Soil water flux density measurements near 1 cm d−1 using an improved heat pulse probe design



This article is corrected by:

  1. Errata: Correction to “Soil water flux density measurements near 1 cm d−1 using an improved heat pulse probe design” Volume 46, Issue 7, Article first published online: 16 July 2010


[1] The heat pulse probe (HPP) technique has been successfully applied for estimating water flux density (WFD). Estimates of WFD have been limited to values greater than 10 cm d−1, except for two recent studies with lower detection limits of 2.4 and 5.6 cm d−1. Although satisfactory for saturated soils, it is recognized that current HPP capabilities are limited for applications in the vadose zone, where WFD values are generally below 1 cm d−1. Since numerical sensitivity analysis has shown that large heater needle diameters may increase HPP capabilities in the lower flux density range, a HPP with a 4-mm-diameter heater needle was developed and tested. WFD values were obtained by fitting temperature data to the analytical solution for a pulsed cylindrical heat source of infinite length. Effective heater-thermistor distance and soil thermal diffusivity values were determined for specific heat input scenarios with zero WFD, prior to imposing water flow across the HPP needles. We showed excellent results in the range of 1–10 cm d−1 and satisfactory results in the range of 10–1000 cm d−1.

1. Introduction

[2] Water flux density (WFD) measurements are required for studying flow and transport processes in environmental and ecosystem applications. In situ estimation of WFD has been successfully demonstrated using the heat pulse technique, applying earlier developments of the heat pulse probe (HPP) [Mori et al., 2005; Mortensen et al., 2006; Ochsner et al., 2005; Ren et al., 2000]. The heat pulse method is relatively simple compared to other available soil WFD methods [Bresler, 1973; Nielsen et al., 1973], which are generally time consuming, mathematically complicated and measurement intensive. Using the HPP method, heat is applied from a cylindrical heat source, and temperature responses are measured at known distances from the heater, using thermistor needles. These temperature responses are used for inversely solving the heat conduction and convection transport equation, optimizing values of the required soil hydrological and thermal parameters. In addition to WFD, the HPP allows other thermal and hydrological properties to be evaluated simultaneously, such as heat capacity, soil water content and thermal diffusivity [Bristow et al., 1993, 1994; Campbell et al., 1991; Tarara and Ham, 1997].

[3] The use of heat as a tracer toward estimation of WFD has progressed enormously in the last decade. However, the HPP method is limited to water flux measurements larger than 2.4 cm d−1 [Mori et al., 2005; Ochsner et al., 2005], whereas the magnitude of vadose zone fluxes are generally 1 cm d−1 or lower. Likely, Byrne et al. [1967, 1968] were the first to use heat tracing for WFD measurements in saturated soil. Their method was based on reaching steady state conditions by applying heat for long periods of time (∼30 min). In unsaturated conditions their method would be limited, as the large heat inputs would cause pore water redistribution and fluid-vapor phase change. Campbell et al. [1991] successfully demonstrated the heat pulse method with a short heat pulse and relatively small heat input. Although Campbell et al. [1991] did not use the HPP to estimate soil WFD, their accurate estimation of soil specific heat showed that soil water redistribution was minimized. The original Campbell et al. [1991] HPP consisted of a 0.813-mm-diameter heater and used a 6-mm needle spacing to satisfy the assumptions of the analytical solution for a pulsed infinite line source [De Vries, 1952; Kluitenberg et al., 1993]. The heating rate for the Campbell et al. [1991] design was 87.5 W m−1 for 8 s, yielding a total heat input per unit length of 700 J m−1. Many others have successfully used this design for estimating soil thermal properties [Bristow et al., 1994] and WFD [Kluitenberg et al., 2007; Mori et al., 2003, 2005; Mortensen et al., 2006; Ochsner et al., 2005; Ren et al., 2000].

[4] Ren et al. [2000] applied low heat input and used the maximum difference between the temperatures at the upstream (Tu) and downstream (Td) needle locations (ΔTud) for indirect soil WFD estimates in the range of 100–545 cm d−1. Wang et al. [2002] and Kluitenberg et al. [2007] improved and simplified the mathematically complex solution of Ren et al. [2000] by using the ratio Tu/Td, instead. However, both solutions using the ratio Tu/Td are extremely sensitive to inevitable variations in distance between the heat source and the upstream (xu) and downstream (xd) needle locations. Furthermore, the ratio (Tu/Td) method was derived from the line source solution, which might not be applicable for a heater needle of larger diameter. In addition, the ratio method still lacks a well-defined temperature sampling time [Ochsner et al., 2005].

[5] We emphasize that the temperature response is controlled by both HPP design and thermal properties of the surrounding soil medium, including the maximum heater temperature increase at the interface between the heater needle and the soil, ΔHTmax. This temperature increase is important, as soil water redistribution by thermal convection and vapor transfer becomes more important as ΔHTmax increases. Thus, minimizing ΔHTmax is essential for accurate soil WFD measurements. Apart from the soil thermal properties, ΔHTmax is also a function of the heat applied per unit area of the cylindrical heater per unit time (W m−2). By using a cylinder of larger diameter, more heat can be applied to the same soil volume, while maintaining a heat input per unit surface area identical to that for a heater needle of smaller diameter. On the basis of numerical solutions, Saito et al. [2007] concluded that distributing a higher heat input per unit length of heater (J m−1) from a larger diameter heater allowed accurate measurement of lower WFD values than previously possible with the original HPP design. However, the larger heater diameter with the same needle distance violates the assumptions of the line source solution, as analyzed by Kluitenberg et al. [1993, 1995]. Therefore, we will instead use the analytical solution for a pulsed infinite or finite cylindrical source [Carslaw and Jaeger, 1959; Kluitenberg et al., 1993, 1995] to match with temperature observations.

[6] The principal objective of this study was to extend the measurement range of WFD to values near 1 cm d−1, using a modified HPP, thereby making the HPP more relevant for vadose zone applications. For that purpose, we developed an alternative HPP with a larger heater diameter and adapted the pulsed cylindrical source solution of Kluitenberg et al. [1993, 1995]. For testing the measurement capability of the modified HPP, laboratory experiments were conducted in the range of 1–1,000 cm d−1 to estimate heat pulse and water flow velocities using an inverse procedure [Hopmans et al., 2002; Mori et al., 2005].

2. Materials and Methods

2.1. Theory

[7] In order to apply the HPP for water flow measurements, we considered the combined transport of heat by conduction and convection. For a homogeneous porous medium through which water is flowing with uniform velocity in the x direction, the equation for two-dimensional heat transfer is [Bear, 1972; Mori et al., 2005]

equation image

where T (°C) is the soil temperature as a function of time, t (s), and spatial positions x and z (m), κ (m2 s−1) is the soil thermal diffusivity, and Vh (m s−1) denotes the heat pulse velocity. The latter term combines both conductive and convective effects, and describes the weighted average of the velocity of the heat convected by the liquid phase and the velocity of heat conducted by the stationary porous medium [Ren et al., 2000], and is related to the Darcy water flux density, Jw (m s−1), by [Mori et al., 2005]

equation image

assuming that the different bulk soil phases are in thermal equilibrium. The bulk volumetric heat capacity, C (J m−3 °C−1), is a function of the volumetric heat capacity of water (Cw), soil mineral material (Cs), and organic matter content, and is determined by the soil porosity, ϕ (m3 m−3) and volumetric water content, θ (m3 m−3). For nonorganic soils, the bulk soil volumetric heat capacity can be estimated from

equation image

[8] When interested in soil thermal properties, equation (1) is solved for heat conduction only, with Vh = 0. The solutions for zero and one-dimensional water flow for a cylindrical heat source of radius a (m) and length 2b (m) in an infinite homogeneous medium [Kluitenberg et al., 1995] are presented in Appendix A. On the basis of equation (A3) and following the notation of Ren et al. [2000], using absolute distances for upstream and downstream locations, xu and xd (both positive in sign), the upstream temperature rise (ΔTu) is defined by

equation image

where the limits of integration are

equation image
equation image

In equation (4a), ΔTu (°C) represents the upstream temperature rise in the plane z = 0, at radial distance xu (m) from the center of the cylindrical source (Figure 1), at time t (s) from initiation of the heat pulse of duration t0 (s), and q (W m−1) is the heat input per time per unit length of heater. In addition, I0 represents the modified Bessel function of the first kind of order zero. Using the same notation, the downstream temperature rise (ΔTd) is defined by

equation image
equation image
equation image
Figure 1.

Domain of the analytical solution for a finite length cylindrical heat source with water flux density in the x direction, where a (m) is the cylindrical heat source radius, 2b (m) is the cylinder length, and xu and xd (m) denote the needle distance between upstream and downstream locations and the heater center, respectively. Apart from b, dimensions are drawn to scale with a = 2 mm and xu = xd = 6 mm. The origin (O) of the coordinate system is at the center of the cylinder.

[9] Kluitenberg et al. [1995] indicated that their solution for a cylindrical heat source represents radial heat conduction into the surrounding soil following the release of heat from the external surface of a hollow cylinder. That physical representation is incorrect. Instead, the solution of Kluitenberg et al. [1995] and equations (4) and (5) assume that the thermal properties inside the cylindrical heat source (x < a) are identical to those of the surrounding soil (x > a). Moreover, it is important to recognize that the solutions do not consider water flow interference by the cylindrical heat source. The results of the presented study will evaluate the importance of the underlying assumptions.

2.2. Experiments

2.2.1. Heat Pulse Probe Design

[10] The large-diameter heater needle was assembled by inserting a cylindrical heating element inside stainless steel tubing (Figure 2). To ensure a prescribed heating rate from the exterior wall of the heater needle, a total of 453 mm long heater wire was uniformly coiled around 0.76 mm thick PVC insulation of internal electrical wire (Table B1, part 1) along its 37.5 mm length, after being threaded with a 6–32 NC die (32 threads per inch). (Instead of including manufacturers in the text, parts information is listed in Table B1.) The enameled heater wire (Table B1, part 2) [Ham and Benson, 2004; Mori et al., 2003] was soldered to the internal electrical wire at the tip of the heating element. Thermal contact with the stainless steel tubing was ensured using thermal conductive epoxy. The heater wire was connected at the other end, just outside the stainless steel tubing base, with 24 AWG shielded instrumentation cable wire using a 26–22 connector (Figure 2). The end of the coiled heater wire was secured to the PVC insulated wire with a small amount of epoxy glue (Table B1, part 3). The coiled heater-PVC element was inserted and secured into the 37.5 mm long, 4.19 mm OD (outer diameter) and 3.68 mm ID (inner diameter) stainless steel tube needle. A high thermal conductive epoxy (Table B1, part 4) was injected into the cavity between the coiled element and the inner wall of the tube to ensure thermal continuity and electrical insulation. A theoretical value of 94.18 Ω was calculated for the total resistance of the 453-mm-long heater wire. After the construction of the heater needle, a total resistance of 96.40 Ω was measured with a voltmeter. For calculating the applied heat input, we used a resistance per unit length of 2,511.5 Ω m−1, as determined from the ratio of the calculated total resistance and the 37.5-mm needle length.

Figure 2.

Schematic of (a) the large-diameter (4.19 mm) heater needle element with the base of the HPP, showing (b and c) cross sections with the different material construction within the heater needle.

[11] As described in detail by Mori et al. [2003], two thermistor needles were constructed from stainless steel tubing and thermistors (Table B1, parts 5 and 6, respectively). The thermistor wires were inserted into the stainless steel tubing and held in place using the same thermal epoxy material as applied for the heater needle, with the thermistor embedded in the tip of the needle. After their assembly, the heater and two thermistor needles were mounted in the same plane through predrilled holes of the HPP Delrin plastic disk (Table B1, part 7) sensor base, and were secured using clear epoxy glue. Although the total needle lengths were 24.5 and 37.5 mm for the thermistor and heater needles, respectively, the corresponding lengths extending from the Delrin base were 13.0 (thermistors) and 26.0 mm (heater). The remaining 11.5 mm included the 10-mm-thick base and an additional 1.5 mm of air space that was needed for the necessary electrical connections (Figure 2), similar to Figure 1 of Saito et al. [2007]. Finally, although the intended center-to-center distance between thermistor and heater needles was 6.0 mm, the true measured distances were 5.93 and 6.29 mm for the upstream and downstream needles, respectively, as measured by a caliper after HPP construction.

[12] The HPP heater and the two thermistors were controlled and logged by a data logger (Table B1, part 8) that was connected to a computer. Thermistor temperatures were measured using a four-wire half-bridge circuit with a 5 kΩ resistors (Table B1, part 9). Using ΔT measurements, computing the temperature rise relative to the time-zero measurement for each thermistor, the temperature measurement repeatability with this equipment was approximately 0.01°C [Mori et al., 2003]. An individual DC regulated power supply (Table B1, part 10) was used to supply the heating power, enabling a maximum heat intensity of 256.2 W m−1, using 31.6 V. The current through the heater element was measured with a four-wire axial precision power shunt (Table B1, part 11). Additional details of the construction and operation of the HPP are provided by Mori et al. [2003].

2.2.2. Experimental Setup

[13] The single HPP was installed horizontally in the center of a 73.3-mm-long and 82.7-mm inner diameter Plexiglas column (Figure 3) using translucent silicon rubber adhesive sealant (Table B1, part 12). The thermistor needles of the HPP were oriented vertically along the direction of induced water flow (x direction in Figure 1), so that temperatures were measured at upstream (u) and downstream (d) locations of the heater needle (H). The lower end of the Plexiglas column was placed into a Tempe cell assembly with a stainless steel porous screen (Table B1, part 13) to support the soil. After installing the HPP, the Tempe cell assembly was filled with saturated Tottori dune sand [Mori et al., 2003, 2005]. After filling the column with a known volume of distilled and deaired water, a predetermined amount of Tottori dune sand was added and mixed, to achieve a uniform and saturated water content equal the total porosity of 0.385 m3 m−3, as computed from bulk and particle density values of 1,630 and 2,650 kg m−3, respectively, according to the 8WP2 wet-packing method of Oliveira et al. [1996]. The HPP calibration was conducted in situ with zero WFD, as described in section 2.3.

Figure 3.

Experimental setup with the 4-mm heater diameter HPP, including the heater (H) and upstream (u) and downstream (d) needles, as set for applying predetermined water fluxes with pumps and measuring them from the drip needle.

[14] Though the main objective of this study was to increase HPP sensitivity to WFD values ≤ 5 cm d−1, we also tested the capability of the new HPP design for WFD values to near 1,000 cm d−1, making the HPP applicable for a wide range of WFD values. In contrast to the work by Mori et al. [2005], water flow was controlled by injecting distilled and deaired water to the top of the column (Figure 3) at a controlled steady state rate, as it is difficult to accurately control flow for small flow velocities using head gradients. A peristaltic pump (Table B1, part 12) was used for the range of 10 to 1,000 cm d−1, while water was injected with a syringe pump (Table B1, part 13) for accurate WFD values <10 cm d−1. However, independent WFD measurements were required because of the formation of small air pockets in the supply and drainage tubing during day-long flow experiments. The reported true WFD values were measured independently from outflow measurements through a drainage needle (Figure 3), averaging over 5-min time intervals.

2.2.3. Heat Pulse Input Scenarios

[15] In response to the recommendations of Saito et al. [2007], we increased the heat input to the heater needle by increasing both heat intensity and heat pulse duration, as compared to the established heat input of about 700 J m−1. The set 1 experiments were conducted using cumulative target heat input (THI) values of 680, 1020, 1,360, 1,700 and 2,040 J m−1. This wide range of THI values was obtained using two scenarios: (1) maintaining a constant heat intensity of approximately 85 W m−1, and changing the heat pulse durations from 8 to 24 s, using 4-s time increments; and (2) by maintaining a constant heat pulse duration of 8 s, and changing the heat intensity from 85 to 255 W m−1, using 42.5 W m−1 increments. Additional heat pulse scenarios II and III were conducted to further increase the sensitivity in the low WFD range. Using a maximum heat intensity of 256.2 W m−1, heat pulse durations of 30-s (set 2) and 40-s (set 3) were used with corresponding total heat input values of 7,686 and 10,248 J m−1, respectively. The exact total heat input values were obtained from continuous current measurements during heat pulse applications as in the work by Mori et al. [2003].

2.2.4. Water Flux Density Scenarios

[16] The experiments were conducted for WFD values in the range of 1–1,000 cm d−1. Approximately constant WFD values were maintained for 1 day, during which all heat pulse experiments for that specific flux density were conducted. Heat pulse scenarios were conducted in sequence, with 45-min time intervals between experiments, to ensure thermal equilibrium in the column prior to heat pulse generation. Each measurement included the temperature response of both the upstream and downstream thermistor needles, for a total duration of 3 min from the start of the heat pulse (t = 0), with 1-s resolution. For set 1, WFD values were in the range of 10–1,000 cm d−1, with one measurement of each THI. For sets 2 and 3, the WFD range was between 1 and 5 cm d−1, with each experiment repeated 3 times. All the experiments were conducted in a controlled temperature room (20 ± 1°C).

2.3. Parameter Estimation of Thermal Properties and Water Flux Density

[17] The analytical cylindrical heat source models for the upstream and downstream locations (equations (4) and (5)) were fitted to the measured temperature responses by minimizing the residuals between measured and fitted temperatures, using an inverse procedure to optimize the relevant parameters of the heat transport equations [Mortensen et al., 2006]. This was done with Matlab (Table B1, part 13) by using the Levenberg-Marquardt nonlinear least squares method of the Optimization Toolbox. Instead of adding an additional fitting parameter, the radius of the cylindrical heater (a) in the analytical solution was fixed with a value of 2.095 mm (OD = 4.19 mm).

[18] Calibration with no-flow conditions was done separately for the upstream and downstream needles, by minimizing

equation image

where OFI is the objective function and N denotes the number of measurement points at times ti. In this study, N was 180 for the 1-s temperature measurements during a 3-min measurement period. ΔT represents the temperature rise for either the upstream (u) or downstream (d) thermistor needle, with superscripts M and O indicating the measured and optimized temperatures, respectively. The calibration was conducted for the column filled with Tottori sand with zero WFD conditions (Vh = 0), assuming known specific heat values of cs = 795.0 and cw = 4181.6 J kg−1 °C−1 (Table 1) [Mori et al., 2003], and q (measured). The vector p contains the distance between the heater center and both upstream and downstream thermistors (xu and xd), and thermal diffusivity (κ). Since parameter values will depend on a multitude of factors, including soil-needle contact, probe geometry and material property, we defined effective values, xeff and κeff. After calibration, these effective values were used as constants for estimation of WFD values in subsequent flow experiments.

Table 1. Average and Standard Deviation Values of Effective Parameters for the Different Target Heat Inputs, as Obtained in the Calibration Procedure, and Differences Between Average Upstream and Downstream Effective Needle Spacinga
Heat Input Set NumberTHI (kJ m−1)UpstreamDownstreamΔxeff (mm)
xeff (mm)κeff (10−7 m2 s−1)xeff (mm)κeff (10−7 m2 s−1)
  • a

    Measured xu and xd physical values were 5.93 and 6.29 mm, giving a difference of 0.36 mm. AVG, average; SD, standard deviation, THI, target heat inputs; xeff, effective needle spacing.


[19] WFD values were evaluated using the simultaneous temperature response of both upstream and downstream thermistor locations, by minimizing

equation image

where the optimized parameter, p, now only includes the optimized heat velocity (Vh) from which the WFD can be computed using equation (2).

3. Results and Discussion

3.1. Heat Pulse Probe Calibration

[20] Effective sensor spacing, xeff, and effective thermal diffusivity, κeff, were determined for each heat pulse scenario of experimental sets 1 through 3 by minimizing equation (6) using known heat input (measured during heat pulse application) and bulk volumetric heat capacity values, estimated from porosity, bulk soil density, and the specific heat and density of water and Tottori sand [Mori et al., 2003]. Figure 4 presents results for the upstream thermistor needle from calibrations with heating durations of 8 and 20 s of set 1. Solutions for both a cylindrical heat source (equation (4)), and the line source [Bristow et al., 1994, equation (5)] for the original HPP design are compared. Both solutions were fit to the data by using equation (6) to optimize xeff and κeff. The solution for the cylindrical heat source clearly provides a better fit than the line source solution, which overestimates the peak and underestimates tail temperatures.

Figure 4.

Temperature response from the calibration (zero water flux) of upstream thermistor needle, during set 1, from 8-s and 20-s heat pulse duration with total heat input of 695 and 1698 (J m−1), respectively. Fits of both line and cylindrical source analytical solutions were evaluated by optimizing xeff and κeff.

[21] Prior to the WFD experiments, effective calibration parameters were determined for each heat pulse input scenario of set 1, thus yielding 10 values of both xeff and κeff from both upstream and downstream temperature responses. For sets 2 and 3, we conducted 9 repeated calibration experiments for each heating scenario, prior to imposing water flow through the column. The average (AVG) and standard deviation (STD) of the xeff and κeff values for sets 1–3 are presented in Table 1. We note that the STD values for set 1 (THI < 2 kJ m−1) are much larger than those for sets 2 and 3. This is not surprising recognizing that set 1 consists of calibration results of 10 different heat pulse scenarios, whereas sets 2 and 3 included repeated measurements of a single THI. Though it is expected that individual calibration for each THI scenario of set 1 would likely increase accuracy, we decided to lump the calibration results of the 10 THI scenarios. Specifically, individual calibration for each THI would create increasing complexity of the calibration process, which did not prove to be necessary because of the separation of calibration results between set 1 and combined sets 2 and 3. Moreover, results to be discussed in section 3.2 showed that the set 1 HPP experiments with the relatively low THI were only sensitive for WFD values larger than 10 cm d−1. Whereas we were more interested in the lower WFD range, we focused on replication of individual heating scenarios for sets 2 and 3.

[22] The results in Table 1 clarify the interpretation of the calibrated effective parameters, in contrast to their true physical meanings. Although they have considerable physical significance, the values of the effective calibration parameters will vary between probe geometries, heat input scenarios and analytical model assumptions, because of expected variations in needle-soil contact and differences between the thermal properties of the HPP sensor base and surrounding soil. Nevertheless, we expect the xeff values between the three sets to be highly correlated to the true physical distance. Measured values for the distances between the center of the heater needle and the upstream and downstream thermistor needles were 5.93 and 6.29 mm, respectively, whereas calibrated effective distance values were in the ranges of 6.197–6.313 mm and 6.549–6.738 mm for the upstream and downstream thermistors (Table 1). Thus, both approaches yielded larger xeff values for the downstream needle. The differences between the upstream and downstream effective distances (last column in Table 1) were 0.352, 0.425 and 0.484 mm, for sets 1, 2 and 3, respectively, whereas the true measured distance between the upstream and downstream needles was 0.36 mm. The magnitudes of the differences in xeff values appear to increase as the heat input increased, which further illustrates that effective calibration parameters are partly determined by the effect of HPP thermal properties and geometry on temperature response. Nevertheless, xeff variations are similar in magnitude as determined by Mori et al. [2005], who used the smaller heater diameter, a different sensor base, and a different analytical solution. For comparison, whereas our calibration resulted in optimized κ values in the range of 6.08 to 7.28 × 10−7 m2 s−1, Mori et al. [2005] estimated effective κ values for the same saturated Tottori sand between 6.5 × 10−7 and 6.7 × 10−7 m2 s−1, for a soil bulk density value of 1,630 kg m−3 and a porosity of 0.371 m3 m−3 (1,630 kg m−3 and 0.385 m3 m−3, in this study).

3.2. Water Flux Density

[23] Values of xeff and κeff (Table 1) were used as constant values to evaluate WFD for subsequent flow experiments by minimizing equation (7). Calibration parameters used were specific for each thermistor needle and THI set. Evaluation of the HPP measurements was conducted through comparisons of the estimated WFD with the independent WFD measurements, as obtained from the outflow measurements.

3.2.1. Set 1 Results

[24] Figure 5 compares measured with estimated WFD values for the range of 10–1,000 cm d−1. The HPP measurements for this range were obtained with the 10 different heat input scenarios of set 1 with each point representing one scenario (10 points per WFD value). Whereas the WFD estimations for ∼10 cm d−1 were in good agreement with the independent measurements, the optimized WFD values in the range of 50–1000 cm d−1 underestimated the corresponding true values. Similar findings were obtained by Ochsner et al. [2005] and can be caused by flow disturbance by the needles and thermal dispersion effects [Hopmans et al., 2002].

Figure 5.

Measured and estimated water flux density in the range of 10–1000 cm d−1 (set 1).

[25] Though we did not use the maximum measured temperature difference between upstream and downstream needles (ΔTud) to estimate WFD, the value of ΔTud can be interpreted as an indicator of WFD sensitivity. In other words, the larger the value of ΔTud, the more significance can be attributed to their temperature difference. The ΔTud values for the experimental data of set 1 are presented in Figure 6, for both variable heat pulse intensity (Figure 6a) and variable heat pulse duration (Figure 6b) THI scenarios. Figure 6 shows that for WFD values larger than 10 cm d−1, ΔTud values are generally equal to or larger than 0.01°C (temperature measurement repeatability), and are positively correlated to total heat input, as would be expected. Instead of using ΔTud values, Wang et al. [2002] proposed methods that make use of the ratio Tu/Td to estimate soil WFD. Although the improved temperature ratio solution of Kluitenberg et al. [2007] allowed for unequal needle spacing, their method is limited, and does not allow for large differences between xu and xd, as for our HPP. Moreover, their solution is based on the line heat source. In our study, using the large-diameter cylindrical heat source and significant differences between xu and xd, the ratio method produced erroneous results.

Figure 6.

Measured ΔTud as function of water flux density and heat input scenarios (set 1) of (a) constant 8-s pulse duration with varying heat intensity (W m−1) and (b) constant 85 W m−1 heat intensity with varying heat pulse duration.

[26] WFD estimation in the lower range of 1–5 cm d−1 was unsuccessful using the various heat pulse scenarios of set 1 (<2 kJ m−1), because values of ΔTud values were near the temperature measurement repeatability of 0.01°C, as marked by the horizontal dashed line in Figure 6. Consequently, ΔTud values were inconsistent, leading to the absence of a positive relation between heat input and ΔTud, as we determined for WFD values larger than 10 cm d−1. Therefore, accurate WFD values <10 cm d−1 require larger heat input values into the soil (sets 1 and 3).

3.2.2. Sets 2 and 3 Results

[27] To evaluate the required heat input for accurate estimation of WFD values below 10 cm d−1, we used equations (4) and (5) to calculate ΔTud with xu = xd = 6 mm, using values of WFD = 1 cm d−1, κ = 6.2 × 10−7 m2 s−1 and q = 256 W m−1. Computed values of ΔTud (Figures 7a and 7c) and ΔHTmax (Figures 7b and 7d) are presented as a function of heat pulse duration (0–40 s) and heater diameter (1.27 and 4.19 mm) in Figure 7 for a range of soil water content values. Figures 7a and 7b show the temperature responses for a heater diameter of 4.19 mm, whereas Figures 7c and 7d present the corresponding temperature responses for a heater diameter of 1.27 mm (the original HPP design). The solid lines represent ΔTud values for the experimental conditions of the saturated Tottori sand, whereas the dashed lines correspond with volumetric water content values in the range from 0.1 to 0.3 m3 m−3. Using 0.02°C as the minimum required temperature difference (ΔTud), the results in Figure 7a show that the minimum heating time is about 30 s. However, as the heating time is increased, the maximum temperature at the heater-soil interface, ΔHTmax, also increases. As determined from the analytical solution, ΔHTmax values are about 25 and 30°C for heat pulse durations of 30 and 40 s, respectively (Figure 7b). Assuming an initial soil temperature of 20°C, the corresponding maximum temperatures at the heater needle would be between 45 and 50°C, which is acceptable from the standpoint that resulting vapor transport is minimal [Saito et al., 2007]. When comparing these results to the original probe design of a 1.27 mm heater needle diameter (Figures 7c and 7d), we note that ΔTud values are very similar to those of the 4.19 mm heater. However, associated soil temperatures rises near the heater needle (ΔHTmax) are much higher, and approach 50°C, leading to soil temperatures of around 70°C and higher. These sensitivity results emphasizes the advantage of using a large heater diameter, enabling the application of larger heat inputs, thereby making possible the estimation of WFD values near the 1 cm d−1 range, without causing significant water redistribution by vapor transport or thermal convection.

Figure 7.

Theoretical (a and c) ΔTud and (b and d) ΔHTmax values as function of heat pulse duration and water content for a 4.19 mm (Figures 7a and 7b) and a 1.27 mm (Figures 7c and 7d) heater diameter HPP. Calculations were made with constants of 1 cm d−1 WFD, 6.2 × 10−7 m2 s−1 thermal diffusivity, and 256 W m−1 heat intensity, where ΔTud was evaluated for xu = xd = 6 mm.

[28] The resulting estimated WFD values in the range of 1–5 cm d−1 for heat input scenarios of sets 2 and 3 are presented in Figure 8. With one exception (5 cm d−1 scenario, set 3), all WFD estimates are in excellent agreement with the independently measured values. We note that WFD estimation was extremely sensitive to the effective calibration parameters (Table 1). In order to successfully estimate WFD in the low range, average effective calibration values using the replicates of sets 2 and 3 were required (Table 1).

Figure 8.

Measured and estimated water flux density in the range of 1–5 cm d−1 (sets 2 and 3), obtained from 30-s and 40-s heat pulse durations with ∼256 W m−1.

3.2.3. Discussion

[29] Our WFD estimation results are surprisingly good, despite various known sources of error. Although the analytical solutions are strictly valid for a domain with identical thermal properties for the heater needle and surrounding soil, by using effective thermal properties, an excellent match between observed and simulated temperature responses was obtained.

[30] Earlier we hypothesized that higher WFD values can cause additional spreading of the heat pulse by thermal dispersion, therefore lowering estimation of WFD [Hopmans et al., 2002]. The variation in optimized WFD for values >10 cm d−1 between selected THI scenarios of set 1 are shown in Figure 9, presenting the data for the 8-s heat input scenarios with varying heat input intensities (W m−1) in Figure 9a and the data for the 85 W m−1 heat input intensity scenarios with varying heat pulse duration in Figure 9b. In general, the results in Figure 9a show that the WFD estimation error decreases as total heat input increases, whereas the opposite result is obtained when varying heat pulse duration (Figure 9b). The effect of thermal dispersion becomes clearer when carefully examining the combined results of Figure 9. In the case of variable heat intensity with constant heating time (Figure 9a), the largest WFD errors are expected to occur with the lowest heat intensities as observed temperature responses are proportional to heat intensity. However, when using a constant heat intensity while increasing heating time (Figure 9b), the heat pulse is spread more with an increase in heating time, causing the expected thermal dispersion effects to dominate for the longer heat pulse duration scenarios. In addition to thermal dispersion, water blockage by the large heater needle can reduce water velocity for both upstream and downstream locations, thereby causing an underestimation of WFD that is proportional to WFD, as observed for WFD values above 10 cm d−1 (Figure 5). We suggest that the underestimations result from a combination of thermal dispersion and water blockage. However, these issues need further investigation.

Figure 9.

Relative error of water flux estimation in the 10–1000 cm d−1 range (set 1) with heat input scenarios of (a) constant 8-s pulse duration with varying heat intensity (W m−1) and (b) constant 85 W m−1 heat intensity with varying heat pulse duration.

[31] When comparing WFD estimates of set 1 (Figure 5) with those of sets 2 and 3 (Figure 8), we note increasing uncertainty of the estimated water flux data of set 1. Specifically, variation in the WFD estimates of set 1 ranged from 50% (10 cm d−1) to 15% (1,000 cm d−1), as defined by the difference between minimum and maximum water flux values relative to the mean (Figure 5), whereas uncertainty values for sets 2 and 3 were at maximum only about 7%. One may expect this to be caused by the averaging of the effective calibration parameters between the various heating scenarios of set 1, whereas separate calibration parameters were obtained for each of the two distinct heating scenarios in sets 2 and 3. However, after changing the calibrated effective distances values of set 1 by adding or subtracting the listed STD values in Table 1, corresponding ranges in WFD estimation varied by only 30% (10 cm d−1) and 1% (1,000 cm d−1). Therefore, the uncertainty of the estimated WFD values is not only caused by the averaging of the calibrated xeff values, but is also determined by violation of model assumptions.

[32] In this study, the experimental HPP measurements were conducted for saturated conditions only. Conducting controlled unsaturated water fluxes requires a complex experimental setup and complicates procedures for determining the actual WFD [Mori et al., 2005; Mortensen et al., 2006]. However, the unsaturated flow experiments of Mori et al. [2005] and Mortensen et al. [2006] confirmed that both analytical and numerical solutions for heat transport apply equally well to unsaturated conditions. Mori et al. [2005] concluded that major differences in heat transport are mainly due to changes in bulk soil heat capacity, and much less because of changes in soil thermal diffusivity. Additionally, the theoretical results in Figure 7 for the unsaturated Tottori sand show that the associated lower soil heat capacity increases the WFD sensitivity because of corresponding larger ΔTud values. Thus, unsaturated soils would require lower heat input values as applied here for the saturated Tottori sand, thereby lowering ΔHTmax. Therefore, the analyzed saturated conditions provide for a worst-case scenario, requiring the maximum heat input.

4. Conclusions

[33] Water flux density (WFD) was successfully estimated in the range from 1 to 1,000 cm d−1 with a 4-mm heater needle diameter HPP. So that water flux density values near 1 cm d−1 could be measured, we developed a large heater needle and applied the analytical solution for an infinite cylindrical heat source [Carslaw and Jaeger, 1959; Kluitenberg et al., 1995]. The temperature results of the analytical solution for both upstream and downstream thermistors were fitted to the measured temperature values using the inverse procedure of Hopmans et al. [2002]. Before estimating WFD, the HPP was calibrated with zero water flow conditions, optimizing effective needle spacing and thermal diffusivity values. Once calibrated, the WFD was inversely estimated by optimizing heat pulse velocity and corresponding WFD values [Mori et al., 2005]. We conclude that the 4-mm heater HPP provides for accurate WFD values between 1.0 and 10.0 cm d−1 and that the underestimation for WFD values larger than 10 cm d−1 was likely caused by thermal dispersion and flow disturbance by the heater needle. We find that WFD estimates are highly sensitive to the effective calibration parameters, with optimized values of needle spacing and thermal diffusivity depending on heat input scenario. To extend the presented method to unsaturated conditions, we must show that the modified HPP with the larger heater diameter and longer heat pulse duration is applicable to estimate unsaturated WFD values below 1 cm d−1. This is the subject of ongoing investigations. Despite uncertainties about the underestimation at water flux densities larger than 10 cm d−1 and the application to unsaturated conditions, we demonstrate the potential of the modified HPP for accurate in situ water flux density measurements.

Appendix A:: Analytical Solution

A1. Zero Water Flux Density

[34] In the absence of water flow, the solution for a cylindrical heat source of radius a (m) and length 2b (m), in an infinite homogeneous medium is [Kluitenberg et al., 1995]

equation image

where the limits of integration are

equation image
equation image

and where the heater is coincident with the z axis, lying between z coordinates −b and +b (Figure 1). Here, ΔT (°C) represents the temperature rise in the plane z = 0, at radial distance r (m) from the center of the cylindrical source at time t (s) from initiation of the heat pulse of duration t0 (s), and the q (W m−1) is the heat input per time per unit length of heater. In addition, I0 represents the modified Bessel function of the first kind of order zero, and erf represents the error function. When writing in Cartesian coordinates, the coordinate r in equation (A1) is defined as r = equation image. If the locations of interest are located at the x axis with y = 0, then r = ∣x∣.

A2. Water Flow in the x Direction

[35] For one-dimensional water flow in the x direction, water flux density is defined positive in the positive x direction (x takes on negative and positive values for upstream and downstream positions, respectively). A solution is obtained by treating the cylindrical source as a moving heat source in the manner of Carslaw and Jaeger [1959]. This is accomplished by substituting for x the expression (xVht). Furthermore, if one is interested in temperature along the principle upstream and downstream directions from the center of the heat source, then y = 0 (Figure 1), and now r = ∣xVht∣ and equation (A1) becomes

equation image


equation image
equation image

For large values of the argument bequation imagexVht∣, the error function term in equation (A2a) is approximately unity. In this investigation, the cylindrical heater was of sufficient length to satisfy this condition, thus allowing equation (A2a) to be written in the form

equation image

using the same limits of integration as in equation (A2).

Appendix B:: Parts and Manufacturers

[36] The heat pulse probe and experimental setup of this study were built using many different parts. Instead of including manufacturers and part details in the text, we refer to Table B1.

Table B1. List of Parts and Manufacturers
Part NumberPartManufacturer
1High-temperature 16 AWG internal electrical wire that can withstand 105°CBelden, Inc., St Louis, MO
2Enameled 79 μm-diameter heater wire with 207.9 Ω m−1 resistance, Nichrome 80 alloyPelican Wire Company, Inc., Naples, FL
32-ton clear epoxy glueITW Devcon Corp., Danvers, MA
4Omegabond 101 high thermal conductive epoxyOmega Engineering, Stamford, Connecticut
524.5-mm-long, 1.27-mm-OD and 0.84-mm inner diameter (18 gauge) stainless steel tubingSmall Parts, Inc., Miami Lakes, Florida
6Thermistor (10 kΩ resistance at 25°C with ±0.2°C tolerance; model 10K3MCD1).Betatherm Corporation, Shrewsbury, Massachusetts
725.4-mm-diameter and 10-mm-thick acetal DelrinMcMaster-Carr Supply Company, Los Angeles, California
8CR10XCampbell Scientific Inc., Logan, Utah
95 kΩ precision resistors (±0.01% tolerance, S102 series Bulk Metal Foil)Vishay Intertechnology, Inc., Malvern, Pennsylvania
10DC regulated power supply (Tenma model 72-6628)MCM Electronics Inc., Centerville, Ohio
114-wire axial precision power shunt (PLV7, 7 Watt at 25°C and 1Ω ± 0.02% tolerancePrecision Resistor Company, Inc., Largo, Florida
12Silicone sealantGE Silicones, Waterford, New York
13SS porous screen (mesh 100)Small Parts, Inc., Miami Lakes, Florida
14Masterflex L/S 7524-40 peristaltic pumpCole-Parmer Instrument Company, Vernon Hills, Illinois)
15Double syringe infusion/withdrawal pump (model 210)KD Scientific Inc, Boston, Massachusetts
16Matlab version 2006bThe Mathworks Inc.


[37] This work was supported by the National Science Foundation Biocomplexity program (proposal 0410055). We acknowledge comments and discussions by many of the occasional visitors in our lab.