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WATER RESOURCES RESEARCH, VOL. 43, W07408, doi:10.1029/2006WR005320, 2007

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ArticleNumerical evaluation of alternative heat pulse
probe designs and analyses
Hirotaka Saito,1,2 Jiri Šimůnek,1 Jan W. Hopmans,3 and Atac Tuli3
Received 5 July 2006; revised 14 April 2007; accepted 24 April 2007; published 11 July 2007.

[1] The heat pulse probe (HPP) has received more attention as it allows in situ,

simultaneous, and automated measurements of soil hydraulic and thermal properties, aswell as soil water fluxes. Although the currently used design allows many applications,changes in HPP design and analyses are needed to increase the sensitivity to smaller
water fluxes. In this study, numerical simulations were used to evaluate the effects of(1) sensor locations and thermal properties of the HPP sensor body material, (2) the heater
diameter and heat pulse intensity, and (3) vapor flow on HPP performance. A numerical
study was carried out using the HYDRUS code for the HPP consisting of threeparallel needles, spaced 6 mm apart. A heater of varying diameter in the center issurrounded by a pair of thermistor needles. Our results show that significantly different
temperature responses will be obtained depending on the axial location of thermistors, andthat only temperature measurements near the middle of the 33-mm-long heater fulfill
the assumption of an infinite line heat source. It is shown that larger heater needle
diameters allow larger heat pulses, leading to larger temperature differences between
upstream and downstream thermistor needles and a higher sensitivity to water flux
measurements. For example, for a standard 1-mm-diameter heater needle, ten times greater
heat pulse input results in ten times greater sensitivity, but with the maximum temperature
at the heater exceeding 100°C. This maximum temperature can be reduced to 80°C
by increasing the heater diameter to 4 mm, while maintaining equal sensitivity.
Consideration of vapor transport significantly reduced temperature increases near the heater
caused by latent heat of vaporization. Larger heat pulses are beneficial for estimation ofliquid water fluxes in the 1 cm d1 range, provided vapor transport is considered.
Citation: Saito, H., J. Šimůnek, J. W. Hopmans, and A. Tuli (2007), Numerical evaluation of alternative heat pulse probe designs andanalyses, Water Resour. Res., 43, W07408, doi:10.1029/2006WR005320.

1. Introduction[2] Soil hydraulic and thermal properties are of greatinterest in many scientific and engineering applications
where accurate predictions of soil water content and temperature are required as these significantly control many soil
physical, chemical, and biological processes. Knowledge ofboth sets of properties is also required by models simulating
simultaneous movement of liquid water, water vapor andheat in the vadose zone, playing a critical role in the overallwater and energy balance, especially in arid or semiarid
regions with generally low soil moisture content values. The
heat pulse probe (HPP) has received increasing attention
over the last decade as it allows in situ, direct, simultaneous,
rapid, and automated measurements of both soil hydraulicand thermal parameters, as well as soil water fluxes.

1
Department of Environmental Sciences, University of California,
Riverside, California, USA.
2
Now at Department of Ecoregion Science, Tokyo University ofAgriculture and Technology, Tokyo, Japan.
3
Department of Land, Air and Water Resources, University of California,
Davis, California, USA.

Copyright 2007 by the American Geophysical Union.
0043-1397/07/2006WR005320$09.00[3] The dual-needle HPP allows rapid estimation of thesoil thermal properties from measured temperature changes,
at a fixed distance from a heater needle that approximates aline source. The original design of the dual heat pulse probe
consists of two parallel 28-mm-long and 0.8-mm-diameter
needles mounted to the HPP sensor body with a 6-mm
separation distance [Campbell et al., 1991]. In their analysis, a short known 8-s heat pulse was generated in one of theneedles. An analytical solution for an infinite line source
derived with an instantaneous (Dirac-type) heat pulse wasused to estimate the heat capacity and volumetric watercontent of the surrounding soil. As the instantaneous heatpulse solution was found inadequate to estimate soil thermal
diffusivity and soil thermal conductivity, Bristow et al.
[1994] developed instead an approach on the basis of theanalytical solution for a short-duration heat pulse. Theirapproach allowed simultaneous estimation of all three soil
thermal properties; i.e., the thermal diffusivity, heat capacity, and thermal conductivity, as well as soil water content[Bristow, 1998].
[4] More recently, Ren et al. [2000] used a tri-needle HPP
to indirectly estimate also the saturated water flux densityfrom the difference in temperature changes measured downstream and upstream of the heating needle. The tri-needle
HPP consists of three parallel needles positioned in a

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common plane and mounted to the HPP sensor body with a
6-mm separation distance between the heater needle andboth thermistor needles. If the three needles are aligned
along the water flow direction, the difference in temperature
signals between the two thermistors is used to estimate
convective heat transport. Thus the basic idea is to extract
information related to soil water flux from the asymmetry of
the temperature responses, as measured from the downstream and upstream thermistors. Ren et al.’s [2000] analytical solution depended on the maximum temperature
difference between the two thermistor needles to estimate
water flux density. While measured and predicted temperature signals agreed well for the sandy soil, significant
deviations occurred for the sandy loam and clay loam that
could not be explained, especially if water flux densities
were small. Wang et al. [2002] later developed a simple
mathematical analysis using the ratio of temperaturechanges recorded at the upstream and downstream sensors.
Although their approach eliminated the complicated numerical integration [Ren et al., 2000], the ratio method systematically underestimated saturated soil water fluxes [Ochsner
et al., 2005]. Gao et al. [2006] recently hypothesized that
this underestimation at high water fluxes may be caused bybypass water flow along the column walls (i.e., the wall flow
effect). Gao et al. [2006] developed a correction factor thatconsiderably improved predictions of saturated water flow.
[5] Several recent studies demonstrated the applicabilityof the HPP method to analyze unsaturated water flow
[Hopmans et al., 2002; Mori et al., 2003, 2005; Mortensen
et al., 2006]. While Hopmans et al. [2002] considered
the tri-needle HPP, Mori et al. [2003, 2005] and Mortensen
et al. [2006] used a multifunctional heat pulse probe(MFHPP), which was developed to evaluate simultaneously
coupled water, heat, and solute transport in unsaturated
soils. Unlike most other studies that relied on analyticalsolutions of heat transport, Hopmans et al. [2002] used anumerical model to simultaneously evaluate the soil
hydraulic and thermal properties, as well as water fluxes.
Mortensen et al. [2006] later extended this numericalapproach to estimate thermal, hydraulic, and solute transport
properties using the MFHPP. Hopmans et al. [2002] also
investigated the possible effects of dispersive heat transport
due to local pore water variations to explain discrepancies
between measured and predicted temperature responses forincreasing water flux densities. They suggested including an
additional thermistor in the transverse direction to waterflow to account for the dispersive heat flux. Although the
MFHPP developed by Mori et al. [2003] was designed tohave four thermistor needles around the heating probe, theyused the same analysis as used with the tri-needle HPP.
[6] According to Mori et al. [2005], horizontal thermistors can be used to estimate the thermal diffusivity, if thewater flux is small enough so that the convective heattransport can be neglected. In the work of Mori et al.
[2005], unsaturated water fluxes were estimated using themultistep outflow experiment combined with the analyticalsolution of Ren et al. [2000] that considers convective heattransport. Despite obtaining accurate predictions of saturated water fluxes, unsaturated water fluxes were generally
overestimated. Differences between measured and predictedvalues increased specially for water fluxes below 0.1 m d1.
Unlike Mori et al. [2003, 2005] who relied only on analy-

W07408tical solutions to estimate soil thermal and hydraulic parameters, Mortensen et al. [2006] used a numerical inverse
analysis to analyze MFHPP measurements. Mortensen etal. [2006] used the flexibility of the numerical inverse
analysis to combine different types of information to simultaneously estimate thermal, hydraulic, and solute transport
properties. In general, the estimated transport properties andsaturated/unsaturated water flux densities were in good
agreement with independently measured values. Their study
showed that the MFHPP combined with the (inverse)
numerical approach is a promising methodology to simultaneously estimate soil hydraulic, thermal, and solute transport properties. An important advantage of the numericalapproach in analyzing data collected with the HPP is theelimination of constraints on needle geometry as imposedby analytical solutions. With user-friendly and powerful
numerical codes simulating coupled water, heat, and solutetransport, it is expected that an increasing number of studies
will adopt numerical approaches in analyzing HPP(or MFHPP) and other measurements.
[7] Although significant advances occurred during thelast decade in applications of the HPP [Ren et al., 2000;
Mori et al., 2003, 2005; Ochsner et al., 2005; Mortensen etal., 2006] and in understanding of the underlying theory[Kluitenberg et al., 1995; Knight and Kluitenberg, 2005], itis rather surprising that the HPP design and operation (i.e.,
about 30-mm-long and 1-mm-diameter probes aligned parallel with 6-mm spacing distance, using a heat pulse ofabout 60 W m1 for 8 s) have not changed much from thoseoriginally presented by Campbell et al. [1991]. With thecurrent design and operation, the lower limit of accurate
estimations of the water flux density is 0.06 m d1, for atemperature measurement resolution of about 0.01°C [Renet al., 2000; Mortensen et al., 2006]. To increase thesensitivity of HPP to smaller water fluxes, changes in
HPP design, operation, and analysis are needed. Additionally, the error analysis by Kluitenberg et al. [1995] showed
the sensitivity of soil thermal properties on needle spacings:
2% error in the spacing between heater and thermistors led
to 4% error in the estimated thermal diffusivity and heat
capacity. Therefore it is very important to use rigid probes tominimize changes in spacing distances during their installation. For that purpose, shorter thermistor needles were
proposed by Mori et al. [2003] to minimize deflection ofneedles during installation. However, it is expected that
thermistor readings close to the HPP body may be affectedby its thermal properties. While thinner needles will distort
the flow field to a lesser extent, they can create sharp
temperature increases for larger heat pulses, leading toevaporation and boiling of soil water around the heater
needle. In contrast, thicker needles will increasingly distort
the flow field, but will reduce temperature increases aroundthe heater for equal heat pulse intensities. Proposed geometrical changes and changes in HPP operation can affect the
HPP performance, and therefore need to be fully evaluated.
[8] Increased temperatures around the heater needle during the release of the heat pulse will create significant
temperature gradients, inducing both liquid water and water
vapor transport in the direction of decreasing temperatures.
High temperatures may lead to water evaporating aroundthe heater and reducing temperatures generated by thereleased heat, thereby affecting heat transport. Yet, the2 of 14

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W07408Figure 1. Schematic showing the heat pulse probe (HPP) with a total of 16 thermistors. Distances
between the centers of the heater and thermistor needles are 6 mm.
simultaneous movement of liquid water, water vapor, andheat has not been considered in the HPP analysis.
[9] The main objectives of this study therefore are to
numerically evaluate (1) effects of thermistor locations of
the heat pulse probe and thermal properties of the HPP
sensor body material, (2) effects of the heater diameter andheat pulse intensity on measured temperature responses, and(3) effects of vapor flow on HPP performance. While thefirst objective is achieved by numerically solving the heattransport equation by considering thermal conduction only,
the second objective also considers convection of sensible
heat. Finally, the third objective is achieved by numericallysolving coupled movement of liquid water, water vapor andheat. The first objective is achieved using both measured
and numerically generated data.

2. Experimental HPP[10] Prior to numerical evaluation of the effects ofthermistors location and HPP design on measurement
sensitivity, we verified the presented numerical approach
experimentally after construction of a multithermistor HPP.
Figure 1 shows a schematic diagram of the specific five-needle HPP constructed to evaluate influence of thermistor
position and sensor body material. With the exception of thelarger number of thermistors and the elimination of theWenner array electrodes, the fabrication of the multithermistor HPP is similar to the MFHPP presented by Moriet al. [2005] and Mortensen et al. [2006]. In addition to a
26-mm-long central heater needle (LN-2), this specific HPP
includes two short 13-mm (SN-5 and SN-6) and two long
26-mm thermistor needles (LN-1 and LN-3). Each thermistor needle incorporates three SN or four LN thermistors.
Additional thermistors were installed on the outside of theheater needle (5) and in the acetal Delrin base of the sensor
body (16), 6 mm away from SN-6 (Figure 1). All needles
were constructed from 1.27-mm O.D. and 0.84-mm I.D.
stainless steel hollow tube [Mori et al., 2003]. The multithermistor HPP included a total of 16 0.46-mm-diameter
thermistors (10 kW resistance at 25°C with 0.004°C precision; Model 10K3MCD1, Betatherm Corporation, Shrewsbury, Massachusetts). After insertion of the thermistors in
the needles, the available hollow space was filled with
Omegabond 101 epoxy (Omega Engineering, Stamford,
Connecticut), which has a relatively high thermal conductivity and is a good electrical insulator [Mori et al., 2003].

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Table 1. Measured Thermal Properties of Delrin That Was Usedfor the Body Material of the HPP in Figure 1 (Section 3.1)aW07408tors were measured in agar solutions after a 103.59 W m1heat pulse for 8 s.

DelrinThermal conductivity, W m1 K1Heat capacity, MJ m3 K1
Thermal diffusivityb, 106 m2 s1

23°C

40°C

60°C

80°C

0.364
1.8
0.20

0.372
2.0
0.19

0.378
2.1
0.18

0.380
2.3
0.16

HPP is heat pulse probe. Density of Delrin is 1.41 gm cm3.
Thermal diffusivity is defined as the ratio of thermal conductivity to heat
capacity.
ab

All five needles were secured through predrilled holes in a
50.8-mm-diameter and 7.9-mm-thick Delrin plug. All
34 lead wires were soldered to 34 corresponding wires of
two 24 AWG shielded multiconductor cables (Belden,
electronics Division, Richmond, Indiana). The multiconductor cables were held in place by another 20-mm-long
Delrin plug that closed the air-filled space within the sensor
body. The thermal properties of the Delrin, measured by
TPRL Incorporated (3080 Kent Avenue, West Lafayette,
Indiana 47906) for a range of temperatures using differential
scanning calorimetry (specific heat) and laser flash diffusivity method (thermal diffusivity), are listed in Table 1.
Reasons for using Delrin were its high tensile strength andthe relative ease by which it can be machined. Since allparts were screwed together, it can be easily disassembled,so that electrical circuitry can be checked and needles canbe interchanged in case of malfunctioning.
[11] The heater was constructed by inserting two loops of
79-mm-diameter 209.7 W m1 Nichrome 80 alloy enameledresistance wire (Pelican Wire Company, Incorporated,
Naples, Florida) into the heater needle (LN-2). Heat input
and temperature logging of all thermistors was controlled by
a CR10 datalogger (Campbell Scientific Incorporated), as
presented by Mori et al. [2003].
[12] To test its functionality and for calibration purposes,
the multithermistor HPP was evaluated in a 4 g L1 agar
solution. As was done by Mori et al. [2003], the agar test wasused to determine the effective separation distance (reff) of all
thermistor needles and the heater, assuming that the thermal
properties of the agar solution are equal to those of water.
This was done by fitting measured heat pulse responses forthermistors 2, 6, 10, and 13 to the solution of heat conduction
equation for an infinite line heat source [Mori et al., 2003,
2005]. As in the work of Mori et al. [2003], we also
determined the reff in full saturation of Tottori sand withwater. Values for reff were between 5.90 and 6.30 mm, andwere equal to values determined by Mori et al. [2003].
[13] Temperature responses of a few selected thermistors
were used to validate the sensitivity results with the inverse
modeling module of the HYDRUS model [Šimůnek et al.,2006], as was done by Mortensen et al. [2006]. In thismodule, the Levenberg-Marquardt algorithm is used forparameter optimization. Since no water flow occurred inthis experiment, only heat conduction was considered. The
objective function minimized in the optimization was
defined as the sum of weighted squared residuals between
measured and simulated temperatures [Mortensen et al.,2006]. For this purpose, temperature changes at all thermis-

3. Numerical Analysis[14] Since an axisymmetrical three-dimensional solution
can be achieved with a two-dimensional model, only a twodimensional formulation will be presented. Perfect thermal
contact was assumed in all numerical calculations at interfaces between HPP needles and the surrounding soil.
Although it has been known that an imperfect thermal
contact can have a profound impact on heat transport
through the interface, theoretical solutions for HPP havealways been derived with the assumption of perfect thermal
contact [e.g., Ren et al., 2000]. All numerical calculations
were carried out using the HYDRUS software program[Šimůnek et al., 2006].
3.1. Sensor Locations and Thermal Properties of HPP[15] The first part investigates the effects of thermistor
locations and HPP body material on temperature measurements. This simulation was conducted independently fromthe experimental study discussed earlier in the paper. The
axisymmetric simulation domain had a radius of 20 mm fromthe center of the heater and a height of 68 mm (Figure 2). To
preserve the axisymmetricity of the problem, only thermal
conduction was considered. The governing equation of heattransport in the axisymmetric domain to be solved is
described by
Cp





@T 1 @@T@@T¼
rl0 ðqÞ
þl0 ðqÞ
@t
r @r
@r
@z
@z

ð1Þ

where r is the radial distance from the center of the domain(m), z is the vertical coordinate (m), Cp is the volumetricheat capacity of the moist soil (J m3 K1), defined as thesum of volumetric heat capacities of the soil and water
multiplied by their volumetric fractions, T is temperature(K), and l0(q) is the soil’s isotropic thermal conductivity(diagonal tensor) (W m1 K1), accounting for the
tortuosity and water content of the porous medium, as
described by a simple equation given by Chung and Horton[1987]:
l0 ðqÞ ¼ b1 þ b2 q þ b3 q0:5

ð2Þ

where q defines the soil volumetric water content (m3 m3),
and b1, b2, and b3 are empirical regression parameters(W m1 K1).
[16] Figure 2 shows the finite element mesh for thecomputational domain with gradually increasing size ofthe finite elements away from the heater. The simulation
domain consisted of two materials: the soil and the HPP
body. The soil was considered uniform and homogeneous,
having an initial temperature of 20°C, a saturated watercontent of 0.43 m3 m3, with a solid phase fraction of
0.57 m3 m3 and a thermal conductivity for a loam (b1 =0.243 W m1 K1, b2 = 0.393 W m1 K1, and b3 =1.534 W m1 K1 in equation (2), [Chung and Horton,
1987]). Volumetric heat capacities of the solid phase, Cs,
and liquid water, Cw, were 1.92 and 4.18 MJ m3 K1,
respectively. The thermal properties of the soil are summarized in Table 2. The body of the HPP (lower 18 mm in

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W07408Figure 2. A finite element mesh used to investigate the effects of sensor locations and heat pulse probesensor materials. Seven observation nodes (ON) are placed along the assumed thermistor probe, located
6 mm away and parallel to the heater probe. The transport domain is axisymmetrical around the vertical axis.
Figure 2) was assumed to be made from either thermallyconductive or nonconductive epoxy following Ham andBenson [2004]. Thermal properties of both epoxies arelisted in Table 3. The heat pulse was applied at the boundary
nodes, corresponding with the wall of the heater needle.
[17] Following the standard geometry of the HPP, thetotal length of the heater was 33 mm, of which 5 mm was
imbedded in the epoxy material of the sensor body, and thediameter was 1 mm. To mimic standard experiments, a
60 W m1 heat pulse was applied to the heater for 8 s at thestart of each model simulation. The heat transport module of
HYDRUS was modified by adding a zero-order source term
to represent the heat pulse boundary condition in finite
element nodes at the surface of the heater (similarly as by
Hopmans et al. [2002]). A given heat pulse was thenuniformly distributed along these nodes (Figure 2). The
zero temperature gradient boundary condition was used in
the remaining boundary nodes. Temperature changes werenumerically recorded for about 300 s at seven observation
nodes, located along the assumed thermistor needle, 6 mm
away from the heater. One observation node (ON1) was
positioned 5 mm in the sensor body, one observation node(ON2) was located at the soil-epoxy interface, and additional five observation nodes were placed in the soil (ON3through ON7 at 5, 10, 15, 20, and 28 mm away from thesoil-epoxy interface, respectively: Figure 2). As the heater
needle was inserted 5 mm into the body of the HPP sensor,
we could evaluate the effects of heating the bottom sectionof the heater on temperature changes along the sensor. Thus

a total of four simulations were conducted that consideredthermally conductive epoxy with (case (a)) and without(case (c)) heating of the sensor body, and thermally nonconductive epoxy with (case (b)) and without (case (d))
heating of the sensor body. The numerical solutions forthese four cases were compared with the infinite line heatsource analytical solution that uses the two-dimensionalradially symmetric transport domain perpendicular to theheater probe, as used by Hopmans et al. [2002].
[18] Another factor that may affect the HPP performanceis the axial conduction of heat in the thermistor needle as
they are usually built from a thermally much more conductive material (stainless steel) than the surrounding soil. If thethermal conductivity of the sensor needle was infinite, theheat would be redistributed instantaneously along the needle
and thermistors located at any position would measure theTable 2. Thermal Properties of the Soil Used in This Study(Sections 3.1, 3.2, and 3.3)
Parameter

Units

Value

References

b1
b2
b3
Cw
Cs
bL
bT

W m1 K1
W m1 K1
W m1 K1
MJ m3 K1
MJ m3 K1
M
M

0.243
0.393
1.534
1.92
4.18
0.002210.1 bL

Chung and Horton [1987]

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W07408

Table 3. Thermal Properties of Epoxies Used for the Heat Pulse
Probe Body (Section 3.1), Adapted From Ham and Benson [2004]
Epoxy

ThermalConductivity, W m1 K1Heat Capacity,
MJ m3 K1

Thermally conductive
Thermally nonconductive

0.6
0.2

2.9
1.7

conductivity in the absence of flow, l0(q), with a macrodispersivity term, which is assumed to be a linear functionof velocity [de Marsily, 1986; Hopmans et al., 2002]. The
apparent thermal conductivity for the two-dimensional heattransport equation is expressed as [Šimůnek and Suarez,1993]:lij ðqÞ ¼ l0 ðqÞd ij þ d ij b T Cw jql j þ ðb L  b T ÞCw

same temperature. However, the axial conduction of heat in
the sensor adds a third dimension to the problem and cannot
thus be evaluated simultaneously with the heat transport
from the heater using axisymmetrical calculations. Had anarrow strip with larger heat conductivity been specified at alocation of observation nodes in the domain shown in
Figure 2, this would have represented, because of theaxisymmetricity of the problem, a conductive cylinder, rather
than a needle. To avoid the need for a full three-dimensional
simulation, an axisymmetric simulation domain that consisted only of the thermistor needle and the surrounding soil
was used to investigate the influence of the axial conductionof heat on temperature profiles within the thermistor needle.
Figure 3 shows the finite element mesh for the resulting
computational domain. The small layer of the soil (0.25 mm
thickness) had to be included into the transport domain sothat heat could be redistributed within a needle and back tothe soil. If the boundary temperature was imposed directlyat the surface of the sensor needle, such redistribution
would not have been possible. The thermistor needle wasassumed to have typical dimensions of 1-mm diameter and
33-mm length. The thermal properties of the soil were thesame as above, while the thermal properties of typicalstainless steel, whose thermal conductivity is one order ofmagnitude greater (16.3 W m1 K1) than that of the soil,
were used for the thermistor needle, as HPP needles areusually made from it. Since this numerical experiment was
conducted solely to demonstrate the impact of the axial
conduction, any material with much greater thermal conductivity than that of the soil could be used. Calculated
temperatures at seven observation nodes in Figure 2 for case(a) and the interpolated values were prescribed as boundary
conditions along the outside boundary of the transportdomain. Temperature changes at three locations correspondto ON2, ON4, and ON6 in Figure 2 (5, 15, and 25 mm fromthe bottom of the domain as shown in Figure 3), in thecenter of the thermistor needle were calculated and compared to those at the domain boundary.
3.2. Heater Size and Heat Pulse Intensity[19] The effects of the diameter of the heater needle andheat pulse intensity on HPP performance was studied using
a two-dimensional transport domain perpendicular to theheat pulse probes (Figure 4). Because we consider heat
convection by steady state liquid water flow in this case, thegoverning equation of heat transport is described byCp@T¼ r½lðqÞrT  Cw ql rT@t

W07408ql;j ql;i
jql j

ð4Þ

where dij is the Kronecker delta (dij = 1 if i = j, and d ij 6¼ 0
otherwise), b L and b T are the longitudinal and transverse
thermal dispersivities (m), respectively (Table 2). The
liquid water flux ql (m s1) is calculated using a twodimensional isothermal Darcian equation:kql ¼ Klh rh þ Klh~

ð5Þ

ð3Þ

where ql is the flux density (m s1) of liquid water, Cw is thevolumetric heat capacity of the liquid water (J m3 K1),
and l(q) now denotes the soil’s apparent thermal conductivity (J m1 s1 K1) that combines the soil thermal

Figure 3. A finite element mesh used to investigate theeffects of axial conduction of heat in the thermistor needle.
The transport domain is axisymmetrical around the verticalaxis. Open squares indicate locations where temperaturechanges were observed, while open circles correspond toobservation nodes (ON) in Figure 2.

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Figure 4. A finite element mesh used to investigate theeffects of the heater size and heat pulse intensity. Effects ofthe vapor transport on the HPP performance was studied
using this two-dimensional transport domain. Temperatures
were recorded at three observation nodes (open circles), Ou,
and Od representing upstream and downstream probes,
respectively, as well as near the heater.
where h is the pressure head (m), ~k is the vertical unit vector,
and Klh is the (isothermal) hydraulic conductivity (m s1).
For isotropic conditions, Klh is a tensor with zero nondiagonalterms. The governing equation for the liquid water flow isthen given by the Richards equation ashi@q¼ r Klh rh þ Klh~
k
@t

h

m i2
Klh ðhÞ ¼ Ks Se0:5 1  1  Se1=m
8
>
<

1
m h<0
q  qr
1
þ
ahjn
½
j
¼
Se ðhÞ ¼
qs  qr >:1h 0

where Ks is the saturated hydraulic conductivity (m s1), Se
is the effective saturation (),qs and qr are the saturated andresidual water contents (m3 m3), respectively; and a (m1),n (), and m (= 1  1/n) are empirical shape parameters.
[21] In this numerical experiment the distance between thecenter of the heater and both thermistor needles (upstream
and downstream) was held constant at 6 mm, so that the sizeof the radiated heat pulse at the sensor needle is consistent
between different heater sizes. The duration of the heat pulse
was 8 s, assuming an infinite and uniform heater source. The
initial temperature was 20°C for the entire domain. The bulk
water flux, except around the heater, was assumed uniform
and unidirectional, parallel with the vertical axis, by specifying identical flux boundary conditions at the top andbottom of the transport domain and assuming a zero horizontal flux at the vertical sides of the domain. While waterflowing into the domain at the top was assumed to have a
20°C temperature, the remaining domain boundaries were
simulated with a second-type boundary condition of zerotemperature gradient. A specified heat pulse was distributed
uniformly across all nodes of the heater.
[22] Temperatures were observed at three observation
nodes: Ou, Od, and at the heater itself (Figure 4). The
impact of heater diameter (d) and heat pulse intensity (H)for a range of water fluxes (q) and water contents (q)(Table 4) was investigated. Since the effect of the vapor
phase was not considered, target water fluxes and watercontents were artificially established by assuming saturated
water flow. For example, to obtain a water flux of 0.1 m d1
and the volumetric water content of 0.3 m3 m3, the soil
was assumed to be saturated, having the saturated hydraulicconductivity Ks equal to 0.1 m d1, the corresponding
saturated water content of 0.3 m3 m3, and with all otherremaining parameters unchanged.
3.3. Vapor Transport[23] To evaluate the effects of the coupled movement ofliquid water, water vapor, and heat, selected simulations
from section 3.2 were reevaluated, while considering phasechanges. The governing equation accounting for heat conduction, convection of sensible heat by movement of liquidwater and water vapor, and transfer of latent heat bydiffusion of water vapor is given by [Nassar and Horton,
1992]
@T
@qv
þ L0
¼ r½lðqÞrT  ðL0 þ Cv T Þrqv  Cv qv rT@t@t
 Cw Trql  Cw ql rTCp
ð6Þ

[20] The pore size distribution model of Mualem [1976]
was used to predict the isothermal unsaturated hydraulicconductivity function, Klh, from the saturated hydraulicconductivity and the van Genuchten’s [1980] model parameters of the soil water retention curve (7) and (8):
ð7Þ

ð8Þ

W07408ð9Þ

where L0 is the volumetric latent heat of vaporization ofliquid water (J m3), qv is the volumetric water vapor
Table 4. Values of Variables Used in Numerical Simulations
Investigating the Effects of the Heater Diameter and Heat Pulse
Intensitya
Variables

ValuesDiameter, d, mm
Flux, q, m d1
Water content, q, m3 m3
HPP intensity, H, W m1

1, 2, 40.01, 0.1, 0.5, 1, 10
0.3, 0.4, 0.5, 0.6
60, 150, 300, 600

aAll possible 240 combinations were considered (section 3.2).

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Table 5. Soil Isothermal and Thermal Conductivity Models and Their Parameter Valuesa

KlT

Modelsdg
Klh hGwT g1 dT

Kvh

Mg
D
rw rsv RT Hr

0

Parameters

Description

ValuesGwT ()g 0, g s2
g, g s2
D, m2 s1
rsv, kg m3
M, kg mol1g, m s2
R, J mol1K1
Hr ()

gain factor
surface tension of soil water at 25°C
surface tension of soil water (= 75.6  0.1425T  2.38 104T2)
vapor diffusivity in soil (= tqaDa)saturated vapor density
molecular weight of water
gravitational acceleration
universal gas constant!
relative humidity
qa7=3
tortuosity factor ¼ 2
qs

2 !
T
5
diffusivity of vapor in air ¼ 2:12 10
273:15(   )!
air filled porosityq
2:6 q 4enhancement factor ¼ 9:5 þ 3  8:5 exp  1 þ pﬃﬃﬃﬃ
qs
fc qs
mass fraction of clay

7.0
71.89
...
...
...
0.018015
9.81
8.314
...

t ()
Da, m2 s1
3

KvT

drsv
D
rw hHr dT

3qa, m mh ()fc ()

...
...
...
...
0.02

a

From Saito et al. [2006]; section 3.3.

content (expressed as an equivalent water content, (m3 m3)),
Cv is the volumetric heat capacity of the water vapor(J m3 K1), and qv is the water vapor flux density(m s1). When coupled heat transport is considered, both
liquid water (ql) and water vapor (qv) fluxes are defined as
the sum of isothermal and thermal fluxes, due to pressure
head and temperature gradients, respectively [Philip andde Vries, 1957; Nassar and Horton, 1992]:
k  KlT rT
ql ¼ qlh þ qlT ¼ Klh rh  Klh~ð10Þ

qv ¼ qvh þ qvT ¼ Kvh rh  KvT rT

ð11Þ

where qlh and qlT are isothermal and thermal liquid water
flux densities (m s1), respectively; qvh and qvT are isothermal and thermal water vapor flux densities (m s1),respectively; h is the pressure head (m), ~k is the verticalunit vector, T is the temperature (K), and Klh (m s1) and KlT(m2 K1 s1) are the (isothermal and thermal) hydraulicconductivities for liquid phase fluxes due to gradients in
h and T, respectively; and Kvh (m s1) and KvT (m2 K1 s1)
are the isothermal and thermal vapor hydraulic conductivities (diagonal tensors), respectively. The governing liquidwater and water vapor flow equation is then given byhi@q¼ r Klh rh þ Klh~k þ KlT rT þ Kvh rh þ KvT rT@thi¼ r KTh rh þ Klh~k þ KTT rT

ð12Þ

where KTh (m s1) and KTT (m2 K1 s1) are theisothermal and thermal total hydraulic conductivities,
respectively, and whereKTh ¼ Klh þ Kvh

ð13Þ

KTT ¼ KlT þ KvT

ð14Þ

To calculate values of KlT, Kvh and KvT, standard models,summarized together with associated parameters in Table 5,
were used [Saito et al., 2006].
[24] The simulation domain was identical to that ofsection 3.2 (Figure 4). However, as now the soil domain

includes the air phase (the soil is unsaturated) to allow forvapor transport, the soil hydraulic properties were different.
Instead, simulations were carried out for a sandy soil (Ks =71.3 or 713.0 m d1, qs = 0.43 m3 m3, qr = 0.045 m3 m3,
a = 14.5 m1, and n = 2.6). The water content corresponding
to the unsaturated hydraulic conductivity equal to the target
water flux was then calculated from the van GenuchtenMualem model (7). This water content was used as the initialwater content with the target flux used as a flux boundary
condition for both top and bottom boundaries. Remaining
boundary conditions for water flow and heat transport were
exactly the same as those used in section 3.2. Because of theinduced vapor transport, water content values and both liquidwater and vapor fluxes are affected by the heat pulse magnitude around the heater. Numerical experiments included
simulations with and without vapor transport for different
heater diameters, water fluxes, water contents, and heat pulse
intensities (Table 6). Water contents listed in Table 6 wereuniquely determined from the target water fluxes. For example, if the target water flux is 0.1 m d1, the corresponding
water content calculated from the van Genuchten-Mualem
model (7) is 0.197 (effective saturation of 0.395).

4. Results and Discussion
4.1. Sensor Location and HPP Body Material
4.1.1. Experiment[25] Figure 5 shows observed temperature changes atthermistors 13, 14, and 15 (Figure 1) and those simultaneously fitted to observed values to estimate thermal conTable 6. Values of Variables Used in Numerical Simulations
Investigating the Effects of Vapor Transport (Section 3.3)aVariables

ValuesDiameter, d, mm
Flux, q, m d1
Water contentc, q, m3 m3
HPP intensity, H, W m1

1, 2, 40.01, 0.1, 1, 10b0.128, 0.197, 0.315d
60, 150, 300, 600

aAll possible 48 simulations were conducted.
Flux of 0.5 m d1 used in Table 4 is not considered here.
Water contents were calculated for given soil hydraulic properties from
specified flux.
d
The water content of 0.315 was used twice for the saturated hydraulicconductivity of 71.3 and 713 m d1.

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Figure 5. Temperature changes in agar solution observedat three thermistors (13, 14, and 15) along one of the shortneedles, SN-6, of the HPP of Figure 1, and those inversely
fitted using HYDRUS.
ductivity and heat capacity of the HPP body (Delrin). The
simulation domain was similar to Figure 2 with slightly
different dimensions and different materials. For example,
thermistor 13 of the experimental SN-6 is 13.0 mm from thesensor base (Figure 1), whereas thermistor at ON5 of thesimulation domain used in the sensitivity analysis (Figure 2)
is 15.0 mm from the Delrin base material of the sensor
body. Although the heater wire is looped twice throughmost of the heater needle, there is only a single loop

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embedded in the Delrin sensor body. To account for thisdifference in heater wire length, the heat pulse in the
numerical study was applied to only 4 mm of the HPP
body. While fitted temperatures matched observed temperatures well, especially those in the agar solution (thermistors
13 and 14), temperature changes in the Delrin sensor
body (thermistor 15) did not correspond well. Estimatedthermal conductivity and heat capacity of the Delrin were0.201 W m1 K1 and 1.2 MJ m3 K1, respectively. These
values are about half of the independently measured values(Table 1), though the estimated thermal diffusivity of 0.17 
106 m2 s1 is relatively close to the measured value. We areuncertain at this time what caused this large difference
between measured and fitted thermal properties of the Delrin,
as we did not explicitly calibrate for thermistor 15 in thesensor body, as we did for the needle thermistors. Most likely,
estimated thermal properties of the Delrin were biased,because the heat pulse was applied to only 4 mm of the
HPP body. However, our results suggest that thermal properties of the HPP body need to be carefully determined, if
thermistors close or in the HPP sensor body are used for soil
thermal measurements. In practice though, estimating thermal properties of the HPP body is usually not a goal, but israther done during calibration. The impact of the discrepancy
between observed and fitted thermal properties of the HPP
body observed in this study therefore should not be significant for soil thermal property measurements.
4.1.2. Numerical Simulations[26] Figure 6 shows temperature changes at the sevenobservation nodes (ON’s) along the sensor needle for

Figure 6. Temperature changes at seven observation nodes (ON1 – ON7) along the sensor probe fordifferent HPP sensor body materials, with and without heating of the sensor body. The numerical solution
for the infinite line source is plotted by the open circles.
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W07408[28] Temperatures presented in Figure 6 were calculated
by assuming that the heat transport was unaffected by thepresence of sensor needles. Specifically, calculations did notaccount for the axial heat conduction in the thermistor
needle. The impact of the axial conduction for case (a),Figure 6, is shown in Figure 7 where temperatures in the

Figure 7. Temperatures (open symbols) at three locations
correspond to ON2, ON4, and ON6, respectively, (5, 15,
and 25 mm from the bottom of the domain in Figure 3) atthe center of the thermistor needle. Lines indicate
temperatures at corresponding observation nodes (ON) at
the boundary (using numerical solution of Figure 6a).
different conditions, compared with the solution of theinfinite line heat source. As expected, temperature responses
varied considerably by thermistor location. For all four
scenarios, the temperature curves for the thermistors in thesensor body (ON1) or at the tip of the needles (ON7) werevery different than for the other thermistors. We conclude
that the temperature response in the HPP body (ON1) isvery sensitive to epoxy type and whether the body is heatedor not. Moreover, body type and heating scenario will affectthe temperature responses at the other five locations along
the sensor needle, thus affecting analysis of the temperature
signals for soil characterization. Similarly, Ham and Benson[2004] showed experimentally that temperature measurements are influenced by the difference in thermal propertiesof the HPP body. Also temperatures at the interface between
the HPP sensor body and the soil (ON2) are affected by
epoxy type and heating of the epoxy, but are less sensitive
than at location ON1. In general, by heating the sensor
body, variations in temperature signals along the thermistor
needle at ON’s 4 and 5 are small and close to the ideal
infinite line heat source solution, especially when thermallyconductive epoxy is used. Temperature responses at the tipof the thermistor needle (ON7) differ significantly from theother observation nodes because of the heat loss by conduction at the tip of the needle, thus violating the infinite
line heat source assumption. Placing thermistors near theend of the sensor needle, when its length is the same as thatof the heater needle, will underestimate predicted temperatures leading to erroneous estimates of thermal properties
and water fluxes.
[27] In conclusion, as long as the thermistors are locatedat or near the center of the heater needle, temperatures are
suitable for HPP analysis. Similar conclusions were drawn
experimentally by Ham and Benson [2004] as well as Moriet al. [2003, 2005]. This result suggests that sensor needles
can be about half the length of the heater element of the
HPP to reduce the risk of deflecting needles when insertingthem into soils. The thermistors should then be installed at
the tip of the sensor needle where measurements deviate
least from theoretical values for the infinite line source.

Figure 8. The calculated maximum temperature difference
between downstream and upstream thermistors (DTMax) asa function of the maximum temperature reached at theheater (TMax) for different heat pulse intensities, water
fluxes, and water contents for each probe diameter(Table 2). An arrow indicates increasing water content
from 0.3 to 0.6 m3 m3. Vertical dashed lines separate data
points for different heat pulse intensities. Open symbols for
q = 10.0 m d1 show results for case where the distance
between the heater probe and the sensor probe remained thesame (5.5 mm), irrespective of heater diameter.

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Figure 9. The maximum temperature difference between
downstream and upstream thermistors (DTMax) as afunction of the water flux for four heat pulse intensities(60 to 600 W m1) with fitted regression lines. Heater
needle diameter is 4 mm, and water content is 0.3 m3 m3.
center of the thermistor needle are compared to thoseobtained in Figure 6a. Calculated temperatures in thethermistor needle at three locations corresponding to threeONs (ON2, ON4, and ON6 in Figure 2) are all very close to
those imposed at the soil boundary. The overall effect of theaxial conduction of heat on temperature changes in thecenter of the thermistor needle is small enough so that
results presented in Figure 6 can be used in further analyses
without directly accounting for axial conduction. The radial
distance of the center of the needle to its surface issignificantly smaller than the axial distances. Since the heat
capacity of the soil is significantly larger than that of thesensor, the effect of the axial conduction compared to theradial conduction is relatively small.
4.2. Heater Diameter and Heat Pulse Intensity[29] Accurate water flux estimations from HPP experiments require accurate temperature differences between
downstream (Od) and upstream (Ou) thermistors (Figure 4)
For example, a measurement precision of 0.01°C is needed
to estimate a water flux density of 0.06 m d1 [Ren et al.,2000]. Whereas higher heat pulse intensities can be applied
to induce larger temperature differences between the Od andOu thermistors, the higher temperatures may cause boilingof soil water near the heater. It is therefore that we evaluatedthe effects of increasing heater diameters and heat pulse
intensities, collectively. Figure 8 summarizes the maximum
temperature differences between the Od and Ou thermistors(DTMax) as a function of the maximum temperature at theheater boundary (TMax) for different heater diameters, heatpulse intensities, water fluxes, and water content values.
Results were obtained with the assumption of perfectthermal contact between the sensor needles and the surrounding soil, realizing that TMax is affected by the contact
resistance. Vapor transport was not taken into account, sothat temperatures may increase above the boiling point ofwater [Ham and Benson, 2004]. Evaluation of the effects ofthe coupled water, vapor and energy transport on HPP
operation is the subject of section 4.3.
[30] Simulations were conducted for four different watercontents (Table 4) for each heat pulse intensity, water flux,

W07408and probe diameter. As the results in Figure 8 show, boththe maximum temperature, TMax, and the maximum temperature difference, DTMax, decrease as the water content ishigher for all possible combinations. This is caused by thelarger heat capacity of water, as compared to that of the soilmaterial [Campbell, 1985]. Also, as shown by Ren et al.
[2000] in their analytical solution, for the same heat pulse
intensity and water content, the maximum temperature
difference (DTMax) increases linearly as the water flux isincreased (Figure 9). As expected, an increase in the water
flux or heat intensity results in larger temperature differences between Ou and Od thermistors due to heat convection
by liquid water.
[31] As the results in Figure 8 indicate, smaller fluxes canbe measured at lower water content or higher heat pulse
intensity values. Thus the heat pulse intensity needs to be
increased to increase the sensitivity of HPP measurements
for smaller water fluxes. However, the higher heat pulse
intensity may cause vapor flow or temperature-induced
liquid water flow, because of the higher soil temperaturesgenerated by the heat pulse. For example, as the results
show, applying 600 W m1 heat pulse using the 1-mmdiameter needle can easily boil the soil water around theheater. This means then when larger heat pulses are used in
the experimental set up, coupled water, vapor, and energy
transport must be considered.

Figure 10. Effects of different diameter heater probes on
TMax and DTMax for different water content values. Water
flux is 0.1 m d1, and heat pulse intensity is 300 W m1.

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Figure 11. The maximum temperature difference between
downstream and upstream thermistors (DTMax) as afunction of the maximum temperature reached at the heater(TMax), calculated with HYDRUS, with (solid symbols) andwithout (open symbols) consideration of vapor flow. For
each water flux and each heater probe diameter, four
different heat pulse intensities of Table 6 are compared. The
arrows indicate increasing heat pulse intensity from 60 to
600 W m1.
[32] The value of TMax can also be reduced by usinglarger diameter heater needles (Figure 8). Figure 10 shows
the impact of changing the heater diameter on DTMax andTMax as a function of the water content for a heat pulse
intensity of 300 W m1 and a water flux of 0.1 m d1.
While TMax drops significantly, when changing the heater
diameter from 1 to 4 mm, DTMax remains almost the same(the difference is 0.02°C or less) for a wide range of watercontent values. This clearly demonstrates the benefit ofusing a larger diameter heater to reduce TMax, while main-

W07408taining the required DTMax resolution. We should mention
that larger diameter heater needles can distort water flow
[e.g., Hopmans et al., 2002, Figure 6]. This distortion could
lead to larger errors when soil hydraulic and thermal properties are estimated using analytically derived solutions. An
additional benefit of using the presented numerical approachis that distortion of water flow is taken into account.
[33] The 10 m d1 results of Figure 8 also show the
impact of maintaining a constant 5.5-mm distance between
the heater and thermistor needle surfaces, irrespective ofheater diameter needle (compare open with solid diamondsymbols). If this distance is held constant for the threeconsidered heater needle configurations, the heater-thermistor distance between their centers increased from 6.0 mm
to 6.5 and 7.5 mm, respectively. While TMax was notaffected, DTMax slightly dropped for all considered watercontent and heat pulse intensity values, indicating that anincrease of needle distance leads to decreasing sensitivity. In
other words, to increase the HPP sensitivity, one may decideto reduce spacing between heater and thermistor needles.
4.3. Vapor Flow[34] The latent heat of vaporization of water may have asignificant effect on predicted temperature responses andestimated soil properties by the HPP method. Mainly
because of a lack of available tools, vapor transport was
rarely taken into account in HPP studies. Only a few studies[e.g., Bilskie, 1994] attempted to use the coupled model ofwater and energy transport to analyze HPP performance.
Recent updates of HYDRUS allow consideration of vapor
transport coupled with movement of liquid water and heat insoils [Saito et al., 2006].
[35] The effects of our vapor transport analysis are
summarized in Figure 11, where the maximum temperature
differences between Od and Ou thermistors (DTMax) areplotted as a function of the maximum temperature at theheater-soil interface (TMax) for different liquid water fluxes,
heater diameters and heat flux intensities (Table 6), with andwithout consideration of temperature-induced vapor transport (compare open with solid symbols). Since the soil
system has to be unsaturated for vapor flow to occur, different
water contents and water flux values than in section 4.2 wereused. Water fluxes and water contents were uniquely linked

Figure 12. Changes in TMax ((TMax)no vap  (TMax)vap),
after accounting for vapor flow as a function of the heatpulse intensity for the 1-mm-diameter heater probe
predicted with HYDRUS. Subscripts ‘‘vap’’ and ‘‘no vap’’
indicate consideration of vapor transport.

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Figure 13. Changes in DT Max ((DT Max ) no vap (DTMax)vap) after accounting for vapor flow as a functionof the heat pulse intensity for the 1-mm-diameter heater
probe predicted with HYDRUS. Subscripts ‘‘vap’’ and ‘‘novap’’ indicate consideration of vapor transport.
through the soil hydraulic properties. The impact of considering vapor transport is especially important for a lower
water flux (e.g., q = 0.01 and 0.1 m d1) at corresponding
lower water contents. Changes in TMax and DTMax after
accounting for vapor flow as a function of the heat pulse
intensity for the 1-mm heater needle are summarized in
Figures 12 and 13. Results for fluxes of 0.01, 1.0, and10.0 m d1 at a heat pulse of 60 W m1 in Figure 13 are notshown as the effect of temperature responses were near 0°C,
and below the precision of the numerical calculations.
Similar inconsequent results were obtained for heater diameters of 2 and 4 mm (not shown). Figures 12 and 13
demonstrate that while consideration of vapor flow had asignificant effect on reducing TMax, the effect on DTMax was
relatively minor. It is important to realize that DTMax was
not as much affected by inclusion of vapor transport, assome analytical solutions use temperature differences between upstream and downstream thermistors only [e.g., Ren
et al., 2000]. For the smaller heat pulse intensity values,
differences between results obtained with the coupled modelcompared to the standard model were nearly identical forlarger water flux values of 1 and 10 m d1 (Figure 11). Not
only is vapor diffusion less important for the larger water
fluxes, the associated larger water contents decreased theeffective vapor diffusivity, because of the correspondinglarger tortuosity and lower air content values.
[36] The impact of the needle diameter on changes in
TMax and DTMax, due to vapor flow for a range of heat pulse
intensity values, is shown in Figure 14 for a water flux of
0.1 m d1. Similar to results presented in Figure 10, whilechanges in TMax decrease as the heater diameter increases(Figure 14, top), changes in DTMax remain the same fordifferent heater diameters (Figure 14, bottom), irrespectiveof heat pulse intensity. Thus the sensitivity of the HPP towater fluxes is maintained, whether coupled water andenergy transport is considered or not, demonstrating thebenefits of using larger diameter heating needles.

5. Summary and Conclusions[37] This study investigated the effects of (1) sensor
location and thermal properties of HPP sensor body mate-W07408

rial, (2) the heat pulse intensity and heater diameter, and(3) vapor flow on heat pulse probe (HPP) performance. In
addition, the effects of axial heat conductance throughthermistor needles were investigated. The following conclusions can be drawn. First, best results are obtained ifthermistors are placed away from the sensor body and endof heater needle, preferably midway along the length of theheater needle. If temperatures at locations close to the HPP
sensor body are used for the analysis, thermal properties ofthe HPP sensor body must be carefully taken into consideration. Our results suggest that the length of thermistor
needles can be much shorter, as also demonstrated byMortensen et al. [2006], leading to a more rigid heat pulse
probe design. Second, using a standard 1-mm heater needle
diameter, the temperature difference between the downstream and upstream thermistors increases linearly with anincrease in input energy (i.e., heat pulse intensity), potentially leading to a maximum temperature near the heater
exceeding 100°C. Soil temperatures can be significantly
reduced by using larger diameter heating needles (e.g., 2 or
4 mm), while maintaining the minimum required temperature differences between downstream and upstream sensor
needles. Consequently, a higher sensitivity to flux measurements can be achieved when applying larger heat intensities
and larger diameter heater needles. Third, consideration of

Figure 14. The impact of the heater diameter on changes in
TMax and DTMax, after accounting for vapor flow as afunction of the heat pulse intensity, predicted with HYDRUS.
Subscript ‘‘vap’’ and ‘‘no vap’’ indicate considerationof vapor transport. Water flux density is 0.01 m d1.

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vapor transport showed that the latent heat of vaporization
can significantly reduce temperature increases near theheater surface, especially when water flux densities are
small. Thus larger heat pulse values can be used, if vapor
transport is considered in the HPP analysis. In addition, we
also demonstrated that axial heat conductance through thethermistor needles is insignificant for the range of conditions explored in the presented sensitivity analysis.
[38] Another approach to increase HPP sensitivity may beachieved by reducing the distance between heater andthermistor needles or by increasing the heat pulse duration.
While decreasing the needle spacing is expected to increase
sensitivity, it may increasingly distort water flow, though
this can be taken into consideration using numerical solutions. The longer heat pulse duration will increase DTMax,
while keeping the increase in TMax to an acceptable temperature level. Both numerical and experimental studies will
be conducted in the near future to further investigate these
additional options for increasing the HPP performance.
[39] Acknowledgments. This work was supported in part by the
Terrestrial Sciences Program of the Army Research Office (Terrestrial
Processes and Landscape Dynamics and Terrestrial System Modeling andModel Integration), by the National Science Foundation Biocomplexity
Program (award 0410055), and by Sustainability of Semiarid Hydrology
and Riparian Areas (SAHRA) under the STC Program of the National
Science Foundation, agreement EAR-9876800. The first author acknowledges the financial support by Japan Society for Promotion of Science
Postdoctoral Fellowship. We are indebted to excellent comments to ourdraft manuscript by three reviewers; in particular we acknowledge thethorough review by G. Kluitenberg, largely improving the clarity of ournumerical results.

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J. W. Hopmans and A. Tuli, Department of Land, Air and Water
Resources, University of California, Davis, CA 95616, USA. (jwhopmans@
ucdavis.edu; atuli@ucdavis.edu)
H. Saito, Department of Ecoregion Science, Tokyo University of Agriculture
and Technology, Fuchu, Tokyo 183-8509, Japan. (hiros@cc.tuat.ac.jp)
J. Šimůnek, Department of Environmental Sciences, University ofCalifornia, Riverside, CA 92521, USA. (jiri.simunek@ucr.edu)

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