Determining Soil Water Flux and Pore Water Velocity by a Heat Pulse Technique. T. Ren, G. J. Kluitenberg, and R. Horton*

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1 552 SOIL SCI. SOC. AM. J., VOL. 64, MARCH APRIL 2000 Raney, W.A., and T.W. Edminster Approaches to soil compac- C.J. Beauchamp Impact of de-inking sludge amendment on tion research. Trans. ASAE 4: agricultural soil quality. p In Proc. of the TAPPI Environ. SAS Institute The SAS system for Windows. Release SAS Conf. Tappi Press. Inst., Cary, NC. Watson, K.K An instantaneous profile method for determining Sullivan, L.A Soil organic matter, air encapsulation and water- the hydraulic conductivity of unsaturated porous materials. Water stable aggregation. J. Soil Sci. 41: Resour. Res. 2: Tisdall, J.M., and J.M. Oades Organic matter and water stable Webber, L.R Incorporation of nonsegregated, noncomposted aggregate in soils. J. Soil Sci. 33: solid waste and soil physical properties. J. Environ. Qual. 7:397 Trépanier, L., J. Gallichand, J. Caron, and G. Thériault Environ mental effects of de-inking sludge application on soil and soil water Weil, R.R., and W. Kroontje Physical condition of a Davidson quality. Trans. ASAE 41: clay loam after five years of heavy poultry manure applications. Trépanier, L., G. Thériault, J. Caron, J. Gallichand, S. Yelle, and J. Environ. Qual. 8: Determining Soil Water Flux and Pore Water Velocity by a Heat Pulse Technique T. Ren, G. J. Kluitenberg, and R. Horton* ABSTRACT Several approaches are available for estimating soil water flux indirectly (Nielsen et al., 1973; Bresler, 1973), A method is presented for measuring soil water flux density (J ) with a thermo-tdr (time domain reflectometry) probe. Constant but these approaches can be time consuming, mathemat- heat input during a small time interval (15 s) is used to emit a heat ically complicated, and measurement-intensive. pulse from a line heat source. Asymmetry in the thermal field near the Byrne et al. (1967, 1968) first applied heat as a tracer heat source is quantified by computing the maximum dimensionless to measure soil water flux. Their instruments consisted temperature difference (MDTD) between upstream and downstream of temperature sensors positioned symmetrically with locations. Heat transfer theory was used to relate MDTD to J. A respect to point or line heat sources. Water flux was thermo-tdr probe was used to obtain measurements of MDTD in measured by characterizing distortion in the thermal water-saturated soil materials of different textures (sand, sandy loam, and clay loam) with imposed water flux densities ranging from 1.16 field around the instruments. Several limitations have 10 prevented these instruments from being used as practi- 5 to m 3 m 2 s 1. A nearly linear relationship between measured MDTDs and fluxes was observed for all soil materials. cal tools for characterizing soil water flux. One limita- Measured and predicted MDTDs agreed well for flow experiments tion is that they require constant heat input for relatively in sand. Greater discrepancies were observed for flow experiments long periods of time (30 min for average flow rates) in sandy loam and clay loam. Despite the lack of universal agreement before reaching thermal equilibrium. Thus, these instrubetween measured and predicted MDTDs, the experimental results ments will have limited applicability in unsaturated soil indicate that the proposed method may provide a useful means of where thermal gradients will result in soil water redistrimeasuring J. The method presented herein improves upon earlier bution. Another limitation is that calibration is required methods by reducing distortion of the water flow field and minimizing to relate flux to instrument response. In addition, the heat-induced soil water redistribution. Because the thermo-tdr probe can be used to make TDR-based measurements of volumetric size of these instruments results in distortion of the water content ( ), the proposed method also may permit measurement soil water flow field in the vicinity of the instrument. of pore water velocity (J/ ). 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 Determining water movement in soil is critical for managing irrigation and drainage and for charac- al., 1968). terizing chemical transport processes. Soil water flux Recent developments in heat-pulse techniques for (flux density) can be measured using a soil water flux measuring soil thermal properties suggest that the inmeter (e.g. Cary, 1970; Dirksen, 1972, 1974); however, struments developed by Byrne et al. (1967, 1968) can these meters are sophisticated and subject to problems, be improved. Campbell et al. (1991) and Bristow et al. including the localized nature of the measurement, distively short (8 s) heating time. This approach for deliv- (1994) used heat-pulse sensors that employed a relaruption of the soil during installation, and interruption of normal patterns of soil water flow (Wagenet, 1986). ering 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 em- T. Ren, Soil and Fertilizer Institute, Hebei Academy of Agricultural ployed by Campbell et al. (1991) and Bristow et al. Sciences, Shijiazhuang, Hebei , China; G.J. Kluitenberg, Dep. (1994) consisted of small needles, each with an outer of Agronomy, Kansas State Univ., Manhattan, KS 66506; and R. diameter of mm. A sensor with small needles Horton, Dep. of Agronomy, Iowa State Univ., Ames, IA Jourobviously would produce less distortion in the soil water nal Paper no. J of the Iowa Agric. and Home Econ. Exp. Stn., Ames, IA; Projects no and 3287, and supported by the Hatch flow field than the instruments of Byrne et al. (1967, Act and State of Iowa Funds. Contribution no J from the 1968). Kansas Agric. Exp. Stn., Manhattan, KS; Western Regional Research Ren et al. (1999) report on the development of a Project W-188. Received 29 Apr *Corresponding author Abbreviations: MDTD, maximum dimensionless temperature difference; TDR, time domain Published in Soil Sci. Soc. Am. J. 64: (2000). reflectometry.

2 REN ET AL.: DETERMINING SOIL WATER FLUX BY A HEAT PULSE TECHNIQUE 553 thermo-tdr probe that permits simultaneous measuresystem, c is the volumetric heat capacity (J m 3 C 1 ) of the multiphase ment of soil water content, electrical conductivity, thermal ( c) is the volumetric heat capacity of the liquid, and conductivity, thermal diffusivity, and volumetric is the volumetric liquid content of the medium. heat capacity. Their probe combines the TDR method This formulation requires the assumption that conductive heat transfer dominates over convective effects. Thus, thermal (water content, electrical conductivity) with the method homogeneity exists between the liquid and the porous meof Bristow et al. (1994) for determining thermal properdium. The convective coefficient in Eq. [2] is V, the heat pulse ties. The thermo-tdr probe consists of three parallel, velocity (Marshall, 1958) or thermal front advection velocity equidistant, hypodermic needles lying in a common (Melville et al., 1985), which is expressed as plane and each containing a heater wire and a thermocouple. We hypothesized that the thermo-tdr probe ( c) V V w J ( c) [3] may provide a means of measuring soil water flux density. c c If a heat impulse is emitted from the center needle, and indicates that the thermal front moves slower than the the outer needles can be used to monitor temperature liquid. The heat pulse velocity may be interpreted as the changes as a function of time. If the probe is aligned so weighted average of the velocities of heat through the liquid that the plane of the needles is parallel to the direction phase and through the stationary porous medium (Marshall, of soil water flow, the flow field will distort the temperaphase 1958). The heat velocity lags behind the front of the liquid ture responses observed at the outer (now upstream because, under the assumption of thermal homogeneity, and downstream) needles. This temperature asymmetry heat from the liquid phase is absorbed instantaneously by the may provide information regarding the soil water flux. porous medium at the thermal front (Melville et al., 1985). If the thermo-tdr probe can be used successfully in this capacity, it would improve upon the instruments of General Solution Byrne et al. (1967, 1968) by reducing distortion of the In order to obtain a solution of Eq. [2] for an infinite line water flow field and minimizing heat-induced redistribusource, heated for a finite time, we begin with the solution for tion of soil water. an instantaneously heated infinite line source in a stationary The objectives of this study were to (i) develop a medium. Carslaw and Jaeger (1959, p. 258) give the analytical heat transfer model that characterizes the thermal field solution around a line-source heater in soil with a uniform water flow field, (ii) develop an appropriate algorithm for T(x, y, t) Q 4 t exp x 2 y 2 4 t [4] relating water flux density (J) and pore water velocity (V w ) to the measurements obtained with a thermo-tdr for a line source parallel to the z-axis and located at (x, y) probe, and (iii) provide an experimental evaluation of (0, 0). The source strength (m 2 C), Q, is defined as the proposed method. Laboratory experiments were conducted with soil materials of different textures and a range of imposed water flux densities. The parameter Q q c [5] estimation method presented herein is similar to a where q is the heat input per unit length (J m 1 ). method suggested by Melville et al. (1985) for estimating The solution for an infinite line source in an infinite moving groundwater velocity from distortion in the thermal medium can be developed by paralleling the derivation for a field around a sensor which consisted of a point heat point heat source in Section 10.7 of Carslaw and Jaeger (1959). source surrounded by a circular array of thermistors. 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 THEORY length at t is Heat Transfer Equations qdt 4 (t t ) exp [x V(t t )]2 y 2 4 (t t ) [6] 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, where c is the thermal conductivity (W m 1 C 1 ). p. 13) Next, we consider heating of the infinite line source at the rate q dt over the interval 0 t t 0. Here, q is the heat input T per unit length per unit time (W m 1 ), and the corresponding t 2 T x 2 T 2 y 2 T T 2 2 z U [1] x source strength becomes Q (m 2 Cs 1 ). Integrating Eq. [6] where T is temperature ( C); t is time (s); is thermal diffusivsubstitution s (t t ) yields a general solution for the (Carslaw and Jaeger, 1959, p. 261) and making use of the ity (m 2 s 1 ); U is velocity (m s 1 ); and x, y, and z are space coortemperature at (x, y, t) dinates. Equation [1] is valid only for a single-phase system. For a T multiphase system, for example, an incompressible porous T(x, y, t) 1 (x, y, t); 0 t t 0 [7] medium with a liquid moving uniformly through it with pore T 2 (x, y, t); t t 0 water velocity V w, Eq. [1] becomes where T t 2 T x 2 T 2 y 2 T 2 z V ( c) T 2 w [2] c x T 1 (x, y, t) 4 q t 0 where is the thermal diffusivity of the multiphase system, s 1 exp (x Vs)2 y 2 ds [8]

3 554 SOIL SCI. SOC. AM. J., VOL. 64, MARCH APRIL 2000 and T 2 (x, y, t) q 4 t t t 0 s 1 exp (x Vs)2 y 2 Temperature Difference Algorithm For the temperature distribution at a distance x d directly downstream from the line source, Eq. [7] becomes where and T(t) T 1 (t); 0 t t 0 T 2 (t); t t 0 [10] Equations [10] and [13] can then be used to compute the difference between the temperature distributions at the downstream and upstream positions: ds T 1 (t) T 1 (t) T 1 (t) [16] [9] T 2 (t) T 2 (t) T 2 (t) [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 4 T1(t) 1 (t); 0 t t 0 4 T(t) q q 4 T 2 (t) T 2 (t); t t 0 1 (t) 4 q t s 1 exp q 0 (x d Vs) 2 ds [11] [18] T 2 (t) q 4 t t t 0 s 1 exp (x d Vs) 2 For the temperature distribution at a distance x u directly upstream from the line source, Eq. [7] becomes where and T (t) T 1 (t); 0 t t 0 T 2 (t); t t 0 [13] T 1 (t) 4 q t s 1 exp (x u Vs) 2 0 where ds [12] t 1 (t) s 1 exp (x d Vs) 2 and 2 (t) t 0 exp (x u Vs) 2 ds [19] s 1 t t 0 exp (x d Vs) 2 ds [14] exp (x u Vs) 2 ds [20] Equation [18] indicates that the dimensionless temperature T 2 (t) q difference is a function of both and V. Graphical examination 4 t s 1 exp t t 0 (x u Vs) 2 ds [15] of this solution with x d x u (Fig. 1) reveals that the MDTD is insensitive to variations in, but nearly proportional to variations in V (Fig. 2). Mathematically, the MDTD is expressed as MDTD t m s 1 t m t 0 exp (x d Vs) 2 Fig. 1. Transient dimensionless temperature difference between the downstream and upstream sensors (Eq. [18]) for a range of thermal diffusivities with V ms 1, x d x u m, and t 0 15 s. Fig. 2. Transient dimensionless temperature difference between the downstream and upstream sensors (Eq. [18]) for a range of heat pulse velocities with m 2 s 1, x d x u m, and t 0 15 s.

4 REN ET AL.: DETERMINING SOIL WATER FLUX BY A HEAT PULSE TECHNIQUE 555 the voltage drop across a precision resistor. Voltage drop was exp (x u Vs) 2 used to determine the current applied to the heater. ds [21] Measurements in agar-stabilized water (Campbell et al., 1991) were used to determine the apparent distances between where t m represents the time at which the dimensionless tem- the center needle (heater) and the outer needles. Values for perature difference reaches a maximum. This result is obtained x u and x d were adjusted so that readings from the upstream by substituting t m for t in Eq. [20] and letting MDTD represent and downstream positions returned the published value of 2 (t m ). MathCad (ver. 7.0) was used to evaluate the integral ( c) 4.18 MJ m 3 C 1. in Eq. [21]. Graphical evaluation of Eq. [21] with x d x u Columns (Fig. 4) were packed with soil material collected reveals a nearly unique relationship between MDTD and V from the A horizons of a Hanlon soil (coarse-loamy, mixed, (Fig. 3). This relationship suggests that measurements of superactive, mesic Cumulic Hapludolls); a Clarion soil (fine- MDTD may provide a useful means of estimating V. Equation loamy, mixed, superactive, mesic Typic Hapludolls); and a [3] then could be employed to compute J, provided that c Harps soil (fine-loamy, mixed, superactive, mesic Typic and ( c) are known. If is known, Eq. [3] also could be used Calciaquolls) (Table 1). Soil material was air-dried, ground, to compute V w. However, it is clear from the left-hand side sieved through a 2-mm screen, wetted to water content of of Eq. [18] that MDTD is inversely proportional to. Thus, approximately 0.1 kg kg 1, and mixed. It then was packed in practice, a relationship between MDTD and J (or V w ) can uniformly into a paper cylinder (0.069 m in diam. and 0.27 m be expected only if the thermal conductivity is known. 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 MATERIALS AND METHODS upon the stopper tightly. A syringe then was used to inject The Thermo-TDR probe consisted of three parallel hypo- liquid wax into the annular gap between the pipe and the soil. dermic needles, each enclosing a line heater and a thermocou- The wax seal was used to prevent the possibility of water flow ple. The rods were 1.3 mm in diameter and 40 mm in length along the column boundary. The top of the column was sealed and spaced 6 mm apart. The heaters were made from 75- m- with another stopper. Four layers of cheesecloth were placed diameter enameled Evanohm wire (Wilbur B. Driver Co., between the stoppers and the soil to ensure uniform flow Newark, NJ), and the thermocouples were chromel constantan across the entire soil cross-section. The thermo-tdr probe type. After the heaters were pulled into the rods and the was inserted into the soil horizontally from a precut slot, and thermocouples were placed at the midpoint length, high-ther- the space between the probe and the pipe was filled with wax. mal-conductivity epoxy was drawn into the rods to provide a Experiments were conducted in a constant temperature water-resistant, electrically insulated probe. See Ren et al. room (20 C). Each soil column was saturated by introducing (1999) for additional details regarding probe design and con- water at the bottom and then gradually increasing the hydraustruction. lic head of the water supply. Soil thermal properties were A heat pulse was generated by applying constant current determined in the absence of water flow (J 0). A nonlinear to the central heater for 15 s with a direct current supply regression method (Welch et al., 1996) was used to estimate (Model 1635, B & K-Precision, Maxtec International Corp.,, c, and from the temperature-by-time data. Chicago, IL). A datalogger (Model 21X, Campbell Scientific, A range of soil water fluxes was obtained by imposing Logan, UT) was used to control the heat input through a relay different hydraulic gradients. When flux was constant, a heat and record the upstream and downstream temperatures at 1-s pulse was applied and temperature-by-time data were reintervals for 120 s. The datalogger also was used to record corded. The same protocol was used to obtain measurements for all soil materials and water flux densities. Heat pulse duration was fixed at t 0 15 s, and measured heat inputs fell in the range q W m 1. 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 x d 5.75 mm and x u 6.01 mm for measurements in the sand and sandy loam, and x d 6.01 mm and x u 5.75 mm for measurements in the clay loam. Measurements of column outflow were used to determine J. Fig. 3. Maximum dimensionless temperature difference (MDTD, Eq. [21]) as a function of heat pulse velocity for a range of thermal diffusivities with x d x u m, and t 0 15 s. RESULTS AND DISCUSSION Thermal properties of the saturated soil materials were measured with J 0 (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

5 556 SOIL SCI. SOC. AM. J., VOL. 64, MARCH APRIL 2000 Fig. 4. Schematic view of the experimental setup (not drawn to scale). sandy loam clay loam. The thermal conductivity and diffusivity of quartz are known to be approximately twice those of other soil minerals. For all soil materials, temperature at the upstream and downstream positions rose rapidly in response to heating and then decreased gradually after reaching a maximum (Fig. 5). Differences between the temperature changes at the upstream and downstream positions for J 0 are due to the fact that there were slight differences in the upstream and downstream probe spacings (x u x d ). For example, in the sand with J 0 (Fig. 5a), the maximum change in dimensionless temperature was greater at the downstream position than at the upstream position. This resulted from the downstream probe spacing (x d 5.75 mm) being smaller than the upstream probe spacing (x u 6.01). Maximum changes in dimensionless temperature of 0.52 (sand), 0.45 (sandy loam), and 0.32 (clay loam) at the downstream position correspond to temperature changes of 1.04, 1.00, and Table 1. Soil physical properties. Bulk Coarse Fine Organic Soil Texture density Sand silt silt Clay matter Mg m 3 % Hanlon Sand Clarion Sandy loam Harps Clay loam 0.93 C, respectively. Comparing curves for J 0 with curves for J 0 reveals that water flow did not cause increased fluctuation in the temperature signals. As anticipated, increases in flux resulted in greater tempera- ture rises at the downstream position and smaller tem- perature rises at the upstream position for all soil materials. Temperature rises at both upstream and downstream positions also appeared to be proportional to water flux. Separation of the temperature curves indi- cates that distortion of the thermal field by water flow can be detected satisfactorily with the sensor arrangement provided by the thermo-tdr probe, at least for the range of soil water fluxes examined in this study. Inasmuch as differences in temperature rise appear to be greatest near the maxima of the curves, it seems 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). c Soil Measured Calculated MJ m 3 C 1 Wm 1 C m 2 s 1 Hanlon 3.07 (0.08) (0.02) 6.33 (0.06) Clarion 3.19 (0.03) (0.04) 5.54 (0.06) Harps 3.24 (0.09) (0.03) 4.23 (0.01) Means and standard deviations (in parentheses) of six observations three from upstream sensor and three from downstream sensor.

6 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.

7 558 SOIL SCI. SOC. AM. J., VOL. 64, MARCH APRIL 2000 Fig. 8. Transient dimensionless temperatures at the downstream (x d 6.01 mm) and upstream (x u 5.75 mm) positions as influenced by soil water flux for the clay loam. Symbols and lines indicate mea- sured and predicted values, respectively. Fig. 6. Transient dimensionless temperatures at the downstream (x d 5.75 mm) and upstream (x u 6.01 mm) positions as influenced by soil water flux for the sand. Symbols and lines indicate measured and predicted values, respectively. logical to use some measure of maximum temperature rise to estimate flux. Differences in the magnitude of temperature rise between soil materials at J 0 (Fig. 5) are direct results of differences in between soil materials (Table 2). Higher values resulted in greater temperature rises. Thus, maximum temperature rise alone, either at the upstream or downstream position, cannot be related Fig. 7. Transient dimensionless temperatures at the downstream (x d 5.75 mm) and upstream (x u 6.01 mm) positions as influenced by soil water flux for the sandy loam. Symbols and lines indicate measured and predicted values, respectively. uniquely to flux. As discussed earlier (see Fig. 1), the maximum of the difference between the temperature traces (MDTD) must be used to remove the influence of. 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 m 3 m 2 s 1 and for the clay loam with J m 3 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. The dimensionless temperature measurements pre-

8 REN ET AL.: DETERMINING SOIL WATER FLUX BY A HEAT PULSE TECHNIQUE 559 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. 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 mate- rial 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 4 T/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 condi- tion 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 Bryne 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 mi- nor 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 sented in Fig. 5 were used to compute MDTDs between the downstream and upstream positions, as a function of J (Fig. 9). It is encouraging that the relationships between flux and measured MDTDs are nearly linear for the three soil materials. Predicted MDTDs were computed as a function of V, heat pulse velocity, using Eq. [21] and estimated values of (Table 2). The results were plotted as a function of J (Fig. 9, lines) by using Eq. [3] to obtain J from V. Also required in Eq. [3] were ( c) 4.18 MJ m 3 C 1 and measured values of c (Table 2). Although reasonable agreement between measured and predicted MDTDs was observed for the sand, predicted MDTDs overestimated the measured MDTDs for the sandy loam and clay loam (Fig. 9). Melville et al. (1985) and Feldkamp (1996) also observed that theoretical results overpredicted the measured responses of heat-based groundwater flow sensors. Their analysis was based upon heat transfer models similar to the one presented herein. The discrepancies between predictions and measurements in Fig. 9 are directly related to the discrepancies noted in Fig. 7 (sandy loam) and Fig. 8 (clay loam). Lack of fit in Fig. 7 and 8 increased with increasing flux, and the same trend is observed in Fig. 9 for both soil materials. On the other hand, good agreement between predictions and measurements in Fig. 6 (sand) resulted in the reasonable agreement between predicted and measured MDTDs (Fig. 9). Despite the lack of universal agreement between measured and predicted MDTDs, the results of these initial experiments provide encouraging evidence that MDTD may prove to be a useful means of estimating soil water flux and pore water velocity. 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 x d x u. We used Eq. [21] to plot

9 560 SOIL SCI. SOC. AM. J., VOL. 64, MARCH APRIL 2000 lower limits of flux detection that can be achieved with and an assessment of the practical upper and lower this method. This is revealed by Eq. [3]. The value of limits for quantifying flux. c, which varies strongly with, will determine how the The proposed method improves upon the approaches upper and lower measurement limits of V will translate suggested by Bryne et al. (1967, 1968) by reducing disto upper and lower measurement limits for J. Failure tortion of the water flow field and minimizing heatof thermal homogeneity will determine the upper limit induced soil water redistribution. Inasmuch as soil ther- for measuring V. The lower limit of detection for V will mal properties can be measured with the thermo-tdr be determined by the resolution with which measurethermal properties) can be achieved easily in the ab- probe, it appears that calibration (determination of soil ments of MDTD can be obtained. Additional experisence of water flow. Thus, it may be possible to avoid mentation with a broader range of water fluxes will be required to define practical upper and lower limits; the calibration problems encountered with the instru- however, a simple analysis of Eq. [21] yields insight ments of Bryne et al. (1967, 1968). regarding the lower limit. A lower limit of V ms 1 is predicted if MDTD can be resolved to within REFERENCES 0.01 C. If MDTD can be resolved to within C, the Bilskie, J.R Dual probe methods for determining soil thermal lower limit decreases to V ms 1. These properties: Numerical and laboratory study. Ph.D. diss. Iowa State Univ., Ames (Diss. Abstr ). values of V correspond to fluxes of J and Bresler, E Simultaneous transport of solute and water under J ms 1, respectively, for the water-saturated transient unsaturated flow conditions. Water Resour. Res. clay loam with c 3.24 MJ m 3 C 1. And the lower 9: limits for J decrease slightly for unsaturated conditions Bristow, K.L., G.J. Kluitenberg, and R. Horton Measurement of soil thermal properties with a dual-probe heat-pulse technique. because of the dependence of c on (see Eq. [3]). Soil Sci. Soc. Am. J. 58: Byrne et al. (1967, 1968) reported a lower limit of J 1 Byrne, G.F., J.E. Drummond, and C.W. Rose A sensor for 10 6 ms 1 in water-saturated soil with their instruments. water flux in soil. Point source instrument. Water Resour. Melville et al. (1985) reported a lower limit of 3.5 Res. 3: Byrne, G.F., J.E. Drummond, and C.W. Rose A sensor for 10 7 ms 1 for the detection of groundwater seepage water flux in soil. 2. Line source instrument. Water Resour. velocity with their heat-pulse sensor. Our calculations Res. 4: and the results of Byrne et al. (1967, 1968) and Melville Campbell, G.S., C. Calissendorff, and J.H. Williams Probe for et al. (1985) suggest that heat-pulse methods may be measuring soil specific heat using a heat-pulse method. Soil Sci. Soc. Am. J. 55: useful for only a limited range of the water fluxes typi- Carslaw, H.S., and J.C. Jaeger Conduction of heat in solids. cally encountered in unsaturated soils. 2nd ed. Oxford Univ. Press, London. Cary, J.W Measuring unsaturated soil moisture flow with a meter. Soil Sci. Soc. Am. Proc. 34: SUMMARY AND CONCLUSIONS De Vries, D.A Thermal properties of soils. p In W.R. van Wijk (ed.) Physics of plant environment. North-Holland Publ. We have presented a method for determining soil Co., Amsterdam. water flux using the thermo-tdr of Ren et al. (1999). Dirksen, C A versatile soil water flux meter. p In Proc. 2nd Symp. on Fundamentals of Transport Phenomena in The thermo-tdr probe permits temperature measure- Porous Media, Vol. 2. IAHR, ISSS, Guelph, Ontario, Canada. ments at locations upstream and downstream from a Dirksen, C Field test of soil water flux meters. Trans. ASAE line heat source, which is used to emit a heat pulse 17: of finite duration. Measured temperatures are used to Feldkamp, J.R Theoretical analysis of an impervious, heated- cylinder groundwater velocimeter. Int. J. Numer. Anal. Methods compute the MDTD between upstream and down- Geomech. 20: stream locations. Theory was developed to relate Kluitenberg, G.J., K.L. Bristow, and B.S. Das Error analysis MDTD to soil water flux and pore water velocity. Inasand conductivity. Soil Sci. Soc. Am. J. 59: of heat pulse method for measuring soil heat capacity, diffusivity, much as the thermo-tdr probe can be used to make Marshall, D.C Measurement of sap flow in conifers by heat TDR-based measurements of volumetric water content, transport. Plant Physiol. 33: the proposed method also may permit measurement of Melville, J.G., F.J. Molz, and O. Güven Laboratory investigation pore water velocity. and analysis of a ground-water flowmeter. Ground Water 23: Experiments with three water-saturated soil materials Nielsen, D.R., J.W. Biggar, and K.T. Erh Spatial variability of and fluxes ranging from to m 3 field-measured soil-water properties. Hilgardia 42: m 2 s 1 revealed a nearly linear relationship between Noborio, K., K.J. McInnes, and J.L. Heilman Measurements measured MDTDs and fluxes. Reasonable agreement of soil water content, heat capacity, and thermal conductivity with was observed between measured and predicted MDTDs a single TDR probe. Soil Sci. 161: Ren, T., K. Noborio, and R. Horton Measuring soil water for a sand, but greater discrepancies occurred for a sandy content, electrical conductivity and thermal properties with a loam and a clay loam. For all soil materials, deviation thermo-tdr probe. Soil Sci. Soc. Am. J. 63: between predictions and measurements increased with Wagenet, R.J Water and solute flux. p In A. Klute increasing flux. Although further refinement of the pro- (ed.) Methods of soil analysis. Part 1. 2nd ed. Agron. Monogr. 9. ASA and SSSA, Madison, WI. posed method is needed, this initial experimental evi- Welch, S.M., G.J. Kluitenberg, and K.L. Bristow Rapid numeridence suggests that it holds promise. Further testing of cal estimation of soil thermal properties for a broad class of heat the method must include unsaturated flow conditions pulse emitter geometries. Meas. Sci. Technol. 7: