Cell Size for Water Content-Dielectric Constant Calibrations for Time

Suchatvee Suwansawat 1 and Craig H. Benson 2

Cell Size for Water Content-Dielectric Constant Calibrations for Time Domain Reflectometry

REFERENCE: Suwansawat, S. and Benson, C. H., ‘‘Cell Size for Water Content-Dielectric Constant Calibrations for Time Domain Reflectometry,’’ Geotechnical Testing Journal, GTJODJ, Vol. 22, March 1999, pp. 3–12.

Although the results presented by Topp et al. (1980) suggest that the relationship between Ka and u is practically unique (i.e., it is independent of soil type or fabric), more recent studies show that composition of the solids and surface electrical conductivity can significantly affect the Ka-u calibration (Herkelrath et al. 1991; White et al. 1994; Suwansawat 1997; van Loon et al. 1997; Benson and Bosscher 1999). Thus, when using volumetric water contents measured with TDR in a quantitative manner, a soil-specific Kau calibration is recommended (Timlin and Pachepky 1996; Benson and Bosscher 1999). The objective of this study was to size a calibration cell that can be used efficiently for determining the Ka-u relationship. Three primary criteria were considered when sizing the calibration cell: (1) the cell should be small enough so that large volumes of soil do not have to be processed; (2) the cell must be large enough so that its boundaries do not affect the calibration; and (3) the calibration must be applicable to longer probes (e.g., 250 to 300 mm long) that are used in the field. The cell was also required to be inexpensive and easy to construct. Another factor that needed to be considered before sizing the cell was selection of the probe configuration. This paper describes the probe configurations that were considered and the tests conducted to determine the size of the calibration cell.

ABSTRACT: Time domain reflectometry (TDR) is a nondestructive electromagnetic technique used to measure the volumetric water content of soil. A key component of the method is the calibration equation relating the apparent dielectric constant (Ka ) to the volumetric water content (u). In this study, tests were conducted to evaluate dimensional requirements for a TDR calibration cell. The results show that a PVC cylinder having the same dimensions as a standard compaction mold (diameter 4 102 mm, height 4 116 mm) is a suitable calibration cell for two-rod TDR probes having diameter 4 4 mm, center-to-center spacing 4 30 mm, and length 4 80 mm. The cell can also be used for three-rod probes having the same dimensions as the two-rod probe, and a center-to-center rod spacing of 20 mm. Calibrations made with this small cell are essentially identical to calibrations made in a much larger cell where boundaries are unlikely to be important. KEYWORDS: reflectometry, cell size, calibrations

The relationship between volumetric water content (u) and dielectric constant of soils was initially investigated by Thomas (1966), Cilhar and Ulaby (1974), Selig and Mansukhani (1975), and Davis and Annan (1977). Subsequently, Topp et al. (1980) showed that the apparent dielectric constant (Ka ) of soil can be measured readily using time domain reflectometry (TDR), a technique that has been used to measure the dielectric constant of liquids and to interrogate faulty transmission lines (Fellner-Feldegg 1969; Andrews 1994). More importantly, Topp et al. (1980) show that TDR can be used to determine u via a Ka-u calibration equation. Since this seminal work, TDR has become used widely as a nondestructive technique to make repetitive measurements of u, particularly in situ. In such applications, TDR probes are buried at various depths to determine spatial and temporal variations in water content nondestructively. Key advantages of the TDR technique are that the analysis is fairly simple and quick, the Ka-u relationship for a given soil is nonhysteretic and insensitive to drift, and the equipment is commercially available for reasonable cost. Moreover, the method can be automated readily for remote application in the field (Baker and Allmaras 1990; Heimovaara and Bouten 1990; Herkelrath et al. 1991; Benson et al. 1994; Jong et al. 1998).

Background Measuring Dielectric Constant with TDR Topp et al. (1980) and Dalton and van Genuchten (1986) provide thorough discussions of the theory and method of using TDR to measure Ka of soil and ultimately u. The theory and method are summarized briefly here. The basis of the TDR method is the relationship between the velocity of an electromagnetic pulse (Vp ) displaced along a transmission line and the dielectric constant of the material between the conductors. This relationship is defined by Topp et al. (1980): Vp 4

Ï

1`

(1)

Ï1 ` tan2 d

Kr 2 where c is the speed of an electromagnetic wave in free space (3 2 108 m/s), Kr is the real part of the complex dielectric constant, and tand is the ‘‘loss tangent,’’ which is written as:

1 Graduate student, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, formerly at Department of Civil and Environmental Engineering, University of Wisconsin, Madison, WI 53706. 2 Associate professor, Department of Civil and Environmental Engineering, University of Wisconsin, Madison, WI 53706.

q 1999 by the American Society for Testing and Materials

c

Kim ` tand 4

Kr

1ves 2 e

(2)

In Eq 2, se is the direct-current (dc) electrical conductivity, Kim is 3

4

GEOTECHNICAL TESTING JOURNAL

the imaginary part of the complex dielectric constant, v is the frequency, and e is the permittivity of free space. When tand ! 1, Vp >

c

Ï Kr

(3)

For most soils, Kim is small relative to Kr at the high frequencies used in TDR (Davis and Annan 1977). Thus, Eq 3 is a good approximation (i.e., tand ! 1) except when the conductors are buried in electrically conductive media such as dense compacted clays, conductive minerals, or highly saline soil (White et al. 1994; Robinson et al. 1995; van Loon et al. 1997; Suwansawat 1997). Nevertheless, when Eq 3 is used, some electrical losses are ignored, and thus Kr is generally replaced with Ka and referred to as the ‘‘apparent’’ dielectric constant of the medium. When TDR is used to measure u, an uninsulated set of conductors referred to as a ‘‘waveguide’’ or ‘‘probe’’ is buried in the soil. Soil between the conductors is the dielectric material, and its apparent dielectric constant controls Vp . A ‘‘reflectometer’’ is used to measure the time between fixed locations (i.e., the beginning and end of the probe), from which Vp and then Ka can be determined. A reflectometer is a combination of an electrical pulse generator, a timer, a sampler, and an oscilloscope (Nissen and Moldrup 1995). The most commonly used reflectometer is the Tektronix 1502B3, but other reflectometers are commercially available and are becoming used more frequently (Benson and Bosscher 1999). The Tektronix 1502B produces a square wave pulse with a 0.225 V amplitude in a rise time , 200 ps. The pulses are 10-ms long and are emitted at 60-ms intervals (Nissen and Moldrup 1995). TDR can be used to determine u because the Ka of soil is a function of the volumetric fractions of the solid, liquid, and gas phases, each having different dielectric constant: 80 (water), solids (3–10), air (1) (Selig and Mansukhani 1975; Topp et al. 1980; Look and Reeves 1990). Because of its relative magnitude, the water component tends to dominate Ka (i.e., as u increases, Ka increases significantly, and vice versa). In addition, because higher u results in higher Ka , an electromagnetic pulse travels slower through soils with higher water content (see Eq 3). A reflectometer can be set to display a graph of voltage versus distance along the conductors (referred to as a ‘‘waveform’’) on its oscilloscope (Fig. 1). The shape of the waveform changes when the pulses displaced along the conductors encounter changes in impedance. Thus, distinct points along the conductor can be identified where the impedance changes (Dowding et al. 1996). A cable technician uses these points in the waveform to locate cable defects. In soil, the waveform contains a distinct bump (upward voltage spike) followed by a large drop where the probe begins and the conductors enter the soil (Topp et al. 1980; Zegelin et al. 1989; Timlin and Pachepky 1996). A large monotonic increase in voltage begins at the end of the probe (Fig. 1). These points on the waveform are used to measure the ‘‘apparent length’’ of the probe (La ), i.e., the probe length reported by the reflectometer. A reflectometer actually measures time rather than distance. Thus, the distance reported on the oscilloscope depends on the setting that defines Vp for the reflectometer. For measuring u, Vp is set on a fixed value for all measurements, which is usually c for consistency. When Vp is set at c, the apparent length of the probe determined from the waveform corresponds to Ka 4 1. Consequently, La . Lp unless the probe is suspended in air. The true length of the probe (Lp ) is known from physical measurements. 3

Mention of trade names and manufacturers is for information only and does not imply endorsement by the authors.

FIG. 1—TDR waveform from a two-rod probe showing La and Lp .

Thus, from Eq 3 and distance-time relationships, Ka can be determined from La displayed on the waveform: Ka 4

1 2 La Lp

2

(4)

TDR Probes TDR probes usually consist of two or more slender metal rods (Fig. 2), or two slender metal straps bound to a dielectric core material (Hook et al. 1992). The latter design is referred to as a blade probe. Probes employing rods are more common, and threerod probes are the most common. The three-rod probe is common because it more closely approximates a coaxial condition and eliminates the need for a balancing transformer called a ‘‘balun’’ (Fig. 2) (Spaans and Baker 1993). From a practical perspective, however, both two- and three-rod probes are acceptable for measuring u. The rods are typically 70 to 300 mm long and spaced 10 to 50 mm apart. Usually the rods are made from stainless steel to prevent corrosion. Water Content Calibrations The calibration relating Ka and u is typically obtained by preparing specimens of soil at various u and then measuring Ka of the soil using a probe and reflectometer. Topp et al. (1980) developed such a calibration using a coaxial soil container 50 mm in diameter and 0.33 m to 1.0 m long. Four ‘‘mineral’’ soils were tested that had a range of textures and organic matter contents. Topp et al. (1980) assumed that these materials represented the chemical and physical extremes of pore sizes and surface properties expected in practice. Eighteen different experiments were conducted to assess the effects of texture, dry density, temperature, salinity, and hysteresis on the Ka-u relationship. In all cases, the Ka-u relationship was similar. From these results, Topp et al. (1980) fit a thirddegree polynomial to the Ka-u data that is now known as ‘‘Topp’s Universal Equation’’: u 4 0.053 ` 0.0292Ka 1 0.00055K 2a ` 4.3 2 1026 K 3a (5)

SUWANSAWAT AND BENSON ON TIME DOMAIN REFLECTOMETRY

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FIG. 2—Two-rod probe: (a) two-rod probe with balun (b) and three-rod probe (c).

A similar Ka-u relationship is presented by Ledieu et al. (1986). Equation 5 is used widely to determine u from TDR measurements of Ka . Although Eq 5 works well for soils primarily composed of quartz, a unique relationship between u and Ka does not exist for all soils (Roth et al. 1992; Dasberg and Hopmans 1992; Dirksen and Dasberg 1993; Malicki et al. 1994; Jacobsen and Schjonning 1995; Suwansawat 1997). For example, Benson and Bosscher (1999) report on a case history where a generic calibration like Eq 5 resulted in an overestimate in u of 0.05, on average, for a silty sand containing a significant fraction calcite, illite, and smectite. Thus, for soils not primarily composed of quartz, soil-specific calibrations should be considered. Sampling Volume Soil-specific calibrations are normally conducted by measuring Ka of specimens prepared in the laboratory in a calibration cell. The

cell must be large enough so that its boundaries do not influence the measurement of Ka , yet small enough for calibration to be efficient. That is, the cell should be just large enough to contain the zone of influence of the probe, i.e., the region surrounding the probe that contains most of the energy in the electric field. Several authors have made recommendations regarding the size of the influence zone. Most are based on theoretical analyses, or measurements made with materials other than soil. Topp and Davis (1985) suggest that the influence zone is approximately a cylinder with diameter 1.4 times the rod spacing. De Clerck (1985) suggests that 94% of the energy in the electric field is contained within a cylinder with diameter equal to twice the rod spacing. Knight et al. (1994) suggest that the signal is practically unaffected by material a distance approximately one-rod spacing from the center of the probe because 95% of the field energy is contained within a distance of one rod spacing. A theoretical analysis by Suwansawat (1997) using theory presented by Zegelin et al. (1994) and Seshadri

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(1971) suggests that the zone of influence for most common tworod configurations (rod spacing #60 mm) is less than 100 mm. In this study, the minimum cell size was determined by conducting tests at various water contents with rods of different length and spacing, various distances from the rods to the edges of the cell, and cells of different size. Waveforms were collected in each case, and Ka was determined. The waveforms and Ka data were then analyzed to determine the minimum cell size that would not affect the Ka-u calibration. Materials and Methods Soil A broadly-graded quartz sand composed of 2% gravel, 96% sand, and 2% fines (particles finer than 0.075 mm) was used for analysis. Quartz sand was used so a comparison could be made with Topp’s equation (Eq 5). A particle-size distribution curve for the sand is shown in Fig. 3. The sand is classified as SP by ASTM Classification of Soils for Engineering Purposes (Unified Soil Classification System) (D 2487). The specific gravity of the sand is 2.65, and its dry unit weight ranges between 16 to 17.5 kN/m3 (dry or bulk density 4 1.63 to 1.78 Mg/m3 ). Sample Preparation A PVC cylinder with a diameter of 300 mm was used for containing specimens of the sand. A 20-mm-thick PVC plate was fastened to the bottom of the cylinder. Sand was tamped into the PVC cylinder in layers 50-mm thick to a void ratio (e) of 0.54 until a specimen 500 mm high was obtained. The cylinder is shown in Fig. 4. Water contents ranging from air-dry to near saturation were used. The desired water content was obtained by gradually spraying a predetermined amount of water on the sand while the sand was continuously mixed. The moistened sand was then tamped into the cylinder and allowed to equilibrate approximately 24 h before testing with TDR. The PVC cylinder was sealed during equilibration to prevent evaporation.

FIG. 4—Schematic diagram of two-rod probe in PVC cylinder with s 4 rod spacing, l 4 distance from base, and e 4 distance from wall.

tific, Inc. (CSI) Model No. PB-30-8-L] is inserted between a parallel line connected to the probe and the coaxial line connected to the reflectometer. A balun is not necessary when using three-rod probes. Schematics of the probes are shown in Fig. 2. Each probe consisted of stainless-steel rods 4 mm in diameter. One end of each rod was ground to a sharp point to facilitate insertion into the soil. The other end was threaded so that a nut and washer could be used to attach the cable. A thin (10-mm) acrylic handle was used to maintain a fixed separation distance between the rods. Length of the rods was varied from 60 to 300 mm. The rod spacing was 30 mm for the three-rod probe, except in a final set of tests conducted in a calibration cell where a rod spacing of 20 mm was used. Rod spacing for the two-rod probes was varied from 10 to 50 mm. Data Collection

TDR Probes Two- and three-rod probes were used, and the two-rod probes were used with and without a balun. The balun [Campbell Scien-

FIG. 3—Particle-size distribution curve for sand used in study.

A TDR probe was inserted into the center of the cylinder after moisture equilibration. Waveforms were then recorded by a Tektronix 1502B reflectometer and collected from the reflectometer digitally using a CSI SDM 1502 TDR Communications Interface. The SDM 1502 was connected to a CSI CR10 datalogger, which was connected to a personal computer via an optically isolated interface (CSI model SC32A). Each waveform was digitized into 250 discrete array elements of voltage and time. The digital waveforms were then graphed (e.g., Fig. 1), and La was determined manually using the tangent method (Topp et al. 1980; Timlin and Pachepky 1996). In the tangent method, the ends of the probe are defined by the intersection of the two tangents to the waveform at each end point of the probe. Three waveforms were collected for each test, and essentially the same La was determined in each case (Suwansawat 1997). After testing, the water content of the sand was measured using a volumeter. A volumeter is a precision cylindrical sampler that can sample soil volumes up to 30 000 mm3 (Suwansawat 1997). Three samples were collected per test at various locations in the cylinder to determine the actual volumetric water content. Water contents of the replicate samples obtained using the volumeter were nearly identical.


7

Probe Configuration Analysis The first step in the testing program was to determine the type of probe to be used, i.e., the number of rods (two or three), the spacing of the rods, and the length of the rods. Each of these factors potentially has an effect on the size of the calibration cell. Tests were conducted at water contents corresponding to dry, moist, and wet (nearly saturated) conditions using a two-rod probe, a two-rod probe with a balun, and a three-rod probe. Waveforms obtained with each probe were then compared for clarity. Subsequently tests were conducted with probes having different rod spacings and lengths. Number of Rods

FIG. 7—Waveforms for a two-rod probe, two-rod probe with a balun, and a three-rod probe for u 4 0.30.

Waveforms—Waveforms for different probe configurations in dry sand are shown in Fig. 5. The waveform obtained using the two-rod probe with a balun is the clearest. Distinct inflection points exist in the waveform at the beginning and end of the probe. Noise is evident in the waveform obtained with the two-rod probe without a balun, which can cause errors in determining La and Ka . The TDR trace from the three-rod probe has little noise, but the beginning of the probe is highly rounded and not as clearly defined as that from the two-rod probe with a balun. Waveforms for the moist condition are shown in Fig. 6. For intermediate water contents, the waveform obtained by the tworod probe without a balun still has noise. For the three-rod probe, the waveform is clearer than in the dry condition, and the beginning of the probe is distinct. The beginning of the probe is clearest in

FIG. 8—Relationships between Ka and u for sand using two-rod probe, two-rod probe with a balun, and three-rod probe.

the waveform from the two-rod probe with a balun due to the steep rise and sudden decline in the signal. Waveforms for the wet condition are shown in Fig. 7. In this condition, all three probes give similar waveforms. The beginning of the probe is clear, and noise is minimal. Thus, the apparent length (La ) can be identified precisely with any of the probes when the soil is wet. FIG. 5—TDR waveforms for a two-rod probe, two-rod probe with a balun, and a three-rod probe for u 4 0.11.

FIG. 6—TDR waveforms for a two-rod probe, two-rod probe with a balun, and a three-rod probe for u 4 0.19.

Water Contents—The Ka-u relationships obtained with the three probe configurations are shown in Fig. 8. Apparent dielectric constants obtained from the two-rod probe and two-rod probe with a balun are similar, which is consistent with the findings of Stein and Kane (1983). Apparent dielectric constants from the two-rod probes are slightly lower than those obtained from the three-rod probe and fall below the curve corresponding to Topp’s universal equation. The slightly lower Ka s from the two-rod probes are expected. Zegelin et al. (1989) show that the electric field surrounding a two-rod probe deviates from that in the true coaxial cell used by Topp et al. (1980), whereas the three-rod probe simulates Topp’s coaxial condition fairly well. From a practical perspective, the data shown in Fig. 8 suggest any of the three probe configurations can be used. The three-rod probe yields Ka closest to the coaxial condition (i.e., as measured by Topp et al. 1980), but all three probes yield a distinct Ka-u relationship that is needed for measuring water content with TDR. However, because the two-rod probe with a balun provided the

8


clearest waveforms for all water contents and a distinct Ka-u relationship, it was selected for use in the remainder of the study. There are also some disadvantages to the two-rod probe without a balun and the three-rod probe. When a balun is not used there is greater risk of encountering stray voltages and currents (Topp and Davis 1985). Such conditions cause noise in the waveform, particularly at low water contents (Fig. 5). Noise can be particularly problematic when automated algorithms for waveform analysis are employed. The waveform from the three-rod probe is more difficult to interpret in dry soils (Fig. 5) because the initial portion of the waveform is very rounded, rather than peaked. Thus, errors in the apparent length (La ) are more likely in dry soils when a three-rod probe is used. Rod Spacing As mentioned earlier, the Ka-u calibration obtained with small probes in a small cell must apply to longer and more widely spaced probes used in the field. Thus, tests were conducted with probes having different rod spacing to determine the minimum spacing that can be used before the Ka-u relationship is affected. Tests were conducted for five different rod spacings (10 to 50 mm) in sand prepared at u 4 0.06 (dry), 0.13 (moist), and 0.30 (wet). Waveforms from the tests at u 4 0.30 are shown in Fig. 9. Similar waveforms were obtained at the other water contents. The voltage drop is greater when the rod spacing is smaller, but a clear waveform is obtained in each case. Similar results have been reported by Zegelin et al. (1989). Apparent dielectric constant versus rod spacing and water content are shown in Fig. 10. In either case, Ka is only slightly affected by rod spacing. At the nearly saturated condition (u 4 0.30), the Ka s for all probe spacings are identical (i.e., Ka ∼ 15.5). For the moist and dry condition Ka changes slightly with rod spacing, with the 10-mm spacing giving the lowest dielectric constant. For spacings between 20 and 50 mm, however, essentially the same dielectric constant is obtained.

FIG. 10—Apparent dielectric constant versus rod spacing (a) and volumetric water content (b).

Rod Length Theoretically rod length should have no effect on the water content calibration if the dielectric constant of the material is uniform and no loss occurs. However, electrical dispersion causes rounding of the waveform (Zegelin et al. 1989) and can affect the measured

FIG. 11—Waveforms for a two-rod probe having different rod lengths (Lp ).

FIG. 9—Effect of rod spacing (s) on TDR waveforms for u 4 0.30.

apparent length (La ). Consequently, there is a minimum length below which the Ka-u calibration will be affected. In addition, attenuation can prevent very long rods from being used (Dalton and van Genuchten 1986). Two-rod probes with a balun were used to assess rod length. Rod lengths of 290, 140, and 60 mm were used and tests were conducted at u 4 0.06, 0.19, and 0.30. Rods 250 to 300 mm long are commonly used in field applications. Waveforms obtained from the probes with different length rods are shown in Fig. 11 for u


4 0.30 (nearly saturated). Similar results were obtained for the other water contents (Suwansawat 1997). The beginning of the probe is unaffected by rod length, since a constant length of transmission line was used to connect the probe and reflectometer. The width of the waveform within the probe (i.e., La ) increases as the rod length increases because the conductors are longer. In addition, the amplitude at the lowest point in the waveform decreases with increasing rod length as a result of increasing cumulative attenuation. That is, longer rods have greater conduction losses, as shown by Dalton and van Genuchten (1986). The relationships between dielectric constant and rod length and water content are shown in Fig. 12. Practically the same dielectric constant is obtained for each volumetric water content (Fig. 12a). In addition, the calibration curves (Fig. 12b) obtained for each length are nearly identical. Although the same Ka-u relationship was obtained for all Lp used here, deviations would have occurred for probes shorter than 60 mm. In Fig. 11, the end of the 60-mm probe falls just after the end of the dispersion region. Thus, 60 mm is probably the shortest length probe that can be used without affecting the Ka-u relationship. This finding is similar to that reported by Heimovaara (1993). He suggests that 50-mm-long rods are too short to measure Ka accurately. Cell Boundary Effects The objective of this portion of the study was to determine how close the probe could be placed to the cell wall or base before the

9

FIG. 13—Waveforms for a two-rod probe having different wall-to-rod ( e ) spacings for u 4 0.30.

waveform or Ka-u relationship is affected. Tests were conducted with wall-to-probe spacings (e) of 0, 10, 20, and 40 mm and spacings from the base to the end of the probe (l) of 10 to 80 mm in 10-mm increments (Fig. 4). Wall-to-Probe Spacing Tests to evaluate the wall-to-probe spacing (e) were conducted at u 4 0.02, 0.19, and 0.30. Waveforms for u 4 0.30 are shown in Fig. 13. These waveforms showed the greatest effect of e. Much smaller differences existed in the waveforms for u 4 0.02 and 0.19 (Suwansawat 1997). As shown in Fig. 13, similar waveforms are obtained for each e. The end of the probe occurs at a slightly shorter distance when e 4 10 mm. In contrast, the waveforms for e 4 40 and 80 mm are nearly identical. Apparent dielectric constant versus e and u are shown in Fig. 14. For u 4 0.19 and 0.30, Ka increases as e increases from 10 to 30 mm (Fig. 14a). This occurs because the PVC wall (dielectric constant ∼ 3 in the MHz to GHz frequency range) should have less influence on the waveform (and Ka ) as e increases. For e $ 30 mm, however, Ka is essentially unaffected by e for all u. The calibration curves in Fig. 14b show a similar result; essentially the same curve is obtained for e $ 30 cm. Base-to-Probe Spacing

FIG. 12—Apparent dielectric constant vs. rod length (a) and volumetric water content (b).

Tests to evaluate the effect of base-to-probe spacing (l) were conducted at u 4 0.05, 0.13, and 0.30. Waveforms for different l are shown in Fig. 15 for u 4 0.13. Greatest variations in the waveform occurred at u 4 0.13. For u 4 0.05 and 0.30, waveforms obtained for each l were identical (Suwansawat 1997). As with the rod spacing and wall-to-probe tests, the beginning of the probe is the same for each l. In contrast, the end of the probe occurs at slightly longer distances as l decreases. Relationships between Ka and l and u are shown in Fig. 16. A slight drop in Ka occurs for u 4 0.13 as l increases from 0 or 40 mm, although only a small change in Ka occurs as l increases beyond 20 mm (Fig. 16a). Higher Kas are obtained for l 4 10 and 20 mm because the apparent distance to the end of the probe is slightly larger (Fig. 16). As a result, larger La is obtained, which yields larger Ka (i.e., from Eq 4). At u 4 0.05 and 0.30, Ka is unaffected by l, which is consistent with the nearly identical waveforms obtained for these u (Suwansa-

10


FIG. 14—Apparent dielectric constant vs. wall-to-rod ( e ) spacing (a) and volumetric water content (b).

FIG. 15—Waveforms for a two-rod probe having different base-to-end of rod ( l ) spacings for u 4 0.13.

wat 1997). Similar results are evident in the Ka-u relationship. For l $ 20 mm, the calibration curves are essentially the same. Calibration Cell The test results shown in Figs. 5 to 16 suggest that a two-rod probe with 4-mm-diameter rods that are 60 mm long and have a

FIG. 16—Apparent dielectric constant vs. base-to-end of rod ( l ) spacing (a) and volumetric water content (b).

center-to-center spacing of 30 mm can be used for developing Ka-u calibrations. A balun should be used to join twin lead cable from the probe to the coaxial cable connected to the reflectometer. Walls of the cell surrounding the soil being tested should be at least 30 mm from the edge of the rods, and the base of the cell should be at least 20 mm from the end of the rods. A convenient cell meeting this requirement can be made from a PVC cylinder having the same dimensions as a standard compaction mold (e.g., ASTM D 698 Method A). Because PVC is less stiff than most metals used for compaction molds, compaction curves obtained from a PVC mold may differ from those described in ASTM Test Method for Laboratory Compaction Characteristics of Soil Using Standard Effort (12,400 ft⋅ lbf/ft3 (600 kN⋅m/m3 )) (D 698). If this deviation is a concern, compaction curves should be prepared using metallic and PVC molds to determine if the PVC mold has a significant effect on the compaction curve. However, the authors have found that PVC molds yield compaction curves essentially identical to those obtained from metallic molds. A schematic of this type of cell is shown in Fig. 17. Rods 90 mm long (80 mm in soil) are used to ensure that rod length has no effect on the calibration. The cell contains a 10-mm-thick acrylic cap with two holes (4.1-mm diameter) into which the rods are inserted. The holes ensure the rods are installed vertically, and the cap prevents evaporation from the soil. A calibration curve obtained from this cell is shown in Fig. 18a, along with a curve


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FIG. 17—Calibration cell used for determining Ka-u calibrations.

obtained using the 300-mm-diameter cylinder and a two-rod probe with 290-mm-long rods that had no acrylic cap. The calibration obtained using the small PVC ‘‘calibration cell’’ is nearly identical to the calibration obtained using the large 300-mm-diameter cylinder, and both data sets can be represented by: u 4 100.577 ` 0.0375Ka 1 0.00141K 2a ` 3.22 2 1025 K 3a

(6)

with an R 2 4 0.996. This calibration for the two-rod probe falls slightly below Topp’s equation (Eq 5) in Fig. 18a, which is consistent with the data in Fig. 8. The similarity of the two data sets also suggests that the plastic cap introduced negligible bias into the measurements. Additional tests were conducted with a three-rod probe in the same cell using a center-to-center rod spacing of 20 mm and length of 90 mm. Results of these tests are shown in Fig. 18b. Again, nearly identical results are obtained from the small PVC ‘‘calibration cell’’ as with the large 300-mm-diameter cylinder. In addition, the data fall directly on the curve corresponding to Topp’s equation (Eq 5). The three-rod probe can be used without wall effects because the electric field around the three-rod probe is approximately three times more concentrated than that surrounding the two-rod probe (Knight et al. 1994). Thus, from a practical perspective the cell can also be used for calibrating three-rod probes. In some cases, particularly with finer-grained soil, disturbance may occur if the rods are pressed into the soil. The disturbance may locally affect the water content, which will affect the calibration curve. If disturbance is a concern, holes can be drilled into the soil using a long drill bit and a drill press as suggested by Meerdink et al. (1996). However, the authors have found that predrilling holes for the rods is rarely needed. Conclusions The objective of this study was to determine the minimum cell size for determining the relationship between the apparent dielectric constant (Ka ) (measured with TDR) and volumetric water content (u). Initially a suitable probe configuration (number of rods, rod spacing, and rod length) was determined. Then, tests were conducted where distances were varied from the cell base to the end of the rods and from the cell wall to the edge of the rods.

FIG. 18—Apparent dielectric constant-volumetric water content calibrations determined using ‘‘calibration cell’’ and 300-mm-diameter PVC cylinder: (a) two-rod probe with balun and (b) three-rod probe.

Tests to evaluate the probe configuration showed that two or three rod probes can be used, but the waveform is clearest for a two-rod probe with a balun. Both probes provide a distinct Ka-u relationship, although the relationship from the three-rod probe is closer to the relationship presented by Topp et al. (1980) based on tests in a coaxial cell. The cell size tests show that a suitable calibration cell is a PVC cylinder having the same dimensions as a standard compaction mold (diameter 4 102 mm, height 4 116 mm), although a slightly smaller cell can be used without affecting the Ka-u calibration. The calibration cell is simple, inexpensive, and only a small volume of soil (0.944 L) needs to be processed for each water content. In addition, although the cell was designed for use with a two-rod probe, tests with a three-rod probe resulted in nearly identical results when the calibration cell and a large cell were used. Tests with the three-rod probe in the calibration cell also yielded a Ka-u relationship consistent with Topp’s equation.

Acknowledgment The National Science Foundation (NSF) provided a portion of the financial support for this study through Grant No. CMS9157116. However, the information presented is that solely of the authors and does not represent the policies or opinions of NSF.

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