Performance of different types of time domain reflectometry probes for water content measurement in partially saturated rocks

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2005wr004643, 2006 Performance of different types of time domain reflectometry probes for water content measurement in partially saturated rocks Toshihiro Sakaki 1,2 and Harihar Rajaram 1 Received 5 October 2005; revised 1 March 2006; accepted 23 March 2006; published 8 July [1] In applying TDR for water content estimation in rocks, the special considerations required include (1) elimination of inaccuracies resulting from gap effects, (2) difficulties in installation of TDR probes, and (3) the need for individual calibration functions between the apparent dielectric constant (K a ) and water content (q) specific to each rock type. Using seven different rock types, we demonstrate that carefully applied conductive silicone fillings can eliminate gap effects in the case of the penetration-type probes. We also propose two designs for surface probes, which can be installed relatively easily and are found to be free of errors attributable to gap effects. Measurement accuracy of three probe types is comparable to conventional water content measurement in soils. Error due to the difference in K a -q calibration functions obtained with different probe types is more significant than error due to variation in a single measurement or spatial variability in rock properties. Citation: Sakaki, T., and H. Rajaram (2006), Performance of different types of time domain reflectometry probes for water content measurement in partially saturated rocks, Water Resour. Res., 42,, doi: /2005wr Introduction [2] Fluid flow and contaminant transport through the unsaturated zone are receiving increased attention as disposal of waste in such media is being seriously considered. To determine site suitability and anticipated performance of such facilities, it is necessary to characterize and model unsaturated flow and transport properties and processes. Fluxes of water and solutes in the unsaturated zone depend strongly on water content and hydraulic properties (which also vary with water content). Thus water content estimation/monitoring is an important and integral component in characterizing and predicting hydrologic processes in the unsaturated zone. [3] A number of methods are available for measuring the water content in soils. The gravimetric method is the simplest and most widely used, but is destructive. Neutron and gamma-ray attenuation methods are nondestructive but require the use of a radiation source, which involves specialized safety protocols. Detailed reviews on techniques for measuring moisture content are available in the literature [e.g., Schmugge et al., 1980; Stafford, 1988; Raats, 2001; Topp and Ferré, 2002]. Since the early 1980s, time domain reflectometry (TDR) has become a widely used method for water content measurement in soils [Topp et al., 1980]. TDR is a dielectric method that can be used in the field, without involving the hazards of radiation-based methods. The TDR equipment used in this study (TDR100 system, Campbell Scientific Inc.) is based on determining the propagation velocity of an electromagnetic step pulse with a voltage of 250 mv and a 10 90% risetime of <170 pico seconds through the material of interest. Typical TDR step pulse with similar characteristics has a bandwidth around 20 khz to 1.5 GHz [Heimovaara, 1994]. It is known that the propagation velocity v is related to the apparent p ffiffiffiffiffi dielectric constant K a by a simple equation, v = c/ K a, where, c = speed of light (= m/s). Because K a is strongly dependent on the volumetric water content q, the volumetric water content may then be determined from the apparent dielectric constant. Dielectric behavior of rock has been studied extensively in the context of petroleum engineering [Rau and Wharton, 1980; Wharton et al., 1980; Shen et al., 1985], geophysics [Sen, 1981, 1984; Sen et al., 1981; Lange, 1983; Kenyon, 1984; Knight and Nur, 1987; Knight and Endres, 1990; Knight and Abad, 1995], radioactive waste management [Nelson, 1993; Schneebeli et al., 1995; Sakaki et al., 1998a, 1998b], and hydrology [Hokett et al., 1992; Lin et al., 1999]. Various models have been proposed for describing the relationship between K a and q. Typical models are summarized and discussed, for example, by Jacobsen and Schjønning [1995], Topp and Ferré [2002], and Robinson et al. [2003]. [4] Topp et al. [1980] studied a wide range of soil samples in the laboratory. They showed that a third-order polynomial function was adequate to describe the relationship between K a and q (equation (1)) for mineral soils. q ¼ AK 3 a þ BK2 a þ CK a þ D ð1þ 1 Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, Colorado, USA. 2 Now at Center for Experimental Study of Subsurface Environmental Processes, Environmental Science and Engineering, Colorado School of Mines, Golden, Colorado, USA. Copyright 2006 by the American Geophysical Union /06/2005WR where, A = , B = , C = , D = Although individual calibration for each actual soil may be necessary for precise measurements of water content [Jacobsen and Schjønning, 1995], this empirical relationship is still the most widely used in soil science and has been found to have broad applicability [Topp and Ferré, 2002]. 1of15

2 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS [5] An alternative to empirically derived calibration functions is often based on dielectric mixing models. Many dielectric mixing models have been proposed for different applications and geometries. The three-phase a-mixing model (equation (2)) [Dobson et al., 1985] (also known as the complex refractive index method, CRIM, when a = 0.5 [Wharton et al., 1980]) is one such mixing model, in which the effect of bound water is not considered. Ka a ¼ ðf qþka air þ qka water þ ð1 fþka solid ð2þ In equation (2), a is the empirical parameter referred to as the geometry factor, f is the porosity of the porous medium, K air, K water, and K solid are the dielectric constant of air (=1), water (=81), and solid minerals in soils and rocks, respectively. Unlike the empirically derived model (equation (1)), use of mixing models requires varying levels of prior knowledge of the soil properties, i.e., porosity, dielectric constant of each component, and geometry factor. One of the difficulties with using the a-mixing model is that the dielectric constant of the solid phase K solid is often unknown and fitted to the data. This semiempirical model relates K a to the dielectric constants and volume fractions of each component but does not refer to any of the microstructure of a porous or granular media [Robinson et al., 2003]. [6] On the basis of the summary by Ferré and Topp [2000], a large number of previous studies showed that a = 0.5 best described the relationship between soil water content q and dielectric constant K a although others found that the value of a falls in a range of By considering the relationship between the real part of the refractive index (which determines the propagation velocity of an electromagnetic wave) and the complex dielectric constant, Whalley [1993] pointed out that if imaginary component of the dielectric constant is assumed to be negligible, the theoretical value of a is 0.5. The effectiveness of modern TDR instruments for the determination of water content depends upon the relatively constant real component of the dielectric constant and the corresponding low value of the imaginary component over the frequency range in which they operate [Ferré and Topp, 2000]. The a- mixing model with a = 0.5 has been found to provide reasonably accurate estimate of the dielectric properties of rock materials [e.g., Rau and Wharton, 1980; Wharton et al., 1980; Shen et al., 1985; Nelson, 1993; Knight and Endres, 1990]. [7] Robinson et al. [2005] proposed a sample scale model for the K a -q relationship in coarse-grained medium in which presence of bound water can be neglected. They considered a situation where the wetting/draining of the material leads to distinct water-saturated and air-dry layers and a TDR probe is inserted perpendicular to the wet-dry front. On the basis of refractive index mixture theory, the two-point mixing model they proposed has the following form: pffiffiffiffiffi K a ¼ q f pffiffiffiffiffiffiffiffi q K sat þ 1 p ffiffiffiffiffiffiffiffi K dry f where, K sat is the water-solid mixture dielectric constant of water-saturated material, K dry is the air-solid mixture dielectric constant of air-dry material, q is the mean ð3þ volumetric water content. Robinson et al. [2005] also showed that this model was still valid, to some extent, when water content in the sample was distributed without a sharp wetting or drying front. The advantage of this model is that no fitting parameter is involved and two extreme values K sat and K dry can be easily measured or derived from a twophase grain-scale mixing model [e.g., Sihvola and Kong, 1988]. [8] Using the a-mixing model, two extreme values for water-saturated (K sat ) and air-dry (K dry ) media can be obtained in terms of K air, K water, and K solid by setting q = f and q = 0 in equation (2), respectively. Rearranging these expressions, substituting them into equation (2), and some algebraic operations yield; Ka a ¼ q f Ka sat þ 1 q f K a dry When a = 0.5 is used, equation (4) is equivalent to equation (3). Although the model proposed by Robinson et al. [2005] (equation (3)) was derived for a two-layer system on the basis of refractive index mixture theory, it can be considered as a special case of the semiempirical a-mixing model with a = 0.5, where the dielectric constants of three phases are written in terms of two extreme values. Therefore it is expected that the two-point mixing model (equation (4)) with a = 0.5 is also reasonably valid for samples with no distinct wetting/drying front as observed by Robinson et al. [2005]. It should be noted that the two-point mixing model (equations (3) and (4)) is more practical than equation (2) because K sat and K dry can be easily measured or derived and no fitting parameter is involved. Other mixing models are summarized in literature [e.g., Robinson et al., 2003]. [9] Introducing medium densities in place of porosity and setting a = 0.5 and K air = 1 convert equation (2) to the following equation [Whalley, 1993]: pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi K a ¼ q K water 1 þ r b r s pffiffiffiffiffiffiffiffiffiffi K solid 1 þ 1 where, r b and r s are bulk and solid densities of the medium, and f =1 r b /r s. For soils whose major component is quartz, it may be assumed that r s = 2.65 g/cm 3 and K solid = 4.7 [Robinson and Friedman, 2003]. For rocks with various minerals, these must be measured or fitted. More models are summarized in literature [e.g., Robinson et al., 2003]. Because of the simple forms, we apply the polynomial function (equation (1)), a-mixing model (equation (2)), and physically based two-point mixing model (equation (3)) to describe the K a -q relationship for the rocks obtained in this study. [10] Accurate estimation of q using TDR requires that the probe must be in a good contact with the material of interest. A probe consists of two or more electrodes (usually metal rods) that are connected to a coaxial cable. A slight void space (hereinafter referred to as gap) between the electrode and the material causes significant measurement error, because the estimated K a is sensitive to the dielectric properties of the material closest to the probe [e.g., Knight, 1992; Knight et al., 1995; Sakaki, 1996; Ferré et al., 1998]. Annan [1977], Ferré et al. [1996], and Knight et al. [1997] quantified the effect of the gaps around the electrodes ð4þ ð5þ 2of15

3 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS analytically and numerically. They showed that the gaps filled with air caused a significant underestimation of K a, whereas water-filled gaps result only in a slight overestimation of K a. [11] The potential of TDR for water content measurement in rocks was first addressed by Hokett et al. [1992]. They hammered penetration-type rod probes (hereinafter referred to as the penetration probe) into slightly undersized guide holes (to minimize gap effects) and attempted to measure water content in sandstone and tuff blocks. To convert the dielectric constant values into water content, they used the well-known Topp s curve (equation (1)) obtained for soils. Although Topp s curve did not lead to accurate water content estimates, individual K a -q calibrations for each rock type improved the measurement accuracy. The low porosity of rocks amplifies the influence of K solid (see equation (2)) or equivalently K dry (equation (3)), thus necessitating individual calibrations. [12] Sakaki et al. [1998a] measured K a -q relationships for nine rock types with porosity values ranging from 1 to 54%. Although they used slightly undersized guide holes, their measured K a -q data showed an abrupt jump at full saturation, which is indicative of a gap effect for the following reason: At low saturations, the gap is filled with air as water tends to be retained in the rock matrix, leading to significant underestimation of the true K a ; whereas close to full saturation, the gap is water-filled and the true K a is overestimated only slightly. Thus the experimentally determined K a -q relationships are not smooth, but show an abrupt jump near full saturation. The measured K a -q curves for rocks deviated substantially from Topp s curve. Sakaki et al. [1998a] developed individual calibration functions for nine different rock types, on the basis of the a-mixing model (equation (2)). Development of satisfactory K a -q calibration functions required fitting parameters in (equation (2)) and a gap thickness, to be consistent with the jump near saturation. The gaps thus lead to many problems in the context of both developing K a -q calibrations, and estimating water contents in the field, where the gap is of unknown thickness and cannot be controlled. Sakaki et al. [1998a] suggested that TDR would be accurate for applications in rocks only when gaps are eliminated. Modifications of the basic penetration probe to eliminate gap effects using conductive silicone coatings have been proposed [Schneebeli et al., 1995; Lin et al., 1999]. However, these studies were only partially successful. Sakaki et al. [1998b] attempted to use different porous fillers (cement paste, plaster and bentonite) in gaps, which also did not satisfactorily eliminate gap effects. [13] Unlike in soils, installation of penetration probes in rocks requires a lot of effort. On the other hand, it is relatively easy to install probes on rock surfaces (hereinafter referred to as the surface probe). Schneebeli et al. [1995] also used surface probes on their granodiorite samples; however, the performance of these probes could not be fully evaluated because of the low porosity. Surface probes have also been used for measuring water content at the soil surface by Selker et al. [1993], who showed that by simply placing the probe wires against the soil surface and recalibrating the K a versus q response, the surface probe matched the accuracy of conventional penetration probes. Surface probes are potentially attractive for use in rocks because of their ease of installation. Previous studies [Schneebeli et al., 1995; Sakaki et al., 1998b] employing surface probes in rocks have glued them onto the rock surface. The dielectric constant of typical adhesives is small (around 2.0, assuming a pulp-like material [von Hippel, 1995]), and may result in underestimation of the true K a, in the same manner as an airfilled gap. However, the influence of the glue layer thickness on the accuracy of K a estimates has not been evaluated systematically to date, though it is an important factor influencing the accuracy of surface probes. [14] In this paper, we have two primary objectives: the first is to demonstrate the ability of conductive silicone fillings to eliminate gap effects in the case of penetration probes. The second is to propose two designs for surface probes in rocks (a conductive silicone adhesive applied as a strip and an aluminum foil tape) and compare the K a -q relationships obtained with these probes against those obtained with the more commonly used penetration probe, for a range of rock types. Our results demonstrate that conductive silicone fillings successfully eliminate gap effects in the case of penetration probes. Both types of surface probes provided smooth K a -q calibrations that led to reasonably accurate water content estimates. The K a -q relationships obtained with the different probe types on the same rock samples exhibited significant differences; we explain these differences on the basis of probe design, and the characteristics of TDR traces obtained with the different probe types. The secondary objective of the paper is to evaluate the accuracy of water content estimates in a simple absorption experiment, considering different probe types and variability in properties within a single block and across different samples of the same rock type. [15] The paper is organized as follows: The rock samples, probe types and installation details, and experimental methods for calibration and absorption experiments are described in section 2. In section 3, we present the K a -q relationships obtained for the different samples using different types of probes. The differences between the different probe types are evaluated and discussed. Section 3 also includes results from the absorption experiment and an evaluation of accuracy in water content estimates obtained with different K a -q relationships. We conclude by summarizing the main results and discussing their implications in the context of applying TDR for field measurements of water content in rocks. 2. Materials and Methods 2.1. Rock Types [16] Seven rock types including a limestone, three sandstones, and three tuffs were used for the laboratory experiments. The sandstone and tuff samples were similar to those used by Sakaki et al. [1998a]; their detailed characteristics are described by Sakaki et al. [1998a]. The limestone sample (Indiana limestone) is a calcite cemented grain stone formed from fossil fragments and oolites [Churcher et al., 1991]. It is made up predominantly of mineral calcite (99%) with a small amount of quartz (1%). All the rock samples used in our study are relatively homogeneous at the sample scale (slight layering can be seen on Tako sandstone samples, relatively large hole-like pores (5 mm) on Ogino tuff samples). This avoids the potential problems in interpretation of TDR response arising from nonuniform water 3of15

4 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Table 1. Physical Properties of Samples Rock Type Specimen Code Effective Porosity f Dry Bulk Density r b, g/cm 3 Solid Density r s, g/cm 3 Indiana limestone IL Indiana limestone IL Kimachi sandstone KS Kimachi sandstone KS Shirahama sandstone SS Shirahama sandstone SS Tako sandstone TS Tako sandstone TS Shirakawa welded-tuff SW Shirakawa welded-tuff SW Tage tuff TT Tage tuff TT Ogino tuff OT Ogino tuff OT content distributions. For each rock type, two samples with dimensions of cm were cut from a single larger block of rock and prepared for analysis. The physical properties of the samples are summarized in Table 1. The effective porosity and dry bulk density for the two samples are almost identical for each rock type, presumably because they were cut from a single block Probe Configurations and Installation Methods [17] A two-wire/strip probe was chosen because it was easier to install than other probes with more wires/strips. On each sample, three types of probes were installed as shown in Figure 1. A modified penetration probe, designed to eliminate gap effects, was installed in the middle of each sample and conductive silicone and aluminum tape surface probes were installed on the opposite surfaces. The performance of the probes was evaluated by first measuring K a -q relationships for seven rock types and then examining whether the relationships show potential errors due to gaps. The K a -q relationships obtained with different probe types were then compared Modified Penetration Probe [18] A penetration probe is usually inserted into the material of interest and the electrodes are surrounded by the material. This is the most widely used configuration. A perfect installation is achieved when gaps between the electrodes and rock are completely eliminated. The modified penetration probe was constructed by employing conductive silicone filling to eliminate gap effects (Figure 2). An electrically conductive room temperature-vulcanized adhesive sealant SSP-779-Silver (Specialty Silicone Products Inc., New York, volumetric resistivity = 0.01 Wcm, hereinafter referred to as conductive silicone) was used to fill the gaps. The procedure for installing the probes is modified from Lin et al. [1999] and described below. [19] The first step in the installation of the penetration probes was to drill guide holes with approximately 7 mm diameter. The guide holes were then filled with conductive silicone. A hollow brass pipe (O.D. = 6.35 mm, I.D. = 3.97 mm) was then slowly inserted with a twisting motion to help the conductive silicone completely fill the space between the pipe and rock. A solid brass rod (diameter = 3.97 mm) was then hammered slowly into the pipe, until silicone flowed out of the gaps. Returned and excess 4of15 silicone was removed, and the solid rod was slowly removed. The sample was then set in an oven for the silicone to cure. The inside of the brass pipes was partially threaded so that the probe head could be attached. The probe head consisted of a cm acrylic plate and two screws onto which a 2.5 m coaxial cable (RG-58A/U) was screwed. The 2.5 cm of the coaxial cable was stripped and split, and open terminals were soldered onto the center conductor and the shield (Figure 2 and Figure A1 in Appendix A). With this configuration, the offset of the probe head was 6.5 cm Surface Probes [20] Installation of the penetration probe requires a lot of time and effort. An alternative approach is to place the electrodes onto the rock surface. Maheshwarla et al. [1995] and Knight et al. [1997] analytically studied the characteristics of such a surface probe. They showed that for negligible probe thickness, the dielectric constant K 0 a obtained directly from the surface probe (hereinafter referred to as the effective dielectric constant) is the arithmetic mean of the apparent dielectric constant of the rock K a and that of air (K air = 1). Therefore K a can be calculated from K 0 a by the following simple relationship. K a ¼ 2K 0 a 1 The range of K a was 4 25 for the rocks used in this study. Because the rocks have larger dielectric constant than that of air, most of sampling volume should be in the rocks [Robinson et al., 2003]. [21] As mentioned above, the surface probes used for rock water content measurement in the previous studies [Schneebeli et al., 1995; Sakaki et al., 1998b] were glued onto the rock surface. The potential influence of the glue layer on measurement accuracy has not been evaluated to date. In this paper, we examined the performance of surface probes that were made of two different materials: conductive silicone and aluminum foil tape. The installation Figure 1. Schematic view of a sample with a penetration probe and two surface probes. ð6þ

5 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure 2. Schematic of the penetration probe installation. The brass pipe and conductive silicone layers together serve as electrodes. procedures for these surface probes are described below. The probe head was similar to that of the penetration probe except that the screws extended through the 1.5 cm air layer between the rock surface and the 12-mm-thick acrylic plate (Figure A2 in Appendix A). This was done to be consistent with the block experiment to be explained in section 2.4. With this configuration, the offset of the probe head was 8.2 cm Conductive Silicone Surface Probe [22] The electrically conductive silicone was directly applied onto a smooth surface of the sample to form a pair of strips. A 0.3-mm-thick template from which the shape of the strips had been cut out was fixed at the appropriate position on the rock surface. The conductive silicone was carefully applied so that the thickness of the silicone was roughly the same as the template, and then the template was carefully peeled off. [23] The silicone strips were 10 cm long, 0.5 cm wide, 2.5 cm apart, and approximately 0.3 mm thick. Applying the conductive silicone directly onto the surface of the sample should eliminate the gaps completely, as the silicone is also expected to fill in the pores on the surface. The SSP- 779-silver silicone cures faster with exposure to high humidity and high temperatures. To reduce the curing time, the sample was moderately wetted before applying the silicone, and then placed in an oven at 103 C for a week. The resistivity of the silicone decreased drastically in the early stages of curing and did not change much after one day, suggesting that a curing time of the order of a day may be sufficient in typical applications of this probe Aluminum Tape Surface Probe [24] An aluminum tape surface probe used was constructed using commercially available kitchen foil tape. The tape consists of aluminum and adhesive layers. The thicknesses of the aluminum and adhesive layers were both 0.05 mm. The tape was cut to the predetermined size and applied at the appropriate position on the rock surface of the dry sample. After some trial and error, it was found that the aluminum foil tape stayed on the rock surface more securely if the sample was placed in an oven at 103 C for a few days after initial installation of the tape strips. This was probably because the adhesive layer melted onto the rock surface at higher temperatures. The tape strips were 10 cm long, 0.5 cm wide, and 2.5 cm apart Measurement of K a -Q Relationships [25] The TDR equipment used in this research was the TDR100 system [Campbell Scientific, Inc., 2000]. For each sample, K a was measured at 11 water content levels at roughly 10% saturation increments. The samples were first placed in a water-filled vacuum desiccator for more than two weeks to achieve full saturation. It was assumed that the specimen was fully saturated if no more weight change was observed. When full saturation was achieved, TDR traces were digitally recorded and K a values were calculated using the PCTDR software package [Campbell Scientific, Inc., 2000] designed for the TDR100 system. The effective dielectric constant directly obtained from the surface probes (K 0 a) was converted into the apparent dielectric constant (K a ) by equation (6). Only for Indiana limestone, K a was also measured through a sequence of increasing water content in a wetting cycle to investigate if the K a -q relationship was hysteretic. It should be noted that every K a measurement reported in this paper is an average of five individual measurements taken at the same water content. Taking an average over five measurements reduces the associated measurement error. For one of the rock types (Indiana Limestone), the measurement error was estimated on the basis of a statistical analysis of 100 individual measurements on the dry samples and at full saturation. For the other six rock types, similar analysis was done only on the dry samples. [26] The water content was reduced stepwise by allowing evaporation from the samples. At lower water contents, where natural evaporation is too slow, the samples were placed in an oven at 103 C until a predetermined mass was achieved. The samples were wrapped in polyethylene sheets and left for a certain period to allow the water to redistribute within the samples. In preliminary tests, K a was measured every day during redistribution until no significant change in K a was observed between measurements. The redistribution period varied from a few days at high saturation where water could redistribute quickly, to a few weeks for low saturations (<30%) where redistribution was slower. On the basis of these preliminary experiments, we decided to use a redistribution period of roughly 20 times the drying interval before each measurement. This procedure was repeated until the samples were dry. All the measurements were taken with the samples wrapped with a mm-thick polyethylene sheet so that no evaporation occurs during the measurement Evaluation of Probes and K a -Q Calibrations in a Simple Absorption Experiment [27] We also present an illustrative evaluation of the accuracy of one of the estimated K a -q relationships (for Indiana 5of15

6 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure 3. Schematic view of the apparatus for the water absorption experiment in an Indiana limestone block. The block is sealed within an acrylic case to prevent surface evaporation. limestone) based on a simple absorption experiment in an Indiana limestone block with dimensions of 7.8(H) 14(W) 23(L) cm. The effective porosity of the block was [28] The block was contained in a sealed acrylic case and the left-end surface (x = 0) was subjected to water absorption (Figure 3 and Figure A3 in Appendix A). The hydraulic head at the supply boundary was kept constant (at the top surface of the block) using a Mariotte tank. The masses of the water reservoir and the block were monitored continuously to measure the total amount of water absorbed by the block (hereinafter referred to as the gravimetric method). On the top surface of the block, aluminum tape surface probes were installed as shown in Figure 3. The probes were aligned so that the water content distribution in the direction of water absorption could be measured. The probe strips were 10 cm long, 0.4 cm wide, and 1 cm apart. Between the top surface and the acrylic lid, a 1.5 cm-thick Styrofoam packer was placed to avoid evaporation. The dielectric constant of Styrofoam is the same as that of air, so that equation (6) is still valid for calculating K a. By using every two adjacent strips, the water content was measured at 22 points within the block at 1 cm intervals. The total amount of water absorbed by the block was obtained by integrating the TDR-measured water content profile within the block, and was compared to the results from the gravimetric method. 3. Results and Discussion 3.1. K a -Q Relationships for Seven Rock Types [29] In Figures 4a 4g, the measured apparent dielectric constants K a are plotted against volumetric water content q for each rock type. As noted previously, the values shown are averages based on five measurements each from two samples for each rock type. The variation in K a -q data between the two samples of each rock type was insignificant, largely because they were cut from a single larger block and had almost identical physical properties (Table 1). The K a values reported in Figures 4a 4g for the surface probes were converted from K 0 a according to equation (6). In Figure 4a, the K a -q data in both wetting and drying cycles are shown. The data from Sakaki et al. [1998a] using penetration probes without silicone fillings are plotted in Figures 4b 4g for comparison. A third-order polynomial (equation (1)), a- mixing model (equation (2)), and two-point mixing model (equation (3)) were applied to the K a -q data obtained by each probe type. The fitted coefficients in the third-order polynomial (A, B, C, and D) and parameters in the a-mixing model (K solid and a) are summarized in Table 2 together with the measured parameters (f, K sat, and K dry ). In the a-mixing model, a higher weight was awarded to a = 0.5 so that a was practically fixed to 0.5. The coefficients of determination (r 2 ) are also given. The mean r 2 values are 0.994, 0.975, and for the third-order polynomial, a-mixing model, and two-point mixing model, respectively, showing that all models describe the K a -q data very well. The third-order polynomials provided slightly better fits to the data, especially in the case of surface probes because the model has four fitting parameters and thus the highest flexibility. The a-mixing model had less flexibility as K solid was practically the only fitting parameter and resulted in slightly smaller r 2 values. The two-point mixing model includes no fitting parameters but only two easy-to-measure parameters. Despite the least flexibility of the two-point mixing model among three calibration models used in this study, the mean r 2 value of means that this model can describe K a -q data for rocks with reasonable accuracy on the basis of actually measured K sat and K dry. The measured values of K sat and K dry for rocks used in this study varied over a range of and , respectively, in the case of penetration probe. The variation in the K sat and K dry values of the rocks is due to the K solid and f as inferred from equation (2). Although not shown here, the measured values of K sat and K dry and corresponding values estimated from equation (2) using the fitted K solid values in Table 2 agreed reasonably well. We did not perform any further theoretical evaluation of K sat and K dry based on a two-phase grain-scale mixing model [e.g., Sihvola and Kong, 1988] because it was not the main focus of this study. In addition, for quartz-dominant soils for which K dry = 2.8 is typically assumed, the two-point mixing model can be further simplified to one-point model [Robinson et al., 2005]. However, for rocks with a wide range of K dry values, the onepoint model is not appropriate Modified Penetration Probe [30] The K a -q relationships obtained using the modified penetration probe were smooth increasing curves. The abrupt jump near full saturation seen in the data of Sakaki et al. [1998a], which is indicative of a gap, did not appear for any of our samples. Elimination of the jump suggests that the gaps had been successfully eliminated by the conductive silicone and improved measurement accuracy was achieved. The differences between the calibration Figure 4. (a g) Experimentally measured variation of K a (apparent dielectric constant) with q (volumetric water content) for different rock types measured with different types of TDR probes. The type of rock is listed inside each plot. The symbols in each plot correspond to different probe types (circle indicates penetration probe, cross indicates conductive silicone surface probe, and triangle indicates aluminum tape surface probe). For Figures 4b 4g, the K a -q data of Sakaki et al. [1998a] using penetration probes without conductive silicone coatings on different samples are also shown for comparison (solid circles). For Indiana limestone in Figure 4a, K a -q relationships were measured in both drying and wetting cycles. No significant hysteresis was observed. At zero and full saturations, confidence intervals (±2 standard deviations estimated from statistical analysis of 100 measurements) are also shown. 6of15

7 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure 4 7of15

8 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Table 2. Fitted Parameters and Coefficients for K a -q Relationships Rock Type Probe Type a Effective Porosity, f b a-mixing Model, Third-Order Polynomial Model, Equation (1): q = AK 3 a + BK 2 a + CK a + D Equation (2): K a a =(f q) K air + a a q K water +(1 f) K solid A B C D r 2 c K solid Two-Point Mixing Model, p ffiffiffiffiffi Equation pffiffiffiffiffiffiffi (3): K a = +(1- q f ) pffiffiffiffiffiffiffiffi q f a r 2 K sat b K sat K dry b K dry r 2 IL p e e e e IL s e e e e IL t e e e e KS p e e e e KS s e e e e KS t e e e e SS p e e e e SS s e e e e SS t e e e e TS p e e e e TS s e e e e TS t e e e e SW p e e e e SW s e e e e SW t e e e e TT p e e e e TT s e e e e TT t e e e e OT p e e e e OT s e e e e OT t e e e e Mean a Probe types are as follows: p, penetration probe; s, silicone surface probe; t, tape surface probe. b Measured values were used. For quartz-dominant soils, K dry = 2.8 can be assumed [Robinson et al., 2005]. c [Shen et al., 1985] for most rocks and for limestone [Wharton et al., 1980], 4.7 for quartz sand, 6.0 for tuff, and 8.9 for calcareous seashell fragments [Robinson and Friedman, 2003]. relationships of Sakaki et al. [1998a] and our results here are not entirely because of gap effects in the data of Sakaki et al. [1998a]. Additional differences result because we did not use the exact same samples as Sakaki et al. [1998a]. Note that in some cases, there are differences in porosity between our samples and those used by Sakaki et al. [1998a]. This suggests that variation in porosity and other physical properties within the same formation can potentially lead to variations in K a -q relationships. [31] The third-order polynomial and the a-mixing model both gave reasonable fits to the data when appropriate parameters were chosen. With only two easy-to-measure parameters, the two-point mixing model also described the K a -q relationships well. Although not shown in Figures 4a 4g, Topp s curve does not adequately represent the K a -q relationships for these rocks. This is because there is a greater influence of K solid on K a for low-porosity rocks [Sakaki et al., 1998a]. Because the regression relationships have a very high r 2 (see Table 2), potential errors in q estimates arises largely from measurement errors in K a.to quantify the error in estimating q, we carried out 100 measurements of K a on the Indiana limestone samples at each of two water contents (dry and fully saturated). On the basis of a statistical analysis of the 100 measurements of K a, resulting q estimates converted using the K a -q relationship resulted in ±2 standard deviation bounds (hereinafter referred to as the ±2s bounds) around individual q estimates of ± at dry end and ± at the fully saturated end. The K a -q data obtained by Topp et al. [1980] using four mineral soils with a wide range of textures and varying organic matter contents showed that their ±2s bound in q estimates was ± for four soils and ± for a sandy loam soil. Our data for Indiana limestone thus show that the accuracy of water content measurement in rock with the modified penetration probe is comparable to that of TDR measurements in soils. Because all the q estimates reported in this paper are averages taken from five individual measurements, the ±2s bounds at the dry and saturated end for Indiana limestone are reduced to ± and ± respectively. Similar statistical analysis on the other six rock types using 100 measurements of K a on dry samples showed that the ±2s bounds are of the same order of magnitude (Table 3). The measurement error is thus largely due to the fact that the TDR traces are influenced by noise in the signal, which leads to slight variations in estimated K a values Surface Probes Conductive Silicone Surface Probes [32] The K 0 a values obtained by the conductive silicone probe were converted into K a values using equation (6). The K a -q relationships were smooth and no jump was observed near full saturation, demonstrating again that by applying the conductive silicone directly on the rock surface to form the electrodes, a perfect contact between rock and electrode had been achieved. It should be noted that the K a values estimated using the conductive silicone surface probes were smaller than those estimated by the modified penetration probe (by up to 20% in some cases). Possible reasons for this discrepancy will be discussed below in section The ±2s bounds for the conductive silicone probe shown in Figure 4a are, especially at full saturation, significantly wider (± in q when taking an average of five measurements) than those for the other two probe types. It is still comparable to the measurement accuracy 8of15

9 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Table 3. ±2s Bounds for Seven Rock Samples at 0% Saturation a Rock Type Penetration Silicone Tape IL KS SS TS SW TT OT a The numbers are ±half width in q around the mean. for soils. We suspect that aging of the conductive silicone is a possible cause for these wider ±2s bounds. For long-term water content monitoring in the field, durability of the conductive silicone needs to be investigated further Aluminum Tape Surface Probe [33] The aluminum tape surface probe also yielded smooth K a -q relationships with no jumps near saturation. The adhesive layer used with this probe has a small dielectric constant and could hence have the same influence as an air gap. Since the adhesive layer cannot be filled with water near full saturation, unlike an air gap, there is no jump in the K a -q relationship. For the same value of q, the K a values obtained with the aluminum tape probe were smaller than the values obtained with the penetration probe (by up to 25% in some cases) and nearly identical to the values obtained with the conductive silicone surface probe. The ±2s bounds for the aluminum tape probe shown in Figure 4a are slightly wider than those for the modified penetration probe. The measurement error for water content estimate (when taking an average of five measurements) is ± at most in q around the mean value near the dry end and ± near the wet end. Thus the aluminum tape surface probe also yields measurement accuracy comparable to that of TDR water content measurement for soils. [34] The ±2s bounds shown for three calibration functions in Figure 4a clearly show that the calibration curve for the penetration probe falls well outside the ±2s bounds for calibration functions for the conductive silicone and aluminum tape surface probes. This suggests that significant errors may arise if different probe types are used for calibration versus field measurement. We consider this issue further below Comparison of K a -Q Relationships Estimated From Different Probe Types [35] In this section, we evaluate the sources of discrepancies between the K a -q relationships obtained with different probe types. We begin by considering surface probes. When converting K 0 a values to K a using equation (6), systematic errors result from the asymmetry in the geometry of surface probes. They are placed entirely on the air side of the system, rather than symmetrically (with half the thickness in air and half in the sample) as assumed in (equation (6)). To estimate the systematic error that results from this source, numerical simulations were carried out to estimate the apparent dielectric constant K a for an asymmetric probe, on the basis of the procedure of Knight et al. [1997]. In these simulations, the true K a was varied from 5 to 25 and K 0 a was computed numerically from a solution of the Laplace equation, with electrodes 2.5 cm apart and 0.5 cm wide, and asymmetrically placed, as in the experiments. The 9of15 thickness of the probe (in the case of the conductive silicone probe) and adhesive layer (in the case of the aluminum tape probe) were the same as in the experiment, and the results are summarized in Figure 5. [36] In the case of the conductive silicone probe with a thickness of 0.3 mm corresponding to the experimental probes, K a is underestimated by about 4 5%. In the case of the aluminum tape probes, both the tape and the adhesive layer contribute to the error. The adhesive layer has a low dielectric constant of the order of 2, typical of pulp-based adhesive materials. It is evident from Figure 5 that the systematic error in estimating K a increases with water content. With a thickness of 0.05 mm for both the tape and adhesive layers (as in our measurements), K a is underestimated by about 2 11%. Note that this effect should depend on the thickness and dielectric properties of the adhesive material. However, because the source of error is systematic, we may correct/adjust K a estimates obtained with surface probes on the basis of the results shown in Figure 5. All the data measured with the surface probes were adjusted using the error estimates shown in Figure 5. The adjusted K a values estimated with the aluminum tape probe are plotted in Figure 6 against the adjusted K a values estimated with the conductive silicone probe. The data roughly fall on the 1:1 line given in Figure 6. [37] In Figure 7, the difference (dk a ) between the K a values estimated with the penetration probe and the adjusted values estimated using surface probes for all rocks is plotted. The difference dk a varies with K a values estimated using penetration probe (thus water content) in a cyclic manner. This cyclic variation was systematic across all rock types and we thus suspected that it results from the TDR traces and the trace analysis procedure. A detailed examination of the TDR traces and their derivatives revealed that multiple reflections [e.g., Yanuka et al., 1988] were affecting the trace shapes in the case of penetration probes. It had not been anticipated prior to the experiments. Multiple Figure 5. Influence of asymmetric electrode placement on K a estimates with surface probes. The conductive silicone probe is 0.3 mm thick. The aluminum tape probe involves an aluminum tape and an adhesive layer, each 0.05 mm thick. Differences between the K a estimates obtained with surface probes and true K a were quantified using finite element simulations.

10 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure 6. K a (tape) versus K a (silicone) for all rocks, with both data adjusted to account for asymmetric probe placement (dashed lines correspond to ±10% deviations). reflections caused by the probe head are independent of rock type as the head is always in air; their magnitudes and locations are specific to probe structure (head type, probe length, etc.). The K a values are calculated on the basis of finding the intersection of two tangent lines on a TDR trace, and the position of the intersection is considered to represent pulse reflection [e.g., Baker and Allmaras, 1990]. With the penetration probes used in this study, multiple reflections near the main pulse reflection led to slight changes in postreflection slope on most traces at low saturations. The variation of postreflection slope with water content was found to be cyclic, matching the trend observed in Figure 7. For example, the overestimation of K a due to the postreflection slope alteration was estimated to be approximately 0.7 (8% relative error for K a = 8 9) for Indiana limestone in the 10 20% saturation range where the largest effect of multiple reflections was observed. In the case of surface probes used in this study, however, no significant multiple reflections were seen near the pulse reflection, as a result of which postreflection slopes did not vary with water content. The typical deviation of surface probe estimates from the penetration probe estimates for Indiana limestone (Figure 4a) is Although the true K a values are not known, the deviations (e.g., for Indiana limestone) of individual surface probe estimates from the true K a are partially (e.g., from Figure 5, 1 for Indiana limestone which showed K a = 8 15) explained by probe asymmetry and thickness. The deviations of penetration probe estimates from the true K a explained by slope alteration resulting from multiple reflections (e.g., 0.7 for Indiana limestone) is relatively insignificant comparing to the total deviations between K a estimates obtained using different probe types. The cause for the rest of the deviations (of surface probe estimates from the penetration probe estimates) remains unknown. Since the effect of the multiple reflections at the point of pulse reflection decay with length, use of longer penetration probes may reduce this effect significantly and yield more accurate K a values. A better design of the probe head would also reduce the multiple reflections. In practical applications of TDR for water content estimation in rocks, the above deviations may not be very critical if the same probe type, dimensions (length, separation, etc.) and TDR system are used in the development of K a -q calibrations and for field measurement. This is because, similar deviations from the true K a occur in both calibration and measurement, and the calibration already accounts for these deviations Accuracy of Water Content Estimation in a Simple Absorption Experiment Estimation of Water Absorption Into Block [38] The results described in section 3.1 suggest that different K a -q calibration functions may be appropriate for different probe types. A question that then arises is whether the differences in K a -q calibrations can lead to significant differences in water content estimates. For instance, what is the error resulting from using the calibration function developed using the penetration probe, for interpreting field or laboratory TDR measurements involving surface probes? To address this issue, we evaluated the accuracy of water content estimates in a simple absorption experiment. The absorption experiment involved a larger Indiana limestone block (7.8(H) 14(W) 23(L) cm, average effective porosity of 0.155, hereinafter referred to as absorption block) with slightly different properties from the samples used for calibration (effective porosity = 0.135). The absorption block was obtained from the same quarry as the calibration samples. However, we have no information on their relative locations within the quarry. This experiment also offers some insights on the implications of natural heterogeneity within the same rock type for water content estimation using TDR. A K a -q calibration function was developed for the absorption block in a uniform drying experiment. The uniform drying experiment was conducted prior to the absorption experiment, following a procedure similar to that described in section 2.3. The K a values were estimated on the basis of 22 aluminum tape surface probes, uniformly spaced over the length of the block. At each probe, five measurements were taken and averaged. [39] During the uniform drying experiment, although the water content was assumed to be uniform within the block Figure 7. Difference (dk a ) between K a estimated with penetration probe and adjusted surface probe values for all rocks. Note the cyclic variation of the dk a with K a (thus water content). 10 of 15

11 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure 8. K a -q relationships for Indiana limestone (calibration A, absorption block with aluminum tape probe; calibration B, calibration sample with aluminum tape probe; and calibration C, calibration sample with penetration probe). For calibration A, ±2s bounds estimated from the 22 sets of measurements using 22 probes are shown by circles. The ±2s bounds for calibrations B and C are estimated from statistical analysis of 100 measurements on the dry sample and at full saturation. after each redistribution period, K a values estimated from different pairs of probes exhibited some scatter (±2s bounds for estimating q are calculated on the basis of this scatter as explained below), potentially due to spatial variation in rock properties within the block. Therefore, at each water content level, the K a values measured with the 22 probes (each of which is an average of five measurements) were averaged and a third-order polynomial, which provided a better fit than the a-mixing and two-point mixing models in this case, was fitted to the mean. The resulting function for the absorption block is as shown in equation (7) and shown in Figure 8 as calibration A. content within the block and thus water absorption into the block can be estimated accurately. The aluminum tape surface probe can thus be considered as a reliable tool for estimating water content in rock, if calibrated appropriately Uncertainty in Water Content Estimates Due to Variations in K a -Q Relationships [42] In the previous section, the water content distribution in the block was estimated using the K a -q calibration relationship obtained on the same block. In real-world situations, calibration relationships may need to be developed using representative samples obtained from the site. This may lead to an additional source of error as the rock properties influencing K a -q relationships (e.g., porosity) vary between different measurement locations. Furthermore, different probe types or probe dimensions may be used at different locations on the basis of installation considerations. To illustrate the potential inaccuracies of water content estimates in such a setting, we converted K a values in the absorption block to q using the K a -q relationships (calibrations B and C shown in Figure 8), from the small calibration samples (IL-1 and IL-2). Calibration B was estimated from the small calibration samples (IL-1 and IL-2) using the aluminum tape probe. Calibration C was obtained from IL-1 and IL-2 using the modified penetration probe. To be consistent with calibration A (equation (7)), third-order polynomials were used (the coefficients are provided in Table 2, calibration B: ILtape, calibration C: IL-penetration). [43] It may be noted from Figure 8 that calibrations A and B agree very closely at q < and at q > 0.130, but deviate at intermediate water contents. Calibration C will lead to uniformly lower water content estimates in comparison to calibrations A and B. The cumulative absorption per unit area and water content distributions estimated with the different calibration functions are shown in Figures 9 and 10, respectively, and are consistent with the expectations based on Figure 8. Calibration A leads to a q estimate of q ¼ 0: K 3 a 0:0139 K2 a þ 0:159 K a 0:570 ð7þ [40] Also shown in Figure 8 are ±2s bounds estimated from the 22 sets of measurements, each of which is an average of five measurements. These bounds reflect error in water content estimates due to two sources: (1) spatial variation in rock properties (e.g., porosity) within the block and (2) measurement error as presented in Figure 4a. The ±2s bounds are ± in q and are of the same order of magnitude as the ±2s bounds observed for the calibration samples (±0.0075). [41] In the absorption experiment, the cumulative absorption per unit area I(t) into the block was calculated by first integrating the water content profile measured by the surface probes over the entire block length and then dividing it by the absorption area. In Figure 9, the results (using calibration A) are plotted against time together with the I(t) measured by the gravimetric methods (computed from the mass gained by the block). The estimated amount of water absorbed obtained using the TDR measurements agrees nearly perfectly with the gravimetric estimate. This demonstrates that with an appropriate K a -q relationship, the water 11 of 15 Figure 9. Cumulative absorption I(t) versus time. The gravimetric estimate is compared to estimates obtained by integrating TDR measurements of water content distribution (calibration A was obtained from the absorption block with aluminum tape probe, calibration B was obtained from the calibration sample with aluminum tape probe, and calibration C was obtained from the calibration sample with penetration probe, see Figure 8).

12 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS may appear as though the a-mixing model or two-point mixing model, which explicitly involves the porosity (f), may be able to capture some of the variability within the same rock type. However, using values for K solid, K sat, K dry, and a from IL-1 and IL-2 together with the different porosity value for the absorption block did not appreciably improve water content estimates. In fact, the water content values estimated with this approach were less accurate than obtained with calibration B in Figure 10. Figure 10. Comparison of water content distributions estimated from different calibration functions (at t = 24 hours) (calibration A was obtained from the absorption block with aluminum tape probe, calibration B was obtained from the calibration sample with aluminum tape probe, and calibration C was obtained from the calibration sample with penetration probe, see Figure 8). about at the absorption face, while Calibrations B and C yield estimates of and respectively. Calibration B yields a higher water content estimate than A over almost the entire region behind the wetting front. Correspondingly, the cumulative amount of water absorbed is overestimated, e.g., by about 16% at over a 24-hour period. With calibration C, the water content in the block is significantly underestimated everywhere, including physically meaningless negative values in the driest regions. [44] Considering the ±2s bounds shown for the various calibration functions in Figure 8, it is clear that the calibration function C for the penetration probe falls well outside the ±2s bounds for calibration functions A and B. Calibrations B and A agree quite closely, except in the range < q < which is the typical range of water content behind the wetting front. In this range, unfortunately, calibration B falls outside the ±2s bounds of calibration A. It leads to an overestimation of q by behind the wetting front as seen in Figure 10, which is still within the range of measurement errors observed in soils as mentioned in section The width of the ±2s bounds for calibration A (mostly representing error due to spatial variation in rock properties within the block) is roughly as same as that for calibration B (representing error in a single K a measurement), Thus both errors in a single K a measurement and spatial variation in rock properties can lead to uncertainty in estimating q. However, in the region behind the wetting front, the water content values are in the range where the difference in K a -q relationships between the calibration samples and the absorption block is the more important source of error. These results suggest that K a -q calibrations obtained with the same probe type are preferable for estimating water content in the field. It will be also helpful to take as many measurements as possible and average them so that the ±2s bounds are narrowed. Even with the same probe type, spatial variation in rock properties can lead to inaccuracies in estimating water content. At first glance, it 4. Summary and Conclusions [45] Application of TDR for water content measurement in rocks requires careful attention to three important issues: (1) elimination of inaccuracies resulting from gap effects, (2) difficulties in installation of TDR probes, and (3) the need to develop individual K a -q calibrations specific to each rock type. In this paper, we have addressed each of these issues. We have demonstrated that conductive silicone fillings, if carefully applied, can successfully eliminate gap effects in the case of penetration probes. The use of conductive silicone fillings led to smooth K a -q calibrations on seven different rock types, without a jump near saturation (caused by a gap) as has been observed in previous studies [e.g., Sakaki et al., 1998a]. The elimination of gap effects and hence the jump near saturation will significantly improve the accuracy of both K a -q calibrations and field water content measurements. Surface probes are an attractive alternative to penetration probes, because they are much easier to install and do not require guide holes. We have proposed two potential surface-probe designs for use in rocks (a conductive silicone probe and an aluminum tape probe) and evaluated their performance carefully. Smooth K a -q calibrations were obtained with both types of surface probe on all the seven rock types investigated, indicating very little uncertainty attributable to gap effects. These surface probes can be installed relatively easily on smooth rock surfaces, such as drift walls. With a modified installation procedure, they may also be installed on the walls of boreholes. [46] The K a -q relationships obtained with the conductive silicone and aluminum tape probes differed slightly, and both exhibited systematic differences from the corresponding relationships obtained with the penetration probes on all the samples. These differences were partially explained on the basis of the probe thickness (conductive silicone probes), probe and adhesive layer thickness (aluminum tape probes) and multiple reflections associated with the probe structure (penetration probes). In the case of surface probes, accuracy can be improved by adjusting estimated K a values to account for the asymmetry induced by probe thickness. While the occurrence of minor multiple reflections near the main pulse reflection in our penetration probes have led to insignificant errors in K a estimates, it should be noted that this feature is specific to the probe with the head structure presented in Figure 2 and an electrode length of 10 cm used here. A better design of the probe head and use of a longer electrode (20 30 cm long) would reduce such effects. In practical applications of TDR for water content measurement in rock, systematic deviations of K a values estimated with each of the probe types from the true K a value should not be a matter of concern, as long as the same probe type with the same dimensions is used for 12 of 15

13 SAKAKI AND RAJARAM: DIFFERENT TYPES OF TDR PROBES FOR ROCKS Figure A1. Probe head for the penetration probe consisting of a 9-mm-thick acrylic plate and two metal screws. The brass pipes installed onto the sample are partially threaded. A RG-58A/U cable is attached as shown. The length of the split is 2.5 cm. The offset value is 6.5 cm. both developing Ka-q calibrations and field measurements. With such an approach, systematic error likely to be incurred at a measurement location is already partially Figure A2. Probe head for the surface probe consisting of a 12-mm-thick acrylic plate and two screws. The head is pushed onto the surface of the sample. A 1.5-cm-thick layer of air between the head and rock surface gives a welldefined boundary condition (half rock, half air), which is consistent with the block shown in Figure A3. A RG-58A/U cable is attached as shown. The length of the split is 2.5 cm. The offset value is 8.2 cm. Calibration was performed for two separations (2.5 cm and 1.0 cm, the latter is consistent with the block experiment in Figure A3). Both showed identical Ka-q relationships. accounted for in the calibration, as long as gap effects are either eliminated or uniform across all water contents. Each of the three probe types considered here is thus capable of providing reasonably accurate water content estimates in field situations (although for a long-term water content monitoring in the field, durability of the conductive silicone probe needs to be investigated further). [47] In a simple water absorption experiment, we demonstrated the accuracy of the aluminum tape surface probe for water content measurements. The estimated cumulative amount of water absorbed using the aluminum tape probe agreed nearly perfectly with the gravimetric estimate. Evaluation of water content estimates from the absorption experiment further highlighted the need for using the same probe type during calibration and field measurement. [48] Another important feature revealed by our investigations is that heterogeneity within the same rock formation is a potential source of error in TDR measurements of water content. The two calibration samples cut from the same piece of rock (Table 1) had almost identical physical properties for all the rock types, and did not exhibit significantly different Ka-q relationships. However, there were differences in physical properties and Ka-q relationships between our samples and corresponding samples of Sakaki et al. [1998a], which were from the same formation. The Ka-q relationship of the Indiana limestone block used in the absorption experiment was also different from that of the calibration samples at high water contents. The semiempirical a-mixing model and physically based two-point mixing model, that explicitly involve the porosity as a parameter, could not explain this difference in Ka-q relationships on the basis of porosity variation alone. From the viewpoint of applying TDR for water content estimation in rock, it thus appears that small variations in Ka-q relationship within the same rock type/formation can lead to error in estimates of hydrologic fluxes over long time periods. This is an issue that warrants further systematic investigation. In field settings, it may thus be appropriate to develop calibrations on several samples to quantify both the mean and variability in Figure A3. Absorption block with 22 probes on the top surface. The 1.5-cm-thick layer above the surface was filled with Styrofoam during the experiment. The top lid is a 12-mmthick acrylic plate. Cables are not shown. The electrode length and spacing are 10 cm and 1.0 cm, respectively. 13 of 15