Effect of gaps around a TDR probe on water content
measurement:
Experimental veriﬁcation of analytical and numerical solutions
Toshihiro SAKAKI 1

Abstract: When installing TDR probes to rock, void
spaces (gaps) between the probe rods and rock may be encountered. In this study, the dielectric constants of waterand air were measured with various controlled gaps and effects of the gaps were experimentally investigated. We focused on a two-rod probe for which analytical and numerical solutions were available. Symmetric longitudinal gaps
(identical uniform gaps along both rods), non-symmetriclongitudinal gaps (uniform gaps along only one rod), andend gaps beyond the end of the rods were considered. For
the symmetric cases, measured dielectric constant valuesof water and air were compared with analytical and/or numerically simulated values, and a close agreement was observed. For the non-symmetric cases, such agreement wasseen only when the gap width was relatively small. As
the gap widths increased, the dielectric constant of water was underestimated compared to the numerically estimated values. Such underestimation was more signiﬁcant
when the gaps were along the rod connected to the center
conductor, suggesting that the sensitivities of the two rods
were uneven. The effect of the end gaps was very limited whereas the longitudinal gaps, e.g., due to loose guideholes or soil shrinkage can lead to signiﬁcant measurement
errors.
Key Words : water content, dielectric constant, time domain reﬂectometry, gap effects, measurement errors

1. Introduction
After the work by Topp et al. (1980), time domain reﬂectometry (TDR) has been widely used since the early 1980’s
for soil water content measurement in various problems insoil science, agriculture, environmental and geotechnicalapplications. The standard device consists of a pulse generator, a probe and a data logger. The method requires
the probe to be embedded into the material of interest. The
probe is usually made of two or three metal rods connected
to a coaxial cable. The transmission velocity of an electromagnetic step pulse along the probe is measured and converted into water content (e.g., Topp et al., 1980).
1 Environmental Science and Engineering Division, Colorado Schoolof Mines, Golden, Colorado, U.S.A. Corresponding author: Toshihiro
Sakaki2011 年 1 月 27 日受稿 2011 年 3 月 8 日受理
土壌の物理性 117 号, 3–9 (2011)

When applying TDR to rock, guide holes must be drilled
prior to the probe installation. Hokett et al. (1992) reported
that the probe must be in good contact with rock in orderto obtain accurate measurements of moisture content. Even
when slightly undersized guide holes are used, it is difﬁcult
to eliminate the longitudinal and end gaps around the probe
completely, because the guide holes are not drilled with aperfectly-matching diameter, nor are they exactly as longas the probe rods. Sakaki et al. (1998) showed a couple ofphotographs in which these gaps are clearly visible.
Annan (1977) considered the case of non-concentricgaps, which coincided with a bipolar coordinate system(Morse and Feshbach, 1953) centered at the probe locations, and derived an analytical expression for the apparent dielectric constant (Ka ). Knight (1992) analyticallyshowed that the highest measurement sensitivity occurs in
the vicinity of the probe rod surface. This implies that if
slight void spaces or gaps exist between the rod and the material, the measurement accuracy may be affected. This isespecially true if the gaps are ﬁlled with material that has
a smaller dielectric constant than that of the surrounding
medium. Ferré et al. (1996) and Knight et al. (1997) usednumerical simulations of the Laplace equation to estimate
the inﬂuence of gaps on apparent dielectric constant. They
showed that for small gaps, the analytical results of Annan(1977) were valid even for concentric gaps. Sakaki et al.
(1998) and Sakaki and Rajaram (2006) presented extensive
data on the application of TDR to water content (θ ) measurement on rocks. The data of Sakaki et al. (1998) wereobtained using probes hammered into slightly undersized
guide holes. The Ka -θ data could not be explained without invoking the potential role of air-ﬁlled gaps around theprobe rods, which was eliminated later by ﬁlling the gaps
with electrically conductive silicone (Sakaki and Rajaram,
2006). Despite the above mentioned works for identifying
the effect of gaps, to our best knowledge, the gap effects
have never been experimentally “measured” in a quantitative fashion under well-controlled conditions, and the analytical and numerical approaches for quantifying the gap
effects have never been veriﬁed against experimental data.
The objectives of this study were; (1) to experimentally

4

土壌の物理性demonstrate the inﬂuence of gaps on errors in estimatesof Ka under well-controlled conditions, (2) to verify the
analytical and numerical solutions for the gap effects and(3) to quantitatively illustrate the magnitude of potential
errors in moisture content estimates, resulting from gaps
around TDR probe rods in soils/rocks based on analytical/numerical calculations for some practically relevant situations. We considered gaps along the rod surfaces (hereafter referred to as the “longitudinal gaps”) and gaps beyond the end of the rods (hereafter referred to as the “endgaps”). Because the analytical and numerical solutions
were available for the longitudinal gaps along a two-rod
probe, we focused only on the two-rod probe. The experimental results were compared to theoretical estimates,
where available, obtained using the analytical solution as
well as numerical solution of the Laplace equation, withspeciﬁed potential on the rod surfaces, and incorporating
the inﬂuence of longitudinal gaps.

2. Material and methods
We considered uniform longitudinal gaps around therods and end gaps beyond the end of the rods to mimic thegaps that are expected in the ﬁeld applications as shown in
Fig. 1(a). When fully saturated, the gaps are ﬁlled withwater whereas the gaps are likely to be ﬁlled with air whenunsaturated. The typical pore size in the rock matrix is
usually smaller than the size of the gaps, causing a larger
suction in the matrix. Therefore, water tends to be retainedin the matrix due to its larger suction, which consequently
makes the gaps ﬁll with air.

Fig. 1 Two-rod TDR probe; (a) Longitudinal and end gaps
when installed into rocks, (b) Tested conﬁguration in the experiments.

第 117 号

(2011)

2.1 Probe conﬁguration
The TDR probe used in this study was the two-rod typeas shown in Fig. 1(b) and is a practical conﬁguration for
application to rocks. A couple of spade tongues were soldered to the split end of a 50 Ω coaxial cable (RG58/AU),
which were screwed to an acrylic plate having dimensionsof 60 × 100 × 10 mm. Brass rods with a diameter d =10 mm, center-to-center spacing s = 30 mm, and lengthl = 200 mm, were attached to the screws above. The cable
was, in turn, connected to a commercially available time
domain reﬂectometer TDR100 (Campbell Scientiﬁc, Inc.,
Logan, Utah). The reﬂectometer emits an electromagnetic
step-pulse with a voltage of 250 mV and a 10 ∼ 90 % risetime of 200 picoseconds (frequency spectrum roughly up
to 1.75 GHz, Tektronix, Inc.,1987), which is transmitted
along the coaxial cable and probe. The TDR traces were
processed by software PCTDR (Campbell Scientiﬁc, Inc.,
Logan, Utah) and apparent dielectric constant Ka was calculated.
2.2 Longitudinal gaps
2.2.1 Experimental setup for longitudinal gaps
In reality, the longitudinal gaps may vary in space as inFig. 1(a) but those cannot be quantiﬁed. Therefore, inthe context of quantifying the effects of longitudinal gaps
around a TDR probe, we considered equivalent uniform
widths. In order to mimic air-ﬁlled gaps with thin but
uniform thickness in water, polyvinyl chloride (PVC) tapewith a uniform thickness was used. The dielectric constantof PVC is substantially smaller than that of water (KPVC =2.9  81) (e.g., Converting Technology Institute, 1997)
thus, its effect is analogous to air-ﬁlled gaps in unsaturated
rock, which is considered to be the most likely situationin the ﬁeld. The PVC tape was carefully wound to therod surface to create uniform-thickness gaps. The PVC
gap thicknesses (δ ) considered were; 0.0 (no gap), 0.107,
0.160, 0.317, 0.475, 0.636, 0.792 and 0.950 mm. The PVC
tape was applied up to 6 layers so that the gap width (thickness of the entire PVC layer) was approximately 1 mm (10% of the rod diameter), which seemed to be large enoughfor the investigation of gap effects in a practical sense. The
values of gap width were obtained by measuring the diameter of the rod with and without the PVC tape using a
digital caliper, taking the difference of the two, and wereconsidered the “actual” thickness of the PVC tapes whenthe measurements were taken. For each gap width, uniform identical gaps along both rods (hereafter “symmetric cases”) and uniform gaps only along one rod (hereafter “non-symmetric cases”) were considered. The nonsymmetric cases were further divided into two categories;
uniform gap along the rod connected to the center conductor (hereafter “center rod” or CT), and uniform gap only

論文：Effect of gaps around a TDR probe on water content measurement

along the rod connected to the shield (hereafter “shieldrod” or SH).
2.2.2 Analytical solution for longitudinal gap effectUsing the bipolar coordinate system (Morse and Feshbach, 1953), Annan (1977) ﬁrst derived the analytical solution for the longitudinal gap effect for a two-wire probewith non-concentric gaps around the rods. Based on his
solution, the analytical solution for the apparent dielectric
constant Ka of water measured by a two-rod probe with
non-concentric circular gaps is;
Ka =Kgap ξ0
Kmaterial
Kmaterial (ξ0 − ξ1 ) + Kgap ξ1(1)

where, Kgap is the dielectric constant of the gap material,
Kmaterial is the true dielectric constant of the material ofinterest, ξ0 and ξ1 are the radial bipolar coordinates thatcan be calculated from rod radius, gap width, and the halfdistance between bipolar foci (see Annan, 1977 for details).
2.2.3 Numerical solution for longitudinal gap effect
Unfortunately, use of equation (1) is limited to symmetric cases where identical uniform gaps exist along bothrods. If the gaps exist only partially, or more generally,
if the medium surrounding the probe is heterogeneous, anumerical method must be employed.
Assuming that the TDR probe rods have an inﬁnite
length, the electrostatic potential ﬁeld in the plane perpendicular to the rods is described by the Laplace equation.
Knight et al. (1997) showed that the apparent relative dielectric constant Ka could be calculated by the following
equation.
 

∂φ
Kgap
ds
∂n
S

(2)
Ka = 1 
∂ φ0
ds
∂n
S1
where, Kgap is the dielectric constant of gap material, φ isthe electrostatic potential for non-homogeneous case (withgaps), φ0 is the electrostatic potential for homogeneouscase (without gaps), n is the outward pointing unit normal
vector on the boundary, S1 is the internal boundary for arod.
Knight et al. (1997) showed that concentric gaps completely surrounding both rods could be modeled reasonably
accurately using the analytical solution for non-concentric
gaps. Thus, in this study, “concentric” gaps were deﬁned
in the ﬁnite element mesh around the rods of a two-rod
probe. The dimensions of the domain were approximately
20 s in the x-direction and 12 s in the y-direction, wheres was the center-to-center rod separation that was 30 mm

5in this case. The diameter of both rods was d = 10 mm.
A no-ﬂux boundary was imposed on the outer boundary(S3 ) on the domain. The nodes on one rod (S1 ) were set
to a constant potential of −1, the potentials on the otherrod (S2 ) were set to 1 (see Knight et al., 1997 for details).
PVC-ﬁlled gaps were incorporated by assigning KPVC =2.9 to appropriate elements whereas Kwater = 81 was given
to all the other elements. The ﬁnite element mesh was constructed so that the thickness of the gaps could be varied
from 0.01 mm to 10 mm. To conﬁrm the accuracy of themodel, the symmetric cases were computed ﬁrst and theresults were compared to the analytical solution. Then, thenon-symmetric cases were simulated. It should be notedthat in the numerical simulation, both rods were identicaland the possible uneven contribution between the centerand shield conductors was not taken into account in themodel.
2.3 Experimental setup for end gaps
For the investigation of the end gap effect, a solid PVC
rod was used to establish gaps at the end of rods. A cylindrical piece of PVC rod with a diameter equal to the rod diameter of 10 mm sliced to have a pre-speciﬁed length was
directly glued to the end face of each rod. The thickness ofthe glue was kept as thin as possible so that its effect was
minimized. Since PVC has a dielectric constant of 2.9, it
should act like air when measured in water. The end gap
lengths considered were; 0, 1, 1.5, 2, 3, 4, 5, 7.5, 10, 15
and 20 mm.

3. Results and discussion
3.1 Effect of longitudinal gaps
Fig. 2 shows the experimental, analytical, and numericalresults of the dielectric constant of water with the longitudinal PVC gaps. For each gap thickness, ﬁve measurements were taken and averaged. In the analytical and numerical calculations, Kwater = 81 was used (von Hippel,
1995). For the symmetric cases, the experimental results(large solid circles) agreed closely with the analytical andnumerical results. As can be intuitively inferred, the larger
the gap widths δ , the larger their effects. In this case, since
Kgap < Kwater , the dielectric constant of water was more
underestimated as δ increased. For comparison, the effects
of “air”-ﬁlled gaps were also calculated using equation (1)and shown in Fig. 2 by a gray solid line. It caused more
underestimation than PVC-ﬁlled gaps. This was expected
because the dielectric constant of air was less than that of
PVC. For the typical gap width range of 0.1 ∼ 0.3 mm(Sakaki et al., 1998), PVC-ﬁlled and air-ﬁlled gaps caused
up to 46 % and 71 % underestimation in the Ka values, respectively, when measured in water with the probe conﬁguration tested. In this study, a relatively large rod diameter
was used so that we were able to investigate gap effects for6

土壌の物理性第 117 号

(2011)

calculated by equation (1) are denoted by a solid line.
Although the measured dielectric constant values wereslightly higher than the analytical values, their behavior
was well explained by equation (1). For the typical gap
width of 0.1 ∼ 0.3 mm as Sakaki et al. (1998) observed,
the overestimation of the dielectric constant in air due tothe PVC-ﬁlled gaps was up to 2 % and not as signiﬁcant as
the 46 % underestimation observed in water.
3.2 Effect of end gaps

Fig. 2 Effect of PVC- ﬁlled gaps in water (d = 10 mm, s = 30mm). The thick solid line denotes the analytical dielectric constant calculated using equation (1) when gaps are ﬁlled with PVC(KPVC = 2.9). The dashed lines are the results from numericalsimulation and equation (2) for the symmetric and non-symmetriccases. Data for the no gap are plotted at δ = 0.001 mm. Note thatthe uneven contribution between the CT and SH rods was nottaken into account in the numerical model.

Fig. 3 Effect of PVC-ﬁlled gap in air (d = 10 mm, s = 30 mm).
The solid line denotes the analytical dielectric constant calculatedusing equation (1) when gaps are ﬁlled with PVC (KPVC = 2.9).
The result for the no gap case is plotted at gap = 0.001 mm.

small δ /d ratios. However, it must be noted that equation(1) implies that for a probe with a smaller rod diameter, theunderestimation will be even more signiﬁcant.

The results for the end gap effects are plotted in Fig. 4.
As there are no analytical and numerical solutions available for this case, only experimental results are presented.
For each end gap length, ten measurements were taken.
The effects were relatively small compared to those for thelongitudinal gaps presented in the previous section. The
obtained Ka values were normalized by 81 so that the end
gap effects were seen in terms of relative errors. Baker andLascano (1989) reported that sensitivity ends abruptly atthe end of the probe, i.e., changes in water content just beyond the end of the rods have no effect on the signal. Ourdata showed that the presence of void space ﬁlled with aircaused a slight underestimation of the apparent dielectric
constant and that the effect was almost constant regardless the size of the void space. For an inﬁnite transmission line, the electromagnetic pulse transmits in the simplest mode referred to as TEM (transverse electromagneticwave), where the electric ﬁled, magnetic ﬁeld, and direction of propagation are perpendicular to each other (Kraus,1991). Near the end of rods, the wave is transmitted inhigher modes, which has possibly been affected by thepresence of air and caused the slight underestimation of
the dielectric constant. This is probably limited to a verysmall region at the end of the rods and thus the effect is as
much as ∼ 2 % regardless the size of the void space. Although the results are probably unique for the probe dimensions shown here, it is recommended that the void length
be minimized as much as possible.

For the non-symmetric cases, the Ka values were closer
to 81 than those obtained for the symmetric cases. This observation was expected because only one rod had the gaps,
thus, the effect was less. Two SH cases with δ = 0.107
and 0.160 mm on the shield rod, where δ was small andwithin the typical range, agreed well with the numericalresults. In other SH cases with δ = 0.317 ∼ 0.950 mm on
the shield rod, the experimental results tend to show lower
Ka values than numerical results. All CT cases with gaps
on the center rod showed smaller Ka values than those in
the SH cases.
Fig. 3 shows the results for PVC-ﬁlled gaps measured
in air (gaps on both rods only). The analytical values

Fig. 4 Normalized dielectric constant of water measured withvarious end gap lengths.

論文：Effect of gaps around a TDR probe on water content measurement

7

Fig. 5 Calculated water content and relative error with water- and air-ﬁlled gaps in a hypothetical rock with true water content of 0.1(squares), 0.25 (triangles), and 0.4 (circles). Probe dimensions; d = 10 mm, s = 30 mm.

3.3 Gap effects on water content in soils androcks
In the previous sections, the gap effects were investigated using limited materials such as water, air, and PVC.
The combination of these materials resulted in somewhat
different contrast of dielectric constants than that of ourinterest, i.e., rock and air-ﬁlled gaps, rock and water-ﬁlled
gaps and so on. Similar relationships for apparent dielectric constant Ka vs. gap width δ were calculated using
equation (1) for both air- and water-ﬁlled gaps in a hypothetical rock material. The dimensions of the two-wire
TDR probe were assumed to be d = 5 mm, s = 30 mm
which are typical for commercially available probes (d =2.4 ∼ 4.8 mm, s = 14 ∼ 40 mm, TRIME system, IMKO,
Inc., Ettlingen, Germany; d = 3.2 ∼ 4.8 mm, s = 30 ∼45 mm, Campbell scientiﬁc, Inc., Logan, Utah; d = 3 mm,
s = 12.5 ∼ 37.5 mm, TRASE system, Soilmoisture Equipment Co., Santa Barbara, California). For demonstration
purposes, it was assumed that Topp’s equation (Topp et al.,1980) to relate Ka and θ was valid for the hypothetical rockmaterial.

showed good agreement with the analytical and numericalsolutions. For the non-symmetric cases, where the longitudinal gaps exist along one rod only, the results were
not as straightforward. The cases with gaps on the centerrod showed more effect than those with gaps on the shieldrod. When an electromagnetic step pulse propagates alonga coaxial transmission line, the center conductor carries
the signal and the shield conductor is ground; i.e., “unbalanced” signal. In a transmission line consisting of twoparallel conductors, both conductors carry a signal, equal
in magnitude but opposite in sign with the ground potential in the center plane; this is referred to as “balanced”
with respect to ground. For a two-rod TDR probe, the electrical ﬁeld will gradually change from unbalanced to balanced (without using a balanced-unbalanced transformer,
e.g., Spaans and Baker, 1993), which often leads to higher
sensitivity on the rod connected to the center conductor.
The uneven contribution of each rod leads to a difference inthe trace data depending on whether the gaps are on the CTor the SH rod as shown in Fig. 6. If the balanced electricalﬁeld cannot be obtained within the length of the probe, theThe apparent dielectric constant Ka of the rock (including air and water phases) was assumed to be 5.9, 13.4, and25 that correspond to water content of 0.1, 0.25 and 0.4
using Topp’s equation. For the probe conﬁguration mentioned above, the apparent dielectric constant was computed using equation (1), and converted into water content.
The calculated water content as well as relative errors areplotted in Fig. 5.

4. Discussion and conclusions
The effects of longitudinal and end gaps were experimentally investigated and, where available, the results
were compared to the analytical and numerical solutions.
Although the end gaps showed very little effects, the longitudinal gaps lead to signiﬁcant errors. For the symmetric longitudinal gaps, where uniform identical longitudinal
gaps were on both rods, the experimentally measured data

Fig. 6 Example traces with δ = 0.636 mm obtained in water.
Longitudinal gaps are on CT rod only, SH rod only, and both rods.
Trace with no gap is also shown for comparison.

8

土壌の物理性CT rod will carry more signal than the SH rod. Therefore,
when measured in water, the gap effect is expected to be
more distinct when PVC-ﬁlled gaps are present along theCT rod, which resulted in more underestimation in Ka in
this case. The trace with gaps on the CT rod shows two
reﬂections; one from the CT rod, and another from the SH
rod without gaps that indeed corresponds to the reﬂection
point of the no gap case. For the trace with gaps on the
SH rod, the second reﬂection disappeared. Overall, theexperimental data revealed that the analytical and numerical solutions were valid to quantify the effect of symmetric longitudinal gaps. Because of the uneven contributionof the rods, the numerical solution tends to somewhat underestimate the effects when the longitudinal gaps are notsymmetric.
When applying TDR to unsaturated rocks, it is likelythat the gaps are ﬁlled with air rather than with water due
to larger suction in the surrounding matrix. Thus, the results imply that much attention should be paid to minimize
gaps around the probe rods when applying TDR to rocks,
as air-ﬁlled gaps are the potential cause for signiﬁcant underestimation of Ka values of the rocks, thus, water content. An electrically conductive material may be used toﬁll such gaps to minimize their effects on the measurement
accuracy (Sakaki and Rajaram, 2006). Moreover, gaps canarise not only for rocks but also for soils if sufﬁcient care
is not taken when packing soil around the probes (Decagon
Devices, 2010), if the sensor is dislocated inside the soil,or when soil shrinkage occurs. Therefore, maximum attention has to be paid to ensure good contact between theprobe and the material of interest when installing probes insoils or rocks.
We have experimentally shown that the center rod ismore sensitive than the shield rod. Limsuwat et al. (2009)
showed similar non-symmetric characteristics in the commercially available capacitance sensors. Even when there
is no gap, the non-symmetric characteristics can lead tosome implications. For example, if a probe is aligned suchthat the rods experience a large water content difference
between the two, the measurement may be biased. Whenmeasuring a wetting front migration and one of the rods
is in the wet region and the other is in the dry region, theresulting Ka is likely to be affected more when the centerrod is in the dry region.

Acknowledgements
This research was partially supported by Japan Nuclear
Cycle Development Institute (JNC Cooperative Research
Scheme on the Nuclear Fuel Cycle).

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論文：Effect of gaps around a TDR probe on water content measurement

要旨TDR プローブを岩石に設置する際にはプローブと岩石試料との間に隙間が発生することがある．本研
究では，既知の隙間を持つプローブを用いて水と空気の比誘電率を計測し隙間の影響を評価した．隙間
の影響に関する理論解および数値解が存在する 2 ロッド型プローブについて，両ロッドに対称に隙間がある場合，片方のロッドのみに隙間がある場合，そしてロッド先端以深に隙間がある場合について検討した．対称な隙間のケースでは，水と空気の比誘電率の計測値は理論解および数値解とよく一致した．
非対称のケースでは隙間幅が比較的小さい場合は数値解と一致したが，隙間幅が大きくなるとともに実験結果の方が数値解より大きな影響を示した ．実験結果および数値解の差異は隙間が同軸ケーブルの芯線に接続したロッドにある場合により顕著に見られ，ロッド間で感度が異なることがわかった．ロッド
先端部以深の隙間の影響は小さかったが，ガイド孔が緩い場合やプローブ周辺の土壌で乾燥収縮が発生するような場合には含水率が過小評価される可能性があることが示された．
キーワード：含水率, 比誘電率, 時間領域反射率法, 隙間の影響, 計測誤差9