Published online 8 March 2006
Published in Vadose Zone J 5:459-468 (2006)
DOI: 10.2136/vzj2005.0053
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Water Content Reflectometer Application to Construction Materials and its Relation to Time Domain Reflectometry
Klas Hanssona and
Lars-Christer Lundinb,*
a Dep. of Land and Water Resources Eng., Royal Institute of Technology, Stockholm, Sweden
b Air and Water Science, Dep. of Earth Sciences, Uppsala Univ., Uppsala, Sweden
* Corresponding author (lars-christer.lundin{at}hyd.uu.se)
Received 4 April 2005.
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ABSTRACT
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Moisture content measurements using time domain reflectometry (TDR) or water content reflectometry (WCR) are basic in many research areas. The goals of this study were to establish a relation between apparent dielectric number and WCR sensor output, to compare TDR and WCR calibration equations for two coarse road construction materials, and to investigate the influence of sampling volume for horizontally installed sensors. Measurements were performed in fluids of known dielectric number and in two incrementally saturated coarse materials. The effect of sampling volume was evaluated using an electrostatic finite-element model. A two-parameter equation was determined relating the apparent dielectric number to Campbell CS616 (Campbell Scientific, Logan, UT) WCR output (r2 > 0.99). A simple calibration can adapt the equation to individual CS616, or similar, sensors. For the finer, coarse material, a three-phase mixing model proved best, while for the coarser material no equation adequately described the measurements. Numerical simulations indicated that limited capillary rise, creating a rapid transition from wet to dry close to saturation, was the explanation, warranting caution when interpreting measurements in nearly saturated coarse materials.
Abbreviations: TDR, time domain reflectometry • WCR, water content reflectometry
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INTRODUCTION
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MOISTURE CONTENT of porous media is a fundamental property in a number of applications in soil science, climate studies, agriculture, and civil engineering. Examples include drainage, irrigation, tillage, energy balance studies, compaction of agricultural soils as well as construction materials and assessment of strength of constructions. Various techniques based on dielectric material properties are widely used to monitor moisture in soils and other porous media. One of the most popular techniques is TDR, where the moisture sensor consists of a probe that is inserted into the porous material, after which a control unit transmits pulses of electromagnetic waves into the probe. The dielectric properties of the material surrounding the probe affect the electromagnetic wave, which can be captured by an oscilloscope for analysis (Nissen and Møldrup, 1995), or interpreted directly using an electronic circuit designed for this purpose (Bilskie, 1997). The output may thus be a trace captured by an oscilloscope, or a value derived by the electronic circuit. In either case, the output is a function of the effective dielectric number of the material. Considering that the dielectric number is significantly higher for water (
80) than for air (1) and solid soil constituents (3–5) (e.g., Ledieu et al., 1986; Wolfarth, 2005), the output is most sensitive to variations in water content, which makes the technique suitable for continuous monitoring of pore water content (e.g., Roth et al., 1990). The TDR instrument (the control unit) is relatively costly, and the signal is somewhat difficult to interpret, which motivated the development of the WCR. The WCR probe features electronics, embedded in the probe head, which generate and analyze the electromagnetic waves directly, eliminating the need for an external TDR instrument (Bilskie, 1997). The probe itself is more expensive than the traditional TDR probe, but for systems consisting of approximately 10 sensors or less, a WCR system is cheaper. There have been some uncertainties about whether the WCR probe is a capacitance-type or of TDR-type sensor since the WCR output depends on an oscillator frequency (typical for capacitance probes) governed by the travel time along the probe (typical for TDR probes) and thus contains elements of both techniques (Seyfried and Murdock, 2001). However, the oscillator driver is a very high speed internal circuit that generates high speed edges, which makes the measurement closer to a traditional TDR measurement than to a capacitance measurement (A. Sandford, unpublished data, 2004). The Tektronix 1502 cable tester (Beaverton, OR), which is commonly used together with conventional TDR probes creates signal rise times of 200 ps (5 GHz) and the circuit in CS616 accomplishes minimum rise times of 2 ns (500MHz) (J. Bilskie, unpublished data, 2005). In this paper, the WCR sensor will be regarded as a modified TDR probe.
Considering the amount of scientific work that has been undertaken during the last 25 yr or so, an extensive knowledge base has been formed regarding the theory and application of TDR systems (e.g., Robinson et al., 2003). A number of calibration equations have been developed—some general, some more suitable for specific soil materials (e.g., Topp et al., 1980; Ledieu et al., 1986; Roth et al., 1990; Jacobsen and Schjønning, 1993, 1995; Kellner and Lundin, 2001; Ekblad, 2004). Considering that the WCR is a fairly new instrument, having been on the market for only a couple of years, a relationship between WCR and TDR outputs would be very useful so the experiences of using TDR could be applied to use of WCR sensors. The main reason why this is not presently possible is that the electronics of the WCR sensor produces a thus far nonquantified time-delay, making a direct comparison impossible (J. Bilskie, unpublished data, 2004).
In addition to water content monitoring, TDR can be used to detect solute transport (e.g., Persson, 1997, 2001). Furthermore, the TDR technique is useful in providing data suitable for validation of simulated water content profiles since its vertical resolution when inserted horizontally is approximately 1 to 3 cm for three-rod probes (Johnsson and Lundin, 1991). Time domain reflectometry and WCR techniques have also become increasingly popular for monitoring of water content in roads (e.g., Hanek et al., 2001; van der Aa and Boer, 1997; Svensson, 1997). The granular materials used to build roads are generally much coarser than what is typical for agricultural soils. Tokunaga et al. (2003) suggested that the water in unsaturated coarse materials (gravel) is associated with the particle surfaces rather than the macroscopic pore network, which is the case for more fine-grained soils. The capillary rise in coarse materials like gravel is typically very limited, which leads to a drastic change in water content close to saturation (e.g., Tokunaga et al., 2003; Bigl and Berg, 1996) and concomitantly an abrupt change in dielectric number. Of particular relevance to this study are the papers dealing with sampling volumes in proximity to sharp dielectric boundaries by Nissen et al. (2003) and Ferré et al. (2003). They demonstrated by allowing the water level in a container to rise incrementally, thus passing the horizontally positioned probe rods, that the sampling volume becomes deformed as the sharp boundary approaches and passes the level at which the sensor was positioned. In conclusion, given that coarse materials exhibit a rapid decrease in water content for pressures slightly below saturation, resembling the sharp boundary studied by Nissen et al. (2003) and Ferré et al. (2003), we hypothesize in this paper that this, in turn, will result in a sampling volume that is vertically asymmetrical.
The objectives of this study were to (i) establish a relationship between the output of conventional TDR probes (signal travel time) and the output obtained using the WCR method (period of a square wave), (ii) evaluate previously reported calibration equations against data from laboratory measurements with the WCR technique in coarse materials, and (iii) investigate whether the sampling volume of the sensor is a factor to be considered when performing measurements in coarse materials.
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THEORY
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Time Domain Reflectometry
Water content measurements using TDR are based on the fact that the water content strongly affects the dielectric number of the investigated porous material. The dielectric number, K, is described by
 | [1] |
where Kr and Kc are the real and imaginary parts of the dielectric number respectively, 
(Sm–1; T3 A2 L–2 M–1) is the electrical conductivity of the material at zero frequency, 
0 (T4 A2 L–2 M–1) is the dielectric number of vacuum (=8.85 x 10–12 F m–1), and f is the measurement frequency (T–1) (e.g., Ledieu et al., 1986; Lin, 2003). At high frequencies the dielectric loss, Kc, is much smaller than Kr, and as seen in Eq. [1], the influence of pore water solutes (affecting electrical conductivity) decreases (Ledieu et al., 1986). As a consequence, TDR measurements are generally performed using very high frequencies to ensure that the complex parts of the dielectric number can be neglected. When performing measurements in practice, the exact composition of the material in which the measurement takes place is generally unknown. In a natural material one can expect a complex structure of geometrically highly irregular solid particles and pores, the latter containing variable fractions of air and water. To precisely compute the composite dielectric number for a natural material is obviously difficult, and as a result the apparent dielectric number, Ka, has been introduced as an approximation to K. Ka is sometimes referred to as the apparent dielectric permittivity or the apparent dielectric constant, but we will use the term apparent dielectric number, since Ka is not constant in a system with varying moisture content.
In most practical applications of TDR in soil science the volumetric water content is estimated based on the time required for an electromagnetic wave to travel up and down the metal rods of the probe. The velocity, v (L T–1), of an electromagnetic wave propagating through a medium is given by
 | [2] |
where c is the speed of light in vacuum (= 3 x 108 m s–1) (e.g., Topp et al., 1980). Thus, using the relationship between distance, velocity, and time, the apparent dielectric number can be calculated using
 | [3] |
where t is the signal travel time (T) and L is the length of the probe metal rods (L). The value of the dielectric number depends on temperature as observed by many researchers, for example, Wraith and Or (1999) and Persson and Berndtsson (1998). Or and Wraith (1999) also suggested a physical explanation for this dependence. The influence of temperature and solutes in the pore water for the WCR sensor was investigated by Seyfried and Murdock (2001), who concluded that both factors may need to be considered in cases of large temperature variations or high solute concentrations.
Linking Water Content Reflectometry and Time Domain Reflectometry
The relationship between travel time and dielectric number was defined in Eq. [3] for TDR probes. The WCR probe head contains embedded electronics, which replace the external TDR instrument (control unit) needed to operate the TDR probes. The electronic circuit generates the electromagnetic wave that propagates along the rods and derives, from the travel time, a square wave that is the output signal of the WCR sensor. However, the electronics create a time delay, thus adding time to the travel time consumed along the rods themselves. Bilskie (1997) suggested the general relationship for the square wave period, P:
 | [4] |
where tcircuit represents the time delay, and the factor 2 accounts for the fact that the signal travels the length of the probes four times in WCR sensors as opposed to twice for TDR sensors. The square wave period P is scaled by the electronic circuit in the sensor head to make it compatible with conventional data acquisition devices. The output as provided by the sensor, Pop, and subsequently registered by a data logger, for example, is thus Pop = CP, where C is a probe–model dependent constant. The time delay also depends on the dielectric number of the material being probed (J. Bilskie, unpublished data, 2004). Consequently, using Eq. [4], TDR calibration equations can be applied to estimate the water content based on WCR output.
Sampling Volume of the Water Content Reflectometry Sensor
The WCR sensor output is influenced by the dielectric number in a volume surrounding the two rods. The same applies to conventional TDR probes. If end effects are neglected, the relevant geometry to study is an area in a plane perpendicular to the rods, referred to as "the sample area" by Ferré et al. (1998). We believe, however, that sampling volume is a better term because sample may also refer to the soil sample used for the actual study. "Volume" is reflecting that we are dealing with three-dimensional measurements, although variations are assumed to occur only in two dimensions. The sampling volume is thus the volume of material in which the main part of the sampling takes place. The sampling volume (sample area) has been evaluated experimentally (Baker and Lascano, 1989; Suwansawat and Benson, 1999; Kellner and Lundin, 2001; Nissen et al., 2003; Ferré et al., 2003) and theoretically (Knight, 1992; Knight et al., 1997). To compute the sampling volume, the electrostatic potential in the vicinity of the probe rods needs to be determined. The electrostatic potential, 
, is governed by
 | [5] |
where the electric field intensity, 
(= –
), provides the basis for the evaluation of the sampling volume (Knight et al., 1994). Previously, Knight (1992) defined a weighting function, w(x,y), in two dimensions
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which yields the relative weight of every point value (or element value) in the evaluated domain, 
. Note that Eq. [6] is also valid for the sampling volume since the electrostatic force field is assumed constant along the sensor rods. Using the weighting function the apparent dielectric number, Ka can be determined from
 | [7] |
As described by Eq. [7], Ka is a weighted average of the actual dielectric values, K(x, y). The apparent dielectric number is the value that the probes effectively sense when performing the measurement.
Calibration Equations
Most equations for TDR to this date have been published having volumetric water content, 
, as a function of apparent dielectric number, while published calibration equations for the WCR probe relate water content to the scaled wave period, Pop. The equations evaluated in this study are briefly introduced below.
Equations Relating Apparent Dielectric Number and Volumetric Liquid Water Content
To evaluate the water content from the measured dielectric number, different approaches may be used including, amongst others, various mixing models and a widely used empirical equation presented by Topp et al. (1980).
Empirical Equations
The commonly used Topp equation was empirically determined from water content and dielectric number measurements in four different mineral soils through regression analysis:
 | [8] |
Topp et al. (1980) showed that Ka is strongly affected by changes in but not particularly sensitive to changes in soil density, texture, salt content, or temperature. Since the equation is simple to apply, it has played an important role in making the TDR method as widespread as it has become in the last 20 yr or so (Yu et al., 1999). Jacobsen and Schjønning (1993) obtained different coefficients for a third degree polynomial regressed on data from 10 mineral soils
 | [9] |
Ekblad (2004) presented the following third-degree polynomial for coarse granular materials,
 | [10] |
and Ledieu et al. (1986), performing measurements in one loam soil, proposed a regression equation based on the square root of the apparent dielectric number:
 | [11] |
Mixing Models
The principle of mixing models is to compute the value of the apparent dielectric number of a porous medium from the average of the dielectric number values of the respective constituents of the material investigated. The weights are simply the volumetric fractions of each constituent. An example of how the water content and the apparent dielectric number, Ka, are related in a mixing model is given by the three-phase model:
 | [12] |
where n is porosity and (1 – n) is the volume fraction of solids. The different Ks are the dielectric numbers for the respective medium (w = water, s = solid), and 
is a geometrical shape factor depending on the orientation of the surrounding soil layers. The value of varies from –1, when the probe rods are perpendicular to layers having different dielectric properties, to 1 for the case when the probe is parallel with the layers (Roth et al., 1990). For Eq. [12], Roth et al. (1990) found by optimization that = 0.46 was the best value when applying the equation to a wide range of soil types, while Ponizovsky et al. (1999) concluded that the variations in between soils calls for calibration to individual soils. In our analysis, we used both = 0.46 as originally suggested by Roth et al. (1990) and = 0.66 as suggested by Jacobsen and Schjønning (1995). Furthermore, to be consistent with the original works, when = 0.46 we used Ks = 3.9 (Roth et al., 1990), and when = 0.66, we used Ks = 3.5 (Jacobsen and Schjønning, 1995). In both cases it was assumed that Kair = 1, and that Kw = 79.43 since the laboratory temperature was approximately 22°C (Wolfarth, 2005).
Equations Relating Wave Period and Volumetric Liquid Water Content
The manufacturer of the WCR probes used, Campbell Scientific Inc., provides three equations of empirical type that translate the probe output, the scaled wave period, Pop, to volumetric water content. The choice of equation depends on the salinity of the pore water of the material. For nonsaline conditions, the recommended empirical equation is
 | [13] |
where Pop should be expressed in microseconds (Campbell Scientific, 2003).
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MATERIALS AND METHODS
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The Water Content Reflectometry Probe
The water content reflectometer used was the CS616 manufactured by Campbell Scientific, which is an improved version of CS615 (Campbell Scientific, 2003). The output from the CS615 is scaled using the factor C = 256 x 128 (J. Bilskie, unpublished data, 2004) instead of C = 1024 as used in CS616 (A. Sandford, unpublished data, 2004), which has implications when linking WCR and TDR. The probe is a robust construction, consisting of two stainless-steel rods inserted into an epoxy head containing electronics on a printed circuit board. A coaxial cable connects the probe head to a data logger. The probe head is 63 mm wide, 18 mm thick, and 85 mm long. The steel rods are 300 mm long, 3.2 mm in diameter, and their centerlines are separated a distance of 32 mm.
Examined Construction Materials
The two materials examined in this study were provided by VTI (The Swedish National Transport Research Institute) where they are considered as reference materials used in various experiments. One consists of natural gravel from a quarry named Olivehult (Material A), and the other of crushed rock from the quarry Skärlunda (Material B). Both materials are typically used as base layers immediately beneath the asphalt layer in a road and are very coarse compared with most soils (Fig. 1
).
Experimental Set-Up and Procedures
Dielectric Fluids Experiment
Measurements in fluids of known dielectric number were performed in a rectangular aluminum container (40 by 10 by 10 cm) to investigate how the time delay in Eq. [4] depends on the dielectric number. The values of the dielectric numbers were adjusted to the temperature of the liquid (Wolfarth, 2005) as measured by thermometer during the experiment (Table 1).
Evaluation of Calibration Equations
The WCR measurements were performed in an acrylic container 20 cm wide, 40 cm long, and 20 cm high (Fig. 2
). Two holes were drilled in one of the vertical sides, 2.5 cm above the inside bottom of the container, to fit the WCR rods. The holes were countersinked such that only 1 or 2 mm of acrylic plastic separated the probe head from the sample material to minimize the influence from the acrylic wall. About 20 cm from the sensor side of the container, a thin plastic support was glued to the bottom of the container. Two holes in the support ensured that the height and parallel position of the steel rods were retained. In preparing the container for the experiment, the granular material was added in intervals, first filling up to the sensor fitting. Then it was packed by gently allowing a 2-kg weight, with bottom diameter eight centimeters, to repeatedly fall onto the surface of the material, making it flat enough for the probe to be properly mounted. After the probe was in place, more material was added and subsequently packed in two rounds until reaching a final depth of approximately 6 cm in both experiments. During the filling process, small amounts of water were added to fix the fines, which otherwise tended to end up near the bottom of the container.
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Fig. 2. The acrylic container and balance used to evaluate the calibration equations. The two-rod water content reflectometry sensor is inserted through holes in the side of the container.
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During the actual experiments, water was supplied to the container using a hose connected to a bottle slightly elevated above the container to provide a suitably slow flow rate. To reach an equilibrium state of moisture in the soil as quickly as possible, a dishwashing cloth (Wettex) was placed on top of the soil material to enable an even infiltration across the surface. Water was dripped onto the cloth in a rectangular grid pattern. After the addition of water to the container, the cloth was removed and weighed to account for water absorbed into the cloth. The water content in the container as given by the WCR sensor was continuously monitored and stored by a CR10 data logger (Campbell Scientific Ltd., Loughborough, UK). It was required that the WCR-registered water content stabilized before water was added again, a time period ranging from 2 d to >1 wk. The amount of water added was quantified by weighing the bottle of water before and after the filling. Additionally, the container was placed on a Mettler PE16 balance (Mettler Instrumente AG, Zürich, Switzerland) with a resolution of 0.1 g that enabled a continuous monitoring of the weight of the container with contents. Water was added incrementally until the whole sample was practically saturated, upon which the entire sample was dried in an oven to establish the final water content gravimetrically. Any remaining water in the container was accounted for by absorbing it into a cloth that was subsequently weighed.
Keeping the container on a balance during the entire experiment enabled control of losses from the system due to leaks of liquid water or evaporation. As a representative example, 6 g of water vanished in 5 d from the system after one particular addition of water in spite of several measures taken to reduce losses. The top of the acrylic container was sealed with plastic film during ongoing experiments. A board of expanded polystyrene was used to insulate the container from any potential heat source in the balance causing vapor flows. Furthermore, the container was covered with cardboard to reduce any disturbances caused by variations in radiation during the day. This setup was used for the determination of the calibration equations and the sampling volume validation example.
Numerical Simulation of Sampling Volume
The calculation of the sensor sampling volume required Eq. [5] to be solved using a nonlinearly varying coefficient (the dielectric number). The dielectric number was calculated as a function of water content using Eq. [8]. Choosing any of the equations listed in this paper obviously introduces uncertainties to the procedure, and Eq. [8] was chosen only because it is the most widely used. The boundary condition was in all simulations a zero potential across the outer boundary and a potential of –1 and 1 V on the inner boundary representing the rods of the WCR sensor. The mathematical problem thus formulated was solved using the commercial finite-element software package FEMLAB (COMSOL AB, Stockholm, Sweden). The numerical domain consisted of 14000 elements representing a real area 10 cm wide and 8 cm high.
To investigate the potential effect of very coarse materials on the sampling volume, an attempt was made to simulate typical measured calibration curves using the electrostatic numerical model and making some crude assumptions in terms of material hydraulic properties. First, we needed to know the water content distribution in the sample after every addition of water to the container. We assumed that the material was fully homogeneous and thus that the water content showed no horizontal variations, but varied continuously in vertical direction as specified by the hydraulic properties and the amount of water in the container. To find the water content distribution, given that the mass of water in the container was known, the pressure head at the bottom of the container was computed by solving the following equation iteratively (Appendix)
 | [14] |
where V is the volume of water in the container, A the hori zontal cross-sectional area, r the residual water content, H the height of the soil sample, s is the saturation water content, hl the pressure head at the bottom of the container, b the pore-size distribution of the Brooks and Corey (1964) water characteristic equation, and he the air entry pressure of the same equation. Five different hydraulic parameter sets (variations of the relatively coarse material sand) were used to investigate the effect on the modeled response (Table 2). After having determined hl, the vertical water content distribution in the container was calculated using the Brooks and Corey equation assuming equilibrium conditions. In the computation of the electric field caused by the interactions of the WCR probe and the porous material, the dielectric number in every node of the domain must be given as input to the model. As an approximation, the dielectric number was calculated based on the water content using Eq. [8]. Consequently, the procedure was repeated after every additional filling of water to obtain a new vertical dielectric number profile. The procedure was then as follows for each simulation as specified by Table 2: (i) calculate the apparent dielectric numbers using the derived dielectric profiles; (ii) calculate the water contents using Eq. [8]; and finally (iii) calculate the wave periods using Eq. [16].
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RESULTS
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Linking Time Domain Reflectometry and Water Content Reflectometry
The results of the experiment showed that an excellent prediction of the tabulated dielectric number values could be obtained (Fig. 3
) by inserting
 | [15] |
into Eq. [4] where the coefficient values, a1 = 5.36 x 10–9 s and a2 = –2.41 x 10–10 s were determined using nonlinear optimization (r2 > 0.99). The apparent dielectric number, Ka, was used in Eq. [15] rather than the dielectric number to be consistent with Eq. [3] even though the measurements were performed in pure liquids (homogeneous). Fortunately, since the time delay of the electronics obeys the same power law as the original Eq. [3] between travel time and dielectric number, Eq. [4] could easily be rearranged to
 | [16] |
which subsequently can be used to obtain the apparent dielectric number from the WCR output. Considering that the general form of the expression has been determined, calibration to individual sensors should be easily performed by, for example, one measurement in air and one in water. We may now use equations originally developed for TDR also for WCR. The established relation with the presented coefficient values was used to compute the apparent dielectric number values, which were used in the evaluation of calibration equations.
Validation of the Numerical Model and the Sampling Volume Theory
The electrostatic numerical model of the WCR probe described above was used to compute the expected apparent dielectric number as measured by the probe. The grid used in the numerical computations was designed such that the dielectric number could be adjusted in discrete vertical steps to represent 17 different water levels. When the water level rose in the container, the apparent dielectric number and the sampling volume changed accordingly (Fig. 4
). The agreement between the modeled dielectric number and the measured dielectric number was excellent (Fig. 5
), which gave confidence in the numerical electrostatic model as a tool that indeed provides an accurate description of the WCR–porous medium system. Additionally, Eq. [16] was used to convert the probe wave period to an equivalent dielectric number and, once again, the equation provided results that seemed correct.
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Fig. 4. The absolute value of the electric field intensity around the two-rod probe when the water depth in the container was 1.5 cm (left), and 3.5 cm (right). The water level is indicated by a black horizontal line.
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Fig. 5. Modeled and measured apparent dielectric number in an open container with a rising water level. The sensor center is located 2.5 cm above the bottom of the tank where the sharp increase in dielectric number occurs.
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Evaluation of Calibration Equations
The measured output period, Pop, was directly inserted into the Campbell equation (Eq. [13]), while it was first divided with C (= 1024) to obtain P, and then converted to the apparent dielectric number using Eq. [16] before use in the Ka-based equations (Eq. [8]
–[12]). The computed volumetric water contents were then compared with the measured volumetric water content (Fig. 6
and 7)
. To evaluate the equations, for the two different materials, the RMSE was computed for every equation and material (Table 3). Evidently, the three-phase mixing model with = 0.66 as suggested by Jacobsen and Schjønning (1995) provided the best prediction for Material A, while the equation suggested by the manufacturer was best for Material B. However, the measured curve for material B is clearly different compared with all calibration equations, which predict a smooth increase in water content with output (Fig. 7).
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Fig. 6. Measured volumetric water content as a function of dielectric number, and calibration curves for Material A (see Table 3 for references).
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Table 3. The root mean square error in the prediction of volumetric water content for various calibration equations.
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Fig. 7. Measured volumetric water content as a function of dielectric number, and calibration curves for Material B (see Table 3 for references).
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The results from all five simulations using the electrostatic numerical model, where an attempt was made to illustrate the influence of hydraulic properties, are plotted in Fig. 8
to render possible a comparison with the measured values of Material B.
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Fig. 8. The measured average volumetric water content vs. the wave period, as well as the simulated response of the same system for different hydraulic properties (see Table 2).
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DISCUSSION
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Equation [16] may not be used directly for sensors other then the one investigated, since a new empirical relation is likely to be needed. However, having established the general form and functional dependence of Eq. [16], only a simple calibration in air and water is needed to adapt it to individual CS616, or similar, sensors. The equation may need to be extended to incorporate a temperature dependence before application in the field considering that the dielectric number of liquid water changes quite considerably with temperature (e.g., Wolfarth, 2005).
The best calibration equation for Material A was the three-phase mixing model using = 0.66 as suggested by Jacobsen and Schjønning (1995), while the equation suggested by the manufacturer was best for Material B. The reason why the three-phase mixing model was more accurate when using = 0.66 rather than = 0.46 may be associated with the experimental method used. It is possible, and perhaps even likely, that due to the filling and light packing of material in rounds, horizontal structures were created in the container. The particles were thus aligned more in parallel with the sensor rods then would have been the case for a relatively undisturbed sample from nature. On the other hand, the procedure was in principle very similar to how roads are built. Such a structure could explain why the model having the larger value corresponded better with the measurements. Ekblad (2004) used a coarse material from the Material B quarry but did not observe a behavior such as the one reported here (Fig. 7). The calibration equation of Ekblad (2004) underpredicts the water content, and one explanation can be that the material was packed harder in his experiment, decreasing the air fraction and increasing the solid fraction, thus leading to a decrease in predicted water content. In contrast, the traditional TDR equations produce overpredictions of water content. The reason is likely that they were developed for finer, agricultural type materials having larger fractions of air and bound water for a given water content. The water considered as bound has a smaller dielectric number (e.g., Yu et al., 1999), resulting in a smaller apparent dielectric number for a certain water content. In contrast, for a given apparent dielectric number, the water content will be overpredicted if there is no or little bound water in the material for which the prediction was done. The coarse materials examined in this study probably have negligible fractions of bound water.
Material B exhibited a behavior that deviates from typical for fine-grained materials. Due to this, no equation fitted the measurements particularly well, certainly not in the general appearance. It appears that the form of the curve, the measured water content exhibiting a reluctance to increase despite the increased average water content, was similar in shape to the curve in Fig. 5, which shows the effect of a sharp dielectric boundary. Furthermore, the volumetric water content was deduced assuming that the mass of water, determined gravimetrically by weighing, was uniformly distributed throughout the sample. Thus, it appears that two factors explain the unusual appearance of the measured –Ka relationship of Material B. First, the gravimetric method fails in providing the actual water content at the height of the sensor probe. Second, the sampling volume is distorted such that the sampling volume is not centered between the probe rods.
The –Ka curve measured in Material B was qualitatively reproduced using the developed numerical model, and thus the hypothesis that the behavior was caused by the limited capillary rise could not be rejected. Therefore, it is likely that the deformed sampling volume caused by the sharp change in water content and apparent dielectric number close to saturation was the cause. The finding suggests caution in laboratory calibrations of coarse materials and exemplifies that water content in coarse material (e.g., road materials) may increase, or decrease, very fast close to saturation. In the field, the same issue emerges in connection with infiltration, close to the groundwater table, or due to heterogeneities. To use TDR or WCR measurements to capture properly the dynamics of a wetting front in the shoulder of a road, as generated by rainfall where the surface runoff caused by the paved surface leads to a focused, large infiltration, measurements need to be taken with very small time intervals using horizontally inserted sensors where all rods lie in the same horizontal plane (Ferré et al., 2002). The time-intensive sampling strategy argument is further supported by the high saturated hydraulic conductivity of these materials.
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CONCLUSIONS
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An empirical two-parameter equation enabling the use of TDR calibration equations for the CS616 WCR sensor was determined. A simple calibration in air and water is needed to establish parameters for individual CS616, or other similar sensors.
A three-phase mixing calibration equation (–Ka curve) was found to be the best for Material A, whereas for the slightly coarser Material B, no calibration equation was satisfactory. Suggested reasons for the inability to reproduce the behavior of Material B were a distorted, asymmetric, sampling volume and shortcomings in using the gravimetric method to determine local water content around the probe. A numerical electrostatic model, solving the Laplace equation, was developed and supported the former suggestion.
The findings point at the need for caution in interpretation of laboratory calibrations of coarse porous materials and suggest that local water content, as measured at the sensor depth, may vary rapidly also in situ.
Measurements are frequently used to validate or reject numerical models since measurements are generally considered to represent the truth. However, it was demonstrated in this study that numerical models can improve the interpretation and explanation of measurements. It can thus be concluded that both measurements and numerical computations benefit from each other, and that the two disciplines are likely to become more intimately linked in the future given the advent of new numerical multiphysics tools.
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APPENDIX
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Assuming hydrostatic equilibrium, the water content distribution, (z), in a container of arbitrary shape accommodating a certain volume of water, V, can be found by solving the following equation
 | [A1] |
where A(z) is the horizontal cross-sectional area at elevation z; hl = h(zl) is the pressure head at the lowest point, zl, in the container; and zu is the elevation of the uppermost point of the container. There are plenty of water characteristic equations, (h), in the literature, but for simplicity, we will use the equation of Brooks and Corey (1964)
 | [A2] |
where r is the residual water content, s the saturation water content, he the air-entry pressure, and b the pore-size distribution index. Using Eq. [A2], and applying Eq. [A1] to a rectangular container of height H, where A is constant, yields
 | [A3] |
Integration of Eq. [A3] results in
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| [A4] |
If the volume of water in the container, with geometry specified as above, is known, we can solve Eq. [A4] iteratively to find the pressure head at the bottom of the container. Having this value, the water content distribution as a function of elevation in the container can easily be calculated using Eq. [A2].
See Fig. 9
for examples solving Eq. [A4], using s = 0.28, r = 0.02, and the remaining hydraulic properties as in Simulations 1 (sand) and 3 (coarse material) (Table 3), for mass of water = (0.1, 0.3,..., 1.4) kg, the resulting water content distributions.
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Fig. 9. The water content distribution in a sand (left) and a coarse material (right) for variable mass of water in the container. The black horizontal line indicates the vertical position of the water content reflectometry probe in the container.
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ACKNOWLEDGMENTS
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The authors are grateful for the valuable assistance provided by Jim Bilskie and Andrew Sandford of Campbell Scientific, and by Erik Kellner. Sören Karlsson manufactured the containers and Inger Påhlsson assisted the main author in the laboratory. The manuscript benefited from the comments of an anonymous reviewer. Furthermore, we are grateful to VTI for providing us with materials and lending the balance. The work was supported by the Swedish National Road Administration (Agreement no. AL 90 A 99:8683).
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ABSTRACT
INTRODUCTION
