisymmetric components of soil heterogeneity. As a result we have obtained a three-dimensional-cartography of electrical potential V(x, y, z) in the medium surrounding the electrodes and the effective dielectric constant of the media observed. The model was implemented to address such questions as the influence of the air/soil boundary when making measurements near the surface and the probe's volume of influence. It is shown that the volume of influence of the probe can be derived from a weighting function based on the density of electromagnetic energy and can be computed for a homogeneous medium. It is numerically observed that the probe's volume of influence is quite independent of the permittivity of the medium in which the probe is embedded. It is then verified that dielectric perturbations located outside the so defined volume of influence have little effect on the calculation of the effective dielectric constant seen by the capacitance probe.
Abbreviations: 
, dielectric constant
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INTRODUCTION
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CAPACITANCE PROBES are now widely used to measure soil moisture. One of the critical factors in determining an accurate measurement of soil moisture is to establish the effective dielectric constant sensed by the probe. The soil volume seen by the probe, the impacts of heterogeneities such as stones or cracks, and the depth of the probe below the air–soil interface, are among the factors that one needs to consider performing measurements. The effective dielectric constant is a weighted average of the dielectric constant () in the vicinity of the electrode. One method to analyze the relationship between the effective and the actual spatial distribution is to use a numerical model that allows an explicit representation of both the probe geometry and spatial distribution.
A number of previous studies have dealt with numerical modeling of soil moisture sensors based on soil measurement (Knight et al. [1997], Ferré et al. [1998], Chambarel and Ferry [2000] for the time domain reflectometry [TDR], and Straub [1994] and De Rosny et al. [2001] for the capacitance probe). De Rosny et al. (2001) successfully used a Finite Element Method to represent the response of a capacitance probe to heterogeneity in the vicinity of the sensor. However, they solved Laplace's equation in a two-dimensional space, which restricted their approach to cases in which the probe and the soil dielectric spatial distribution have an axial symmetry. This constraint may be inappropriate for many sensors on the market and for most types of soil heterogeneity.
The goal of this study is to extend the approach of De Rosny et al. (2001) to a three-dimensional representation of the probe–soil system using the Finite Element Method. As a first step we tested the three-dimensional model against measurements made with well-defined medium-probe system geometry. Then the model was used in selected cases as a tool to improve our interpretation of the measurements. In particular, we addressed some practical problems such as the positioning of the probe near the surface and the volume of influence of the probe.
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THEORY
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The frequencies used in capacitance probes are generally in the region of several tens of MHz. The wavelength of the electromagnetic field (about several meters) is then much larger than the dimensions of the probe, allowing one to apply the quasi-static approximation and so ignore propagation along the electrodes (Feynman, 1979).
The mathematical model of the electrical potential [V(x, y, z)] is given by Laplace's equation:
 | [1] |
where (x,y,z) is the electrical permittivity of the medium surrounding the two electrodes or that of the probe's structure. We assume that the annular electrode is grounded (V1 = 0) whereas the central electrode is excited by a voltage V2. The probe capacitance C can be computed from the charge Q of the central electrode given by:
 | [2] |
Q can be computed by applying the Gauss theorem
 | [3] |
where S is any surface in the dielectric medium enclosing the central electrode, and 
is an outward-oriented elementary surface vector.
Equation [1] was solved in a finite domain (
), where the elements were discrete. The Finite Element Method was implemented using the classical Weighted Residual Method (Dhatt and Touzot, 1981).
To solve Laplace's equation the boundaries 
of () were divided into two parts: 1 that corresponds to the external surface of the electrodes and 2 that limits the finite domain ().
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MATERIALS AND METHODS
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Description of the Capacitance Probe
A HMS9000 (SDEC, France) capacitance probe was used in the study (Chanzy et al., 1998; De Rosny et al., 2001). It has two electrodes located along the same axis (Fig. 1)
. The electrode set to ground potential is an annular ring whereas the excitation electrode is a rod centered on the axis and positioned below the ring. The electrodes are held together by a polyvinyl chloride (PVC) structure. The capacitance probe delivers an analog signal, which can be calibrated using a reference media (air, ethanol).
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Fig. 1. Geometrical shape of the capacitance probe used in the simulations (Fig. 1 is not drawn to scale to improve clarity of the drawing).
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Measurements
Measurements were performed either to validate the model or to address the question of the minimum depth above which the soil–air boundary affects measurement. For validation we used the data presented in De Rosny et al. (2001). We selected the experiment with the low dielectric constant sheath surrounding the electrodes as an example of a case with a strong perturbation of the measurements. In this experiment the rod of the central electrode was coated with thermo-retractable polymers of different thicknesses. This approach simulated the case in which an air gap separates the soil and the electrodes, as may occur in shrunken soil.
To test the ability of the model to represent slight variations in the probe signal we selected the experiment with a toroidal perturbation of the dielectric constant. This experiment consisted in locating a PVC torus with a square cross-section (3 by 3 mm) and a 20-mm outer diameter in different locations around the central electrode. The measurements were made using ethanol, whose dielectric constant falls within the range observed for soil and makes the control of the torus position easy. Two experiments were conducted to establish the influence of the air–soil boundary. To analyze the effect of the surface on a probe embedded vertically in the soil, the capacitance probe was positioned at various depths in ethanol, as described by De Rosny et al. (2001). In a second experiment designed to analyze the effects of a horizontal installation, the probe was inserted horizontally at a depth of 12 cm from a trench in a nonsaline silt loam soil having a moisture of 0.15 m3 m–3. Then, soil layers from the surface were successively removed down to 2 cm above the central electrode.
Numerical Aspects
To solve Eq. [1] we used an efficient C++ Object-Oriented Programming for the finite element code called FAFEMO (Fast Adaptive Finite Element Modular Object) (Chambarel and Ferry, 2000; Dhatt and Touzot, 1981).
The modeled system consisted of a cylindrical volume (200-mm diameter and 300-mm height), which included air and soil and the capacitance probe. It is large enough to minimize all boundary effects (De Rosny et al., 2001). We divided the whole domain (measurement medium and probe structure) into linear tetrahedral elements (e.g., Dhatt and Touzot, 1981). Depending on the different examples, we usually had more than 200000 tetrahedral elements and 48000 nodes. The mesh was finer in the vicinity of the electrodes because of the sharp potential gradients in this area: the elementary volume was about 5 mm3 near the electrodes and about 100 mm3 at the periphery of the cylindrical volume. The structure of the probe was included in the model with air inside the annular electrode whereas the PVC elements were characterized by a relative of 3.
The following boundaries conditions were applied:
- Dirichlet's conditions on the electrodes (1) with a potential set to 0 V for the annular electrode and 1 V for the central electrode.
- Neuman's conditions on (2). The domain () is not infinite and it is difficult to prescribe the potential condition on 2. However, we assumed that the domain was large enough to consider that the electric field was tangential to the surface (2). So, the normal electric field component on 2 was null.
As mentioned above, the capacitance probe delivered an analog signal, which was scaled by making measurements in media with known dielectric constants. A similar scaling was done with the simulations. An empirical relationship was established between the computed capacitance and the of the different homogeneous media. With heterogeneous media, this calibration provided an effective dielectric constant. In this study, the effective dielectric constant was used to compare the measurements and simulations. If the electrodes were embedded in a nonhomogeneous dielectric medium, the effective dielectric constant of this medium was then that of a homogeneous medium giving the same probe response.
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RESULTS
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Equipotential Surfaces in the Dielectric Medium
In Fig. 2
, we present an example of equipotential surfaces simulated in a case where the probe was located near the surface of the soil in a medium with r equal to 25 as in wet soil. We observe a very fast decrease in potential in the vicinity of the lower end of the central electrode, leading to a quasi-uniform potential at a distance of a few centimeters from the electrode. The large potential gradient produces an intense electric field and a very high density of electromagnetic energy in the vicinity of the central electrode. It is also noticed that the potential gradient is even more intense in the PVC structure holding the electrodes together which shows that this zone makes a very significant contribution to the measurements.
Effective Dielectric Constant
Calibration of the numerical simulation was done with homogeneous media having a ranging from 1 to 81. We obtained a linear relationship (r2 = 0.999) between the capacitance of the probe and the effective of the medium:
 | [4] |
Linearity is consistent with the measurements made by the capacitance probe. Coefficients of this linear relationship depended on the geometry of the probe. The contribution of the instrument to the total capacity is large. However, the probe is very sensitive to the dielectric constant of the media measured.
Model Validation
Figure 3
shows the variations of the effective in ethanol (r = 25)—with a PVC toroidal inclusion (r = 3)—as a function of the distance L between the central electrode tip and the top surface of the torus (L is negative when the torus is placed below the central electrode tip). Variations of the measurement were well reproduced by the model. The observed shift of 0.1 is within the absolute calibration accuracy of the probe. It had been noticed previously that manipulation of the cable to the probe could produce variations of about ± 0.1 in the dielectric constant.
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Fig. 3. Effective dielectric constant in ethanol with a toroidal inclusion. The dots correspond to the values measured, and the curve corresponds to simulation.
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Figure 4
shows the effective in ethanol (r = 25) as a function of the thickness of the sheath made of thermo-retractable material (r = 3) around the central electrode. Simulation results showed the same fast decrease in the effective in relation to the thickness of the sheath as observed experimentally. Similar to the two-dimensional model of De Rosny et al. (2001), the three-dimensional model developed in this study provides a good representation of the influence of the heterogeneity on capacitance probe measurements.
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Fig. 4. Effective dielectric constant in ethanol with a thermoretractable sheath around the central electrode. The dots correspond to the values measured, and the curve corresponds to simulation.
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Necessity of a Three-Dimensional Model
In De Rosny et al. (2001) the axial symmetry of the probe was used to reduce the dimensionality of the problem, but the investigations were limited to a distribution of the soil that depended only on the distance to the axis of symmetry and the position along this axis. In the case of a nonaxially symmetrical inclusion in a homogeneous medium, or in different implementations of the probe according to the air–soil interface, the dimensionality of the problem cannot be reduced.
As three-dimensional simulations are much difficulty to implement than the two-dimensional ones (meshing operations, simulation time), we were interested to assess to what extent we could use a two-dimensional approach to estimate the influence of a nonaxisymmetric perturbation in the observed medium. Therefore, we assumed that such heterogeneities could be represented by a fraction 
of a torus. Furthermore, we supposed that the measurement variation 
r of such a torus fraction is equal to t where t is the measurement variation related to the whole torus. We could test these assumptions by comparing the results thus obtained with the three-dimensional simulation of a medium with a partial torus embedded near the electrode (the torus had a 20-mm outer diameter with a 9 mm2 square cross-section; the distance between the top surface of the torus and the central electrode tip being 15 mm). With the exception of the two ends ( = 0 and = 1), it is shown that significant differences occurred between the two approaches (Fig. 5)
. In a homogeneous medium ( = 0) or a medium comprising a toroidal inclusion ( = 1) the axial symmetry of the medium dielectric constant was preserved and the electric field lines have the same symmetry. For the other cases (
]0 –1[), the electric field lines no longer have axial symmetry: they are tighter in the zone where the torus fraction (1 – ) was removed (if the homogeneous medium has a permittivity superior to that of the inclusion). Such an asymmetry affects the electric charge distribution on the electrodes and thus the capacitance. In this case study, we have shown that the impact of heterogeneities addressed either by two-dimensional or three-dimensional simulations leads to relative differences of about 20%. Thus a three-dimensional approach is required to tackle general problems of perturbations on capacitance probe measurements.
Recommendations for Probe Installation when Located near the Soil Surface
Vertical and horizontal installations were investigated. The effective of the medium was recorded as a function of the depth of the sensor. The depth was defined from the central electrode tip in the case of a vertical installation and from the axis of symmetry for the horizontal installation (Fig. 6)
. It is shown that measurements were well simulated by our model with a maximum relative difference of 5% (Fig. 6). With the vertical installation, a depth of 5.2 cm was required when applying a severe criterion ( < 0.1). In such a configuration, the upper level of the visible part of the annular electrode (not covered by PVC) should be placed at least 1 cm below the surface. With a horizontal installation, measurements could be made from a depth of 4 cm if we allowed the same severe criteria on the dielectric constant as that used for the vertical installation. These measurements remained acceptable when moving the probe up to 2 cm. In both cases measurements could be made very close to the soil surface.
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Fig. 6. Effective dielectric constant as a function of dipping depth in ethanol in the two cases of vertical and horizontal installation of the capacitance probe. The dots correspond to the values measured, and the curve corresponds to simulation.
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Definition of the Probe Volume of Influence
The goal was to determine which part of the ground surrounding the probe played a significant role in the measurement of effective .
To define the volume of influence of the probe, we used a method similar to that used by Knight (Knight, 1992), which was based on the spatial distribution of energy. The density of electromagnetic energy is defined by (Feynman, 1979):
 | [5] |
with:
The total energy within the domain of calculation is given by:
 | [6] |
Assuming that the electrostatic energy contribution of the space falling outside of the domain of calculation is negligible, Wtot can also be expressed by:
 | [7] |
We can divide the domain of calculation into soil and probe where probe represents the probe (electrodes, PVC, air) and soil the media of measurement. Equation [6] can be rewritten by:
 | [8] |
In the domain of measurements, we can define a weighting function w(x,y,z), which relates the spatially distributed dielectric constant (x,y,z) to the effective dielectric constant (eff):
 | [9] |
From Eq. [5] and [8], we obtain:
 | [10] |
By using the simulated calibration relationship (Eq. [4]) that relates the probe capacitance and the effective dielectric constant of the medium, Wtot can also be written by:
 | [11] |
Equation [11] suggests that the probe capacitance is equivalent to two capacitances in parallel. We numerically demonstrated that the second term of Eq. [12] is approximately equal to Wprobe (approximately 1.7 x 10–12 J) integrated over the probe domain. When the dielectric constant of the measured media varied from 1 to 81, Wprobe relative variations were lower than 4%. This shows that Wprobe and Wsoil are decoupled and so, we can use the first term of Eq. [11] to compute the Wsoil:
 | [12] |
By combining Eq. [8], [9], and [12] and recognizing that s in Eq. [4] is equivalent to eff, we obtain the weighting function w(x,y,z) in the measured media:
 | [13] |
This weighting function can be calculated in any medium but will vary according to the spatial distribution of (x,y,z) that has an influence on the local amplitude of the electric field. The integral of Eq. [13] can be used to define the volume of influence l of the probe. We define this volume as the smallest one allowing the integral term to be equal to a given threshold (T).
 | [14] |
Practically, the smallest volume l can be obtained by sorting the elements according to their weighting function calculated in the center of gravity of each element. Starting with the element with the highest weighting function, we sum the numerical value of the product w(x,y,z) d associated to every element until we reach value T.
In Fig. 7
, we represent the probe volume of influence for different threshold values (T) in case of a homogeneous medium. In all cases the volume of influence is small. It is roughly equal to 120 cm3 with T = 0. 95. Above 0.9, the T value has a strong impact on the probe volume of influence. In Fig. 8
, it is shown that the probe volume of influence is not affected by the dielectric constant in the measured medium.
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Fig. 7. Domains in the XZ plane, corresponding respectively to the integral of the weighting function equal to 0.90, 0.95, and 0.99 in the case of a homogeneous medium (r = 1). (Fig. 7 is not drawn to scale to improve clarity of the drawing).
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Fig. 8. Domains in the XZ plane, corresponding respectively to the integral of the weighting function equal to 0.99 in the case of different homogeneous medium (r = 1, 25, 81).
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To assess the importance of heterogeneities on the volume of influence, we display in Fig. 9
the results with a probe installed vertically near the surface at different depth. When the annular electrode was partly inserted in the media (L = 32 mm), the shape of the volume of measurement was modified compared with the case in which the probe was more fully inserted. However the extent of the volume remained in the same order of magnitude. We also examined the case of the horizontal insertion of the probe in a homogeneous medium below the minimum depth to prevent the influence of the soil–air boundary (see Fig. 6). The volume of influence of the probe is affected by the installation with a slight asymmetry (result not shown).
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Fig. 9. Domains in the XZ plane corresponding to the integral of the weighting function equal to 0.95, in the case of a vertical insertion of the probe with two different values of dipping depth in the soil.
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The results displayed in Fig. 7 and 8, confirm that the shape of volume of influence depends on the spatial variability of soil dielectric constant. However, it is also shown the volume of influence remains roughly in the same order of magnitude. So, to normalize the volume of influence characterization, we suggest defining it as the one obtained with a homogeneous medium.
We now needed to check whether the volume resulting from the weighting function obtained with a homogeneous medium truly represented the probe's sphere of influence. In other words we had to check that any perturbation located outside the sphere of influence had only a small influence on the measurement. We evaluated the influence of a toroidal inclusion (90-mm o.d., square section 36 mm2, distance between the top surface of the tore and the central electrode tip 20 mm) on the probe. This inclusion was then positioned just close to the volume of influence corresponding to T = 0.99. We tested the cases of inclusions with r = 25 (or 3), in a homogeneous medium where r = 3 (or 25). The relative variation of the capacity is only equal to 0.1% in both cases. These variations are thus negligible when the distance from inclusion to the central electrode exceeds 39 mm. The results are in agreement with those presented in Fig. 7: the influence of the external medium is very weak immediately beyond the volume of measurement. In the case of a vertical insertion of the probe, the relative variation of the capacity obtained for –2.6 cm < L < 10 cm is weak (approximately 1%). In the case of a horizontal insertion, the relative variation when the air–soil boundary is beyond the volume of influence is much smaller (approximately 0.5%).
These results demonstrated that the proposed normalization procedure used to determine the volume of influence, regardless of the heterogeneity of the media gave a relevant indication of the probe's volume of influence.
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CONCLUSION
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This study shows that the response of a capacitance probe to spatial variations of the dielectric constant can be well represented by a numerical model to solve Laplace's equation. The results obtained with the axially symmetrical two-dimensional model of De Rosny et al. (2001) are extended in this study to a three-dimensional model. Adding the third dimension enlarges the domain of application of numerical modeling significantly. For instance, it is a very efficient tool in determining the minimum depth of insertion above which the probe is influenced by the air–soil boundary. We also demonstrated that the influence of nonaxially symmetrical heterogeneities on the response of the probe could only be addressed using a three-dimensional model. Finally this study provides a practical method to define the volume of influence of the probe. The three-dimensional model allows us to compute a weighting function representing the contribution of each elementary volume to the probe response. We define the volume of influence as the smallest volume in which the integral of the weighted function w(x,y,z) is equal to a threshold value T (typically 0.9, 0.95, or 0.99). Although the weighting function seems to be complicated and very sensitive to the structure of the medium in the vicinity of the electrodes, the volume of influence established for T = 0.95 seems to be reasonably stable, even when the medium measured is not homogeneous. In a number of cases we verified that the perturbations of the medium located outside the volume of influence have only a very limited effect on the measurements. In the cases considered, which correspond to severe heterogeneities, we found that the relative error on the dielectric constant never exceeded 1%.
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ACKNOWLEDGMENTS
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This work was funded by the French " Programme National de Recherche en Hydrologie".
Received for publication October 16, 2002.
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REFERENCES
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- Dhatt, G., and G. Touzot. 1981. Une présentation de la méthode des éléments finis. (In French.) Editions Maloine S.A., Paris.
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- Knight, J.H., P.A. Ferré, D.L. Rudolph, and R.G. Kachanoski. 1997. A numerical analysis of the effects of coatings and gaps upon relative dielectric permittivity measurement with time domain reflectometry. Water Resour. Res. 33:1455–1460.
- Straub, A. 1994. Boundary element modelling of a capacitive probe for in situ soil moisture characterization. IEEE Trans. Geosci. Remote Sens. 32:261–266.
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ABSTRACT
INTRODUCTION
) when the fraction