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

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 ) but these approaches can be time consuming, mathematwith a thermo-TDR (time domain reflectometry) probe. Constant 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, field around the instruments. Several limitations have and clay loam) with imposed water flux densities ranging from 1.16 3 prevented these instruments from being used as practi1025 to 6.31 3 1025 m3 m22 s21. A nearly linear relationship between cal tools for characterizing soil water flux. One limitameasured MDTDs and fluxes was observed for all soil materials. tion is that they require constant heat input for relatively Measured and predicted MDTDs agreed well for flow experiments 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 size of these instruments results in distortion of the probe can be used to make TDR-based measurements of volumetric soil water flow field in the vicinity of the instrument. water content (u), the proposed method also may permit measurement Experimental results showed poor agreement between of pore water velocity (J/u). 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 D water movement in soil is critical for al., 1968). managing irrigation and drainage and for characRecent developments in heat-pulse techniques for terizing chemical transport processes. Soil water flux measuring soil thermal properties suggest that the in(flux density) can be measured using a soil water flux struments developed by Byrne et al. (1967, 1968) can meter (e.g. Cary, 1970; Dirksen, 1972, 1974); however, be improved. Campbell et al. (1991) and Bristow et al. these meters are sophisticated and subject to problems, (1994) used heat-pulse sensors that employed a relaincluding the localized nature of the measurement, distively short (8 s) heating time. This approach for delivruption of the soil during installation, and interruption ering the heat impulse has been shown to cause minimal of normal patterns of soil water flow (Wagenet, 1986). soil water redistribution in unsaturated soil (Noborio et al., 1996; Bilskie, 1994). In addition, the sensors emT. Ren, Soil and Fertilizer Institute, Hebei Academy of Agricultural ployed by Campbell et al. (1991) and Bristow et al. Sciences, Shijiazhuang, Hebei 050051, 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 0.813 mm. A sensor with small needles Horton, Dep. of Agronomy, Iowa State Univ., Ames, IA 50011. Jourobviously would produce less distortion in the soil water nal Paper no. J-18360 of the Iowa Agric. and Home Econ. Exp. Stn., flow field than the instruments of Byrne et al. (1967, 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 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. 1999.*Corresponding author (rhorton @iastate.edu). Abbreviations: MDTD, maximum dimensionless temperature difference; TDR, time domain reflectometry. Published in Soil Sci. Soc. Am. J. 64:552–560 (2000). REN ET AL.: DETERMINING SOIL WATER FLUX BY A HEAT PULSE TECHNIQUE 553 rc is the volumetric heat capacity (J m2 C2) of the multiphase thermo-TDR probe that permits simultaneous measuresystem, (rc), is the volumetric heat capacity of the liquid, and ment of soil water content, electrical conductivity, theru is the volumetric liquid content of the medium. mal conductivity, thermal diffusivity, and volumetric This formulation requires the assumption that conductive heat capacity. Their probe combines the TDR method 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 V 5 uVw (rc), rc 5 J (rc), rc [3] may provide a means of measuring soil water flux density. 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, 1958). The heat velocity lags behind the front of the liquid of soil water flow, the flow field will distort the temperaphase because, under the assumption of thermal homogeneity, ture responses observed at the outer (now upstream heat from the liquid phase is absorbed instantaneously by the and downstream) needles. This temperature asymmetry porous medium at the thermal front (Melville et al., 1985). 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 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) 5 Q 4pat exp 12 2 1 y 2 4at 2 [4] relating water flux density (J) and pore water velocity (Vw) to the measurements obtained with a thermo-TDR for a line source parallel to the z-axis and located at (x, y) 5 probe, and (iii) provide an experimental evaluation of (0, 0). The source strength (m 8C), Q, is defined as the proposed method. Laboratory experiments were conducted with soil materials of different textures and Q 5 q rc [5] a range of imposed water flux densities. The parameter estimation method presented herein is similar to a where q is the heat input per unit length (J m2). 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 dt9 at time t9, qdt9 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 qdt9 released per unit THEORY length at t9 is Heat Transfer Equations qd t9 4pl( t 2 t 9) exp 52 [x 2 V(t 2 t 9)]2 1 y 2 4a(t 2 t 9) 6 [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 l 5 rca is the thermal conductivity (W m2 8C2). p. 13) Next, we consider heating of the infinite line source at the rate q9d t9 over the interval 0 , t # t0. Here, q9 is the heat input per unit length per unit time (W m2), and the corresponding ]T ]t 5 a9 1 2T ]x2 1 ]2T ]y2 1 ]2T ]z2 2 2 U ]T ]x [1] source strength becomes Q9 (m 8C s2). Integrating Eq. [6] (Carslaw and Jaeger, 1959, p. 261) and making use of the where T is temperature (8C); t is time (s); a9 is thermal diffusivsubstitution s 5 (t 2 t9) yields a general solution for the ity (m s2); U is velocity (m s2); 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 multiphase system, for example, an incompressible porous T(x, y, t) 5 5 y, t); 0 , t # t0 T2(x, y, t); t . t0 [7] medium with a liquid moving uniformly through it with pore water velocity Vw, Eq. [1] becomes where