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DIVISION S-1-SOIL PHYSICS

Errors of Dual Thermal Probes Due to Soil Heterogeneity across a Plane Interface

^a CSIRO Land and Water, G.P.O. Box 1666, Canberra, ACT 2601, Australia
^b Dep. of Agronomy, Kansas State University, Manhattan, KS 66506 USA

gjk{at}ksu.edu

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
The dual thermal probe enables estimation of the thermal diffusivity, volumetric heat capacity (C), and thermal conductivity of soils. It has been much employed for moisture estimation because of the strong dependence of C on soil water content. The present study investigates the effectiveness of dual probes near heterogeneities. Exact solutions are found for four probe/soil configurations involving heterogeneity. In one configuration the heating probe is on the plane interface between different soils. For the three other configurations it is located close to three different discontinuities: near a soil surface and outside, and behind, a wetting front. Heterogeneity errors are found to be small provided the heterogeneity is no closer to the probes than probe separation (typically 0.006 m). Estimates of C may provide good resolution of soil water content in critical regions such as near surfaces and fronts.

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
THE DUAL-PROBE HEAT pulse method for measuring soil volumetric heat capacity was proposed by Campbell et al. (1991). Because of the direct relation between soil heat capacity and volumetric moisture content, dual thermal probes offer means of estimating soil water content (Bristow et al., 1993; Tarara and Ham, 1997; Bremer et al., 1998; Bristow, 1998; Song et al., 1998). Dual probes also enable estimates of soil thermal diffusivity and conductivity (e.g., Kluitenberg et al., 1993; Bristow et al., 1994; and Kluitenberg et al., 1995). Kluitenberg et al. (1993, 1995) investigated the errors that were incurred in the simplified analysis (Campbell et al., 1991) in which the probes are treated as infinite and heat input as an instantaneous line source.

These various studies have left unexamined the question of dual-probe error due to heterogeneity, such as spatial variation of soil thermal properties. We have in mind not only variation of intrinsic soil properties, but also of moisture content within a homogeneous soil. We observe, further, that the various theoretical analyses treat the dual probe as embedded in an infinite soil mass. This assumption cannot be valid, for example, when the probes are located very close to the soil surface. Propinquity to the surface is, from this viewpoint, one more form of heterogeneity. The greatest fluctuations of soil moisture, and hence of thermal properties, occur at and near the soil surface. The question of the performance of dual probes close to the surface is therefore of considerable interest and practical importance. Because of their small size, dual probes may be useful near heterogeneities; but it is clearly desirable to test this expectation.

In the present study we develop exact solutions for four probe/soil configurations involving heterogeneity. We limit our analyses to the simple model of Campbell et al. (1991) with infinite probe lengths and the heat pulse an instantaneous line source. Introducing the complications of the nonzero radius and finite length of the probes, and the nonzero duration of the heat pulse, would leave unaffected the order of magnitude of the errors arising from heterogeneity. Kluitenberg et al. (1993) have shown that these complications cause only minor modifications to the simple Campbell model for typical probe geometries and heating times.

	Mathematical models

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
Heat Equation
The heat conduction equation for a soil region of uniform properties is

(1)

where T is temperature (K), t is time (s), is the thermal diffusivity (m² s^-1), 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 by 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). We shall also use polar coordinates (r,) such that

(3)

The heating probe is thus at r = 0.

We shall examine instantaneous source solutions of Eq. [1] for which and C (and hence, in general, also) have different values in different regions.

Heating Probe on Interface
We consider first the case where the line z = 0 in the (x,z) plane is the interface between two soils with different thermal properties. The heating probe at r = 0 lies on this interface. For z > 0, = ₊, C = C₊; for z < 0, = _-, C = C_-. The sensor probe is at (x,z) = (0,), or (r,) = (,/2) as shown in Fig. 1 . We put

(4)

so the ratio of diffusivities is

(5)

View larger version (60K): [in this window] [in a new window]   Fig. 1 The heating probe H at (x,z) = (0,0) on the interface between soil in z > 0 with = +, C = C+ and z < 0 with = -, C = C-. The sensor is at (x,z) = (0,)

(6)

where signifies ₊ and _- in the two regions. We take the initial condition as

(7)

and the boundary condition at infinity as

(8)

Generality is not lost by taking T = 0 in Eq. [7] and [8]. The principle of superposition permits application of the solutions to cases with an arbitrary initial uniform temperature T₀. In our various solutions T then becomes T - T₀, the temperature rise above the initial T₀. Solutions in the two regions are connected by the two further conditions of continuity of temperature and heat flux across the interface between the two regions. These are displayed in reduced form as Eq. [21] below.

The dimensional structure of Eq. [6] implies an instantaneous source solution of the similarity form

(9)

with

(10)

Furthermore, all the conditions on Eq. [6] are expressible in terms of F, , and , so the similarity solution is indeed the required solution. The substitution of Eq. [9] and [10] in Eq. [6], [7], and [8] reduces the number of independent variables by one, and we get

(11)

with

(12)

We seek the solution corresponding to the instantaneous source of strength q at (r,t) = (0,0). (Carslaw and Jaeger [1947, 1959] call their Q the source strength. The physical source strength, q in the present work, is CQ in their notation for a homogeneous region.) In the limit as 0 the heat flow is radial and approaches that for separate instantaneous sources in the two half-regions z > 0 (0 < < ) with thermal properties ₊, C₊, ₊ and z < 0 (- < < 0) with _-, C_-, _-. If the source strengths in the two half-regions are ¹₂q₊ and ¹₂q_-, the relevant solutions (Carslaw and Jaeger, 1947, p. 218; 1959, p. 257) are

(13)

(14)

But the two regions have the same value of T at r = 0. This requires that

(15)

and, since the total source strength is

(16)

(17)

The solutions [13] and [14] thus become, in terms of F,, and ,

(18)

(19)

It is readily shown that Eq. [18] and [19] are solutions of Eq. [11] with the dependence on (the second term on the right) dropped; but these solutions give a discontinuity of F on > 0, = 0 and . Therefore they are not exact solutions of [11] but, as we shall see, they give an approximation which is good and useful for present purposes.

Reverting to the full Eq. [11], we see that since Eq. [18] and [19] are exact at = 0, they provide the further condition

(20)

Continuity of both temperature and heat flux across the interface requires that

(21)

Exploratory numerical solutions of the full Eq. [11], subject to Eq. [12], [20], and [21], show that solutions [18] and [19] are excellent approximations to the true solution except when || is small. Table 1 compares the true (numerical) solution for the case = 0.65, ß = 0.4225 with solutions [18] ( > 0) and [19] ( < 0). We see that the error in Eq. [18] and [19] for 3 is less than 1% for || = ₄ and ³₄, and is miniscule for || = ₂

View this table: [in this window] [in a new window]   Table 1 Comparison of exact (numerical) and approximate (Eq. [18] and [19]) solutions for 2F(+ + -)/q

(22)

We use Eq. [22] to estimate heterogeneity errors of the dual probe in what follows. But first we develop solutions relevant to three other types of heterogeneity.

Probe Near Soil Surface
The greatest fluctuations of water content and other physical conditions occur in the immediate surface layer of the soil. This region is of great interest, but presents the most serious instrumental difficulties. The question to be asked immediately about the accuracy of dual probes is: How close to the soil surface can they be located before interaction with the surface produces significant errors?

Though a small quantity of heat will escape from the soil surface into the atmosphere and vegetation above, we here adopt the no-flux boundary condition at the surface. This is conservative inasmuch as the largest possible deviation from infinite medium behavior occurs when no heat escapes.

As before, we place the heating probe at (x,z) = (0,0), but we consider three locations of the sensor probe, as shown in Fig. 2 . For Configuration (a) the sensor coordinates are (,0), for (b) (0,-), and for (c) (0,). Note that for all three configurations z₀ is the depth below the soil surface of the closer probe (of both probes for Configuration [a]).

View larger version (40K): [in this window] [in a new window]   Fig. 2 Dual probes near the soil surface or wetting front. Configurations (a), (b), and (c) of heating probe H and sensor S are presented. For each configuration H is at (x,z) = (0,0)

(23)

We find in this way the required solutions for the time course of sensor temperature T(x,z,t): For Configuration (a)

(24)

For Configurations (b) and (c)

(25)

With the dual probe mounted vertically, the sensor temperature is the same regardless of which probe is the upper one.

Probe Near Wetting Front
A second question of practical interest concerns the behavior of a dual probe in dry soil when a sharp wetting front approaches it during infiltration. In these circumstances the thermal conductivity of the dry soil is small, whereas that of the wetted soil is large. In the analysis that follows we assume that, for the duration of the probe observation (at most a few tens of seconds), the front may be treated as stationary (and convective heat transfer as negligible).

We gain some insight into the errors induced in the probe by the approaching wetting front through the following idealized problem: The wetting front is represented by a step-function increase in water content, and the value behind the front is treated as effectively infinite relative to the small of the dry soil. Under these conservative assumptions (giving upper error bounds) the wetting front boundary condition is T = 0.

The heating probe is again at (x,z) = (0,0) and we consider again the three locations of the sensor probe in Fig. 2. This time they are relative to the wetting front, not the soil surface. The T = 0 condition at z = z₀ is realized here by applying the Method of Negative Images to the standard solution [23]. The solutions we seek are then: For Configuration (a)

(26)

For Configurations (b) and (c)

(27)

Here and are the values for the dry soil.

Probe Behind Wetting Front
Similar considerations apply after the front has engulfed the dual probe, and we pursue a similar idealization. Here the conservative assumption (giving upper error bounds) is that, relative to the large value of for the wet soil, we may take = 0 for the dry region. Then the no-flux boundary condition applies at the wetting front, and the relevant solutions are Eq. [24] and [25], with the values of and those for the wet soil. As before, our analysis treats the front as stationary and convective heat transport as negligible. Configurations (a), (b), and (c) still apply, but z is now positive downwards.

	Analysis of heterogeneity errors

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
C, , and in the Homogeneous System
Lubimova et al. (1961), Jaeger (1965), and Campbell et al. (1991) showed that the standard solution [23] allowed inference of C, , and from observed values of t_m and T_m. Here t_m is the time interval between the instantaneous heat pulse and the instant when the sensor reaches its maximum temperature T_m. Differentiating Eq. [23] with respect to t and setting the result to zero yields the relations

(28)

(29)

(30)

These results are for an instantaneous line source of vanishingly small radius, released into a homogeneous region extending to ± in all three space dimensions. Note that it follows from Eq. [23] and [30] that the dependence of T_m on t_m for this homogeneous system is

(31)

C, , and in Heterogeneous Systems
We proceed to calculate t_m and T_m for the heterogeneous systems that yield Eq. [22], [24], [25], [26], and [27]. We shall designate these various "heterogeneous" values t_mh and T_mh. We then establish the errors when these values are used in Eq. [28]–[30].

Heating Probe on Interface
It is readily shown from Eq. [22] that the sensor temperature maximum occurs for

(32)

Putting this t_mh for t_m in Eq. [30] gives

(33)

Now ₊ is exactly the thermal diffusivity of the soil between the two probes, so this form of heterogeneity produces no error in the estimate of diffusivity.

On the other hand, Eq. [22] gives

(34)

Comparison with Eq. [31] shows that

(35)

since here t_mh = t_m. It then follows that, with the heating probe on the interface, Eq. [29] gives

(36)

But the correct value is ₊, so the fractional error is

(37)

Since the observation gives = ₊ correctly, it follows that Eq. [37] also gives the fractional error in C.

Probes near Soil Surface
Configuration (a).
We find t_mh by differentiating Eq. [24] with respect to t and putting the result equal to zero, obtaining

(38)

Using Eq. [30] and introducing the dimensionless quantity

(39)

we reduce Eq. [38] to the simple form

(40)

This explicit relation is then readily translated, with the aid of Eq. [39], into the dependence on the dimensionless ratio z₀/. We use Eq. [40] to evaluate t_mh/t_m for various values of Z; then Eq. [39] gives z₀/ corresponding to various pairs of values of t_mh/t_m and Z. Figure 3 graphs the resulting relationship.

View larger version (18K): [in this window] [in a new window]   Fig. 3 Dependence of the ratio tmh/tm on z0/ for dual probes near a surface or front. In the heterogeneous system the sensor maximum temperature occurs at t = tmh. For a homogeneous system it occurs at t = tm. Here is the distance between probes, z0 the distance between the surface or front and the probes (Configuration [a]) or the nearer probe (Configurations [b] and [c]). The near-surface curves apply also behind a wetting front

(41)

Combining this with Eq. [31] and using Eq. [30] and [39], we find that

(42)

Since Z is a known function of z₀/, Eq. [42] leads immediately to the dependence of T_mh/T_m on z₀/. This is shown in Fig. 4 .

View larger version (16K): [in this window] [in a new window]   Fig. 4 Dependence of the ratio Tmh/Tm on z0/ for dual probes near a surface or front. Tmh is the sensor maximum temperature in the heterogeneous system. For a homogeneous system it is Tm. Other symbols are the same as in Fig. 3

(43)

The fractional error in estimating C is therefore

(44)

With _h the value of obtained by putting t_m = t_mh and T_m = T_mh in Eq. [29], we have

(45)

so the fractional error in estimating is

(46)

Finally, putting _h for the value of found with t_m = t_mh and T_m = T_mh in Eq. [30] gives

(47)

The fractional error in estimating is then

(48)

With Z a known function of z₀/, we translate Eq. [44], [46], and [48] into the fractional errors in C, and as functions of z₀/, as shown in Fig. 5 .

View larger version (24K): [in this window] [in a new window]   Fig. 5 Bounds on fractional errors in thermal diffusivity , thermal conductivity , and volumetric heat capacity C for dual probes near a surface when heterogeneity is ignored. The label on each curve indicates , , or C, together with the configuration. The curves apply also to probes behind a wetting front

(49)

Using Eq. [30] and introducing the dimensionless quantity

(50)

we retrieve Eq. [40] with Z₁ in place of Z. Previously we converted dependence on Z into dependence on z₀/ through use of Eq. [39]. Here Eq. [50] is a quadratic equation in z₀/ and the relevant positive root is

(51)

Equations [40] through [48] carry over except that Z is replaced by Z₁ and we now use Eq. [51]. Figures 3, 4, and 5 include the results for Configurations (b) and (c).

Probes near Wetting Front
Solutions here follow closely those for near-surface probes. The difference is that the second term on the right of Eq. [26] and [27] is negative, whereas it is positive in Eq. [24] and [25]. The consequence is that all results carry over, except for appropriate sign adjustments. We present them below without further comment.

Configuration (a)

(52)

(53)

(54)

(55)

(56)

Configurations (b) and (c)
The solutions are as Eq. [52] through [56] with Z₁ in place of Z. Equation [51] translates the dependences on Z₁ into dependences on z₀/.

Figures 3 and 4 include these near-front results analogous to those for the near-surface probes. Figure 6 graphs the bounds on fractional errors in C, , and as functions of z₀/ for near-front probes. Note that in this case the bounds on fractional errors in C and become infinite as z₀/ 0. We therefore limit the graphs to the fractional error range 0 to 1.

View larger version (22K): [in this window] [in a new window]   Fig. 6 The same as for Fig. 5 for dual probes near a wetting front. Bounds on fractional errors in , , and C are 1, , and respectively as z0/ 0

	Discussion

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

Probe near Soil Surface
Figure 5 shows that for Configurations (a), (b), and (c) the fractional error in is at most about -0.28. On the other hand, the fractional error of and C may reach -0.5 for small values of z₀/. In all cases, the magnitude of errors in and C is less for Configurations (b) and (c) than for (a).

All errors become small for z₀/ 1. The errors in C with z₀/ 1 are smaller than those in and for all three configurations. This is important when C is used to estimate water content (Bristow et al., 1993; Tarara and Ham, 1997; Bremer et al., 1998; Bristow, 1998; Song et al., 1998).

It is of some interest that Configurations (b) and (c) give the same results, not only here but also for probes near a wetting front. It turns out that it doesn't matter which probe is closer to the surface or front.

Probes near Wetting Front
Figure 6 shows that, for all three configurations, fractional errors are very large as z₀/ 0. The error in approaches 1.0 and for and C it approaches infinity. Errors for Configurations (b) and (c) are less than for (a). In all cases errors decrease rapidly as z₀/ increases and are small for z₀/ 1. The fractional errors of C for z₀/ 1 are smaller than those of and . This is useful when water content is estimated from C.

Probes behind Wetting Front
We reiterate that our results for probes near the soil surface hold also for probes behind a wetting front. Figure 5 applies to that case as well. We note that, for z₀/ = 0, our analysis of probes behind a wetting front (Configuration [b]) corresponds to the case in which the heating probe lies on the interface and = . Figure 5 shows fractional errors of -0.5 for C and , precisely the result obtained from the limit of Eq. [37] as . As discussed earlier, no error in occurs for the case of the heating probe on an interface. This result also agrees with Fig. 5 for z₀/ = 0.

Oblique Configurations
The methods developed here apply also to probe configurations oblique to plane interfaces. It is readily shown that, for a given value of z₀/, oblique configurations give errors intermediate between those for Configuration (a) and those for (b) and (c). It is notable that errors are the same whichever probe is nearer the interface.

	Conclusions

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
This study suggests that heterogeneity errors of dual thermal probes are small provided the heterogeneity is no closer to the probes than probe separation. Since this is of order 0.006 m, it appears that thermal probes offer a means of estimating , , and C close to soil surfaces and also close ahead of, and behind, wetting fronts. In particular the estimates of C may provide good resolution of soil water content in critical regions such as near surfaces and fronts. Our analyses of error near surfaces and fronts are conservative. We emphasize that the fractional errors presented in the various equations and figures give upper bounds on error magnitude.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 
Contribution no. 99-46-J from Kansas Agric. Exp. Stn., Manhattan, KS. Research supported by Western Regional Research Project W-188.

Received for publication August 12, 1998.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION Mathematical models Analysis of heterogeneity errors Discussion Conclusions REFERENCES

 

• Bremer D.J., Ham J.M., Owensby C.E., Knapp A.K. Responses of soil respiration to clipping and grazing in a tallgrass prairie. J. Environ. Qual. 1998;27:1539-1548.[Abstract/Free Full Text]
• Bristow K.L. Measurement of thermal properties and water content of unsaturated sandy soil using dual-probe heat-pulse probes. Agric. For. Meteorol. 1998;89:75-84.
• Bristow K.L., Campbell G.S., Calissendorff K. Test of a heat-pulse probe for measuring changes in soil water content. Soil Sci. Soc. Am. J. 1993;57:930-934.[Abstract/Free Full Text]
• Bristow K.L., Kluitenberg G.J., Horton R. Measurement of soil thermal properties with a dual-probe heat-pulse technique. Soil Sci. Soc. Am. J. 1994;58:1288-1294.[Abstract/Free Full Text]
• Campbell G.S., Calissendorff C., Williams J.H. Probe for measuring soil specific heat using a heat-pulse method. Soil Sci. Soc. Am. J. 1991;55:291-293.[Abstract/Free Full Text]
• Carslaw H.S., Jaeger J.C. Conduction of heat in solids, 2nd ed Oxford, UK: Clarendon Press, 1959.
• Jaeger J.C. Application of the theory of heat conduction to geothermal measurements. In: Lee W.H.K., ed. Terrestrial heat flow. Washington, DC: Am. Geophys. Union Monogr. 8. Am. Geophys. Union, 1965:7-23.
• Kluitenberg G.J., Bristow K.L., Das B.S. Error analysis of the heat pulse method for measuring soil heat capacity, diffusivity, and conductivity. Soil Sci. Soc. Am. J. 1995;59:719-726.[Abstract/Free Full Text]
• Kluitenberg G.J., Ham J.M., Bristow K.L. Error analysis of the heat pulse method for measuring soil volumetric heat capacity. Soil Sci. Soc. Am. J. 1993;57:1444-1451.[Abstract/Free Full Text]
• Lubimova H.A., Lusova L.M., Firsov F.V., Starikova G.N., Shushpanov A.P. Determination of surface heat flow in Mazesta (USSR). Ann. Geophys. 1961;14:157-167.
• Song Y., Ham J.M., Kirkham M.B., Kluitenberg G.J. Measuring soil water content under turfgrass using the dual-probe heat-pulse technique. J. Am. Soc. Hortic. Sci. 1998;123:937-941.[Abstract/Free Full Text]
• Tarara J.M., Ham J.M. Measuring soil water content in the laboratory and field with dual-probe heat-capacity sensors. Agron. J. 1997;89:535-542.[Abstract/Free Full Text]

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (5) Citing Articles via Google Scholar Google Scholar Articles by Philip, J.R. Articles by Kluitenberg, G.J. Search for Related Content PubMed Articles by Philip, J.R. Articles by Kluitenberg, G.J. GeoRef GeoRef Citation Agricola Articles by Philip, J.R. Articles by Kluitenberg, G.J.