WATER RESOURCES RESEARCH, VOL. 34, NO. 5, PAGES 1207–1213, MAY 1998

The dependence of the electrical conductivity measured by time domain reflectometry on the water content of a sand P. A. Ferre´, J. D. Redman, and D. L. Rudolph Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada

R. G. Kachanoski Graduate Studies, University of Saskatchewan, Saskatoon, Canada

Abstract. We present paired measurements of the water content and electrical conductivity collected in a laboratory column packed with a homogeneous, clean sand over a wide range of water content and pore water electrical conductivity (EC) conditions. The EC was determined using the method of Nadler et al. [1991] from waveforms collected with two-rod time domain reflectometry (TDR) probes with and without baluns and with three-rod probes without baluns. Following Heimovaara et al. [1995], we calibrated the probes in saline solutions to account for the series resistance of the cable and connectors. The calibrated EC shows a nonlinear dependence on the water content that is well described by a simple power relationship [Archie, 1942]. Recognizing that calibration in saline solutions is impractical for some TDR probes, we demonstrate that the EC response can be calibrated directly using the results of drainage events, incorporating only a separate calibration of the cable resistance. None of the probe designs shows any clear advantage for EC measurement.

1.

Introduction

Early investigations of the relationship between the electrical conductivity (EC) of a porous medium and the water content of the medium focused on direct current (dc) measurements in consolidated and clean, unconsolidated sands [Archie, 1942]. More recent investigations examined dc measurements in loamy soils [Rhoades et al., 1976]. For a given water content, both investigators found a linear relationship between the EC of a porous medium and the EC of the pore water in the medium. There is also a near-linear relationship between the EC of a solution of a single electrolyte, such as potassium chloride (KCl), and the concentration of that electrolyte [Barthel et al., 1980]. These relationships lead to a near-linear relationship between the EC of a porous medium at a given water content and the solute concentration in the pore water. Direct current conductivity measurement is, therefore, a useful, nondestructive method of monitoring the concentration of simple electrolytic solutions in a porous medium with a constant water content. Archie [1942] and Rhoades et al. [1976] both found that the EC has a nonlinear dependence on the water content. Therefore, to use EC measurements to monitor solute concentrations in variably saturated media, both the water content and the EC must be measured independently. Time domain reflectometry (TDR) measures both the water content and the EC in approximately the same volume of porous medium [Dalton et al., 1984; Topp et al., 1988]. Standard TDR probes comprise two or three parallel metal rods. Two-rod probes are often connected to a cable tester through a balun, while three-rod probes are connected directly through coaxial cables. A common method of TDR water content analCopyright 1998 by the American Geophysical Union. Paper number 98WR00218. 0043-1397/98/98WR-00218$09.00

ysis is generally accepted for all of these probe designs [Topp et al., 1980]. Initial research into the use of the attenuation of TDR pulses to infer the EC of a soil focused on the response of probes to changes in the EC of the pore water in saturated soil samples [Dalton et al., 1984; Topp et al., 1988]. Nadler et al. [1991] compared the EC responses of a two-rod probe with a balun to those of a three-rod probe without a balun in samples of a silty loam mixed with saline solutions to six spatially uniform water contents ranging from 0.07 to 0.28. Nadler et al. [1991] calculated the EC using the Giese-Tiemann (GT) analysis, an analysis based solely on the impedance at late time on the waveform (LTI), and three other methods of EC analysis. In addition, they examined the EC and water content responses of probes inserted through two layers of media of differing water contents. In response to the work of Nadler et al. [1991], Heimovaara [1992] showed that in theory, the GT and LTI methods of analysis are identical. Ward et al. [1994] presented EC measurements in a fine sand packed to four water contents ranging from 0.05 to 0.25. Nadler et al. [1991] showed a linear relationship between the EC response and independent measurements of the bulk EC of the medium; Ward et al. [1994] showed a linear relationship between the EC response of a TDR instrument and the concentration of an electrolytic solute in the pore water under uniform water content conditions. However, neither of these investigations presented sufficient data to define the dependence of the TDRmeasured EC on the water content. Heimovaara et al. [1995] applied a theoretical relationship between the bulk EC and the water content [Mualem and Friedman, 1991] to monitor solute movement under variable water content conditions with TDR. Risler et al. [1996] monitored the EC with TDR during cyclic wetting and drainage of an electrolytic solution, reporting a linear dependence of the TDR-measured EC on the water content.

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and between the electrical conductivity and the water content. Heimovaara et al. [1995] rewrote this relationship in a form that also applies in the absence of surface conductivity, b1 s 5 s wu eff

@1 2 ~1 2 Q 1/m! m# 2 , 1 2 ~1 2 Q 1/q! q

(4)

where ueff is the mobile fraction of the pore water and Q is the reduced water content. The reduced water content is defined by the residual water content u r and the saturated water content u s as, Q 5 ~ u 2 u r!/~ u s 2 u r!.

Figure 1. Locations of voltage measurements for EC analyses on a TDR waveform.

The objective of this experiment was to examine, under controlled laboratory conditions, the relationship between the TDR-measured EC and the water content over a wide range of soil water contents and pore water salinities. We present paired measurements of the EC and water content collected during drainage in a homogeneous sand for a range of pore water EC conditions. This approach provides a complete set of data to define clearly the dependence of the EC measured by parallel-rod TDR probes on the water content while ensuring that the medium properties are uniform throughout the sample volume of the probes. In addition, the performance of two-rod probes both with and without baluns and of three-rod probes without baluns is compared. Finally, we demonstrate that paired measurements made during drainage, coupled with independent measurements of the resistance of the cable and connectors, can be used to calibrate the EC response of probes in situ.

2. Relationships Between the Electrical Conductivity and the Water Content Archie [1942] formed an empirical relationship between the DC electrical conductivity s of consolidated and clean, unconsolidated sands and properties including the water saturation S w , porosity f, and pore water EC, s w ,

s 5 s wS nwf m.

(1)

The constants m and n are soil-specific with typical values for m ranging from 1.3 to 2.0 and n approximately equal to 2 for sandy soils. Substituting for the saturation S w as the ratio of the water content u to the porosity f, equation (1) becomes n

s 5 s wu f

m2n

.

(2)

Rhoades et al. [1976] developed a relationship for loamy soils based on a capillary model,

s 5 s wu ~a Ru 1 b R! 1 s s.

(3)

This relationship includes a contribution due to surface conduction s s , which is generally negligible for soils with low fractions of clay and colloids. Mualem and Friedman [1991] developed a theoretical relationship based on the similarity of the shapes of the relationships between the hydraulic conductivity and the water content

(5)

Including the definition of the reduced water content, equation (4) has six fitting parameters: m, q, u eff, b 1 , u r , and u s . In our analyses we use equation (2) to represent the relationship between the EC of the porous medium and the water content. This form is based on observations made in low clay content soils similar to the sand used in our study. In addition, it has the practical advantage of requiring only three fitting parameters: m, n, and f.

3. Measurement of Electrical Conductivity With TDR In an analysis of the performance of a balun designed specifically for use with TDR probes, Spaans and Baker [1993] presented three methods of EC analysis from TDR waveforms previously reported by Dalton and van Genuchten [1986] and Topp et al. [1988]. They found that the EC calculated using the GT analysis [Topp et al., 1988] from waveforms collected with TDR probes in saline solutions agreed most closely with independent measurements of the EC of the solutions. The GT relationship is of the form

s5

Z0 1 120 p L Z u

S

D

2V 0 21 , Vf

(6)

where V 0 and V f are the voltages measured from the TDR waveform at the locations shown in Figure 1. Z u , the output impedance of the pulse generator, is a constant, equal to 50 V for a Tektronics 1502B cable tester, for example. The characteristic impedance of the probe, Z 0 , is independent of the properties of the surrounding medium [Baker and Spaans, 1993] and can be determined from measurements in a medium of known dielectric permittivity, as was suggested by Zegelin et al. [1989]. Therefore, for a given probe of length L, the GT analysis (equation (6)) can be simplified to include only a single calibration constant,

s 5 c GT

S

D

2V 0 21 . Vf

(7)

Kachanoski et al. [1992] used the LTI analysis to monitor solute transport under steady state flow conditions with TDR. This analysis is based solely on R L , the impedance at late time, shown by the location V f on the waveform in Figure 1,

s 5 c N/R L.

(8)

The LTI analysis also depends on a single calibration constant, c N . To eliminate the influence of small perturbations on the waveform, we used an average of the impedances over a small time window around V f . On the basis of a single reflection definition of the reflection

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coefficient, Heimovaara [1992] showed that the value of V 0 can be taken to be a constant, with the result that the GT and LTI analyses are identical. Therefore our analyses consider only the LTI method. Heimovaara et al. [1995] showed that the series resistance of the cable and connectors leading to a TDR probe must be considered when determining the EC of the medium from the TDR-measured resistance. Including this series resistance, equation (8) becomes 1 RL 1 1 5 5 1 , s c N s medium s cable

(9)

where s is the inverse of the total resistance measured by the TDR instrument, smedium is the conductance of the sample of porous medium surrounding the probe, and scable is the inverse of the series resistance of the cable and connectors leading to the probe. From equation (9) the EC of the medium surrounding the probe is defined by the late time impedance as 1

s medium

4.

5

RL 1 2 . c N s cable

(10)

Experimental Design

We examined the dependence of the TDR-measured EC on the water content in a sand-filled column. The use of a homogeneous medium avoided the complications of reflections from material boundaries seen by Nadler et al. [1991]. We used a

Figure 2. Design of the laboratory column and the pressure– water content relationship of Borden sand.

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Table 1. Configurations of the TDR Probes Probe

Rods Used

Balun

TDR12 TDR13 TDR12n TDR123

1, 2 1, 3 1, 2 1, 2, 3

Yes Yes No No

clean sand to eliminate the contributions of surface conductance to the measured EC and the influence of bound water on water content determinations [Dasberg and Hopmans, 1992]. The fine- to medium-grained sand was collected on Canadian Forces Base Borden, Ontario, Canada, as part of ongoing experiments at the site. To achieve complete drainage of the sand without the need for a pressure plate at the base of the column, we used a 2-m-long polyvinyl chloride (PVC) column with a hanging water table placed 20 cm below its base. On the basis of the drainage curve for the sand [Nwankwor, 1982], shown in Figure 2, the upper half of the column was expected to drain to near-residual water content under these conditions. A sealed end cap fitted with a 0.952-cm (3/8 inch) Swagelock fitting covered the base of the column, and a steel screen placed in the fitting retained the sand. We used three horizontal metal rods for our TDR measurements (Figure 2). Longer rods increase the separation in time of the characteristic reflections from the beginning and end of the rods on a TDR waveform, improving the precision of propagation velocity determinations. Therefore we used a relatively large diameter (20 cm) PVC column to allow for the use of longer horizontal TDR rods than are commonly used in vertical column experiments. Each solid metal rod was 22.5 cm in length, with a diameter of 0.25 cm; the rod separation was 1.5 cm. Four probe configurations were used: TDR12, TDR13, TDR12n, and TDR123. Table 1 summarizes the configurations of the probes. For probe TDR12, rods 1 and 2 connected to the cable tester through a balun (ANZAC TP-103 impedancematching transformer). Twin-wire shielded cable (#9090 Belden) connected the rods to the balun, and the balun was placed directly on a cable tester (Tektronix 1502B). The twin-wire cable was 2.9 m long to separate the balun reflections and the characteristic reflections from the beginning and end of the rods on the waveform. Similarly, rods 1 and 3 were connected to the cable tester through a balun (TDR13). For probe TDR12n, rods 1 and 2 were connected directly to the cable tester through RG-58 C/U coaxial cable without a balun. The coaxial cable was 2.9 m long for direct comparison with the designs using a balun and the twin-wire cable. For probe TDR123, rods 1, 2, and 3 were directly connected to the cable tester through a coaxial cable. To improve the connection between the rods and the coaxial cable for TDR123, we used a small metal plate to connect the outer shield of the cable to rods 1 and 3, while the central conductor of the cable connected directly to rod 2. For probe TDR12n the plate connected rod 2 to the cable shield, and the central conductor of the coaxial cable connected to rod 1. This variety of probe designs, all measuring within nearly the same volume of the porous medium, allowed us to compare the performance of three-rod probes described by Zegelin et al. [1989] with that of standard two-rod probes [Topp et al., 1982] and to assess the impact of baluns on the EC response. Software written by Redman [1995] was used to acquire waveforms on a personal computer via an RS232 cable.

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Figure 3. Calibration of the EC response used for the LTI analysis and determination of the equivalent resistance of the TDR cables for the four probe designs.

5. Calibration of TDR Probes in Saline Solutions The conductivity response of a TDR probe is commonly calibrated in saline solutions because it is a simple method by which the probes can be calibrated for a wide range of EC conditions while maintaining spatially uniform conditions throughout their sample volumes. Initially, we collected waveforms with the rods extending through the far wall of the unpacked column filled with a series of KCl solutions. Equation (10) describes a linear relationship between the late time impedance R L and the inverse of the EC of the calibration solution for a fluid-filled column. The slope of the linear relationship is directly related to the constant c N , and the intercept defines the inverse of the resistance of the cable, balun, and connectors between the cable tester and the probe. Figure 3 shows the late time impedance as a function of the inverse of the EC of the calibration solution for the four TDR probe configurations. Linear regressions of equation (10) to the data, along with their slopes, intercepts, and r 2 values are shown as well. All of the regressions are highly linear. Furthermore, the line resistance introduced by probes with baluns was not significantly different from that seen for probes without baluns. This finding disagrees with the results presented by Spaans and Baker [1993], who found that the type of balun used in this experiment introduced a large intercept to a plot of the TDRmeasured EC as a function of the EC of saline solutions.

6. Dependence of the TDR-Measured EC on the Pore Water EC After calibration in saline solutions, the column was packed with sand. To achieve uniform packing, the sand was dropped through crossed screens held above the surface of the sand pack using a technique similar to that described by Wygal [1963]. After packing, the rods were driven into the column until they were flush with the far column wall. The column was flooded with deionized water by slowly raising the water table from below the base of the column to a point above the surface of the sand. The column remained saturated for 7 days to leach any highly soluble components. Then the column was drained, and flooding was repeated. With the water table near the surface of the column, a saline

solution was ponded at the surface of the saturated column and allowed to infiltrate. Solution was added continuously until the waveform collected with probe TDR123 remained constant in time, indicating that the saline solution had replaced the resident pore water above the base of the sample volume of the TDR probes. Then the hanging water table was lowered to the initial position below the base of the column, and both the EC and the water content were monitored with all of the TDR probes as the column drained. By measuring continuously during free drainage, many paired water content and EC measurements were collected, allowing for a full description of the relationship between the TDR-measured EC and the water content. After each solution drained, the column was reflooded from below, and the procedure was repeated for the next solution, using a total of seven KCl solutions with EC values ranging from 0.06 to 0.63 S m21 (KCl concentrations of 0.38 to 4.08 g L21). The water content was determined from the measured relative dielectric permittivities using the general relationship presented by Topp et al. [1980]. The water contents measured in the column during the experiments ranged from 0.081 to 0.35. Nwankwor [1982] measured water contents ranging from 0.068 to 0.37 in the same site material by gravimetric methods. This general agreement supports the use of the general equation to relate the water content to the relative dielectric permittivity of the medium. Maintaining uniform conditions throughout the sample volume avoids any complications introduced by spatial weighting of variable water contents and EC values within the sample volumes of the probes. For the homogeneous sand in the column, initially saturated with the flushing solution, the water content and EC should be constant with elevation at any given time during drainage over the 3-cm maximum rod separation. Agreement among the water content values measured with the two- and three-rod probes confirmed that the water content was spatially uniform throughout the measurement volume. Combining equations (2) and (10) shows the dependence of the calibrated EC response, 1/R9L , on the pore water EC and the water content, 1 5 R9L

S

RL 1 2 c N s cable

D

21

5 s wu nf m2n.

(11)

For the packed column it is reasonable to assume that the porosity is spatially uniform throughout the measurement volume of the probes. For a given water content condition, equation (11) describes a linear relationship between the calibrated EC and the pore water EC with a zero intercept. Figures 4– 6 show paired values of the calibrated TDR-measured EC and the pore water EC collected at three water contents with probes TDR12, TDR12n, and TDR123, respectively. The results for probe TDR13 are very similar to those shown for TDR12. Linear regressions to the data show near-zero intercepts. An average intercept was determined for each probe, with values of 0.0018, 20.0010, and 20.0003 S m21, respectively. Equation (11) can be rewritten to account for the small, nonzero intercepts seen in Figures 4 – 6, 1 5 R9L

S

RL 1 2 c N s cable

D

21

5 s wu nf m2n 1 b.

(12)

The form of equation (12) suggests that the constant b represents an additional series resistance. However, given the negative fitted values, the physical meaning of this variable is

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Figure 4. Calibrated TDR EC measured with probe TDR12 as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown. unclear, and it may simply indicate some artifact of the method of EC analysis or slight inaccuracies inherent in the measurement of impedance with TDR.

7. Dependence of the EC Response on Soil Water Content Figures 4 – 6 show that the slopes of the linear relationships between the calibrated TDR EC and the pore water EC are dependent on the water content of the medium. Therefore a functional relationship between the slope and the water content is necessary to define the pore water EC from the EC and water content responses obtained from TDR waveforms. From equation (12) the slope S of the relationship between the corrected TDR EC and water content is S5

~1/R9L! 5 u nf m2n. sw

(13)

Taking the logarithm of both sides of equation (13) gives log S 5 n log u 1 log f m2n 5 n log u 1 B.

(14)

Slopes were determined for each probe for nine water content conditions ranging from 0.15 to 0.30 during drainage of

Figure 5. Calibrated TDR EC measured with probe TDR12n as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown.

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Figure 6. Calibrated TDR EC measured with probe TDR123 as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown.

the seven flushing solutions. Figure 7 shows the logarithm of the slopes determined for probes TDR12, TDR12n, and TDR123 as a function of the logarithm of the water content. The results for TDR13 are very similar to those found for TDR12; for clarity they are not included in Figure 7. Linear regressions of equation (14) to the data and the slope, intercept, and goodness of fit for the linear regressions are shown in Figure 7. For each probe, the value of n in equation (2) is defined as the slope of the linear regression in Figure 7. The reported n values clearly demonstrate the nonlinear dependence of the EC on the water content over the full range of water content values. Furthermore, the fitted values of n are consistent with the approximate value of 2 found for direct current measurements in clean sands by Archie [1942]. However, n should represent a property of the porous medium; therefore the differences among the fitted values for the probes indicate that the probe design has some influence on the TDR-derived EC response. The intercept B in equation (14) is defined for each probe by the intercept shown in Figure 7. The inverse of the late time impedance can be calculated for

Figure 7. Paired measurements of the logarithm of the slope of the calibrated TDR EC as a function of the pore water EC and the logarithm of the water content for probes TDR12, TDR12n, and TDR123. Linear regressions to the data are shown.

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any given pore water EC by substitution of the fitted parameters found through linear regression into equation (12), 1 5 RL

S

cN cN 1 s wu 10 B 1 b s cable n

D

21

.

(15)

Figure 8 compares the inverse of the late time impedance measured during drainage of the seven flushing solutions to the value of 1/R L calculated using equation (15) for probe TDR12. Very similar results were found for all of the probe designs.

8.

Probe Calibration Using Drainage Data

The goodness of fit shown in Figure 8 and the linearity of the results shown in Figures 3–7 support the use of a relationship with the form of equation (15) to describe the dependence of the TDR-measured EC on the water content and pore water EC. Following the procedure presented above, two of the variables (c N and scable) were determined with the probes placed in saline solutions, while the remaining constants (b, B and n) were found using measurements in the medium of interest. For parallel-rod probes, the constant c N is dependent on the separation of the rods along their entire length. Given that the configuration of a probe can vary significantly during rod insertion, especially in the field, c N should be calibrated in situ. In contrast, the resistance of the cable and connectors should be independent of the rod configuration. Given that calibration of long, continuous-rod probes in saline solutions can be impractical, the constant scable can be determined by measuring the EC of saline solutions with shorter rod pairs. Taking the value of scable found by calibration of each probe in saline solutions, we fitted equation (15) directly to paired measurements of R L , u , and s w collected during three drainage experiments (pore water EC 5 0.063, 0.142, and 0.631 S m21) to determine n, B, b and c N . Figure 9 compares the inverses of the measured and calculated late time impedance values as a function of the water content for all seven drainage events using the fitted values for the four constants. The close agreement shown suggests that drainage data coupled with the measured value of scable can be used to calibrate the EC response of TDR probes. Similar results were found for all of the probe designs; the values of the four constants found by

Figure 9. Measured and calculated values of the late time impedance response of probe TDR12 for seven flushing solutions as a function of water content. Fitting parameters were found through multivariate regressions to equation (15). multivariate fitting are shown in Table 2. The table also includes the root-mean-square errors of the fits, showing slightly better fits for the probes without baluns, TDR12n and TDR123. Unfortunately, the data collected during the drainage of a single solution were insufficient to define fitting parameters that accurately represented the data over a range of pore water salinities. The procedure requires at least two solutions to determine the dependence of the TDR-measured EC on the pore water EC; best results were found using the data from at least three drainage events. In practice, solutions representing the range of expected solute concentrations should be used to calibrate the probe responses most accurately.

9.

Conclusions

In agreement with Ward et al. [1994], we find a linear relationship between the calibrated TDR-measured EC and the pore water EC under constant water content conditions. In addition, our results show that a small, nonzero intercept is necessary to relate the TDR-measured EC to the EC of a medium containing a saline solution; however, the physical meaning of this intercept is unclear. Measurement of the water content and EC during drainage provided complete data sets that allowed for clear definition of the dependence of the TDR-measured EC on the water content. In contrast to the findings of Risler et al. [1996], our results clearly demonstrate a nonlinear dependence of the TDRmeasured EC on the water content. Although the fitting parameters varied with the probe design, the responses of all of the probes investigated were well described by the simple, nonlinear expression determined by Archie [1942]. Baluns act as high-pass filters, resulting in a slow reduction in the ampliTable 2. Multivariate Fits of the Constants in Equation (15) Using Data From Three Drainage Events

Figure 8. Measured and calculated values of the late time impedance response of probe TDR12 for seven flushing solutions as a function of water content. Fitting parameters were found through linear regressions to equations (10), (12), and (14).

Probe

cN

n

10 B

b

RMS Error

TDR12 TDR13 TDR12n TDR123

1.014 1.018 19.648 3.606

2.119 2.216 2.042 1.953

1.526 1.455 7.612 1.963

0.004 0.004 20.005 20.0004

0.00064 0.00512 0.00022 0.00025

Pore water EC 5 0.063, 0.142, and 0.631 S m21.

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tude of the waveform with travel time. Spaans and Baker [1993] found that the inclusion of standard baluns limits the ability of TDR probes to measure the EC of a medium; for our experimental conditions, our results do not show any deleterious effects of the use of a balun. Measurements in saline solutions can define the equivalent resistivity of the cable, balun, and connectors between the pulse generator and a TDR probe. Then the dependence of the EC response of a TDR probe on the water content and pore water EC can be calibrated using paired measurements collected during drainage without the need for additional information such as the drainage curve required to fit the Mualem and Friedman [1991] model. The focus of this work has been to establish a basic model for the dependence of the TDR-measured EC on the water content of a simple soil. Future work should consider more complex conditions including soils with significant surface conductance and with spatially variable water contents and solute concentrations.

References Archie, G. E., The electric resistivity log as an aid in determining some reservoir characteristics, Trans. Am. Inst. Min. Metall. Pet. Eng., 146, 54 – 62, 1942. Baker, J. M., and E. J. A. Spaans, Comments on “Time domain reflectometry measurements of water content and electrical conductivity in layered soil columns,” Soil Sci. Soc. Am. J., 57, 1395–1396, 1993. Barthel, J., F. Feuerlein, R. Neueder, and R. Wachter, Calibration of conductance cells at various temperatures, J. Solution Chem., 9, 209 –219, 1980. Dalton, F. N., and M. T. van Genuchten, The time-domain reflectometry method for measuring soil water content and salinity, Geoderma, 38, 23–250, 1986. Dalton, F. N., W. N. Herkelrath, D. S. Rawlins, and J. D. Rhoades, Time-domain reflectometry: Simultaneous measurement of water content and electrical conductivity with a single probe, Science, 224, 989 –990, 1984. Dasberg, S., and W. Hopmans, Time domain reflectometry calibration for uniformly and nonuniformly wetted sandy and clayey loam soils, Soil Sci. Soc. Am. J., 56, 1341–1345, 1992. Heimovaara, T. J., Comments on “Time domain reflectometry measurements of water content and electrical conductivity of layered soil columns,” Soil Sci. Soc. Am. J., 56, 1657–1658, 1992. Heimovaara, T. J., A. G. Focke, W. Bouten, and J. M. Verstraten, Assessing temporal variations in soil water composition with time domain reflectometry, Soil Sci. Soc. Am. J., 59, 689 – 698, 1995. Kachanoski, R. G., E. Pringle, and A. Ward, Field measurement of

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solute travel times using time domain reflectometry, Soil Sci. Soc. Am. J., 56, 47–52, 1992. Mualem, Y., and S. P. Friedman, Theoretical prediction of electrical conductivity in saturated and unsaturated soil, Water Resour. Res., 27(10), 2771–2777, 1991. Nadler, A., S. Dasberg, and I. Lapid, Time domain reflectometry measurements of water content and electrical conductivity of layered soil columns, Soil Sci. Soc. Am. J., 55, 938 –943, 1991. Nwankwor, G. I., A comparative study of specific yield in a shallow unconfined aquifer, M.Sc. thesis, Univ. of Waterloo, Waterloo, Ont., Canada, 1982. Redman, J. D., WATTDR user’s manual, 10 pp., Waterloo Cent. for Groundwater Res., Waterloo, Ont., Canada, 1995. Rhoades, J. D., P. A. C. Raats, and R. S. Prather, Effects of liquidphase electrical conductivity, water content and surface conductivity on bulk soil electrical conductivity, Soil Sci. Soc. Am. J., 40, 651– 665, 1976. Risler, P. D., J. M. Wraith, and H. M. Gaber, Solute transport under transient flow conditions estimated using time domain reflectometry, Soil Sci. Soc. Am. J., 60, 1297–1305, 1996. Spaans, E. J. A., and J. M. Baker, Simple baluns in parallel probes for time domain reflectometry, Soil Sci. Soc. Am. J., 57, 668 – 673, 1993. Topp, G. C., J. L. Davis, and A. P. Annan, Electromagnetic determination of soil water content: Measurement in coaxial transmission lines, Water Resour. Res., 16, 574 –582, 1980. Topp, G. C., J. L. Davis, and A. P. Annan, Electromagnetic determination of soil water content using TDR, II, Evaluation of installation and configuration of parallel transmission lines, Soil Sci. Soc. Am. J., 46, 678 – 684, 1982. Topp, G. C., M. Yanuka, W. D. Zebchuk, and S. J. Zegelin, Determination of electrical conductivity using time domain reflectometry: Soil and water experiments in coaxial lines, Water Resour. Res., 24, 945–952, 1988. Ward, A. L., R. G. Kachanoski, and D. E. Elrick, Laboratory measurements of solute transport using time domain reflectometry, Soil Sci. Soc. Am. J., 58, 1031–1039, 1994. Wygal, R. J., Construction of models that simulate oil reservoirs, Soc. Pet. Eng. J., 3(4), 281–286, 1963. Zegelin, S. J., I. White, and D. R. Jenkins, Improved field probes for soil water content and electrical conductivity measurement using time domain reflectometry, Water Resour. Res., 25, 2367–2376, 1989. P. A. Ferre´, J. D. Redman, and D. L. Rudolph, Department of Earth Sciences, Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI. (e-mail: [email protected]). R. G. Kachanoski, Graduate Studies, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A4.

(Received July 29, 1997; revised January 12, 1998; accepted January 14, 1998.)

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The dependence of the electrical conductivity measured by time domain reflectometry on the water content of a sand P. A. Ferre´, J. D. Redman, and D. L. Rudolph Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada

R. G. Kachanoski Graduate Studies, University of Saskatchewan, Saskatoon, Canada

Abstract. We present paired measurements of the water content and electrical conductivity collected in a laboratory column packed with a homogeneous, clean sand over a wide range of water content and pore water electrical conductivity (EC) conditions. The EC was determined using the method of Nadler et al. [1991] from waveforms collected with two-rod time domain reflectometry (TDR) probes with and without baluns and with three-rod probes without baluns. Following Heimovaara et al. [1995], we calibrated the probes in saline solutions to account for the series resistance of the cable and connectors. The calibrated EC shows a nonlinear dependence on the water content that is well described by a simple power relationship [Archie, 1942]. Recognizing that calibration in saline solutions is impractical for some TDR probes, we demonstrate that the EC response can be calibrated directly using the results of drainage events, incorporating only a separate calibration of the cable resistance. None of the probe designs shows any clear advantage for EC measurement.

1.

Introduction

Early investigations of the relationship between the electrical conductivity (EC) of a porous medium and the water content of the medium focused on direct current (dc) measurements in consolidated and clean, unconsolidated sands [Archie, 1942]. More recent investigations examined dc measurements in loamy soils [Rhoades et al., 1976]. For a given water content, both investigators found a linear relationship between the EC of a porous medium and the EC of the pore water in the medium. There is also a near-linear relationship between the EC of a solution of a single electrolyte, such as potassium chloride (KCl), and the concentration of that electrolyte [Barthel et al., 1980]. These relationships lead to a near-linear relationship between the EC of a porous medium at a given water content and the solute concentration in the pore water. Direct current conductivity measurement is, therefore, a useful, nondestructive method of monitoring the concentration of simple electrolytic solutions in a porous medium with a constant water content. Archie [1942] and Rhoades et al. [1976] both found that the EC has a nonlinear dependence on the water content. Therefore, to use EC measurements to monitor solute concentrations in variably saturated media, both the water content and the EC must be measured independently. Time domain reflectometry (TDR) measures both the water content and the EC in approximately the same volume of porous medium [Dalton et al., 1984; Topp et al., 1988]. Standard TDR probes comprise two or three parallel metal rods. Two-rod probes are often connected to a cable tester through a balun, while three-rod probes are connected directly through coaxial cables. A common method of TDR water content analCopyright 1998 by the American Geophysical Union. Paper number 98WR00218. 0043-1397/98/98WR-00218$09.00

ysis is generally accepted for all of these probe designs [Topp et al., 1980]. Initial research into the use of the attenuation of TDR pulses to infer the EC of a soil focused on the response of probes to changes in the EC of the pore water in saturated soil samples [Dalton et al., 1984; Topp et al., 1988]. Nadler et al. [1991] compared the EC responses of a two-rod probe with a balun to those of a three-rod probe without a balun in samples of a silty loam mixed with saline solutions to six spatially uniform water contents ranging from 0.07 to 0.28. Nadler et al. [1991] calculated the EC using the Giese-Tiemann (GT) analysis, an analysis based solely on the impedance at late time on the waveform (LTI), and three other methods of EC analysis. In addition, they examined the EC and water content responses of probes inserted through two layers of media of differing water contents. In response to the work of Nadler et al. [1991], Heimovaara [1992] showed that in theory, the GT and LTI methods of analysis are identical. Ward et al. [1994] presented EC measurements in a fine sand packed to four water contents ranging from 0.05 to 0.25. Nadler et al. [1991] showed a linear relationship between the EC response and independent measurements of the bulk EC of the medium; Ward et al. [1994] showed a linear relationship between the EC response of a TDR instrument and the concentration of an electrolytic solute in the pore water under uniform water content conditions. However, neither of these investigations presented sufficient data to define the dependence of the TDRmeasured EC on the water content. Heimovaara et al. [1995] applied a theoretical relationship between the bulk EC and the water content [Mualem and Friedman, 1991] to monitor solute movement under variable water content conditions with TDR. Risler et al. [1996] monitored the EC with TDR during cyclic wetting and drainage of an electrolytic solution, reporting a linear dependence of the TDR-measured EC on the water content.

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and between the electrical conductivity and the water content. Heimovaara et al. [1995] rewrote this relationship in a form that also applies in the absence of surface conductivity, b1 s 5 s wu eff

@1 2 ~1 2 Q 1/m! m# 2 , 1 2 ~1 2 Q 1/q! q

(4)

where ueff is the mobile fraction of the pore water and Q is the reduced water content. The reduced water content is defined by the residual water content u r and the saturated water content u s as, Q 5 ~ u 2 u r!/~ u s 2 u r!.

Figure 1. Locations of voltage measurements for EC analyses on a TDR waveform.

The objective of this experiment was to examine, under controlled laboratory conditions, the relationship between the TDR-measured EC and the water content over a wide range of soil water contents and pore water salinities. We present paired measurements of the EC and water content collected during drainage in a homogeneous sand for a range of pore water EC conditions. This approach provides a complete set of data to define clearly the dependence of the EC measured by parallel-rod TDR probes on the water content while ensuring that the medium properties are uniform throughout the sample volume of the probes. In addition, the performance of two-rod probes both with and without baluns and of three-rod probes without baluns is compared. Finally, we demonstrate that paired measurements made during drainage, coupled with independent measurements of the resistance of the cable and connectors, can be used to calibrate the EC response of probes in situ.

2. Relationships Between the Electrical Conductivity and the Water Content Archie [1942] formed an empirical relationship between the DC electrical conductivity s of consolidated and clean, unconsolidated sands and properties including the water saturation S w , porosity f, and pore water EC, s w ,

s 5 s wS nwf m.

(1)

The constants m and n are soil-specific with typical values for m ranging from 1.3 to 2.0 and n approximately equal to 2 for sandy soils. Substituting for the saturation S w as the ratio of the water content u to the porosity f, equation (1) becomes n

s 5 s wu f

m2n

.

(2)

Rhoades et al. [1976] developed a relationship for loamy soils based on a capillary model,

s 5 s wu ~a Ru 1 b R! 1 s s.

(3)

This relationship includes a contribution due to surface conduction s s , which is generally negligible for soils with low fractions of clay and colloids. Mualem and Friedman [1991] developed a theoretical relationship based on the similarity of the shapes of the relationships between the hydraulic conductivity and the water content

(5)

Including the definition of the reduced water content, equation (4) has six fitting parameters: m, q, u eff, b 1 , u r , and u s . In our analyses we use equation (2) to represent the relationship between the EC of the porous medium and the water content. This form is based on observations made in low clay content soils similar to the sand used in our study. In addition, it has the practical advantage of requiring only three fitting parameters: m, n, and f.

3. Measurement of Electrical Conductivity With TDR In an analysis of the performance of a balun designed specifically for use with TDR probes, Spaans and Baker [1993] presented three methods of EC analysis from TDR waveforms previously reported by Dalton and van Genuchten [1986] and Topp et al. [1988]. They found that the EC calculated using the GT analysis [Topp et al., 1988] from waveforms collected with TDR probes in saline solutions agreed most closely with independent measurements of the EC of the solutions. The GT relationship is of the form

s5

Z0 1 120 p L Z u

S

D

2V 0 21 , Vf

(6)

where V 0 and V f are the voltages measured from the TDR waveform at the locations shown in Figure 1. Z u , the output impedance of the pulse generator, is a constant, equal to 50 V for a Tektronics 1502B cable tester, for example. The characteristic impedance of the probe, Z 0 , is independent of the properties of the surrounding medium [Baker and Spaans, 1993] and can be determined from measurements in a medium of known dielectric permittivity, as was suggested by Zegelin et al. [1989]. Therefore, for a given probe of length L, the GT analysis (equation (6)) can be simplified to include only a single calibration constant,

s 5 c GT

S

D

2V 0 21 . Vf

(7)

Kachanoski et al. [1992] used the LTI analysis to monitor solute transport under steady state flow conditions with TDR. This analysis is based solely on R L , the impedance at late time, shown by the location V f on the waveform in Figure 1,

s 5 c N/R L.

(8)

The LTI analysis also depends on a single calibration constant, c N . To eliminate the influence of small perturbations on the waveform, we used an average of the impedances over a small time window around V f . On the basis of a single reflection definition of the reflection

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coefficient, Heimovaara [1992] showed that the value of V 0 can be taken to be a constant, with the result that the GT and LTI analyses are identical. Therefore our analyses consider only the LTI method. Heimovaara et al. [1995] showed that the series resistance of the cable and connectors leading to a TDR probe must be considered when determining the EC of the medium from the TDR-measured resistance. Including this series resistance, equation (8) becomes 1 RL 1 1 5 5 1 , s c N s medium s cable

(9)

where s is the inverse of the total resistance measured by the TDR instrument, smedium is the conductance of the sample of porous medium surrounding the probe, and scable is the inverse of the series resistance of the cable and connectors leading to the probe. From equation (9) the EC of the medium surrounding the probe is defined by the late time impedance as 1

s medium

4.

5

RL 1 2 . c N s cable

(10)

Experimental Design

We examined the dependence of the TDR-measured EC on the water content in a sand-filled column. The use of a homogeneous medium avoided the complications of reflections from material boundaries seen by Nadler et al. [1991]. We used a

Figure 2. Design of the laboratory column and the pressure– water content relationship of Borden sand.

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Table 1. Configurations of the TDR Probes Probe

Rods Used

Balun

TDR12 TDR13 TDR12n TDR123

1, 2 1, 3 1, 2 1, 2, 3

Yes Yes No No

clean sand to eliminate the contributions of surface conductance to the measured EC and the influence of bound water on water content determinations [Dasberg and Hopmans, 1992]. The fine- to medium-grained sand was collected on Canadian Forces Base Borden, Ontario, Canada, as part of ongoing experiments at the site. To achieve complete drainage of the sand without the need for a pressure plate at the base of the column, we used a 2-m-long polyvinyl chloride (PVC) column with a hanging water table placed 20 cm below its base. On the basis of the drainage curve for the sand [Nwankwor, 1982], shown in Figure 2, the upper half of the column was expected to drain to near-residual water content under these conditions. A sealed end cap fitted with a 0.952-cm (3/8 inch) Swagelock fitting covered the base of the column, and a steel screen placed in the fitting retained the sand. We used three horizontal metal rods for our TDR measurements (Figure 2). Longer rods increase the separation in time of the characteristic reflections from the beginning and end of the rods on a TDR waveform, improving the precision of propagation velocity determinations. Therefore we used a relatively large diameter (20 cm) PVC column to allow for the use of longer horizontal TDR rods than are commonly used in vertical column experiments. Each solid metal rod was 22.5 cm in length, with a diameter of 0.25 cm; the rod separation was 1.5 cm. Four probe configurations were used: TDR12, TDR13, TDR12n, and TDR123. Table 1 summarizes the configurations of the probes. For probe TDR12, rods 1 and 2 connected to the cable tester through a balun (ANZAC TP-103 impedancematching transformer). Twin-wire shielded cable (#9090 Belden) connected the rods to the balun, and the balun was placed directly on a cable tester (Tektronix 1502B). The twin-wire cable was 2.9 m long to separate the balun reflections and the characteristic reflections from the beginning and end of the rods on the waveform. Similarly, rods 1 and 3 were connected to the cable tester through a balun (TDR13). For probe TDR12n, rods 1 and 2 were connected directly to the cable tester through RG-58 C/U coaxial cable without a balun. The coaxial cable was 2.9 m long for direct comparison with the designs using a balun and the twin-wire cable. For probe TDR123, rods 1, 2, and 3 were directly connected to the cable tester through a coaxial cable. To improve the connection between the rods and the coaxial cable for TDR123, we used a small metal plate to connect the outer shield of the cable to rods 1 and 3, while the central conductor of the cable connected directly to rod 2. For probe TDR12n the plate connected rod 2 to the cable shield, and the central conductor of the coaxial cable connected to rod 1. This variety of probe designs, all measuring within nearly the same volume of the porous medium, allowed us to compare the performance of three-rod probes described by Zegelin et al. [1989] with that of standard two-rod probes [Topp et al., 1982] and to assess the impact of baluns on the EC response. Software written by Redman [1995] was used to acquire waveforms on a personal computer via an RS232 cable.

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Figure 3. Calibration of the EC response used for the LTI analysis and determination of the equivalent resistance of the TDR cables for the four probe designs.

5. Calibration of TDR Probes in Saline Solutions The conductivity response of a TDR probe is commonly calibrated in saline solutions because it is a simple method by which the probes can be calibrated for a wide range of EC conditions while maintaining spatially uniform conditions throughout their sample volumes. Initially, we collected waveforms with the rods extending through the far wall of the unpacked column filled with a series of KCl solutions. Equation (10) describes a linear relationship between the late time impedance R L and the inverse of the EC of the calibration solution for a fluid-filled column. The slope of the linear relationship is directly related to the constant c N , and the intercept defines the inverse of the resistance of the cable, balun, and connectors between the cable tester and the probe. Figure 3 shows the late time impedance as a function of the inverse of the EC of the calibration solution for the four TDR probe configurations. Linear regressions of equation (10) to the data, along with their slopes, intercepts, and r 2 values are shown as well. All of the regressions are highly linear. Furthermore, the line resistance introduced by probes with baluns was not significantly different from that seen for probes without baluns. This finding disagrees with the results presented by Spaans and Baker [1993], who found that the type of balun used in this experiment introduced a large intercept to a plot of the TDRmeasured EC as a function of the EC of saline solutions.

6. Dependence of the TDR-Measured EC on the Pore Water EC After calibration in saline solutions, the column was packed with sand. To achieve uniform packing, the sand was dropped through crossed screens held above the surface of the sand pack using a technique similar to that described by Wygal [1963]. After packing, the rods were driven into the column until they were flush with the far column wall. The column was flooded with deionized water by slowly raising the water table from below the base of the column to a point above the surface of the sand. The column remained saturated for 7 days to leach any highly soluble components. Then the column was drained, and flooding was repeated. With the water table near the surface of the column, a saline

solution was ponded at the surface of the saturated column and allowed to infiltrate. Solution was added continuously until the waveform collected with probe TDR123 remained constant in time, indicating that the saline solution had replaced the resident pore water above the base of the sample volume of the TDR probes. Then the hanging water table was lowered to the initial position below the base of the column, and both the EC and the water content were monitored with all of the TDR probes as the column drained. By measuring continuously during free drainage, many paired water content and EC measurements were collected, allowing for a full description of the relationship between the TDR-measured EC and the water content. After each solution drained, the column was reflooded from below, and the procedure was repeated for the next solution, using a total of seven KCl solutions with EC values ranging from 0.06 to 0.63 S m21 (KCl concentrations of 0.38 to 4.08 g L21). The water content was determined from the measured relative dielectric permittivities using the general relationship presented by Topp et al. [1980]. The water contents measured in the column during the experiments ranged from 0.081 to 0.35. Nwankwor [1982] measured water contents ranging from 0.068 to 0.37 in the same site material by gravimetric methods. This general agreement supports the use of the general equation to relate the water content to the relative dielectric permittivity of the medium. Maintaining uniform conditions throughout the sample volume avoids any complications introduced by spatial weighting of variable water contents and EC values within the sample volumes of the probes. For the homogeneous sand in the column, initially saturated with the flushing solution, the water content and EC should be constant with elevation at any given time during drainage over the 3-cm maximum rod separation. Agreement among the water content values measured with the two- and three-rod probes confirmed that the water content was spatially uniform throughout the measurement volume. Combining equations (2) and (10) shows the dependence of the calibrated EC response, 1/R9L , on the pore water EC and the water content, 1 5 R9L

S

RL 1 2 c N s cable

D

21

5 s wu nf m2n.

(11)

For the packed column it is reasonable to assume that the porosity is spatially uniform throughout the measurement volume of the probes. For a given water content condition, equation (11) describes a linear relationship between the calibrated EC and the pore water EC with a zero intercept. Figures 4– 6 show paired values of the calibrated TDR-measured EC and the pore water EC collected at three water contents with probes TDR12, TDR12n, and TDR123, respectively. The results for probe TDR13 are very similar to those shown for TDR12. Linear regressions to the data show near-zero intercepts. An average intercept was determined for each probe, with values of 0.0018, 20.0010, and 20.0003 S m21, respectively. Equation (11) can be rewritten to account for the small, nonzero intercepts seen in Figures 4 – 6, 1 5 R9L

S

RL 1 2 c N s cable

D

21

5 s wu nf m2n 1 b.

(12)

The form of equation (12) suggests that the constant b represents an additional series resistance. However, given the negative fitted values, the physical meaning of this variable is

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Figure 4. Calibrated TDR EC measured with probe TDR12 as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown. unclear, and it may simply indicate some artifact of the method of EC analysis or slight inaccuracies inherent in the measurement of impedance with TDR.

7. Dependence of the EC Response on Soil Water Content Figures 4 – 6 show that the slopes of the linear relationships between the calibrated TDR EC and the pore water EC are dependent on the water content of the medium. Therefore a functional relationship between the slope and the water content is necessary to define the pore water EC from the EC and water content responses obtained from TDR waveforms. From equation (12) the slope S of the relationship between the corrected TDR EC and water content is S5

~1/R9L! 5 u nf m2n. sw

(13)

Taking the logarithm of both sides of equation (13) gives log S 5 n log u 1 log f m2n 5 n log u 1 B.

(14)

Slopes were determined for each probe for nine water content conditions ranging from 0.15 to 0.30 during drainage of

Figure 5. Calibrated TDR EC measured with probe TDR12n as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown.

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Figure 6. Calibrated TDR EC measured with probe TDR123 as a function of the pore water EC for three water contents: 0.15, 0.22, and 0.30. Linear regressions to the data are shown.

the seven flushing solutions. Figure 7 shows the logarithm of the slopes determined for probes TDR12, TDR12n, and TDR123 as a function of the logarithm of the water content. The results for TDR13 are very similar to those found for TDR12; for clarity they are not included in Figure 7. Linear regressions of equation (14) to the data and the slope, intercept, and goodness of fit for the linear regressions are shown in Figure 7. For each probe, the value of n in equation (2) is defined as the slope of the linear regression in Figure 7. The reported n values clearly demonstrate the nonlinear dependence of the EC on the water content over the full range of water content values. Furthermore, the fitted values of n are consistent with the approximate value of 2 found for direct current measurements in clean sands by Archie [1942]. However, n should represent a property of the porous medium; therefore the differences among the fitted values for the probes indicate that the probe design has some influence on the TDR-derived EC response. The intercept B in equation (14) is defined for each probe by the intercept shown in Figure 7. The inverse of the late time impedance can be calculated for

Figure 7. Paired measurements of the logarithm of the slope of the calibrated TDR EC as a function of the pore water EC and the logarithm of the water content for probes TDR12, TDR12n, and TDR123. Linear regressions to the data are shown.

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any given pore water EC by substitution of the fitted parameters found through linear regression into equation (12), 1 5 RL

S

cN cN 1 s wu 10 B 1 b s cable n

D

21

.

(15)

Figure 8 compares the inverse of the late time impedance measured during drainage of the seven flushing solutions to the value of 1/R L calculated using equation (15) for probe TDR12. Very similar results were found for all of the probe designs.

8.

Probe Calibration Using Drainage Data

The goodness of fit shown in Figure 8 and the linearity of the results shown in Figures 3–7 support the use of a relationship with the form of equation (15) to describe the dependence of the TDR-measured EC on the water content and pore water EC. Following the procedure presented above, two of the variables (c N and scable) were determined with the probes placed in saline solutions, while the remaining constants (b, B and n) were found using measurements in the medium of interest. For parallel-rod probes, the constant c N is dependent on the separation of the rods along their entire length. Given that the configuration of a probe can vary significantly during rod insertion, especially in the field, c N should be calibrated in situ. In contrast, the resistance of the cable and connectors should be independent of the rod configuration. Given that calibration of long, continuous-rod probes in saline solutions can be impractical, the constant scable can be determined by measuring the EC of saline solutions with shorter rod pairs. Taking the value of scable found by calibration of each probe in saline solutions, we fitted equation (15) directly to paired measurements of R L , u , and s w collected during three drainage experiments (pore water EC 5 0.063, 0.142, and 0.631 S m21) to determine n, B, b and c N . Figure 9 compares the inverses of the measured and calculated late time impedance values as a function of the water content for all seven drainage events using the fitted values for the four constants. The close agreement shown suggests that drainage data coupled with the measured value of scable can be used to calibrate the EC response of TDR probes. Similar results were found for all of the probe designs; the values of the four constants found by

Figure 9. Measured and calculated values of the late time impedance response of probe TDR12 for seven flushing solutions as a function of water content. Fitting parameters were found through multivariate regressions to equation (15). multivariate fitting are shown in Table 2. The table also includes the root-mean-square errors of the fits, showing slightly better fits for the probes without baluns, TDR12n and TDR123. Unfortunately, the data collected during the drainage of a single solution were insufficient to define fitting parameters that accurately represented the data over a range of pore water salinities. The procedure requires at least two solutions to determine the dependence of the TDR-measured EC on the pore water EC; best results were found using the data from at least three drainage events. In practice, solutions representing the range of expected solute concentrations should be used to calibrate the probe responses most accurately.

9.

Conclusions

In agreement with Ward et al. [1994], we find a linear relationship between the calibrated TDR-measured EC and the pore water EC under constant water content conditions. In addition, our results show that a small, nonzero intercept is necessary to relate the TDR-measured EC to the EC of a medium containing a saline solution; however, the physical meaning of this intercept is unclear. Measurement of the water content and EC during drainage provided complete data sets that allowed for clear definition of the dependence of the TDR-measured EC on the water content. In contrast to the findings of Risler et al. [1996], our results clearly demonstrate a nonlinear dependence of the TDRmeasured EC on the water content. Although the fitting parameters varied with the probe design, the responses of all of the probes investigated were well described by the simple, nonlinear expression determined by Archie [1942]. Baluns act as high-pass filters, resulting in a slow reduction in the ampliTable 2. Multivariate Fits of the Constants in Equation (15) Using Data From Three Drainage Events

Figure 8. Measured and calculated values of the late time impedance response of probe TDR12 for seven flushing solutions as a function of water content. Fitting parameters were found through linear regressions to equations (10), (12), and (14).

Probe

cN

n

10 B

b

RMS Error

TDR12 TDR13 TDR12n TDR123

1.014 1.018 19.648 3.606

2.119 2.216 2.042 1.953

1.526 1.455 7.612 1.963

0.004 0.004 20.005 20.0004

0.00064 0.00512 0.00022 0.00025

Pore water EC 5 0.063, 0.142, and 0.631 S m21.

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tude of the waveform with travel time. Spaans and Baker [1993] found that the inclusion of standard baluns limits the ability of TDR probes to measure the EC of a medium; for our experimental conditions, our results do not show any deleterious effects of the use of a balun. Measurements in saline solutions can define the equivalent resistivity of the cable, balun, and connectors between the pulse generator and a TDR probe. Then the dependence of the EC response of a TDR probe on the water content and pore water EC can be calibrated using paired measurements collected during drainage without the need for additional information such as the drainage curve required to fit the Mualem and Friedman [1991] model. The focus of this work has been to establish a basic model for the dependence of the TDR-measured EC on the water content of a simple soil. Future work should consider more complex conditions including soils with significant surface conductance and with spatially variable water contents and solute concentrations.

References Archie, G. E., The electric resistivity log as an aid in determining some reservoir characteristics, Trans. Am. Inst. Min. Metall. Pet. Eng., 146, 54 – 62, 1942. Baker, J. M., and E. J. A. Spaans, Comments on “Time domain reflectometry measurements of water content and electrical conductivity in layered soil columns,” Soil Sci. Soc. Am. J., 57, 1395–1396, 1993. Barthel, J., F. Feuerlein, R. Neueder, and R. Wachter, Calibration of conductance cells at various temperatures, J. Solution Chem., 9, 209 –219, 1980. Dalton, F. N., and M. T. van Genuchten, The time-domain reflectometry method for measuring soil water content and salinity, Geoderma, 38, 23–250, 1986. Dalton, F. N., W. N. Herkelrath, D. S. Rawlins, and J. D. Rhoades, Time-domain reflectometry: Simultaneous measurement of water content and electrical conductivity with a single probe, Science, 224, 989 –990, 1984. Dasberg, S., and W. Hopmans, Time domain reflectometry calibration for uniformly and nonuniformly wetted sandy and clayey loam soils, Soil Sci. Soc. Am. J., 56, 1341–1345, 1992. Heimovaara, T. J., Comments on “Time domain reflectometry measurements of water content and electrical conductivity of layered soil columns,” Soil Sci. Soc. Am. J., 56, 1657–1658, 1992. Heimovaara, T. J., A. G. Focke, W. Bouten, and J. M. Verstraten, Assessing temporal variations in soil water composition with time domain reflectometry, Soil Sci. Soc. Am. J., 59, 689 – 698, 1995. Kachanoski, R. G., E. Pringle, and A. Ward, Field measurement of

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solute travel times using time domain reflectometry, Soil Sci. Soc. Am. J., 56, 47–52, 1992. Mualem, Y., and S. P. Friedman, Theoretical prediction of electrical conductivity in saturated and unsaturated soil, Water Resour. Res., 27(10), 2771–2777, 1991. Nadler, A., S. Dasberg, and I. Lapid, Time domain reflectometry measurements of water content and electrical conductivity of layered soil columns, Soil Sci. Soc. Am. J., 55, 938 –943, 1991. Nwankwor, G. I., A comparative study of specific yield in a shallow unconfined aquifer, M.Sc. thesis, Univ. of Waterloo, Waterloo, Ont., Canada, 1982. Redman, J. D., WATTDR user’s manual, 10 pp., Waterloo Cent. for Groundwater Res., Waterloo, Ont., Canada, 1995. Rhoades, J. D., P. A. C. Raats, and R. S. Prather, Effects of liquidphase electrical conductivity, water content and surface conductivity on bulk soil electrical conductivity, Soil Sci. Soc. Am. J., 40, 651– 665, 1976. Risler, P. D., J. M. Wraith, and H. M. Gaber, Solute transport under transient flow conditions estimated using time domain reflectometry, Soil Sci. Soc. Am. J., 60, 1297–1305, 1996. Spaans, E. J. A., and J. M. Baker, Simple baluns in parallel probes for time domain reflectometry, Soil Sci. Soc. Am. J., 57, 668 – 673, 1993. Topp, G. C., J. L. Davis, and A. P. Annan, Electromagnetic determination of soil water content: Measurement in coaxial transmission lines, Water Resour. Res., 16, 574 –582, 1980. Topp, G. C., J. L. Davis, and A. P. Annan, Electromagnetic determination of soil water content using TDR, II, Evaluation of installation and configuration of parallel transmission lines, Soil Sci. Soc. Am. J., 46, 678 – 684, 1982. Topp, G. C., M. Yanuka, W. D. Zebchuk, and S. J. Zegelin, Determination of electrical conductivity using time domain reflectometry: Soil and water experiments in coaxial lines, Water Resour. Res., 24, 945–952, 1988. Ward, A. L., R. G. Kachanoski, and D. E. Elrick, Laboratory measurements of solute transport using time domain reflectometry, Soil Sci. Soc. Am. J., 58, 1031–1039, 1994. Wygal, R. J., Construction of models that simulate oil reservoirs, Soc. Pet. Eng. J., 3(4), 281–286, 1963. Zegelin, S. J., I. White, and D. R. Jenkins, Improved field probes for soil water content and electrical conductivity measurement using time domain reflectometry, Water Resour. Res., 25, 2367–2376, 1989. P. A. Ferre´, J. D. Redman, and D. L. Rudolph, Department of Earth Sciences, Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI. (e-mail: [email protected]). R. G. Kachanoski, Graduate Studies, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A4.

(Received July 29, 1997; revised January 12, 1998; accepted January 14, 1998.)

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