Soil Science Society of America Journal 65:1641-1647 (2001)
© 2001 Soil Science Society of America
DIVISION S-1 - SOIL PHYSICS
A New Perspective on Soil Thermal Properties
Tyson E. Ochsnera,
Robert Horton*,a and
Tusheng Renb
a Dep. of Agronomy, Iowa State Univ., Ames, IA 50011, USA
b Hebei Academy of Agricultural Sciences, 598 W. Heping Road, Shijiazhuang, Hebei 050051, PRC
* Corresponding author (rhorton{at}iastate.edu)
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ABSTRACT
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The soil thermal properties—heat capacity (C), thermal diffusivity (
), and thermal conductivity (
)—are important in many agricultural, engineering, and meteorological applications. Soil thermal properties are largely dependent on the volume fraction of water (
), volume fraction of solids (vs), and volume fraction of air (na) in the soil. In many natural settings , vs, and na vary greatly over time and space, but data showing the effects of these variations on thermal properties are not readily available. We used a heat-pulse method to measure the thermal properties of 59 packed columns of four medium-textured soils covering large ranges of , vs, and na. The measured data reveal the commonly overlooked but dominant influence of na on soil thermal properties. Notably, the measurements show that the of these soils at 20°C can be accurately described as a decreasing linear function of na 
. Good agreement exists between the measured data and common models for and C.
Abbreviations: C, heat capacity • TDR, time domain reflectometry • , thermal diffusivity • , thermal conductivity • , volume fraction of water • vs, volume fraction of solids • na, volume fraction of air
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INTRODUCTION
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SCIENTISTS RECOGNIZED the importance of soil thermal properties as early as the 19th century (Forbes, 1849). Thermal properties of soil influence the partitioning of energy at the ground surface, and are related to soil temperature and the transfer of heat and water across the ground surface. For these reasons, soil physicists, crop scientists, biologists, and micrometeorologists study thermal properties. These properties are also important in engineering applications. For example, the electric current rating for buried cables depends on the thermal conductivity of the surrounding soil, as does the efficiency of heat pump systems. Scientists have long known that soil thermal properties are strongly influenced by the , vs, and na in the soil (Patten, 1909). These volume fractions are highly variable in time and space, particularly near the soil surface. A clear understanding of how these variations affect thermal properties is needed, yet relatively few sets of measured data have been published.
Several researchers have measured the relationships between soil water content, bulk density, and soil thermal properties. Table 1 summarizes some of the published studies in which thermal properties of soil were measured. These studies are representative of the published research on soil thermal properties and volume fractions of soil water, solids, and air. Table 1 does not include experiments using pure sand, crushed rock, gravel, or peat, all of which may have thermal properties drastically different from most soils. The research summarized in Table 1 provides valuable information, particularly about the change of as a function of water content. This relationship was measured and reported in ten of these studies. In five of the studies listed in Table 1, the change of as a function of bulk density was measured and reported. Also, thermal diffusivity was measured and reported as a function of water content in five of the studies and as a function of bulk density in two of the studies.
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Table 1. Overview of previous research regarding the relationships between thermal properties and volume fractions of soil phases.
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Nonetheless, large gaps exist in the published data. For example, we were unable to find a single study reporting any thermal property as a function of air-filled porosity, although some scientists have noted the strong influence of air-filled porosity on thermal conductivity (Bilskie, 1994, p. 45; Campbell and Norman, 1998, p. 123). We were also unable to find any study where two of the thermal properties were measured over a range of and vs for a given soil. (Note that when any two thermal properties are measured the third can be obtained by the definition = 
.) Ghuman and Lal (1985) and Bilskie (1994) did measure two thermal properties for a total of six soils over a range of water contents, but the bulk density was held constant for each soil. Kersten (1949) measured and specific heat, cs, and reported cs as a function of temperature. Other researchers have used the fact that the heat capacity of a soil is the sum of the heat capacities of the soil constituents to estimate C. This estimate is usually based on assumed values for the specific heat of the soil mineral and organic particles. From this estimate of C, and a measurement of either or , the remaining thermal property is estimated (Kolyasev and Gupalo, 1958; Nakshabandi and Kohnke, 1965). Based on our assessment of existing data, we believe that further studies of soil thermal properties and their relationships to volume fractions are needed.
Bristow et al. (1994) presented the first technique capable of simultaneously measuring all three soil thermal properties. Their simple and accurate heat-pulse method permits rapid nondestructive determination of soil thermal properties in field or laboratory settings. This new method greatly simplifies studies of soil thermal properties. Bristow (1998) pointed out the need for thermal property data, and applied the heat pulse technique to study the thermal properties of a sandy soil. He concluded that the heat-pulse technique should be utilized to collect more soil thermal property data.
The purpose of our research was to obtain greater understanding of soil thermal properties by utilizing the heat-pulse method. In particular, we examined the relationships between thermal properties and volume fractions of soil water, solids, and air. Temperature is another factor that influences the thermal properties of soil, but temperature was held constant in this study (20°C) enabling us to focus on the effects of the volume fractions. This paper presents the results of thermal property measurements on four medium-textured soils. The relationships between the measured thermal properties and gravimetrically determined , vs, and na are shown. These relationships are interpreted qualitatively and quantitatively, and the potential significance of these relationships is explained.
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MATERIALS AND METHODS
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Thermal properties of 59 packed columns of four medium-textured soils were measured in a laboratory setting using the Bristow et al. (1994) heat-pulse method. The soils studied were Clarion sandy loam (fine-loamy, mixed, superactive, mesic Typic Hapludolls) and Harps clay loam (fine-loamy, mixed, superactive, mesic Typic Calciaquoll) from Iowa, and a silt loam and silty clay loam from China. Table 2 shows the particle-size distributions, organic matter contents, and particle densities of these four soils. Particle-size distributions were determined by sieving for the sand fraction and by the pipette method for the silt and clay fractions (Day, 1965). Particle densities were determined using the pycnometer method (Blake and Hartge, 1986). Organic C contents were determined by dry combustion for the sandy loam and the clay loam (Nelson and Sommers, 1982). For the silt loam and silty clay loam, organic C contents were determined by the Walkely–Black titration method (Nelson and Sommers, 1982). Organic C contents were then converted to the organic matter values listed in Table 2. The soils ranged in sand content from 12 to 66% and in clay content from 11 to 32%. The soils were air-dried, ground, sieved through a 2-mm screen, and moistened to various water contents. The moist soil was then packed into small columns (7.65 cm in diameter and 7.65 cm high) at carefully controlled bulk densities. The volume and mass of the columns were measured at several stages during packing to ensure uniform bulk density. After packing, the columns were sealed and allowed to equilibrate in a constant temperature room at 20°C for at least 2 wk prior to measurement. All measurements were performed at 20°C.
The thermal properties of each column were measured using a thermo-time domain reflectometry (thermo–TDR) probe (Ren et al., 1999). Thermo–TDR is a new tool that permits determination of soil thermal properties using a heat-pulse method. Thermo–TDR probes are also capable of determining soil electrical properties using TDR, but only the thermal properties determined using the heat-pulse method are discussed in this paper. Thermo-TDR probes have three parallel, hollow stainless steel rods protruding 4 cm from a waterproof epoxy body. The rods are parallel to each other and lie in one plane separated by 
6 mm. Each rod contains a resistance heater and has a chromel-constantan thermocouple located at its midpoint. Further details of probe design and construction are contained in Ren et al. (1999).
The rods of the thermo-TDR probe were inserted into each soil column from the top. Power was applied to the resistance heater in the middle rod from a direct current power supply (Model 1635, B&K-Precision, Maxtec International Corp., Chicago, IL) for 15 s. A data logger (Model 21X, Campbell Scientific, Logan, UT) controlled and recorded the power input, and recorded the resulting temperature increase, T (K), in the outer two rods. De Vries (1952) showed that T as a function of time at a radial distance from the heat pulse source is given by
 | [1] |
where t is time (s), t0 is the duration of the heat pulse (s), r is the radial distance (m), is the soil thermal diffusivity (m2 s-1), q is the amount of heat applied (W m-1), C is the volumetric heat capacity (J m-3 K-1) and -Ei(-x) is the exponential integral. The exponential integral can be evaluated using formula 5.1.53 of Abramowitz and Stegun (1972) for 0 
x 1 and formula 5.1.56 for 1 x 
. We used the nonlinear regression technique presented by Welch et al. (1996) to fit this analytical solution to the temperature increase versus time data collected from the probe. This fitting technique yields the thermal properties of the soil—C, , and (by definition = C). The spacing between the rods must be known to calculate the thermal properties. This spacing was determined by calibrating the probes in agar-stabilized water (6 g agar L-1) at 20°C, assuming the volumetric heat capacity of the agar–water solution is equal to the volumetric heat capacity of water (4.17 MJ m-3 K-1).
After the thermal property measurements, the soil columns were weighed, oven-dried at 105°C, then weighed again, and , vs, and na were calculated. Note that vs is equal to the bulk density divided by the particle density. The particle density for each soil is shown in Table 2. Also, note that once and vs are calculated then na is known because the sum of the volume fractions is 1. The ranges of , vs, and na were chosen to cover most of the conditions under which these medium-textured soils may exist in the field. As shown in Fig. 1
, ranged from 0.02 to 0.46 m3 m-3, vs ranged from 0.33 to 0.66 m3 m-3 
, and na ranged from 0.02 to 0.57 m3 m-3.
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Fig. 1. Volume fractions of water (), solids (vs), and air (na) for 59 packed columns of four medium-textured soils.
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RESULTS AND DISCUSSION
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The most common approach for presenting soil thermal property data is to plot thermal properties as a function of for a single soil at a constant vs. This narrowly-defined approach is used to illustrate the effect of a single variable, , on the thermal properties of that soil. Less commonly, thermal properties are plotted as a function of for a single soil at several different vs values. This broader approach adds one level of complexity by illustrating the effect of and vs (and na indirectly) on the thermal properties of that soil. A third possible approach to the presentation of thermal property data adds one more level of complexity. Thermal properties for different soils can be plotted as functions of , vs, and na across ranges of these volume fractions. Of the approaches described, this third one gives the most comprehensive illustration of the factors affecting soil thermal properties. We use this comprehensive presentation approach to illustrate the primary relationships between volume fractions and the thermal properties of the four soils studied. The results of our heat-pulse thermal property measurements are plotted in Fig. 2
against the gravimetrically determined volume fractions of water, solid, and air.
Thermal conductivity is the thermal property of most interest in many applications. The data in Fig. 2 clearly show that the variation in between these soil samples can be primarily explained by the variation in na between the samples. As na increases, decreases in a generally linear fashion for these soils samples. The relationship between and na is obviously stronger than the relationship between and , which is most commonly reported in the literature. (Note that the versus na relationship is not the mirror image of the versus relationship as might be assumed.) The strong inverse linear relationship between and na (r2 = 0.93, Table 3) shown in Fig. 2 highlights the fact that na exerts a limiting effect on under the conditions of these measurements. This limiting effect is logical since at 20°C the thermal conductivity of air (0.025 W m-1 K-1, Campbell and Norman, 1998; Table 8.2) is one order of magnitude less than the thermal conductivity of water (0.596 W m-1 K-1, Campbell and Norman, 1998; Table 8.2) and two orders of magnitude less than the thermal conductivity of soil minerals (2.5 W m-1 K-1, Campbell and Norman, 1998; Table 8.2). It is worth noting that the thermal conductivity as measured in this experiment is actually influenced by both pure conduction and by latent heat transfer across soil pores caused by water vapor movement (de Vries, 1963). An increase in air-filled porosity would be expected to increase the latent heat transfer within the soil, assuming that the relative humidity in the pore space does not change. Our data suggest that the influence of this increase in latent heat transfer is overwhelmed by the sharp decrease in pure conduction as na increases.
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Table 3. Coefficients of determinations (r2) from linear regression of thermal properties versus volume fractions for 59 samples of four soils.
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We examined the widely used de Vries model for thermal conductivity (de Vries, 1963) to determine if the model supported our finding about the dominant influence of air-filled porosity on soil thermal conductivity. Details of the model are presented in the appendix. The modeled and measured thermal conductivities generally agreed well, r2 = 0.92 (Fig. 3)
, but the modeled values are notably higher than the measured values at low thermal conductivities. We used the model to compare the relative significance of variations in and variations in na. To facilitate this comparison, we modeled the versus relationships for two hypothetical soils. To the first hypothetical soil we assigned a vs of 0.33 m3 m-3 and a thermal conductivity of the soil solids, s, of 3.06 W m-1 K-1. These values correspond to the smallest observed values of the same parameters in our measured data; therefore, the model results for this hypothetical soil represent de Vries model predictions of the minimum possible values of for our actual soils. To the second hypothetical soil, we assigned a vs of 0.66 m3 m-3 and a s of 3.72 W m-1 K-1. These values correspond to the largest observed values of the same parameters in our measured data; therefore, the model results for this hypothetical soil represent de Vries model predictions of the maximum possible values of for our actual soils. We let for these hypothetical soils vary from 0.05 to 0.30 m3 m-3, which is approximately the full range of possible values for a soil with vs = 0.66 m3 m-3. For this water content range the modeled minimum should be less than or equal to all measured values of . Likewise, for this water content range the modeled maximum should be greater than or equal to all measured values of . The de Vries model minimum and maximum predictions, along with the measured data for the real soils, are shown in Fig. 4a
as a function of water content. In general, the modeled maximum and modeled minimum thermal conductivities bound a region which contains the measured values. As expected, all measured values in the specified water content range lie on or below the modeled maximum . Also, the majority of the measured values in the specified water content range lie on or above the modeled minimum . The de Vries model supports the measured data, showing the same general shape and a similar degree of variation in the water content versus thermal conductivity relationship.
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Fig. 3. Modeled thermal conductivity using de Vries model versus measured thermal conductivity for 59 samples of four soils. Solid line is the 1:1 line.
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Fig. 4. Thermal conductivity as a function of (a) water content and (b) air-filled porosity. Circles are measured data. Solid lines represent maximum expected thermal conductivity values based on de Vries model. Dashed lines represent minimum expected thermal conductivity values based on de Vries model.
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We then examined the exact same modeled results by plotting versus na. The de Vries model minimum and maximum predictions, along with the measured data for the real soils, are shown in Fig. 4b as a function of air-filled porosity. Unlike in Fig. 4a, in Fig. 4b the modeled maximum and modeled minimum occur in separate regions of the x-axis. The modeled minimum values occur at high values of na and modeled maximum values occur at low values of na, and the modeled maximum and minimum values appear more like a single function of na than boundaries of a region. The de Vries model and the measured data both show that depends more strongly on na than on . In Fig. 4b most of measured values in the specified water content range lie on or below the modeled maximum . However, a large majority of the measured values in the specified water content range lie below the modeled minimum , i.e., the model appears to overpredict at high values of na. This disagreement between the de Vries model and the measured data suggests that the model may underestimate the limiting effect of na on .
The relationships shown in Fig. 2 between heat capacity and volume fractions of water, solids, and air are also of particular interest. The data show that variations in soil heat capacity can be primarily explained by variations in water content. As increases C increases, and the relationship is linear. In the case of heat capacity, the relationship with na is approximately the mirror image of the relationship with . Variations in na have about the same relevance as do variations in for explaining variations in C. The influence of variations in vs is clearly secondary to the influence of and na, as evidenced by the coefficients of determination in Table 3. The measured relationships between C, , vs, and na are as expected from the additive model of heat capacity. Details of the heat capacity model are presented in the appendix. Since the heat capacity of soil solids (2.0–2.5 106 J m-3 K-1, Table 4) is intermediate to that of water (4.17 106 J m-3 K-1) and air (0.0012 106 J m-3 K-1 at 20°C and 101 kPa, Campbell and Norman, 1998; Table 8.2), we expect variations in C to be less correlated with variations in vs than with variations in or na. The agreement between the modeled and measured heat capacities, r2 = 0.94, suggests that both the model and the measurements are accurate (Fig. 5)
. These findings lend further validity to recent efforts to use heat-pulse measurements of C to determine or change in (Tarara and Ham, 1997; Bristow, 1998; Song et al. 1998). These findings also highlight the fact that heat-pulse measurements of C are equally well-suited for determining na.
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Table 4. Estimated thermal properties of soil solids calculated by fitting de Vries model of thermal conductivity and additive model of heat capacity to measured data. (See appendix for details.)
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Fig. 5. Modeled heat capacity versus measured heat capacity using additive model for heat capacity. Solid line is the 1:1 line.
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Thermal diffusivity, the third thermal property of soil, is by definition the ratio of the thermal conductivity to the heat capacity. Therefore, the physical causes for variations in thermal diffusivity are completely contained within the physical causes for variations in and C. Still it is useful to briefly examine the relationships between (and the volume fractions—, vs, and na. The thermal diffusivity data in Fig. 2 show that variations in na explain much of the variation in between these soil samples. The versus na relationship is similar to, but weaker than, the versus na and C versus na relationships. Also, of note is the fairly strong relationship between and vs. The data show that increases steadily as vs increases except in the driest samples of the silt loam soil. The relationship between and vs is much stronger than the more commonly studied relationship between and .
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CONCLUSION
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We have presented a unique and comprehensive set of soil thermal property measurements over a range of , vs, and na for four medium-textured soils. This is the first study, of which we are aware, in which and C were measured over a range of , vs, and na for a given soil. The unique nature of these measurements permitted us to obtain a new perspective on the associated relationships. This new perspective reveals the commonly overlooked but dominant influence of na on soil thermal properties. The results clearly show that, for these medium textured soils, thermal properties are more strongly correlated with na than with or vs. Therefore, the traditional viewpoint of considering thermal properties as functions of and vs may not be the most useful perspective. In particular, our measurements show that the thermal conductivity of these medium-textured soils at 20°C can be accurately described as a decreasing linear function of the air-filled porosity.
Our measurements compared well with common models for soil thermal properties. The de Vries model for thermal conductivity fits our measured data reasonably well, but the model appears to overestimate at high values of na for these soils. The additive model for heat capacity also fits the measured C data well, giving further evidence that these thermal property measurements are accurate.
We modeled the thermal conductivity of the soil samples using the procedure of de Vries (1963). In this procedure the thermal conductivity of soil is calculated as the weighted average of the conductivities of the various soil constituents according to the formula
 | [A1] |
where xi is the volume fraction of each constituent, i is the thermal conductivity of each constituent, and n is the number of soil constituents. The weighting factors, ki, depend on the shape and the orientation of the granules of the soil constituents, and on the ratio of the conductivities of the constituents. The subscript zero refers to the continuous fluid surrounding the solid particles—air for dry soil and water for moist soil—with k0 = 1. Other values of ki are calculated from
 | [A2] |
where gj represents the shape factors for the i-th constituent, and g1 + g2 + g3 = 1. Assuming g1 and g2 are equal, only one shape factor must be estimated for each constituent.
For the purposes of this study, we modeled our soil samples as a mixture of three constituents, water, solids, and air, and we assumed that water was the continuous fluid in all samples. The thermal conductivity of water is 0.596 W m-1 K-1 at 20°C, and the thermal conductivity of the soil solids, s, was chosen for each soil so that the modeled and measured values were identical for the most saturated sample of that soil. The s values for each soil are shown in Table 4. The thermal conductivity of the air-filled pores is considered to be the sum of a and v, where a is the thermal conductivity of dry air (0.025 W m-1 K-1 at 20°C), and v accounts for heat transfer across the air-filled pores by water vapor. Above some critical water content, the air-filled pores are assumed to be saturated with water vapor, and v is assumed to be 0.074 W m-1 K-1 at 20°C. Below the critical water content, we assumed that v decreases linearly with water content to a value of zero for oven-dry soil. The critical water content for the four soils in this study was taken to be 0.15 m3 m-3, which is within the range of values used by de Vries (1963). The shape factor, g1, for the soil solids was taken to be 0.144. For water contents above the critical water content, the value of g1 for the air-filled pores is given by
 | [A3] |
For water contents below the critical water content, the value of g1 for the air-filled pores is given by
 | [A4] |
where c is the critical water content and g1c is the value of Eq. [A3] at the critical water content.
The heat capacity of soil is modeled as the weighted sum of the heat capacities of the soil constituents (Campbell and Norman, 1998). Omitting the negligible contribution of air in the soil, the equation is
 | [A5] |
where 
s is the particle density of the soil solids (Table 2), w is the density of water (998 kg m-3 at 20°C), cw is the specific heat capacity of water (4182 J kg-1 K-1 at 20°C), and cs is the specific heat capacity of the soil solids. Following Campbell et al., 1991, cs can be calculated from heat pulse C measurements by rearranging Eq. [A5], assuming and bulk density (svs) are known. The average value of cs for each soil is shown in Table 4, and these values were used in Eq. [A5] to produce the modeled results shown in Fig. 5.
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NOTES
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Journal Paper No. J-19021 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA, Project No. 3287. Supported by the Hatch Act and the State of Iowa.
Received for publication September 12, 2000.
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November 1, 2003;
2(4):
580 - 588.
[Abstract]
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T. Ren, T. E. Ochsner, R. Horton, and Z. Ju
Heat-Pulse Method for Soil Water Content Measurement: Influence of the Specific Heat of the Soil Solids
Soil Sci. Soc. Am. J.,
November 1, 2003;
67(6):
1631 - 1634.
[Abstract]
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TOP
ABSTRACT
INTRODUCTION
