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DIVISION S-1-SOIL PHYSICS

Determining Soil Water Flux and Pore Water Velocity by a Heat Pulse Technique

^a Soil and Fertilizer Institute, Hebei Academy of Agricultural Sciences, Shijiazhuang, Hebei 050051, China
^b Dep. of Agronomy, Kansas State Univ., Manhattan, KS 66506 USA
^c Dep. of Agronomy, Iowa State Univ., Ames, IA 50011 USA

rhorton{at}iastate.edu

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
A method is presented for measuring soil water flux density (J) with a thermo-TDR (time domain reflectometry) probe. Constant heat input during a small time interval (15 s) is used to emit a heat pulse from a line heat source. Asymmetry in the thermal field near the heat source is quantified by computing the maximum dimensionless temperature difference (MDTD) between upstream and downstream locations. Heat transfer theory was used to relate MDTD to J. A thermo-TDR probe was used to obtain measurements of MDTD in water-saturated soil materials of different textures (sand, sandy loam, and clay loam) with imposed water flux densities ranging from 1.16 x 10^-5 to 6.31 x 10^-5 m³ m^-2 s^-1. A nearly linear relationship between measured MDTDs and fluxes was observed for all soil materials. Measured and predicted MDTDs agreed well for flow experiments in sand. Greater discrepancies were observed for flow experiments in sandy loam and clay loam. Despite the lack of universal agreement between measured and predicted MDTDs, the experimental results indicate that the proposed method may provide a useful means of measuring J. The method presented herein improves upon earlier methods by reducing distortion of the water flow field and minimizing heat-induced soil water redistribution. Because the thermo-TDR probe can be used to make TDR-based measurements of volumetric water content (), the proposed method also may permit measurement of pore water velocity (J/).

Abbreviations: MDTD, maximum dimensionless temperature difference • TDR, time domain reflectometry

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
DETERMINING WATER MOVEMENT IN SOIL is critical for managing irrigation and drainage and for characterizing chemical transport processes. Soil water flux (flux density) can be measured using a soil water flux meter (e.g. Cary, 1970; Dirksen, 1972, 1974); however, these meters are sophisticated and subject to problems, including the localized nature of the measurement, disruption of the soil during installation, and interruption of normal patterns of soil water flow (Wagenet, 1986). Several approaches are available for estimating soil water flux indirectly (Nielsen et al., 1973; Bresler, 1973), but these approaches can be time consuming, mathematically complicated, and measurement-intensive.

Byrne et al. (1967, 1968) first applied heat as a tracer to measure soil water flux. Their instruments consisted of temperature sensors positioned symmetrically with respect to point or line heat sources. Water flux was measured by characterizing distortion in the thermal field around the instruments. Several limitations have prevented these instruments from being used as practical tools for characterizing soil water flux. One limitation is that they require constant heat input for relatively long periods of time (30 min for average flow rates) before reaching thermal equilibrium. Thus, these instruments will have limited applicability in unsaturated soil where thermal gradients will result in soil water redistribution. Another limitation is that calibration is required to relate flux to instrument response. In addition, the size of these instruments results in distortion of the soil water flow field in the vicinity of the instrument. Experimental results showed poor agreement between theory and measurements for the point-source instrument (Byrne et al., 1967) and a double-valued calibration curve for the line-source instrument (Byrne et al., 1968).

Recent developments in heat-pulse techniques for measuring soil thermal properties suggest that the instruments developed by Byrne et al. (1967, 1968) can be improved. Campbell et al. (1991) and Bristow et al. (1994) used heat-pulse sensors that employed a relatively short (8 s) heating time. This approach for delivering the heat impulse has been shown to cause minimal soil water redistribution in unsaturated soil (Noborio et al., 1996; Bilskie, 1994). In addition, the sensors employed by Campbell et al. (1991) and Bristow et al. (1994) consisted of small needles, each with an outer diameter of 0.813 mm. A sensor with small needles obviously would produce less distortion in the soil water flow field than the instruments of Byrne et al. (1967, 1968).

Ren et al. (1999) report on the development of a thermo-TDR probe that permits simultaneous measurement of soil water content, electrical conductivity, thermal conductivity, thermal diffusivity, and volumetric heat capacity. Their probe combines the TDR method (water content, electrical conductivity) with the method of Bristow et al. (1994) for determining thermal properties. The thermo-TDR probe consists of three parallel, equidistant, hypodermic needles lying in a common plane and each containing a heater wire and a thermocouple. We hypothesized that the thermo-TDR probe may provide a means of measuring soil water flux density. If a heat impulse is emitted from the center needle, the outer needles can be used to monitor temperature changes as a function of time. If the probe is aligned so that the plane of the needles is parallel to the direction of soil water flow, the flow field will distort the temperature responses observed at the outer (now upstream and downstream) needles. This temperature asymmetry may provide information regarding the soil water flux. If the thermo-TDR probe can be used successfully in this capacity, it would improve upon the instruments of Byrne et al. (1967, 1968) by reducing distortion of the water flow field and minimizing heat-induced redistribution of soil water.

The objectives of this study were to (i) develop a heat transfer model that characterizes the thermal field around a line-source heater in soil with a uniform water flow field, (ii) develop an appropriate algorithm for relating water flux density (J) and pore water velocity (V_w) to the measurements obtained with a thermo-TDR probe, and (iii) provide an experimental evaluation of the proposed method. Laboratory experiments were conducted with soil materials of different textures and a range of imposed water flux densities. The parameter estimation method presented herein is similar to a method suggested by Melville et al. (1985) for estimating groundwater velocity from distortion in the thermal field around a sensor which consisted of a point heat source surrounded by a circular array of thermistors.

	Theory

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Heat Transfer Equations
For a homogeneous, isotropic, infinite medium moving with uniform velocity in the x direction, the equation for combined heat conduction and convection is (Carslaw and Jaeger, 1959, p. 13)

(1)

where T is temperature (°C); t is time (s); ' is thermal diffusivity (m² s^-1); U is velocity (m s^-1); and x, y, and z are space coordinates.

Equation [1] is valid only for a single-phase system. For a multiphase system, for example, an incompressible porous medium with a liquid moving uniformly through it with pore water velocity V_w, Eq. [1] becomes

(2)

where is the thermal diffusivity of the multiphase system, c is the volumetric heat capacity (J m^-3 C^-1) of the multiphase system, (c) is the volumetric heat capacity of the liquid, and is the volumetric liquid content of the medium.

This formulation requires the assumption that conductive heat transfer dominates over convective effects. Thus, thermal homogeneity exists between the liquid and the porous medium. The convective coefficient in Eq. [2] is V, the heat pulse velocity (Marshall, 1958) or thermal front advection velocity (Melville et al., 1985), which is expressed as

(3)

and indicates that the thermal front moves slower than the liquid. The heat pulse velocity may be interpreted as the weighted average of the velocities of heat through the liquid phase and through the stationary porous medium (Marshall, 1958). The heat velocity lags behind the front of the liquid phase because, under the assumption of thermal homogeneity, heat from the liquid phase is absorbed instantaneously by the porous medium at the thermal front (Melville et al., 1985).

General Solution
In order to obtain a solution of Eq. [2] for an infinite line source, heated for a finite time, we begin with the solution for an instantaneously heated infinite line source in a stationary medium. Carslaw and Jaeger (1959)(p. 258) give the analytical solution

(4)

for a line source parallel to the z-axis and located at . The source strength (m² °C), Q, is defined as

(5)

where q is the heat input per unit length (J m^-1).

The solution for an infinite line source in an infinite moving medium can be developed by paralleling the derivation for a point heat source in Section 10.7 of Carslaw and Jaeger (1959). In the element of time dt' at time t', qdt' heat units per unit length are emitted from the infinite line source. The temperature at time t at (x, y, t) due to the heat qdt' released per unit length at t' is

(6)

where is the thermal conductivity (W m^-1 °C^-1).

Next, we consider heating of the infinite line source at the rate q'dt' over the interval 0 < t t₀. Here, q' is the heat input per unit length per unit time (W m^-1), and the corresponding source strength becomes Q' (m² °C s^-1). Integrating Eq. [6] (Carslaw and Jaeger, 1959, p. 261) and making use of the substitution yields a general solution for the temperature at (x, y, t)

(7)

where

(8)

and

(9)

Temperature Difference Algorithm
For the temperature distribution at a distance x_d directly downstream from the line source, Eq. [7] becomes

(10)

where

(11)

and

(12)

For the temperature distribution at a distance x_u directly upstream from the line source, Eq. [7] becomes

(13)

where

(14)

and

(15)

Equations [10] and [13] can then be used to compute the difference between the temperature distributions at the downstream and upstream positions:

(16)

(17)

Substitutions of Eq. [11] and [14] into Eq. [16] and of Eq. [12] and [15] into Eq. [17] yield a solution for the temperature difference between the downstream and upstream positions. Using a dimensionless temperature difference, this solution is written

(18)

where

(19)

and

(20)

Equation [18] indicates that the dimensionless temperature difference is a function of both and V. Graphical examination of this solution with (Fig. 1) reveals that the MDTD is insensitive to variations in , but nearly proportional to variations in V (Fig. 2) .

View larger version (23K): [in this window] [in a new window]   Fig. 1 Transient dimensionless temperature difference between the downstream and upstream sensors (Eq. [18]) for a range of thermal diffusivities with , and t0 = 15 s

View larger version (15K): [in this window] [in a new window]   Fig. 2 Transient dimensionless temperature difference between the downstream and upstream sensors (Eq. [18]) for a range of heat pulse velocities with = 6.0 x 10-7 m2 s-1, xd = xu = 0.006 m, and t0 = 15 s

(21)

where t_m represents the time at which the dimensionless temperature difference reaches a maximum. This result is obtained by substituting t_m for t in Eq. [20] and letting MDTD represent ₂(t_m). MathCad (ver. 7.0) was used to evaluate the integral in Eq. [21]. Graphical evaluation of Eq. [21] with reveals a nearly unique relationship between MDTD and V (Fig. 3) . This relationship suggests that measurements of MDTD may provide a useful means of estimating V. Equation [3] then could be employed to compute J, provided that c and (c) are known. If is known, Eq. [3] also could be used to compute V_w. However, it is clear from the left-hand side of Eq. [18] that MDTD is inversely proportional to . Thus, in practice, a relationship between MDTD and J (or V_w) can be expected only if the thermal conductivity is known.

View larger version (19K): [in this window] [in a new window]   Fig. 3 Maximum dimensionless temperature difference (MDTD, Eq. [21]) as a function of heat pulse velocity for a range of thermal diffusivities with xd = xu = 0.006 m, and t0 = 15 s

	Materials and methods

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

A heat pulse was generated by applying constant current to the central heater for 15 s with a direct current supply (Model 1635, B & K-Precision, Maxtec International Corp., Chicago, IL). A datalogger (Model 21X, Campbell Scientific, Logan, UT) was used to control the heat input through a relay and record the upstream and downstream temperatures at 1-s intervals for 120 s. The datalogger also was used to record the voltage drop across a precision resistor. Voltage drop was used to determine the current applied to the heater.

Measurements in agar-stabilized water (Campbell et al., 1991) were used to determine the apparent distances between the center needle (heater) and the outer needles. Values for x_u and x_d were adjusted so that readings from the upstream and downstream positions returned the published value of .

Columns (Fig. 4) were packed with soil material collected from the A horizons of a Hanlon soil (coarse-loamy, mixed, superactive, mesic Cumulic Hapludolls); a Clarion soil (fine-loamy, mixed, superactive, mesic Typic Hapludolls); and a Harps soil (fine-loamy, mixed, superactive, mesic Typic Calciaquolls) (Table 1) . Soil material was air-dried, ground, sieved through a 2-mm screen, wetted to water content of approximately 0.1 kg kg^-1, and mixed. It then was packed uniformly into a paper cylinder (0.069 m in diam. and 0.27 m in height) positioned on a stopper. After the paper was removed, a PVC pipe (0.08 m i.d. and 0.30 m in height) was placed co-axially outside the soil column and pushed down upon the stopper tightly. A syringe then was used to inject liquid wax into the annular gap between the pipe and the soil. The wax seal was used to prevent the possibility of water flow along the column boundary. The top of the column was sealed with another stopper. Four layers of cheesecloth were placed between the stoppers and the soil to ensure uniform flow across the entire soil cross-section. The thermo-TDR probe was inserted into the soil horizontally from a precut slot, and the space between the probe and the pipe was filled with wax.

View larger version (44K): [in this window] [in a new window]   Fig. 4 Schematic view of the experimental setup (not drawn to scale)

A range of soil water fluxes was obtained by imposing different hydraulic gradients. When flux was constant, a heat pulse was applied and temperature-by-time data were recorded. The same protocol was used to obtain measurements for all soil materials and water flux densities. Heat pulse duration was fixed at , and measured heat inputs fell in the range . Apparent spacings between the center needle and the outer needles of the thermo-TDR probe were determined to be 5.75 and 6.01 mm. The probe was oriented so that and for measurements in the sand and sandy loam, and and for measurements in the clay loam. Measurements of column outflow were used to determine J.

	Results and discussion

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Thermal properties of the saturated soil materials were measured with (Table 2) . The magnitude of measured c values follows the order clay loam > sandy loam > sand. Differences in c were largely the result of differences in the bulk densities of the soil materials (Table 1). Measured c values compare favorably with those predicted using the model of De Vries (1963). The magnitude of measured and values follows the order sand > sandy loam > clay loam. These differences were the results of differences in bulk density as well as differences in soil mineralogy. The differences in the sand content of the soil materials (Table 1) indicate that the volume fraction of quartz likely followed the order sand > sandy loam > clay loam. The thermal conductivity and diffusivity of quartz are known to be approximately twice those of other soil minerals.

View this table: [in this window] [in a new window]   Table 2 Volumetric heat capacity (c), thermal conductivity (), and soil thermal diffusivity () measured at zero water flux (J = 0). Calculated values of c were obtained with the model of de Vries (1963)

View larger version (69K): [in this window] [in a new window]   Fig. 5 Transient dimensionless temperature at the downstream and upstream positions of the thermo-TDR probe as a function of soil water flux for the (a) sand, (b) sandy loam, and (c) clay loam

Equations [10] and [13], along with known values of J and estimated values of and (Table 2), were used to predict changes in temperature at the downstream and upstream positions, respectively. Excellent agreement between measured and predicted temperature change was observed for the sand (Fig. 6) . For the sandy loam (Fig. 7) and clay loam (Fig. 8) , predicted and measured values matched well at smaller soil water fluxes. For larger fluxes, the theory overestimated change in temperature at the downstream position and underestimated change in temperature at the upstream position. Deviations between predicted and measured temperature changes exceeded 10% for the sandy loam with J > 2.40 x 10^-5 m³ m^-2 s^-1and for the clay loam with J > 2.73 x 10^-5 m³ m^-2 s^-1. The rather systematic nature of these deviations suggests that the model (Eq. [10] and [13]) may have limitations at higher fluxes. One possibility is that the condition of thermal homogeneity begins to fail as flux increases. It is also possible that the assumed heater geometry becomes inappropriate with increasing water flux. Kluitenberg et al. (1995) determined that an infinite line source provides an excellent approximation of the finite, cylindrical heater of the thermo-TDR probe, but their analysis was restricted to the case of J = 0. Other possible explanations for the deviations between measured and predicted temperature changes are flow distortion due to the needles of the thermo-TDR probe and systematic flow nonuniformity within the soil columns.

View larger version (33K): [in this window] [in a new window]   Fig. 6 Transient dimensionless temperatures at the downstream (xd = 5.75 mm) and upstream (xu = 6.01 mm) positions as influenced by soil water flux for the sand. Symbols and lines indicate measured and predicted values, respectively

View larger version (34K): [in this window] [in a new window]   Fig. 7 Transient dimensionless temperatures at the downstream (xd = 5.75 mm) and upstream (xu = 6.01 mm) positions as influenced by soil water flux for the sandy loam. Symbols and lines indicate measured and predicted values, respectively

View larger version (31K): [in this window] [in a new window]   Fig. 8 Transient dimensionless temperatures at the downstream (xd = 6.01 mm) and upstream (xu = 5.75 mm) positions as influenced by soil water flux for the clay loam. Symbols and lines indicate measured and predicted values, respectively

View larger version (12K): [in this window] [in a new window]   Fig. 9 Maximum dimensionless temperature difference (MDTD) as a function of soil water flux for the sand, sandy loam, and clay loam. Symbols and lines represent measured and predicted (Eq. [21]) values, respectively

Equation [21] was used to plot the predicted MDTDs in Fig. 9. Although computation of this integral presents no special problems, avoiding this computation in practice would be desirable. Herein lies the value of the relationship presented in Fig. 3. The MDTD–V relationship apparently can be approximated with a single curve (slightly nonlinear) for a wide range of . This would eliminate repeated evaluation of Eq. [21]. However, it is important to recognize that use of the relationship depicted in Fig. 3 would require a thermo-TDR probe constructed such that . We used Eq. [21] to plot the MDTDs in Fig. 9 because our thermo-TDR probe had unequal apparent probe spacings (x_d x_u).

A limitation of the instruments suggested by Byrne et al. (1967, 1968) was that calibration was required to relate instrument response to soil water flux. Calibration essentially required determination of for the soil material in which an instrument was placed. The same limitation is encountered in using the thermo-TDR probe along with the theory presented herein. Recall that dimensionless temperatures are given as 4T/q' in Fig. 5. Thus, in order to compute a value of MDTD, must be known in addition to q' and temperature rises at the upstream and downstream positions. In addition, c and (c) are required to obtain the MDTD–J relationship from the MDTD–V relationship. We have shown, however, that the method of Bristow et al. (1994) can be applied to thermo-TDR probe measurements to obtain in situ estimates of and c, provided a zero flux condition can be achieved. Use of the thermo-TDR probe also provides TDR-based measurements of , thus allowing determination of pore water velocity in addition to soil water flux.

The thermo-TDR probe, as employed in our experiments, certainly causes less distortion of the water flow field than the instruments employed by Byrne et al. (1967, 1968). But only the needles of the thermo-TDR probe were placed in the flow pathway in our experiments. Some distortion of the flow field can be expected when the needle housing also lies in the flow pathway. We anticipate, however, that this distortion will be minor in comparison with that caused by the instruments of Byrne et al. (1967, 1968). The needle housing of the thermo-TDR probe is relatively small with respect to the length of the probe needles. And, because the thermo-TDR probe was not originally designed for the purpose of flux measurements, it may be possible to modify the probe design to further minimize flow distortion.

Although our experimental work was limited to measurements in water-saturated soil materials, there appear to be no limitations to using the proposed method for water flux measurements in unsaturated soil. But this remains to be verified experimentally. The degree of saturation most certainly will impact the upper and lower limits of flux detection that can be achieved with this method. This is revealed by Eq. [3]. The value of c, which varies strongly with , will determine how the upper and lower measurement limits of V will translate to upper and lower measurement limits for J. Failure of thermal homogeneity will determine the upper limit for measuring V. The lower limit of detection for V will be determined by the resolution with which measurements of MDTD can be obtained. Additional experimentation with a broader range of water fluxes will be required to define practical upper and lower limits; however, a simple analysis of Eq. [21] yields insight regarding the lower limit. A lower limit of V 9 x 10^-7 m s^-1 is predicted if MDTD can be resolved to within 0.01°C. If MDTD can be resolved to within 0.001°C, the lower limit decreases to V 9 x 10^-8 m s^-1. These values of V correspond to fluxes of J 7 x 10^-7 and J 7 x 10^-8 m s^-1, respectively, for the water-saturated clay loam with . And the lower limits for J decrease slightly for unsaturated conditions because of the dependence of c on (see Eq. [3]). Byrne et al. (1967, 1968) reported a lower limit of in water-saturated soil with their instruments. Melville et al. (1985) reported a lower limit of 3.5 x 10^-7 m s^-1 for the detection of groundwater seepage velocity with their heat-pulse sensor. Our calculations and the results of Byrne et al. (1967, 1968) and Melville et al. (1985) suggest that heat-pulse methods may be useful for only a limited range of the water fluxes typically encountered in unsaturated soils.

	Summary and conclusions

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
We have presented a method for determining soil water flux using the thermo-TDR of Ren et al. (1999). The thermo-TDR probe permits temperature measurements at locations upstream and downstream from a line heat source, which is used to emit a heat pulse of finite duration. Measured temperatures are used to compute the MDTD between upstream and downstream locations. Theory was developed to relate MDTD to soil water flux and pore water velocity. Inasmuch as the thermo-TDR probe can be used to make TDR-based measurements of volumetric water content, the proposed method also may permit measurement of pore water velocity.

Experiments with three water-saturated soil materials and fluxes ranging from 1.16 x 10^-5 to 6.31 x 10^-5 m³ m^-2 s^-1 revealed a nearly linear relationship between measured MDTDs and fluxes. Reasonable agreement was observed between measured and predicted MDTDs for a sand, but greater discrepancies occurred for a sandy loam and a clay loam. For all soil materials, deviation between predictions and measurements increased with increasing flux. Although further refinement of the proposed method is needed, this initial experimental evidence suggests that it holds promise. Further testing of the method must include unsaturated flow conditions and an assessment of the practical upper and lower limits for quantifying flux.

The proposed method improves upon the approaches suggested by Byrne et al. (1967, 1968) by reducing distortion of the water flow field and minimizing heat-induced soil water redistribution. Inasmuch as soil thermal properties can be measured with the thermo-TDR probe, it appears that calibration (determination of soil thermal properties) can be achieved easily in the absence of water flow. Thus, it may be possible to avoid the calibration problems encountered with the instruments of Byrne et al. (1967, 1968).

	NOTES

TOP NOTES ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Journal Paper no. J-18360 of the Iowa Agric. and Home Econ. Exp. Stn., Ames, IA; Projects no. 3262 and 3287, and supported by the Hatch Act and State of Iowa Funds. Contribution no. 99-424-J from the Kansas Agric. Exp. Stn., Manhattan, KS; Western Regional Research Project W-188.

Received for publication April 29, 1999.

	REFERENCES

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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 ISI Web of Science (25) Citing Articles via Google Scholar Google Scholar Articles by Ren, T. Articles by Horton, R. Search for Related Content PubMed Articles by Ren, T. Articles by Horton, R. GeoRef GeoRef Citation Agricola Articles by Ren, T. Articles by Horton, R.