soil water movement in response to temperature gradients: experimental measurements and model...

Soil Water Movement in Response to Temperature Gradients:Experimental Measurements and Model Evaluation

L. B. Bach*ABSTRACT

Temperature gradients may have a significant effect on soil watermovement under certain conditions, but inclusion of these effects addscomplexity to the flow analysis. This study was conducted to helpclarify the significance of nonisotbermal water flow, and to examinetheoretical and numerical descriptions of the transport processes. Atinitial water contents of 0.00, 0.049, 0.099, 0.151, and 0.282 m3 m~\isothermal and nonisothermal laboratory experiments were conductedto provide direct information on soil water movement in response totemperature gradients. These data were used to evaluate the numer-ical simulation model SPLaSHWaTr2, and to examine calculation ofthe thermal conductivity, \(0), the thermal vapor diffusivity, DTv(8),and the temperature coefficient of the matric potential, C$, based onmodifications to a theory proposed by Philip and de Vries in 1957. Astatistically significant effect of the temperature gradient was foundat an initial water content of 0.151 m3 m~3, which corresponds to apressure head of approximately —1.2 m of H2O. No effect of thetemperature gradient was found at initial water contents of 0.00, 0.049,0.099, or 0.282 m3 m-3. Under isothermal conditions, the model pro-vided simulated water-content profiles that were in good agreementwith measured profiles. Under nonisothermal conditions, profiles sim-ulated by the model were in poor agreement with the measured data,using the original values of X(0), £>Tv(9), and C+. Sensitivity analysisshowed that A(B) and OTv(6) bad a negligible influence on nonisoth-ermal water movement. On the other hand, C$ had a significant in-fluence on nonisothermal water movement. Adjusting Q, from theoriginal value of -0.0068 K ' to the value suggested by Philip andde Vries, —0.00209 K ', improved the agreement between simulatedand measured water-content profiles, particularly at an initial watercontent of 0.282 m3 m~3. With the adjustment in Q,, the model sim-ulations were in good agreement with the measured data, indicatingthat the Philip and de Vries theory provides an adequate descriptionof the nonisothermal transport processes.

SOIL WATER MOVEMENT in unsaturated soils hasbeen extensively studied for many years. Most of

the studies have been conducted under isothermal con-ditions, which neglect water movement in response totemperature gradients. In arid and semiarid regions,where temperature gradients can be very large, trans-fer of water from warmer to cooler regions may havea significant effect on soil water status, especially innear-surface horizons. Prediction of water content forthese areas may be in error if the effects of tempera-ture gradients are not considered. However, incorpo-ration of temperature components into theoretical ornumerical models greatly complicates analysis. In someinstances, neglecting thermal effects may not causesignificant errors in estimation of soil water move-ment. Additional experimental data is needed to helpdetermine the conditions under which temperaturegradients are important in soil water movement. At

U.S. Forest Service Umatilla National Forest, 2517 SW Hailey,Pendleton, OR 97801. Research completed at USDA-ARS South-west Watershed Research, Tucson, AZ. Received 27 Feb. 1991."Corresponding author.

Published in Soil Sci. Soc. Am. J. 56:37^16 (1992).

the same time, theoretical and numerical models thatincorporate simultaneous water movement and heatflow must be tested and validated with experimentaldata.

Relatively few studies have been conducted to di-rectly measure soil water movement in response totemperature gradients. Laboratory studies by Gurr etal. (1952), Rollins et al. (1954), and Taylor and Cav-azza (1954) demonstrated that, when a closed soil col-umn is subjected to a temperature gradient, water flowfrom warm to cool regions occurs largely in the vaporphase. Condensation at the cold end results in liquidwater flow back to the warm regions due to the mois-ture gradient that develops. Hutcheon (1958) foundthat the net movement of moisture due to a tempera-ture gradient was greatest at intermediate initial watercontents, e.g., values that were 50 to 60% of the watercontent corresponding to a pressure head of —1.0 mof H2O. Similar studies by Cassel et al. (1969) andJoshua and de Jong (1973) also found that the effectof a temperature gradient on soil water movement wasgreatest at intermediate water contents, values corre-sponding to pressure heads between — 10 and — 150m of H2O. All the above studies employed closed soilcolumns, in which there was no flux of water into orout of the column.

Philip and de Vries (1957) and de Vries (1958)developed a theory to describe water and heat move-ment in porous media under nonisothermal conditions.Their theory accounted for water flow in the liquidand vapor phases in response to both water contentand temperature gradients. The Philip and de Vriesapproach is based on the simultaneous solution of twononlinear partial differential equations containing fourmoisture- and temperature-dependent diffusion coef-ficients. Working with laboratory soil columns, Gee(1966) found that the net fluxes of water calculatedwith this theory were two to three times smaller thanthe observed net fluxes for a silt loam soil at inter-mediate soil water contents, while Cassel et al. (1969)found that the calculated net fluxes of water were ingeneral agreement with the observed net fluxes for afine sandy loam soil at low soil water contents. Jack-son et al. (1974) evaluated the theory on a clay loamsoil under field conditions and found that it adequatelypredicted the measured water fluxes at intermediatesoil water contents, while the isothermal theory betterpredicted the measured water fluxes in both wet anddry soils.

Milly and Eagleson (1980, 1982) and Milly (1982,1984) developed a comprehensive numerical modelfor simultaneous moisture and heat flow using a ma-tric-head-based equation formulation instead of thewater-content-based formulation of Philip and de Vries.They tested the model against analytical solutions andmeasured data for several simplified cases and foundthat the model performed well under these conditions.However, no tests were performed in which the fullmoisture- and heat-flow model was compared with

37

38 SOIL SCI. SOC. AM. J., VOL. 56, JANUARY-FEBRUARY 1992

experimental data that measured water movement un-der the influence of both water-content and tempera-ture gradients.

Additional experimental data is needed to help clar-ify the importance of soil water movement in responseto temperature gradients. The majority of laboratoryexperiments were conducted using closed soil col-umns, in which there was no water movement into orout of the columns. These studies indicate that tem-perature gradients are significant at intermediate initialwater contents, while other studies indicate that tem-perature gradients are important under moist condi-tions (Gary, 1965), or only under dry conditions (Philip,1957). In addition, many numerical models incorpo-rate simultaneous moisture- and heat-flow equations,with little testing to indicate the conditions under whichnonisothermal moisture flow will be significant in themodeling process. Most of these models employ thePhilip and de Vries formulation, which contains stor-age and transport coefficients that are not accuratelyknown, and in some cases require assumptions aboutthe values of the parameters required for their calcu-lation.

The objectives of this study were to (i) evaluate thesignificance of nonisothermal soil water movementusing open laboratory columns, in which a flux ofwater is applied at the same time and in the samedirection as the temperature gradient, and (ii) validatea numerical simulation model for simultaneous mois-ture and heat flow, and examine calculation of noni-sothermal water movement by the Philip and de Vriestheory. The use of laboratory experimental data allowsseparation of isothermal and nonisothermal flux com-ponents, thereby providing direct information on theconditions under which temperature gradients are im-portant in soil water movement. Comparison of themodel output with measured data provides a means ofexamining the theoretical calculation of the storageand transport coefficients by evaluating changes intheir parameter values via sensitivity analysis.

MATERIALS AND METHODSSoil moisture changes due to imposed temperature gra-

dients were directly measured at initial water contents of0.00, 0.049, 0.099, 0.151, and 0.282 m3 m-3. At eachinitial water content, two experimental tests were con-ducted, one under isothermal conditions, and one with animposed temperature gradient. A steady flux of water wasapplied to one end of all 10 soil columns. The experimentswere conducted with the columns in a horizontal positionso that gravity effects were negligible.

Otero sandy loam soil (coarse-loamy, mixed [calcar-eous], mesic Ustic Torriorthent) was packed into polycar-bonate columns with inside dimensions 5 by 5 by 30 cm(Fig. la). Square columns were used to ensure a set pathlength for passage of gamma radiation in the measurementof volumetric soil moisture, as discussed below. Holes weredrilled in the top of the columns at specific intervals toaccept thermocouple wire for measuring soil temperature.Each column had a removable side plate to facilitate soilpacking, and removable end plates to allow placement ofhot and cold temperature controls. The sides and ends ofthe column were fitted with rubber gaskets to prevent waterleakage. All columns were packed to a uniform bulk densityand initial water content (Bach, 1989).

To control ambient temperature conditions during an ex-

a)

0.05m

THERMOCOUPLEHOLES

THREADEDSCREW HOLES

REMOVABLE SIDE PLATENOTE: END PLATES NOT SHOWN

b) STYROFOAMINSULATION

COPPERMANIFOLD

BEADSTYROFOAM

TUBING TOCONSTANT

TEMPERATUREBATH

COARSE SAND

REMOVABLE/" COVER

Fig. 1. Schematics of (a) the experimental soil column and (b)the constant-temperature box used to create isothermal andnonisothermal conditions for t tie experiment.

perimental run, the columns were placed in a 45 by 30 by10 cm constant-temperature box built from 0.635-cmPlexiglas1 (Fig. Ib). Temperature control in the box wasaccomplished by circulating water from a constant-temper-ature bath through a copper manifold located in the bottomof the box. Uniform coarse sand covered the manifold andextended up to the soil column in order to provide goodthermal contact between the manifold and the soil. The areaabove the column was insulated with bead Styrofoam, andthe outside of the box was insulated with 2.54-cm Styro-foam, except for a narrow slit required for passage of thegamma beam used for water-content measurement. The col-umns were placed in the constant-temperature box at least24 h prior to the start of an experimental run to ensure thatthey had reached a constant uniform initial temperature.

A steady flux of water was applied to one end of the soilcolumns using a syringe injection pump. A small hole wasprovided in the center of the inflow end plate to allowaccess for the needle from the syringe pump. The flow ratefrom the pump was held constant for both the isothermaland nonisothermal run at each initial water content (Table1). The rate was chosen so that several days would berequired for the wetting front to reach the end of the soilcolumn, thus allowing sufficieni: time for measuring water-content changes at each position in the soil column. Sincethe syringe pump was located outside of the constant-tem-perature box, a small hole was drilled in the box to allow

'Trade names used in this paper are included for information only,and do not constitute endorsement by the author or the USDA.

BACH: SOIL WATER MOVEMENT IN RESPONSE TO TEMPERATURE GRADIENTS 39

Table 1. Experimental conditions for isothermal (I) and nonisothermal (NI) water movement through soil columns at five differentinitial water contents.

0.000 m3m-3

Water Flux (cm h-1)Temperature (°C)

Average initialHot endCold end

I0.085

23.45

NI0.085

23.4935.518.5

0.049 m'm-3

I0.091

23.68

NI0.091

23.5535.518.5

0.099 m3m~3 0.151 m3m-3

I0.094

23.42

NI I

0.094 0.094

23.07 24.0035.524.0

NI0.094

23.8835.524.0

0.282 m'rn-3

I0.092

23.55

NI0.092

23.1035.524.0

access for the syringe needle. The hole was sealed withspray foam insulation, and the temperature of the injectedwater was maintained at the ambient temperature of theconstant-temperature box. Comparisons of the measuredwater contents at the end of each experiment with the totalamount of water added by the syringe pump indicated thatthere were no losses of water during the experimental runs.Each run was terminated when the wetting front reached0.283 m, thereby providing a zero-flux bottom boundarycondition for the model simulations.

A temperature gradient was applied to the soil columnsduring the nonisothermal runs, using specially designedcolumn end plates. The gradient was applied in the samedirection and was initiated at the same time as the waterflux from the syringe injection pump. The hot end wasconstructed using a series of 10-W resistors attached to aBakelite plate that was covered with brass on the side incontact with the soil. The temperature of the resistors wascontrolled by a proportional temperature controller that re-sponded to a thermocouple located at the interface betweenthe plate and the soil. The cold end was constructed usinga brass plate and polyvinyl chloride reservoir in which waterfrom a constant-temperature bath was continuously circu-lated. A small pin hole was drilled in this plate to maintainatmospheric pressure in the columns. The desired temper-ature gradient for this study was 0.5 °C cm-1; however,mechanical difficulties with the output display on the con-stant-temperature bath caused an error in the setting for thecold end. Thus, the applied gradients were slightly greaterthan 0.5 °C cm-1 for initial water contents of 0.00 and0.049 m3 m-3, and slightly less than 0.5 °C cm-1 for theother initial water contents (Table 1). All columns wereinsulated with 0.013 m of Styrofoam to reduce heat lossfrom the sides and top.

Volumetric water content was measured at 0.01-m inter-vals along the column using the gamma-attenuation method.Error analysis showed that one standard deviation of themeasured gamma counts yielded a probable error of ap-proximately ±0.006 m3 m-3 in the estimate of volumetricwater content, while two standard deviations yielded aprobable error of ± 0.01 m3 m-3. Since the column packingmethod required addition of water to the soil as it waspacked, this single-source gamma apparatus could not pro-vide dry soil count rates. Instead, the initial count rate througheach column after packing was used as a starting measure-ment, and all water-content calculations were made as achange from the initial value. The initial count rate wasalso used to evaluate the uniformity of soil packing bycomparing the count rate at each measurement location.This procedure indicated that the uniformity of packing be-tween the isothermal and nonisothermal soil columns at thesame initial water content was quite good. In addition, theaverage initial temperature and applied water flux for eachset of paired columns were very close (Table 1), indicatingthat the columns were hydraulically similar prior to onsetof the experiments.

Soil temperature was measured using 0.051-cm (24-gauge)copper-constantan thermocouples that were placed in thecolumns at distances of 0.005, 0.065, 0.125, 0.185, 0.245,

and 0.295 m from the water-injection end. Wires were con-nected to a Campbell Scientific data logger (Campbell Sci-entific, Logan, UT), which recorded the temperatures hourly.To improve the accuracy of the temperature measurements,the thermocouples were constructed to read differentialtemperature, using the 0.005-m thermocouple as a refer-ence. The absolute temperature of the 0.005-m thermocou-ple was measured using the data-logger reference junction,and the differential temperature of all other locations wasconverted to absolute temperature using this value. Themeasurement error for absolute temperature was <0.5 °Cfor all experimental runs.

Hydraulic properties of the Otero sandy loam were de-termined in the laboratory for use in estimating the param-eters of the storage and transport coefficients, as well as toprovide inputs to the numerical model. The moisture-re-lease curve was determined at 20 °C using a pressure-plateapparatus (Klute, 1986). Particle-size analysis was basedon USDA size specifications, with silt and clay determinedby the hydrometer method (Gee and Bauder, 1986). Totalamount of sand, silt, and clay was 75, 9, and 16% of thesolid fraction, respectively. Saturated hydraulic conductiv-ity, measured in soil columns using a constant-head method(Klute and Dirksen, 1986), was 0.011 mm s-1. Particledensity (Blake and Hartge, 1986) and organic matter (Nel-son and Sommers, 1982) were 2.66 Mg m-3 and 0.0112kg kg-1, respectively. Saturated water content of the Oterosandy loam soil was 0.47 m3 m~3.

MODEL DESCRIPTION

Governing EquationsThe numerical simulation model employed in this study

is SPLaSHWaTr2, which provides a one-dimensional so-lution to the coupled water- and heat-flow equations pre-sented by Milly and Eagleson (1980, 1982) and Milly (1982,1984). The one-dimensional equation for nonisothermal watermovement is

Pl dtar

where pv is the density of water vapor (kg m-3), p, is thedensity of liquid water (kg m-3), 6 is the volumetric watercontent (m3 m-3), fy is the matric potential (m), 6a is thevolumetric air content (m3 m~3), T is the temperature (K),t is the time (s), x is the horizontal distance (m), K is thehydraulic conductivity (m s-1), D^, is the isothermal vapordiffusivity (m S"1), and D-^ is the thermal vapor diffusivity(m2 s-1 K-1). Gravity effects have been neglected to beconsistent with the experimental procedure. In this equa-tion, K is associated with movement of liquid water due toa matric-potential gradient, D^, is associated with move-

40 SOIL SCI. SOC. AM. L, VOL. 56, JANUARY-FEBRUARY 1992

ment of water vapor due to a matric-potential gradient, andDTV is associated with movement of water vapor due to atemperature gradient. Movement of liquid water due to atemperature gradient is implicitly represented through thetemperature coefficient of the matric potential, Qj, (K-1),which accounts for the fact that the 6(4») curve is temper-ature dependent.

The one-dimensional equation for heat flow is

where the quantities H^ and H2 are given byHt = [L0 + cf (T - ro)]6a

H2 = c,Pl (T - T0) - CpPv (T - T0) - ft,W - LoPv][2b]

and C is the volumetric heat capacity (J m-3 K-1), X is thethermal conductivity of the medium enhanced by vapor dis-tillation in the air-filled pores (W m-1 K-1), L is the latentheat of vaporization of water (J kg-1), c, is the specific heatof liquid water (J kg-1 K-1), qm is the total mass flux ofwater (kg m-2 s-1) L0 is the value of L at an arbitraryreference temperature T0 (K), cp is the specific heat of watervapor at constant pressure (J kg-1 K-1), and W is the dif-ferential heat of wetting (J kg-1).

Storage and Transport CoefficientsThe moisture-retention curve, 9(i|»), was mathematically

described using the relation proposed by Brooks and Corey(1966). For the Otero sandy loam soil, a modification oftheir procedure (Bach, 1989) provided the best fit to theexperimental data relative to several alternative fitting pro-cedures. Figure 2 shows a good fit between the measuredand calculated values of the moisture-retention curve at 20 °C.

The moisture-retention curve is generalized for nonisoth-ermal conditions by the relation

where i|»e is the effective \\i, which adjusts 4» at any temper-ature r to the temperature at which the 9(i|;) curve wasdetermined, T0. The calculated value of J;. is used in the

0.25

0.20

0.15 -

0.10 -

0.05 -

0.00

• MEASURED

CALCULATED

-10 -100

• ( - (m)Fig. 2. Measured and calculated values of the moisture-retention

function for Otero sandy loam at 20 °C, where 6 is thevolumetric water content and t|* is the matric potential.

moisture-retention curve to determine 9. The variable Q, isthe temperature coefficient of the matric potential, whichis defined by Philip and de Vries (1957) using the surface-tension model of temperature dependence

r - ] d(T

* ~ ~ ATcr AT [4]

where a is the surface tension of water (J m-2). There isconsiderable discussion in the literature as to the validityof Eq. [4]. Several investigators have found that the tem-perature dependence of t|; was two to 10 times greater thanthat predicted by the surface-tension relationship (Gardner,1955; Taylor and Stewart, 1960; Wilkinson and Klute, 1962;Jury and Miller, 1974). To account for this, the SPLaSH-WaTr2 model uses a value of -6.8 x 10~3 K-1 for C+,which is more than three times the value predicted by thesurface-tension model.

The hydraulic conductivity (K) is expressed as(9) [5]

where Ks is the measured value: of K at natural saturation,Kr is the hydraulic conductivity relative to its value at sat-uration, and KT is a temperature correction for nonisoth-ermal conditions. In this equation K, (6) is expressed aseffective water content, 6f, raised to a power, e. Since theK(Q) curve was not experimentally measured, the value ofe was determined via model calibration under isothermalconditions, at an initial water content of 0.099 m3 m-3.The best-fit e value under these conditions was 5.3. Thisvalue was then used for calculation of the hydraulic-con-ductivity function for all other initial water contents underboth isothermal and nonisotherrnal conditions.

The isothermal vapor diffusivity is given by (Milly, 1984)

[6]

where Z>atm is the molecular diffusivity of water vapor inair (m2 s-1), v is the mass-flow factor, and a is the tor-tuosity of the air-filled pore space.

The thermal vapor diffusivity is given by (Milly, 1984)

where

/ = i/ = n - 9,,

e < ek

ek < e

[7a]

[7b]

n is the porosity, and 6k is the moisture content at whichliquid flow becomes negligible. The parameter t, is the ratioof the average temperature gradient in the air-filled poresto the overall average temperature gradient. Its evaluationis discussed by Milly (1984).

The thermal conductivity is determined according to themethod of de Vries (1966), which requires estimates of thevolumetric fractions and thermal nxmductivities of the water,air, quartz, other minerals, and organic-matter constituents.Direct measurements of the volumetric fraction of quartzare difficult to obtain. Many studies that have been con-ducted to test the de Vries theory have relied on estimatedvalues for this parameter. Since i:he volumetric fractions ofthe soil constituents are user-supplied inputs to theSPLaSHWaTr2 model, the volumetric fraction of quartzwas estimated as the sand fraction of the soil, as was doneby Hadas (1969). The volumetric fraction of organic matterwas measured in the laboratory, and the volumetric fraction


of other minerals was obtained by difference. The valuesfor the Otero sandy loam soil are: quartz, 0.398; organicmatter, 0.012; and minerals, 0.121. Values of the thermalconductivity for each constituent were taken from de Vries(1966).

Model SimulationsThe finite-element solution algorithm employed in the

SPLaSHWaTr2 model is described by Milly (1982). The0.30-m-long soil column was divided into 120 model ele-ments, with nodes located at data-measurement points. In-itial conditions for both isothermal and nonisothermalsimulations were

T(x,0) = T0

where \\i0 is the initial matric potential, determined fromthe initial water content using the calculated 8(>J>) relation-ship, and T0 is the measured average initial temperature ofthe soil column (Table 1). Boundary conditions on moisturefor both isothermal and nonisothermal simulations were aconstant flux at the water-injection end of the column (Ta-ble 1), and a zero water flux at the other end. Boundaryconditions on temperature for the nonisothermal simulationswere specified as the values of the hot and cold end plates.Model outputs of water and temperature distributions werecompared with the measured data at initial water contentsof 0.052, 0.099, 0.151, and 0.282 m3 m-3. Initial watercontents of 0.00 and 0.049 m3 m~3 could not be simulatedbecause the model cannot compute the 6(i|<) relationship forvalues of 6 less than residual saturation, which is 0.05 m3

m-3 for the Otero sandy loam. Since 0.052 m3 m~3 wasthe lowest feasible value for model simulation, this valuewas compared with the experimental data at 0.049 m3 m~3.

RESULTS AND DISCUSSIONTemperature and water-content measurements were

obtained at a number of different times following ap-plication of the boundary conditions. Figure 3a showsthe final temperature distribution for the five isother-mal runs. The largest temperature difference withinany of the four initially moist soil columns was 0.20 °C,indicating that isothermal conditions were achieved.The initially dry soil column (0.00 m3 m-3) exhibitedslight temperature changes due to the heat of wetting,but these changes caused differences no greater than0.65 °C within the soil column. Figure 3b shows thefinal temperature distribution for the five nonisother-mal runs. For these runs, a steady temperature wasreached within 10 h at all thermocouple measurementlocations. The top three curves had cold-end temper-atures of 24 °C, while the lower two curves had cold-end temperatures of 18.5 °C, as discussed above. Thecurves show that the experiments conducted at initialwater contents of 0.099, 0.151, and 0.282 m3 m-3

resulted in similar temperature gradients, as did theexperiments conducted at initial water contents of 0.00and 0.049 m3 m-3.

The shape of the nonisothermal temperature curvesin Fig. 3b indicates that heat flow within the soil col-umns was not one dimensional, as was assumed inthe SPLaSHWaTr2 model. Figure 4 compares themeasured final temperature distribution within the soilcolumn with the final temperature distributions cal-culated from the SPLaSHWaTr2 model and from a

36

34 -

32 -

30 -

28

26 -

24

22 -

20 -

-6-8,-B- 6

o.ooo0.049

-A- 9^0.099-•- 8.= 0.151

-m- B= 0.282

a)

LLJ<r

iHIQ.

HI

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00 0.05 0.10 0.15 0.20 0.25

DISTANCE ALONG COLUMN (m)

0.30

Fig. 3. Final temperature distribution for (a) the five isothermalruns and (b) the five nonisothermal runs, where 6, is theinitial water content.

two-dimensional heat-flow model for an initial watercontent of 0.099 m3 m-3. The SPLaSHWaTr2 modelincludes secondary terms representing latent and sen-sible heat transfer, while the two-dimensional modelincludes heat transfer due to conduction alone. It canbe seen that the temperature data are very poorly rep-

oomEC

HIQ_

36

34

32

30

28

26

24

22

20

18

— 1-D SOLUTION-- 2-D SOLUTIONO MEASURED DATA

O ~

O

0.00 0.05 0.10 0.15 0.20 0.25 0.30

DISTANCE ALONG COLUMN (m)Fig. 4. Measured and simulated one- and two-dimensional final

temperature distributions for an initial water content of 0.099m3 m"3.


resented by the one-dimensional equation, while thetwo-dimensional equation provides a reasonable fit tothe measured data. A closer fit might be obtained byincluding the secondary heat-flow terms, however so-lution of the equation becomes more difficult. Figure4 also shows that there are differences in the gradientsof temperature throughout the column for one- vs.two-dimensional heat flow. In general, the one-di-mensional temperature gradient is smaller at the hotend of the soil column, and larger at the cold end ofthe soil column, than the two-dimensional gradient.This fact will be considered when evaluating the non-isothermal model predictions.

Figures 5 through 8 show measured and predictedwater-content distributions for initial water contentsof 0.049, 0.099, 0.151, and 0.282 m3

m-3 at timeswhen the wetting front had reached approximately two-thirds of the distance from the inflow end of the col-umn to the outflow end. For all nonisothermal exper-iments, the hot end of the column is at x = 0.0 andthe cold end is at* = 0.3 m. In Fig. 5 through 8, thechange in water content from the initial value is plot-ted on the ordinate, as it was this value that was ob-tained from the gamma apparatus.

The expected effect of the applied temperature gra-dient is to increase the rate of water movement in thenonisothermal columns relative to the rate in the iso-thermal columns. Since the initial water content and

applied water flux are constant for the two columns,the increased rate of water movement will serve toreduce the water content at the hot end and increaseit at the cold end relative to i:he values measured underisothermal conditions. For columns at initial watercontents of 0.00 (not shown), 0.049 m3 m~3 (Fig. 5a),and 0.099 m3 m-3 (Fig. 6a) there was very little effectof the temperature gradient on soil water movement.For the columns at an initial water content of 0.151m3 m-3, the experimental data indicate that there wassome effect of the applied temperature gradient on soilwater movement. At 12 h (Fig. 7a), the wetting fronthad reached a distance of 0.193 m in the nonisother-mal column, compared with 0.153 m in the isothermalcolumn. Similar results were found at other measure-ment times; for example, at 6 h (not shown), the wet-ting front had reached a distance of 0.133 m in thenonisothermal column, compared with 0.093 m in theisothermal column. For the columns at an initial watercontent of 0.282 m3 m-3, the data do not show adistinct wetting front as do the plots for the otherinitial water contents (Fig. 8a). Apparently, at thisinitial water content, the hydraulic conductivity washigh enough that the wetting front reached the end ofthe column quite rapidly, and the water content roseuniformly after that time. Very little effect of the ap-plied temperature gradient can be observed in the graphsat this initial water content.

E,CO

UICD

0.24 -0.20 -0.16 -0.12 -0.080.04 -

00.280.24 -0.20 -0.160.12 -0.08 -0.04

0

O ISOTHERMAL MEASURED

• NON-ISOTHERMAL MEASURED

ISOTHERMAL PREDICTED

0.24 -

0.20 -0.16 -0.12 -

0.08 -0.04 -

n

--- NON-ISOTHERMAL PREDICTED

***--•••.••••""--•*•,

V v

\»c)

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

DISTANCE ALONG COLUMN (m)

Fig. 5. Water-content (0) distributions for an initial watercontent of 0.049 m3 m~3 (a) measured under isothermal andnonisothermal conditions, (b) measured and predicted underisothermal conditions, and (c) measured and predicted undernonisothermal conditions.

U.£O

0.24 -

0.20 -0.16 -0.12 -0.08 -0.04 -

0

0.24 -0.20 -0.16 -0.12 -0.08 -0.04 -

n

3) O ISOTHERMAL MEASURED


'*§»•••§£

° *8**§gg•

«b) - —— ISOTHERMAL PREDICTED

^5^0©— ©Q,-,

^^®KN

0.24 -

0.20 :

0.16 -0.12 -0.08 -0.04 -

0

C) --- NON-ISOTHERMAL PREDICTED

**-*•••%

***%%

•

. . . . . . . . •mmm. .0.00 0.04 0.08 0.1:2 0.16 0.20 0.24 0.28

DISTANCE ALONG COLUMN (m)Fig. 6. Water-content (0) distributions for an initial water

content of 0.099 m3 m-3 (a) measured under isothermal andnonisothermal conditions, (b) measured and predicted underisothermal conditions, and (c) measured and predicted undernonisothermal conditions.


Statistical analysis was performed to determine ifthe effect of the temperature gradient was significantat an initial water content of 0.151 m3 m~3. For thedata collected in this study, which were single valuesof water content at various positions in the soil col-umn, traditional analysis of variance techniques arenot valid (J. Zumbrunnen, Colorado State Univ., 1987,personal communication). In order to analyze this data,each graph of water content as a function of distancewas fitted with a quadratic function. At each selectedtime, the isothermal graph was compared with thenonisothermal graph for differences in their interceptand linear and quadratic terms; statistical differencesin the terms indicate a significant temperature effect.According to this analysis, the effect of the tempera-ture gradient on soil water movement was significantat both 6 and 12 h at an initial water content of 0.151m3 m~3. Additionally, this analysis confirmed thatthere were no differences between the isothermal andnonisothermal experimental graphs at initial watercontents of 0.00, 0.049, 0.099, and 0.282 m3 nr3.

Under isothermal conditions the model simulationsare in general agreement with the experimental data(Fig. 5b-8b). The poorest fits under isothermal con-ditions are found for the simulations conducted at 0.052m3 m-3 (and compared with measurements at 0.049m3 m~3), where the model calculates less water move-ment due to the temperature gradient than is indicated

by the experimental data. Some differences betweenthe simulated and measured values can also be ob-served at an initial water content of 0.099 m3 m~3,where the model does not predict the rapid reductionin water content at the wetting front. Under isothermalconditions, the hydraulic conductivity, K(Q), and theisothermal vapor diffusivity, D^ (0), are the twotransport coefficients that influence soil water move-ment. For an initial water content of 0.049 m3 m~3,comparison of the original simulations with ones inwhich D^ was set to zero showed that the water-content calculations changed by < 0.003 m3 m~3 forall positions within the soil column. For the columnas a whole, the new simulations yielded a change of<2% from the original simulations, indicating that theeffect of D^, on soil water movement was negligiblefor this initial water content under the current set ofexperimental conditions. In addition, for the Oterosandy loam soil, values of D^, are negligible com-pared with values of K for water contents > 0.065 m3

m~3; therefore, this parameter would not have anyeffect on model calculations for an initial water con-tent of 0.099 m3 m~3. This indicates that the differ-ences between the model simulations and the measureddata under isothermal conditions may be due to theapproximate shape of the calculated hydraulic-con-ductivity function. The calibration procedure for K =8e

£ fit the average conductivity in the soil column.

E.CD

o

u.ô

0.24 -0.20 -0.16 -0.12 -0.08 -0.04 -

0

0.280.24 -

0.20 -0.16 -

0.12 -0.08 -0.04 -

n

a) O ISOTHERMAL MEASURED


aa888»»8o*»s«««.

b) —— ISOTHERMAL PREDICTED

OrT)^ 7==^c*yic=>^C>©l<V-\

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0.24 -0.20 -0.16 -0.12 -0.08 -0.04 -

0 , -,- , , - , , -n —— i —— i —— i —— i —— ,--

C) --- NON-ISOTHERMAL PREDICTED

*****-**«•

"*"""*••*^0^

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.1


content of 0.151 m3 m~3 (a) measured under isothermal andnonisothermal conditions, (b) measured and predicted underisothermal conditions, and (c) measured and predicted undernonisothermal conditions.

<rem

ECD

zUJC3Z

i

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0.24 -0.20 -0.16 -0.12 -0.08 -0.04 -

0n ORU.£O

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0 "n OftU.&o

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a) O ISOTHERMAL MEASURED


8ft§J'teftM*8«iQo ••&£ fi &•••™™O™Ô O (^•"' ^••••w* ™

b) —— ISOTHERMAL PREDICTED

rr-—-— -i i ,-i ooo^Qnn no o rsO^ O '-'O (j ^^ O—G —— Q — " —— D

C) —— NON-ISOTHERMAL PREDICTED

***•**_ -**«****^ i*»0***0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28


content of 0.282 m3 m"3 (a) measured under isothermal andnonisothermal conditions, (b) measured and predicted underisothermal conditions, and (c) measured and predicted undernonisothermal conditions.


2.4

2.0 -

j~ 1.6 -*

ov 1.2

0.8

0.4 -

0.0

VOL QUARTZ = 0.3975;8K = 0.0543

VOL QUARTZ = 0.1987;0 = 0.0543

VOL QUARTZ = 0.3975;8 = 0.0871

K

0.0 0.1 0.2 0.3 0.4 0.5

3 -38 (m m )

Fig. 9. Thermal conductivity (\) as a function of water content(0) for changes in the values of the moisture content at whichliquid flow becomes negligible (8J and the volumetric fractionof quartz.

illO

1O

0.24 -

0.20 -

0.16 -

0.12 -

0.08 -

0.04 -

n

- - MODEL RESULTS; C ,= - 2.09 X 1 o"* -3

• • • • • • • • MODEL RESULTS; C =-6.8x10

• MEASURED DATA

-»-•-«-. *•»»•»• , . *•*• v""""""" -» J-».................. •••••••••••••••••-""i . • • • • •0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

DISTANCE ALONG COLUMN (m)Fig. 10. Measured and simulated water-content (6) distributions

under nonisothermal conditions for the temperaturecoefficient of the matric potential (Q,) = -2.09 x 10~3

K- ' at an initial water content of 0.282 m3 m~3. Also plottedare the model calculations with C* = —6.8 x 10~3 K~'.

Lack of fit at lower water contents may be the reasonfor the differences between the model simulations andthe measured data under the above circumstances. Ingeneral, for initial water contents of 0.099, 0.151, and0.282 m3 m~3, the differences between the measuredand simulated values fall within the range of the water-content measurement error of 0.006 m3 m-3.

Under nonisothermal conditions, the model is inpoor agreement with the measured data, particularlyat initial water contents of 0.099 (Fig. 6c) and 0.282m3 m~3 (Fig. 8c). For both initial water contents, thecalculated water contents are lower at the hot end ofthe column and higher at the cold end relative to themeasured water contents, indicating that the modeloverpredicts thermally induced water movement. Foran initial water content of 0.282 m3 m~3, the shape

U.£.O

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0.20 -.

0.16 -

0.12 -

0.08 -

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O

— Isothermal Predicted- - Non-Isothermal

V T = 35.5-24 °C• • • • • • • • Non-Isothermal

V T = 55-25 'C

^^^

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UJo

o

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

DISTANCE ALONG COLUMN (m)Fig. 11. Model simulations at sn initial water content (6) of

0.151 m3 m~3 using a temperature (T) gradient of 1 °C cm-',compared with a temperature gradient of 0.5 °C cm-1 andisothermal conditions.

of the simulated water-content profile is markedly dif-ferent than the measured profile. The movement ofmoisture under nonisothermal conditions depends ontwo components: the gradients of water and temper-ature, and the values of the storage and transport coef-ficients. The temperature gradients are calculated inthe SPLaSHWaTr2 model using the one-dimensionalheat-flow equation. Figure 4 showed that heat flowwas not one-dimensional, and that the calculated one-dimensional temperature gradient was smaller than themeasured gradient for the hot end of the soil columnand larger for the cold end of the soil column. Thus,the calculated water movement should be less than themeasured water movement for distances close to thehot end, causing the calculated data to fall above themeasured data at these locations. This is opposite tothe situation depicted in Fig. 6c and 8c, indicatingthat the differences between model simulations andmeasured data are not due to differences in the cal-culated and measured temperature gradients. Since themodel compared relatively well with the experimentaldata under isothermal conditions, differences betweenthe model simulations and the measured data undernonisothermal conditions are probably due to incorrectpredictions of the transport and storage coefficients bythe model or the theory on which it is based. Undernonisothermal conditions, these coefficients are thethermal conductivity, X(6), the thermal vapor diffu-sivity, DTv(6), and the temperature coefficient of thematric potential, C^. A sensitivity analysis was per-formed in which the value of each thermal variablewas changed, in turn, while the values of the othervariables remained constant. The new model simula-tions were compared with the original results at eachinitial water content.

The value of X.(6) was decreased by independentlychanging two different input variables: (i) 6k, themoisture content at which liquid flow becomes neg-ligible, and (ii) the volumetric fraction of the mineralconstituents. Since most of the evidence presented inthe literature indicates that the de Vries (1966) model


predicts values of X. that are too large (Wierenga etal., 1969; Kimball et al., 1976; Horton and Wierenga,1984), the model was not reevaluated with larger val-ues of X(6). The value of 9k was increased from 0.054to 0.087 m3 m~3, which is the value of 6 at a matricpotential of —0.320 m of H2O. This caused a signif-icant decrease in the values of X up to 9 = 0.087,after which the values are the same as for 0k = 0.054m3 m~3 (Fig. 9). The volumetric fraction of quartzwas decreased by a factor of two, from 0.398 to 0.199,causing an average decrease of approximately 17% inthe values of X across the entire range of 6. Modelsimulations performed with these changes yielded water-content values that were within 0.003 m3 m~3 of theoriginal simulations for all locations within the soilcolumn at all initial water contents. For the columnas a whole, the new simulations yielded a change of< 1% from the original simulations, indicating that theeffect of X on soil water flux was negligible for allinitial water contents. Thus, when water-flux or water-content prediction is the information sought throughuse of a coupled moisture- and heat-flow model, es-timates of the parameters 6k and the volumetric frac-tions of the mineral constituents will yield accurateresults. However, Fig. 9 indicates that the X(0) curveis quite sensitive to the choice of 9k and the volumetricfractions, indicating that, when heat flux is the desiredinformation, experimental measurements of these pa-rameters may be required.

The thermal vapor diffusivity, D^Q), was set tozero, and the new model simulations yielded water-content values that were within 0.003 m3 m"3 of theoriginal simulations for all locations within the soilcolumns at all initial water contents. For the columnas a whole, the new model simulations yielded a changeof <1% from the original simulations, indicating thatthermally induced vapor flow does not contribute sig-nificantly to total water movement in the soil columnsfor this study.

The effect of the temperature coefficient of the ma-tric potential, C^, on soil water flow was evaluatedby comparing the original simulations with those inwhich Q was set to -2.09 x 10"3 K'1, which isthe value predicted by the surface-tension model oftemperature dependence (Philip and de Vries, 1957).For an initial water content of 0.282 m3 m~3, themodel simulations using C^ = —2.09 x 10 "3 K"1

more closely match the measured data in both orderof magnitude of the water-content values and the shapeof the water-content profiles, compared with the orig-inal simulations (Fig. 10). Some improvement be-tween model calculations and measured data was alsofound for initial water contents of 0.099 and 0.151 m3

m~3. The improved prediction at higher water con-tents is consistent with several previous studies, whichfound that the effect of temperature on the moisture-retention function is smaller when the soil is very wet(Taylor and Stewart, 1960; Wilkinson and Klute, 1962;Campbell and Gardner, 1971). Nimmo and Miller(1986) calculated gain factors, which represented theincrease in temperature effects over that predicted bythe surface-tension model. They showed that the gainfactors were a function of water content, having val-ues close to one under wet conditions. We would

therefore expect C^, to be close to - 2.09 x 10~3 K-1

for the simulations performed at an initial water con-tent of 0.282 m3 m-3. Setting Q = -6.8 x 10~3

K~ l resulted in simulated water fluxes in the soil col-umn that were greater than the measured fluxes forthis initial water content. With the adjustment in C^,,the model simulations are in closer agreement withthe measured data. Some differences are still found atan initial water content of 0.099 m3 m~3, particularlyclose to the wetting front; however, some of this maybe due to the difficulties encountered under isothermalconditions. Overall, relatively good predictions areobtained with the SPLaSHWaTr2 model.

The information reported above indicates that theeffect of a temperature gradient is small at very highand very low initial water contents. The maximummovement due to a temperature gradient was found atan intermediate initial water content, 0.151 m3 m"3,which corresponded to a i|/ value of approximately—1.2 m of H2O. Although statistically significant, thetemperature effect was still fairly small; however, theapplied temperature gradient was only 0.5 °C cm"1.In dry soil conditions, temperature gradients can bemuch larger than this, especially in arid areas (Chang,1957; Wierenga and de Wit, 1970; Pikul and All-maras, 1984). To evaluate the effect of a larger tem-perature gradient, model simulations were conductedat an initial water content of 0.151 m3 m~3 with atemperature gradient of 1 °C cm"1, or 55 °C and thehot end and 25 °C at the cold end. Figure 11 showsthat, when the gradient is increased to 1 °C cm"1, theeffect becomes much greater, and use of isothermalpredictive equations alone will result in significant er-rors in the estimation of soil water movement.

SUMMARY AND CONCLUSIONSSoil water movement in response to temperature

gradients was evaluated through a combination of lab-oratory experimentation and numerical modeling. Thelaboratory experimentation provided direct informa-tion on the significance of nonisothermal soil watermovement, while the numerical modeling provided ameans for examining theoretical calculation of thestorage and transport coefficients. The conclusions are:

1. A significant effect of the temperature gradientwas found at an intermediate initial water con-tent of 0.151 m3 m"3, with little or no effectfound at very high and very low initial watercontents. These results are in general agreementwith previous studies; however, the specific watercontents at which temperature gradients are sig-nificant appear to be different depending on ex-perimental conditions.

2. Under isothermal conditions, the model pro-vided good fits to the measured data for initialwater contents of 0.099, 0.151, and 0.282 m3

-3m3. Under nonisothermal conditions, the model pro-

vided poor fits to the measured data using theoriginal values of the thermal conductivity, X(6),the thermal vapor diffusivity, DTv(Q), and thetemperature coefficient of the matric potential,<v

4. The X(0) function was strongly affected by


changes in the input variables 6k and the volu-metric fraction of quartz, which could have im-portant implications for heat flow in soils.Changing the \(6) curve had a negligible influ-ence on nonisothermal water movement underthe conditions of this study.

5. The £>Tv(6) function had a negligible influenceon nonisothermal water movement for all fourinitial water contents under the conditions of thisstudy.

6. The coefficient C$ had a significant influence onnonisothermal water movement. Adjusting CL,from -6.8 x H)-3 K-1 to -2.09 x 10~3 K-'significantly improved agreement between sim-ulated and measured water-content profiles foran initial water content of 0.282 m3 m~3. Inaddition, some improvement was also found forinitial water contents of 0.099 and 0.151 m3 m~3.These results support evidence that there is lessof an effect of temperature on the moisture-re-tention function when the soil is very wet.

7. With the above adjustments in Q, the modelsimulations are in general agreement with themeasured data, indicating that the Philip and deVries theory provides an adequate description ofsoil water movement in response to temperaturegradients for the experimental conditions of thisstudy.

ACKNOWLEDGMENTSI wish to thank Dr. Roger E. Smith (USDA-ARS, Ft.

Collins, CO) and Dr. Arnold Klute (Colorado State Univ.)for their advice and counsel. I also wish to thank Dr. P.Christopher D. Milly (USGS, Princeton, NJ) for use of hisnumerical model, and Mrs. Virginia Ferreira (USDA-ARS,Ft. Collins, CO) for her helpful suggestions and cheerfulassistance with the computer programming.

soil water movement in response to temperature gradients: experimental measurements and model...

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