Soil Science Society of America Journal 63:1585-1591 (1999)
© 1999 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Dual Thermal Probes near Plane Interfaces
G.J. Kluitenberga and
J.R. Philipb
a Dep. of Agronomy, Kansas State University, Manhattan, KS 66506 USA
b CSIRO Land and Water, G.P.O. Box 1666, Canberra, ACT 2601, Australia
gjk{at}ksu.edu
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ABSTRACT
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The dual probe estimates the thermal properties of soils. Because it is strongly dependent on water content, volumetric heat capacity is useful for moisture estimation. A previous study of probe errors due to soil heterogeneity is taken further with the aid of a new solution for instantaneous line sources near a plane interface between two regions. The solution allows for different conductivities and heat capacities in the regions, but requires uniform diffusivities. This condition holds reasonably well near wetting fronts in a range of soils. Previous error bounds are confirmed and sharpened. The conclusion in the previous study that errors are small when the heterogeneity is no closer to the probes than the probe separation (typically 0.006 m) is reinforced. The expectation that the dual probe gives good resolution of water content close to fronts and interfaces is strengthened.
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INTRODUCTION
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CAMPBELL ET AL. (1991) PROPOSED the dual-probe heat pulse method for measuring soil volumetric heat capacity. The method gives means of estimating soil water content (Bristow et al., 1993; Tarara and Ham, 1997; Bremer et al., 1998; Bristow, 1998; Song et al., 1998) because of the direct relation between heat capacity and water content. Dual probes also yield estimates of soil thermal conductivity and diffusivity (Kluitenberg et al., 1993; Bristow et al., 1994; Kluitenberg et al., 1995).
Philip and Kluitenberg (1999) initiated study of probe errors due to spatial variation of soil thermal properties. Here we introduce an improved analysis that avoids some simplifying approximations of the previous work. It gives precise results not previously available that confirm the upper bounds on error and general conclusions of Philip and Kluitenberg (1999).
We find exact solutions for four probe/soil configurations involving heterogeneity. Three of the configurations were studied previously; for the fourth configuration the interface between regions of different thermal properties separates the two probes. The analysis is limited to the model of Campbell et al. (1991) with probes of infinite length and zero radius and the heat pulse an instantaneous line source. Removing these simplifications would greatly complicate the analysis but leave unaffected the order of magnitude of the errors due to heterogeneity.
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Mathematical model
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Heat Conduction Equation
The heat conduction equation for a soil region of uniform thermal diffusivity 
(m2 s-1) is
 | (1) |
where T is temperature (K), t is time (s), and 
2 the Laplacian (m-2). The thermal conductivity 
(Wm-1 K-1) and the volumetric heat capacity C (Jm-3 K-1) are related to by
 | (2) |
We treat the heating probe as an instantaneous line source: infinite in length, of strength q (Jm-1), released at the instant t = 0. The (x,z) plane is normal to the line source, which is located at (x,z) = (0,0).
Solution in the Heterogeneous Region
This study depends on a new solution for Eq. [1]: for an instantaneous line source in an infinite region such that, in z < z1 (z1 
0), the thermal conductivity and the volumetric heat capacity are and C; in z > z1 they are 
and C. We see that, although the diffusivity has the uniform value throughout -
< z < , the conductivity and heat capacity differ between the two half-regions. Without loss of generality we take the initial condition as
 | (3) |
This new solution must give a source of strength q at (x,z,t) = (0,0,0) and ensure continuity of both T and the normal heat flux density across z = z1. We thus require
 | (4) |
and
 | (5) |
where T1 denotes T in z 
z1 and T2 denotes T in z z1.
We find the solution by generalizing to the time-dependent Eq. [1] an ingenious modification of the Method of Images which was applied to the steady Laplace equation by Bear (1972)(p. 311).
In z z1, the solution
 | (6) |
satisfies Eq. [1] and [3] and gives the required source at (x,z,t) = (0,0,0); and in z z1, Eq. [7] satisfies Eq. [1] and [3] also
 | (7) |
We fix the undetermined constants a and b by putting Eq. [6] and [7] in Eq. [4] and [5]. We obtain
 | (8) |
so that
 | (9) |
It follows that Eq. [6] and [7] become
 | (10) |
Character of the Solution
The character of the new Eq. [10] is illustrated in Fig. 1
. The figure shows, for the instant
, the dimensionless isotherm
, with
. Heat flow lines, orthogonal to the isotherms, are sketched in. These flow lines are not streamlines in the standard sense, because the total flow between lines decreases with increasing distance from the source. The dimensionless space coordinates are x/z1, z/z1.
For a homogeneous infinite region with = 1 the isotherms would be concentric circles and the flow lines equally spaced radii. Both would be centered on the source. Figure 1a, for = 1
4, shows characteristic behavior for < 1. Isotherms in the lower region are no longer concentric. Those reaching the interface are refracted there, and become circular arcs centered on the source in the upper region. Flow lines in the lower region are bent away from the interface, with the small number of flow lines reaching the interface refracted into radial lines, also centered on the source.
Figure 1b, for = 4, typifies behavior for > 1. In this case isotherms in the lower region deviate from concentricity in the opposite sense; those that reach the interface are refracted there into concentric circular arcs in the upper region. Flow lines in the lower region are bent toward the interface, with the increased number of flow lines intersecting the interface refracted there into radial lines, in this case in the opposite sense. It is of interest that the solution for z z1 is independent of z1. Also of interest is the fact that the solution for z z1 is the same as the solution for a homogeneous soil with the average conductivity (1 + )/2.
Applications of the Solution
The requirement that have the same value in both regions might appear, at first glance, a severe limitation on the usefulness of Eq. [10]. For many soils, however, tends to remain relatively constant over the 
range (, volumetric water content) from 0.2 s to s, with s the saturated water content. The heat capacity and conductivity, on the other hand, tend to increase roughly linearly with , typically by about threefold over this range. See, for example, Moench and Evans (1970), Jury et al. (1991)(p. 182), Marshall et al. (1996)(p. 372), and Bristow (1998). Data on a sand, a clay, and a peat collected by Kersten (1949) and presented by de Vries (1963) suggest a lower limit of 0.1 s, with a fourfold range in conductivity.
Accordingly, we apply Eq. [10] to circumstances studied by Philip and Kluitenberg (1999) with the dual probe located either close outside, or close behind, a sharp wetting front. In that work the approximation for the probe outside the front was equivalent to taking the present = and that for the probe behind the front to = 0. The earlier analysis could not treat the case with the heating probe and the sensor on different sides of the interface of front. As we show below, the present analysis can. Because the thermal diffusivities of soil and air differ strongly in magnitude, the analysis here has no direct relevance to the case of dual probes close to the soil surface studied by Philip and Kluitenberg (1999).
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Analysis of heterogeneity errors
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Sensor Temperature for Four Probe Configurations
We therefore consider the four probe configurations near an interface or front depicted in Fig. 2
. For each the heating probe is at (x,z) = (0,0) and the interface or front is the line z = z1. Configurations (a), (b), and (c) are as in Philip and Kluitenberg (1999), with z0 the distance from the interface to the closer probe (both probes for Configuration [a]). For (a) the sensor coordinates are (
,0) with z0 = z1. For (b) they are (0, -) with z0 = z1. For (c) the sensor coordinates are (0,) with z0 = z1 - . For Configuration (d) the sensor is in z z1 with coordinates (0,) and z1 . For this configuration we define z0 as the distance of the heating probe from the interface; i.e., z0 = z1. The configuration with the heating probe on the interface, as studied by Philip and Kluitenberg (1999), is the special case of Configuration (d) with z0 = z1 = 0.
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Fig. 2 Dual probes near an interface or wetting front. The Configurations (a), (b), (c), and (d) show positions of heating probe H and sensor S. For each configuration H is at (x,z) = (0,0)
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Equations [10] then give us the time course of sensor temperature for each configuration.
For Configuration (a)
 | (11) |
For Configurations (b) and (c)
 | (12) |
For Configuration (d)
 | (13) |
Equations [11] through [13] are consistent with previous results of Philip and Kluitenberg (1999). Taking = 0 reduces Eq. [11] and [12] to their Eq. [24] and [25]; = reduces them to their Eq. [26] and [27]. In addition, the present Eq. [13] is identical to their Eq. [22].
The Homogeneous System
Putting = 1 in Eq. [10] gives the standard instantaneous line source solution in the homogeneous infinite region (Carslaw and Jaeger, 1947, p. 218; 1959, p. 257)
 | (14) |
Carslaw and Jaeger call their Q the source strength. The physical source strength, q in the present study, is CQ. Equation [14] allows inference of C, , and from observed values of tm and Tm (Lubimova et al., 1961; Jaeger, 1965; Campbell et al., 1991). Here tm is the time interval between the instantaneous heat pulse and the instant when the sensor reaches its maximum temperature Tm. The relevant relations are
 | (15) |
 | (16) |
 | (17) |
and it follows from Eq. [14] and [17] that
 | (18) |
Heterogeneous Systems
We now calculate tm and Tm for the heterogeneous systems with Configurations (a)–(d). We designate these "heterogeneous" values tmh and Tmh and then establish the errors when these values are used in Eq. [15]–[17].
Configuration (a)
Differentiating Eq. [11] with respect to t and equating the result to zero gives tmh. We find
 | (19) |
Using Eq. [17], together with the dimensionless quantity
 | (20) |
as in Philip and Kluitenberg (1999), we reduce Eq. [19] to
 | (21) |
We have, from Eq. [11], that
 | (22) |
Combining this with Eq. [18] and using Eq. [17] and [20], we get
 | (23) |
With Ch the value of C if we put Tm = Tmh in Eq. [15], we find
 | (24) |
The fractional error in estimating C is therefore
 | (25) |
With h the value of obtained by putting tm = tmh and Tm = Tmh in Eq. [16], we find
 | (26) |
with the fractional error in estimating
 | (27) |
Finally, with h, the value of found with tm = tmh in Eq. [17], we get
 | (28) |
and the fractional error
 | (29) |
With Z a known function of z0/, Eq. [25], [27], and [29] translate directly into the dependence of the various fractional errors on z0/.
Configurations (b) and (c)
For these cases, differentiating Eq. [12] gives
 | (30) |
Using Eq. [17] and the dimensionless quantity
 | (31) |
we retrieve Eq. [21] with Z1 in place of Z. Equation [31] is a quadratic in z0/ and the relevant positive root is
 | (32) |
Equations [21]–[29] carry over except that Z is replaced by Z1 and we use Eq. [31]. This procedure for Configurations (b) and (c) is the same as in Philip and Kluitenberg (1999).
Configuration (d)
It is readily found from Eq. [13] that
 | (33) |
so that
 | (34) |
This configuration gives no error in the estimate of the diffusivity. In this case we compare Ch and h with the "true" values of volumetric heat capacity and conductivity, namely their mean values in the region between the probes, 
and 
.
Evidently
 | (35) |
and
 | (36) |
The fractional errors are thus
 | (37) |
We explore the implications of these various results below.
Heterogeneity Errors: Dependence on
Configurations (a)–(c)
The present Eq. [21]–[29] are wholly consistent with Philip and Kluitenberg (1999). Putting = 0 gives their Eq. [40]–[48] which provide upper error bounds for probes behind a wetting front; putting = in the present equations gives their Eq. [52]–[56] for probe error bounds outside a wetting front.
For a given value of z0/, fractional errors in heat capacity, conductivity, and diffusivity all vary smoothly with as it increases from 0 to . The errors in C, , and are negative for 0 < 1 and positive for > 1. It is of interest that in the limit z0/ 
0 the errors are proportional to ( - 1)/2 and as z0/ they are proportional to ( - 1)/( + 1). Figure 3
shows the variation with of these factors.
The curves of functional dependence of fractional error on z0/ for various values of form an orderly family, ranging from = 0 to = . These bounding curves were given as Fig. 5 and 6 in Philip and Kluitenberg (1999). Figure 4
in this study shows the family of curves for fractional error of heat capacity for Configuration (a) with ranging from 0 to . Figure 5 shows the corresponding curves for Configurations (b) and (c). In each figure the curves are for = 0, 0.25, 0.5, 1, 2, 4, and .
Configuration (d)
As noted earlier, taking z0 = 0 gives the heating probe on the interface. This case was analyzed in Philip and Kluitenberg (1999). It gave Eq. [37] with z0 = 0, as it should. Figure 6 shows curves of dependence on z0/ of fractional error of both heat capacity and conductivity, again for = 0, 0.25, 0.5, 1, 2, 4, and . The symmetries are evident.
A Front Moving Toward and Past a Dual Probe
We illustrate this analysis with the following example. We have a dual probe with the heating probe at (0,0) and the sensor at (0,), initially in Configuration (c), with a sharp wetting front at z1(t) such that z1(t) >> . The quantity z1(t) decreases as the front descends slowly toward the sensor, engulfs it, and continues downward further. We seek to calculate the varying probe estimate of soil heat capacity as the front travels downward (i.e., as z1[t] decreases) and to compare the probe estimate with the true value of the heat capacity of the soil between the probes. We assume is uniform for the soil, and that the heat capacity is C in the unwetted soil and 4C in the wetted soil.
Note that the system remains in Configuration (c) with = 4 for z1(t) > . For 0 z1(t) it is in Configuration (d), again with = 4. Finally, for z1(t) < 0, we have an inverted form of Configuration (b) with = 14.
Figure 7
depicts the results of our analysis. The probe error becomes significant and positive for 1.7 z1(t)/ > 0.5 with maximum error when the front encounters the sensor at z1(t)/ = 1. The error is significant and negative for 0.5 > z1(t)/ -0.8, with its greatest magnitude when the front encounters the heating probe at z1(t)/ = 0.
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Discussion
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Our general results, and Fig. 4 and 5 in particular, confirm the error bounds calculated in Philip and Kluitenberg (1999). The present analysis, however, gives closer error estimates for probes both outside and behind wetting fronts. The previous study concluded that probe errors due to heterogeneity are negligible as long as the distance of the nearest probe to the interface or front is not less than the probe spacing (typically 0.006 m). As expected, errors for 0 < < 1 and for 1 < < are less than the conservative values for = 0 and established in the earlier work. We note as examples that, for Configurations (a)–(c) with z0/ of order 1, errors for = 4 are 60% of those for = ; for = 14 they are 60% of those for = 0.
On the other hand, direct application of this analysis to probe errors close to the soil surface is not feasible because the thermal diffusivity of air greatly exceeds that of soil. The thermal conductivity of soil, however, exceeds that of still air by a factor of order 20, and probably also exceeds appreciably that of turbulent air in immediate contact with the ground. We therefore expect that the loss to the air of heat released by the probe pulse is not significant. Accordingly the = results of Philip and Kluitenberg (1999) remain the best estimate of error bounds for probes near the soil surface.
This new analysis of heterogeneous systems is relevant also to layered soils provided the two thermal conductivities differ much more strongly than the diffusivities. Our results strengthen the conclusion of Philip and Kluitenberg (1999) that thermal probes provide a means of estimating C, , and close to wetting fronts. As they noted, the values of C yield estimates of soil water content near wetting fronts; we conclude, however, that reliable measurements of C and water content cannot be expected when the front lies between the two probes.
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NOTES
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Contribution no. 99-187-J from Kansas Agric. Exp. Stn., Manhattan, KS. Research supported by Western Regional Research Project W-188.
Received for publication October 26, 1998.
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TOP
ABSTRACT
INTRODUCTION
. The curves with arrows approximate the heat flow lines orthogonal to the isotherms. As the text explains, these are not streamlines in the standard sense
