soil-water movement in response to imposed temperature gradients1

Soil-Water Movement in Response to Imposed Temperature Gradients1

D. K. CASSEL, D. R. NIELSEN, AND J. W. BIGGAR2

ABSTRACTRedistribution of soil water within insulated, uniformly

packed, horizontal samples of unsaturated Columbia fine sandyloam at several soil-water contents was studied in response toimposed temperature gradients ranging from 0.5 to l.OC/cm.Soil bulk density and initial, transient, and final soil-water-content distributions were determined each 0.5-cm along thecolumn by gamma-radiation attenuation. Initial, transient, andfinal soil temperature distributions were monitored by glass-encased thermistors at 2-cm intervals—both at the center and0.3 cm from the column wall. Apparent thermal and isothermalsoil-water diffusivity values were calculated using transientwater content data. The observed net water flux was found toincrease with decreasing water content throughout the 0.077—0.274 cm3/cm3 range. For Columbia soil at 0.077 cm3/cm3 theobserved mean net water flux across 1-cm sections of the soilshowed acceptable agreement with that predicted by the theoryof Philip and deVries; Pick's law and the modified Taylor-Cary irreversible thermodynamic equation both underpredictedthe observed fluxes.

Additional Key Words for Indexing: thermal soil-water dif-fusivity, isothermal soil-water diffusivity.

PRONOUNCED seasonal and diurnal temperature fluctua-tions in the soil have long been recognized. Seasonal

fluctuations affect soil temperatures to depths greater thanthe rooting zones of most crops while diurnal ones influenceapproximately the surface 30-cm of bare soil (Smith, 1932;and P. J. Wierenga, 1968. An analysis of temperature be-havior in irrigated soil profiles. Ph.D. Thesis. Universityof California, Davis}. The temperature variation withdepth gives rise to thermal gradients which tend to movewater in both the vapor and liquid phases. Lebedeff (1927)conducted a field experiment over the winter months inRussia and concluded that more than 6 cm of soil watermoved upward into the soil profile in response to the sea-sonal thermal gradients. Laboratory investigations havelikewise detected soil water movement in response to ther-mal gradients. Rollins et al. (1954) reviewed existingliterature on the subject.

Before a theory was advanced by Philip and de Vries(1957), many investigators attempted to describe theobserved soil-water movement using Pick's law modified

1 Contribution from the Department of Water Science & En-gineering, University of California, Davis. This investigationwas partially supported by funds received from the Water Re-sources Center, University of California, and from the US ArmyElectronics Research and Development Activity, Ft. Huachuca,Ariz., under Grant no. DAAB G29-07-67-C00034. ReceivedOct. 28, 1968. Approved Feb. 24, 1968.

2 Laboratory Technician, Professor and Associate Professorof Water Science. The senior author is now Associate Professorof Soil Phvsics, Department of Soils, North Dakota StateUniversity, Fargo.

3G. W. Gee, 1966. Water movement in soils as influencedby temperature gradients. Ph.D. Thesis. Washington StateUniversity, Pullman.

for porous media, which necessarily assumed all movementin the vapor phase; Pick's law consistently underpredictedthe observed net water movement. The theory advancedby Philip and de Vries for describing soil-water movementin response to temperature gradients was based partlyupon further refinements of Pick's law. Water was assumedto move in both the vapor and liquid phases in response toboth soil-water pressure and thermal gradients. Dirksen,describing soil-water movement observed in freezing col-umns of soil in the absence of a water table, found accept-able agreement with this theory (C. Dirksen, 1964. Watermovement and frost heaving in unsaturated soil withoutan external source of water. Ph.D. Thesis. Cornell Uni-versity, Ithaca, N.Y.). More recently, Taylor and Gary(1960) proposed a theory based upon irreversible thermo-dynamics for describing soil-water movement in responseto temperature gradients. A strict limitation placed uponthis theory as it was originally proposed is the requirementthat the heat flux through the soil be known. The theorywas later modified by Taylor and Gary (1964) to eliminatethe heat flux requirement. Both the Philip-de Vries and theTaylor-Carv theories involve the use of soil-water-diffu-sivity coefficients. Gee working with 10-cm-long sealedcolumns of Palouse silt loam concluded that the Taylor-Cary theory held for only a limited range of soil-water con-tent.3 The purposes of this investigation are to compareunder rigorous laboratory conditions at a given soil-watercontent the applicability of these two theories for describ-ing soil-water movement in response to imposed tempera-ture gradients and to examine values of the apparent ther-mal and isothermal soil-water-diffusivity coefficients.

THEORYFlick's law of diffusion modified for vapor movement in por-

ous media as reported by Rollins et al. (1954) is given by

= ~D (P/P-p) [1]

where ?Fick (g/cm2day) is the vapor flux, Z>atm (cmVday) themolecular diffusivity of water vapor in air, P (mm Hg) the totalair pressure, p (mm Hg) the partial pressure of water vapor,a the tortuosity, a (cmVcm3) the volumetric air content, andV> (g/cm4) the water vapor density gradient. The value of ain the above equation is taken as 0.66 (Penman, 1940).

The Philip-de Vries theory describes the net water flux(g/cm2 day or cm/day if the density of water is assumed to be1 g/cm3) as

-dev [2]

where Z>0vap and Deii<l (cmVday) are the isothermal soil-waterdiffusivities for vapor and liquid, respectively; -DTVan an^ ^Tiiq(cmVday C) the thermal soil-water diffusivities of vapor andliquid, respectively; V0 (cmVcm3) the soil-water-content gradi-ent; and VT (C/cm) the soil temperature gradient. This theoryallows for the tendencies of water to move in opposing direc-tions and in response to different driving forces. Before this

493

494 SOIL SCI. SOC. AMER. PROC., VOL. 33, 1969

equation can be used to predict the net water flux, however, anaccurate value of D0liq or of the hydraulic conductivity K(cm/day) versus 6 must be known. A discussion of each of thefour diffusivity terms is presented in the original paper by Philipand de Vries (1957). The equations used to calculate theoreticalvalues of these four diffusivity terms from basic physical quan-tities are given in the Appendix of this paper.

The Taylor-Cary theory requires that a linear rate or flowequation can be written for each component of the system

Ji = 2 LikXk (i = 1,2,...Jc = l

[3]

where 7; is the flux of the i'th component, Lik the phenomeno-logical coefficient due to the Xth driving force affecting the/th flux, Xk the driving force or affinity of the £th component,and N the number of driving forces. Since in the realm of irre-versible thermodynamics the flux of one component of a system(heat) influences the flux of a second component (water) ofthe system, use of [3] for predicting soil-water movementrequires measurement of the heat flux in addition to the netwater flux in order to determine values of Lik (LWQ or LQW).For a closed soil-water system having an imposed temperaturegradient and which has attained a steady-state flow condition,[3] was simplified by Taylor and Gary (1964) to

T-C 8 Af [4]

where <?T_C (g/cm2day or cm/day) is the net water flux, x(cm) the distance along the column, T' (degrees Kelvin) thetemperature and /3* = —de/d In T or Lwq/DB whereDg = (£>0liq + £>0var>) and w ar"d 1 represent water andcalorimetric heat, respectively. Assumptions inherent in [4] arediscussed by Taylor and Gary (1964).

Consider a horizontal, isothermal, uniformly packed, closedsoil column of_length L having a uniform, unsaturated soil-water content 60. Upon imposing a different temperature ateach end of the column, water immediately begins to redis-tribute within the soil. If the water content versus distanceB (x, t) changes a small yet measurable amount over somesmall time interval, the mean net water flux [q0t,s(x)] acrossany given cross section during this time interval is equal to thegain or loss of water from the soil on either side of the crosssection divided by the time interval. These experimentallyobserved values of [<?obs(A:)], or their mean value obtainedby integration over all values of x along the column provideestimates of <?FiCk, <?p_dev and ?T-C which can be compareddirectly with values of the right-hand side of equations [1], [2],and [4], containing quantities which can be measured or calcu-lated directly.

On the other hand, equation [2] rewritten to describe themean net water flux at x observed over the time interval A^ is

- Dr [5]

where DeT is the apparent isothermal diffusivity (Dg\ia +

DTvap) at the temperature of the soil T (degrees C), (V0)i themean water content gradient at x for A/j, DT

T the apparentthermal diffusivity (DTliq + DTvap) at T, and (VT)i the meantemperature gradient at x for A^. Similarly, the observed meannet flux [<?obs(;c)]2 at x for the time interval (t2 —^) or Af2is given by

[«„„„(*)], = -DI (vex -DI (VD,. [6]

soil column, a simultaneous solution yields values of the appar-ent isothermal and thermal soil-water diffusivities.

EXPERIMENTAL

Oven-dried Columbia fine sandy loam was packed in 20-cm-long glass cylinders having a 7.60-cm inside diameter. Twoglass encased thermistors—one located at the center of thecolumn (long thermistor) and one 0.3-cm from the column wall(short thermistor)—were inserted at 2-cm intervals along thecolumn. In addition one thermistor was located at the center ofeach end of the column. Columns were selected on the basisof uniform bulk density Db (g/cm3) as determined by gamma-radiation attenuation at 0.5-cm intervals along the column. Thecolumn was saturated with 0.0 IN CaSO4 to which a few crys-tals of phenol had been added to inhibit microbial activity.Desired levels of soil-water content were obtained (i) byremoving solution using pressure differences applied directly toporous plates at each end of the column, and (ii) by passingwarm air at a flux of 0.4 cm/min through the column. Thedirection of warm air flow was reversed periodically to preventexcessive drying at either end. The soil-water content was deter-mined at 0.5-cm intervals along the column by gamma-radiationattenuation. An aluminum end-plate with a reservoir capableof being filled with water was sealed to each end of the columnwith Silastic caulk (Dow-Corning Corp.), and the entire columnassembly was covered with 6-cm of polyurethane insulationapplied as a foaming liquid that expanded to all surfaces andcrevices before it solidified. A minute vent allowed atmosphericpressure to be maintained within the soil column at all times.A uniform temperature near that of the ambient temperaturewas initially established throughout the column for 5 days.

Temperature gradients were established lengthwise along thecolumn by circulating preheated or precooled water (± 0.05C)through the endplate reservoirs. Initial, transient, and final tem-perature distributions were monitored at 2-cm intervals alongthe column by means of a data acquisition system; initial, transi-ent, and final water contents were monitored at 0.5-cm inter-vals. Final distributions were taken as those when no significantwater content change could be detected.

Two soil columns, A and B, were used in this investigation.Column A was studied at mean initial soil-water contents S0 of0.183 and 0.101 cmVcm3; column B at 60 = 0.077 cmVcm3.Each set of initial and boundary conditions is denoted as a runand is identified in Table 1. A more detailed description of themethods is available (D. K. Cassel, 1968. Soil-water behaviorin relation to imposed temperature gradients. Ph.D. Thesis.University of California, Davis).

RESULTSMean bulk density values, based upon the distributions

determined by gamma-radiation attenuation were 1.594and 1.607 g/cm3 for columns A and B, respectively. Themaximum range in Db was 0.055 and 0.053 g/cm3 for

Table 1—Initial and boundary condition temperatures at leftand right ends of experimental soil columns for each

experimental run. The capital letterin each run number identifies

the soil column used

Inasmuch as all quantities in [5] and [6] except Dg and DT canbe measured or computed for different values of x along a given

Experi-mental

runnumber

AlA2A3A4BlB2B3B4

MeanInitialwater

contentcm3 /cm'

0.1010.1010.1010.1010.0770.0770.0770.077

Initialtemperature

Left•C8.75

18.5023.1026.9021.8518.6023.1026.90

Right•c

8.758.758.808.85

23.208.758.708.70

Finaltemperature

Left•c

18.5023.1026.9030.3018.6023.1026.9030.30

Right•C8.758.808.858.858.758.708.708.70

CASSEL ET AL.: SOIL-WATER MOVEMENT 495

•^0.14U

LUI-

§0.06

o:LUH§0.02

o o o 76 hourt* * * 186• •• 352

0.077

•18.60 C

10DISTANCE (CM)

20

Fig. 1—Initial and transient soil-water-content distributionsfor soil column Bl with an average wafer content of 0.077cm3/cm3. Initially, the soil column was at a uniform tem-perature of 23.4C; thereafter, at distances of O and 20 cm,the temperature was maintained at 18.60 and 8.75C, re-spectively.

LU

10 -

10DISTANCE (CM)

20

Fig. 2—Final soil temperature distributions for column B atan average soil-water content of 0.077 cm3/cm3.

"^ 0.15u^

WAT

ER

CO

NTE

NT

(CO o

O g

Run BlRun 82Run 63Run B4

18.60 C Bl 8.79 c23.10 82 8.7026.90 83 8.7030.30 84 8.70

10DISTANCE (CM)

20

Fig. 3—Final soil-water-content distributions for column B atan average soil-water content of 0.077 cm3/cm3 for differentsoil temperatures maintained at x = O and x = 20 cm.

columns A and B, respectively. The extreme values occurredat or near the ends of the columns.

Figure 1 shows the soil-water content versus distance forrun Bl for three times after the boundary temperatures of18.60 and 8.75C were established. Each datum representsthe mean of three 1-min counts and has a random errorless than ±0.003 cm3/cm3. After a period of 76 hours, anappreciable quantity of water had moved from the warmto the cold end of the column. The soil-water-content dis-tribution continued to change with time as shown by thedata. Upon reaching final distribution of both water con-tent and temperature which remained invariant for times> 352 hours, run B2 was initiated by increasing the bound-ary temperature at the warm end of the column to 23.IOCSimilarly, after measurements for run B2 were completed,runs B3 and B4 were made. Figure 2 presents final tem-perature distributions derived from the mean of the longand short thermistor temperature values for the four con-secutive runs. The sigmoidal shape of the smooth curvesdrawn through the data for runs B2, B3, and B4 indicates

30

OLUrrID

< 20rru

10

10

DISTANCE (CM)20

Fig. 4—Final soil temperature distributions for column A atan average soil-water content of 0.101 cm3/cm3.

that heat was lost from the system for column temperaturesabove ambient and vice versa—a point to be discussedlater. The final soil-water-content distributions measuredconcomitantly with those of temperature for these runs areshown in Fig. 3. In general, the water content data becamemore scattered as the differences in column boundarytemperatures were increased.

Nearly identical boundary temperatures (see Table 1)produced the same type of soil-water and thermal responsefor runs Al to A4 as for runs Bl to B4. For equal timeintervals the net water movement from the warm to coldend was not as great for column A with 80 = 0.101cm3/cm3 as for column B at the lesser W0 value of 0.077cm3/cm3 (see Fig. 3 and 4). Transient soil-water-contentdistribution data for runs Al to A4 (not given) manifestedsufficient scatter that they were not amenable to an accu-rate analysis using equations [5] and [6]. Nevertheless,


oro^5U

OO

cc

I

0.15

0.10

0.05

IS.50 C II 8.75 C2S.IO A 2 8.752B.90 A3 8.8030.30 A 4 8.85

DISTANCE (CM)Fig. 5—Final soil-water-content distributions for column A at

an average soil-water content of 0.101 cm3/cm3 for differentsoil temperatures maintained at x = O and x = 20 cm.

0.12

O

00.09

UJt-OO

cc 0.06

0.035.65 5.70

LN KELVIN TEMPERATURE

Fig. 6—Final soil-water-content values plotted as a function ofthe logarithm of temperature of runs Bl and B2.

the final soii-water-content distributions for these runsgiven in Fig. 4 are well-defined and provide ample datacoupled with the final temperature distribution data (Fig.5) to examine the behavior of /?*. Integration of 6(x)over the column length for all runs showed no water was

0.14

O 0.11

OO

CCLU 0.08

I

0.055.65 5.70

LN KELVIN TEMPERATURE

Fig. 7—Final soil-water-content values plotted as a function ofthe logarithm of temperature for runs A3 and A4.

lost at any time. The application of unequal boundary tem-peratures to column A with 60 = 0.183 cnvVcm3 did notcause a measurable change in the soil-water distribution.

To enable the data to be analyzed in light of the Taylor-Cary equation, a plot of water content versus In T' foreach final water content distribution was required. Figure6 which shows these plots for runs Bl and B2 was derivedfrom the final temperature distribution curve and the finalwater content distribution data given in Fig. 2 and 3,respectively. Similarly, Fig. 7 was derived from Fig. 4 and5. The value of /?* (= —d6/d\nT') versus d for all eightruns are presented in Fig. 8 and 9. For values of /?* lessthan 2 an error less than ± 0.1 is expected. As the slopesof the curves in Fig. 6 and 7 increase, the errors in ascer-taining the slopes and thus /3* increase.

Values of the observed mean net water flux [?0i)s(.x)]were computed at each 1-cm interval along the columnfor the 0-76 and 76-186 hour intervals for run Bl. Thesevalues along with the fluxes predicted by Pick's law(<7F i ( . k) , the Philip-de Vries equation (^p_dev). ar|d theTaylor-Cary equation (<?T_C) are presented in Tables 2 and

UJcc

cc 10UQ-

- - " 0.03 008 0.13WATER CONTENT (CM3/CM3)

Fig. 8— f t" versus final soil-water content for runs Al, A2, Bl,and B2.

cc

Ct I0

3Lij

A - 60»0.IOI

B - §.= 0.077

0.03 0.06 0.13WATER CONTENT (CM3/CM3)

Fig. 9—/3* versus final soil-water content for runs A3, A4, B3,and B4.


Table 2—Comparison of observed net water fluxes through 1-cm sections to those predicted by Pick's law, the Philip-de VriesTheory, and the Taylor-Cary Theory for the O to 76-hour interval of run Bl

Columninterval

cm

123456789

10111213

' 141516171819

[q o b s w]xio>cm/day

1. 803.334.645.756.677.448.068.448.728.898.999.029.029 .028.968.898.618.005.66

q F ) o k x l O >

cm/day

2.661.390.8741.521. 111,221.511.661.801.741.501.631.581.801.521.791.982 ,242.18

&

1.480.4170. 1880.2640. 1660. 1640.1870. 1970.2060.1960.1670.1810.1750.1200. 1700.2010. 2300.2800.385

VdeVX 1 0 'cm/day

10.96.934.884.865.855.866.827.808.478.477.568.42

10.39,309.30

10.111.913.614.0

["obsWl

6.062.081.050.8450.8770.7880.8460.9240.9710.9530 8410,9331.141.031.041.141.381.702.47

< » T - C X 1 0 >

cm/day

0.560.140.140.300.120.450.270.550.570.570.570.420.760.670.680.580.880.52

-0. 18

["obsM!

0.3110.0420.0300.0520.0180,0600.0330.0650.0650.0640.0630 0470.0840.0740.0760.0650.1020.065

-0.032

Table 3—Comparison of observed net water fluxes through 1-cm sections to those predicted by Pick's law, the Philip-de VriesTheory, and the Taylor-Cary Theory for the 76 to 186-hour interval of run Bl

ColumnInterval

cm123456789

10111213141516171819

I-*.™ >""

cm/day

1.673.034.285.366.317.147.928.649. 199.619.759.669.368.727.786.424.813.031.31

"Pick* 10S

cm /day2.741.430.8761.521.111.221.511.661.861,791.501.631.581.801.521.791.912.172.18

"rick'"obs Ml

1.640.4720.2050.2840.1760.1710. 1910.1920.2020,1860,1540.1690.1690.2060,1950.2790.3970,7161.66

qp-devx l°3

cm/day

11.06.924.714.735.775.736.767.718.758.347.518.27

10.29.239.009.96

11.213.312.5

'lobs (x)l

6.592.281.100.8820.9140.8030.8540.8920.9520.8680.7700.8561.091.061.161.552.334.399.54

"T-CX I° J

cm/day

0.430.0880.0950. 190.140.410.230.590.400.470.530.450.660.680.280.320.660.35

-0.18

KbTMi0.260.0290.0220.0350.0220.0570.0290.0680.0440.0490.0540.0470.0710.0780 0360.0500.140.12

-0.14

3. Water content gradients used in equations [2] and [4]were mean values averaged over each 1-cm interval andwere derived from Fig. I. The final temperature gradientswere used for the computations since the transient tempera-tures lasted for only a few hours. In general, predictionsbased upon Pick's law were about five times too smallwhile those based upon the Taylor-Cary equation wereabout 10-40 times too small. The Philip-de Vries equationyielded values approximately equal to those measured.

With the aid of equations [5] and [6], the apparent ther-mal and isothermal soil-water-diffusivity values were com-puted at 1-cm intervals along the column for run Bl. The

calculated apparent isothermal diffusivity D0app and ap-parent thermal diffusivity Z)r

app along with the correspond-ing diffusivity values predicted by the Philip-de Vriestheory (ZV'rcd and DT

precl) are presented in Table 4.Values of ZV™' (= D8lkl + D»vap) and D/relJ (= Driiq

+ £>Tvnp) were obtained from the equations in the appen-dix using values of physical constants taken from the 1965Handbook of Chemistry and Physics or the InternationalCritical .Tables and independently measured values ofD0liq, t and de/dt for 6 = 0.077 cmVcm3. Inasmuch asa reliable method is not yet available for measuring Dglivexcept in the very dry range (Jackson, 1965), and that

Table 4—Comparisons of D0app and DTarp calculated with equations [5] and [6] at several distances along the column from

transient water content data of run Bl with those predicted by Philip-de Vries theory with the equations in the Appendix

Distancealong

column

cm23456789

101617

Meantemp

•C1818171717161616151212

Meanwater

content

cms/cms

0. 0730.0740.0740.0750.0750.0760.0760.0760.0770.0770.0078

Dapp>0

cm* /day0.3090.4400.5500.5310.4710.219

-0.313-0. 769-1.192.443.24

predS

cm1 /day0.1360. 1360.1350. 1350. 1350.1340. 1340.1430.1420.1460.157

appT

cm* /day C0.01270.01900.02100.02300.02150. 0203 '0.02000.02000. 008590.009940.0170

predT

cm* /day C0.02010.01980.01970 01870.01970.01960. 01960.01930.01870.01850.0181

D'"

Dpred

2.273.244.073.933.491.63

-2.34-5.38-8.3916.720.6

T

Dpred

0.6320.9601.071.171.091.041.021.040.4590.5370.939

* The symbols DgPP and D^,PP refer to the quantities D. and DI. used In equations [5] and [6] .


estimates of £>0liq are about 10s times greater than Z>«vap,£>0liq was approximated by Dg (= D0liq + Dgmp) usingvalues reported by Cassel et al. (1968) for the same soilfor desorption (drying soil).

DISCUSSION

The magnitude of the net water flux [<?0bS(*)] f°r

Columbia fine sandy loam was observed to vary with watercontent: fluxes for approximately equal time intervals werefound to be greatest for column B with S0 = 0.077cm3/cm3, less for column A with 00 = 0.101 cm3/cm3,and least for column A with 60 = 0.183 cmVcm3. Severalinvestigators (Smith, 1943; Jones and Kohnke, 1952; Gurret al., 1952; Gouda and Winterkorn, 1949; and Globus,1960) observed that a specific water content or a range ofwater content existed at which the maximum net watertransfer for a given soil occurred. The results of this experi-ment indicate that if a maximum net water transfer existsit must be for values of 6 < 0.077 cmVcm3 or possibly for0.077 < e < 0.101 cm3/cm3.

Examination of Tables 2 and 3 indicates that qvick isappreciably less than [<?obs(x)] for both time intervals forColumbia fine sandy loam in a manner similar to thatreported for other soils by Gee4, Taylor and Cavazza(1954), and Gurr et al. (1952). On the other hand, valuesof [qabs(x)] do show acceptable agreement with values of<7p-<iev predicted by the theory of Philip and de Vries.Values of qT_c/[qol>s(x)] show that the Taylor-Cary equa-tion underpredicted the observed flux 10- to 40-fold. Pre-diction of the net water flux with the Taylor-Cary equationsrequired the values of an experimentally determined par-ameter ft* (= —de/d\r\T') which, in the development ofthe Taylor-Cary theory, is assumed constant over somerange of soil-water content value (Taylor and Gary, 1960,1964). A constant value of /?* is indicated by a plot of 6versus InT' being linear over a given water content interval.Figures 6 and 7 indicate that 0(ln!T') is not linear. Figures8 and 9 further illustrate the nonlinearity of 0(lnT') with/?* plotted versus soil-water content for all eight runs.Hutcheon (1958), Jackson et al. (1965), and Gee5 havereported the same nonlinear behavior for different soils.Although the degree of data scatter manifested in Fig. 6and 7 is substantial, the number of data is sufficient toallow a smooth well-defined curve to be drawn—a resultnot obtained in previous investigations.

The apparent isothermal soil-water-diffusivity valuesDgT calculated from the experimental data fall into threegroups as indicated by the ratios of calculated Dgapp topredicted values D0pred in Table 4. The first group, thoseDg values calculated on the interval 2 < x < 7 cm, areapproximately three times greater than those predicted.The change in water content over this interval based uponsmooth curves drawn through the data in Fig. 1 was lessthan 0.02 cnvVcm3, yet the mass of water passing throughthe cross section of the column at each x was large enough

to calculate reliable values for Iq^^x)^ an<^ ["ZobsWtAccumulative errors associated with the integration processin calculating [9obs(x)]i and [<7obs(*)]2 might possibly beresponsible for the negative values of DgT on the interval8 < x < 10. For greater values of x, values of D0r areabout 20 times greater than the predicted values. Thisdiscrepancy may be attributed not only to possible accumu-lative integration errors but to the fact that D0pred wasbased upon data for desorption whereas the soil was actu-ally wetting in that portion of the column. Moreover,extreme values in soil bulk density occurred in that portionof the column. Calculated values of DT

T were in generalagreement with those predicted throughout the entirelength of column.

An obvious consideration in this investigation is thedifficulty in measuring the net water flux. Under perfectsteady-state conditions, the value of q in equations [2] and[4] would be zero. For transient conditions the value of qwas not measured directly but inferred from changes inwater content along the column during specific time inter-vals. Values of [<7obs(*)] are time dependent having maxi-mum values immediately after temperature differences areimposed and decreasing to zero eventually. Thus, whencomparing [qobs(x)] to <?,,_deV or <?T_C, it is necessary touse values of [qobs(x)] obtained early in the experimentand make comparisons over the same time intervals. It isalso necessary to maintain the range in 6 within 1-cmsections as small as possible inasmuch as Dg is extremelysensitive to changes in 6. Dg appears both in equations [2]and [4]. Gee (1966) calculated the mean net water flux fora 10-cm section for the favorable, short time interval fromO to 25 hours, but the soil-water contents measured withinthe sample varied from 0.093 to 0.172 cmVcm3. In run Bl,water contents only varied from 0.061 to 0.113 cm3/cm3

across the entire sample. On the other hand, a small rangein 9 creates potential errors in estimating the net water .flux from measured changes in soil-water-content dis-tributions.

A second consideration which has received insufficientattention is the manner in which boundary temperaturesare established including the provision to have no heatflux in the radial direction. Upon changing the boundarytemperatures at one or both ends of a column, heat movesthrough the column creating transient temperature dis-tributions which are easily analyzed with the one-dimen-sional heat-flow model

_et' ~ a*2 O < x < 20 [7]

where T (°C) is the temperature, t' (hours) the time, andk (cm2/hour) the apparent thermal diffusivity. This anal-ysis is only approximate inasmuch as the apparent thermaldiffusivity is assumed constant even though heat transferoccurs by conduction and convection. The initial andboundary conditions for run Bl, for example, were:

4 Ibid.5 Ibid. T = 23.40 O < x < 20 t' = O [8a]


T = 18.50 x = O

T - 8.80 x = 20

The solution of [7] and [8] is

/' > O [8b]

t' > O [8c]

..O = 15-50 - 0.485, +

sinn = l n

-knW r'/400) [9]

which does not allow for any heat movement that maytake place experimentally in the radial direction.

The mean values of the long and short thermistortemperatures versus distance along the column calculatedfor run Bl using equation [9] are shown in Fig. 10. Ingeneral, the value of the apparent thermal diffusivityk (cm2/hour) given near each calculated curve decreaseswith time. This decrease is a manifestation of heat trans-ferred by mass flow and of heat exchange in the radialdirection. It is evident from Fig. 8 that although constantboundary temperatures were established almost immedi-ately, it was several hours before a stable distribution wasreached within the column. Taylor and Gary (1960) usinga 10-cm long column of Millville silt loam reported com-parable times to attain stable temperature distributions.Hence, during the initial stages of the experiment when themost reliable estimates of the net soil-water flux are avail-

24

20

O

UIrr

QLLL)Q.2LL)

16

12

INITIAL Bl

16.92

o o o o.22 hoursV 7 V 0.28• • • 1.02o o o 2.35* + + 4.50

.077

Thermol d i f f u s i v i t y is shown foreoch co lcu lo ted c u r v e ( .cmVhr)

8O 10

DISTANCE (CM)20

Fig. 10—Initial and transient soil temperature distribution forfive times for run Bl. B0 was 0.077 cm3/cm3; TL and TRwere 18.60 and 8.75C, respectively. Solid lines were calcu-lated using a Fourier series solution of the one-dimensionalheat flow equation. Each datum is the mean of long andshort thermistor temperature measurements.

able, unwanted fluctuations in soil-temperature gradientsare most pronounced and most difficult to handle usingequations [2] and [4]. Hadas (1968) studying thermallyinduced water movement, attempted to use transient tem-perature gradients rather than steady-state values but metwith limited success.

Although extreme care was used to insulate the soilcolumns, radial temperature gradients were observed inall columns, particularly near the ends of the column orwhen soil temperatures differed markedly from the ambienttemperature of the controlled temperature laboratory. Forexample, differences in temperature between the long andshort thermistors at x = 2-cm for runs Bl to B4 were 0.05,0.35, 0.6, and 0.8 C, respectively. For the largest differ-ence of 0.8C for run B4, this corresponds to an estimatedradial heat loss from a 1-cm length of column (centeredat x ~ 2 cm) of 9.1 cal/hour compared with an estimatedlongitudinal heat flow of 200 cal/hour. Even though thisrepresents only a 4% radial heat loss, radial temperaturegradients must give rise to significant vapor pressure gradi-ents which in turn causes vapor to move radially. Further-more, condensation of this vapor in the cooler outer regionof the column must give rise to soil-water-pressure gradi-ents causing liquid water movement to the center. Theconverse of these processes must occur for column tem-peratures below the ambient temperature. A shortcomingof most previous investigations was the failure to measureor even acknowledge radial temperature gradients. Al-though a "cell-constant" could account for the net heatloss or gain from the column, its use would not preventunwanted radial fluxes of both heat and water withinthe column.

Experiments designed to measure soil-water transfer inthe presence of temperature gradients are difficult toconduct—even the selection of a suitable sample containerpresents a difficulty. Glass cylinders were selected to housethe soil samples for this experiment. Glass is easily pene-trated by gamma radiation and is visually transparent.Even though the thermal conductivity of glass is slightlylarger than that of acrylic plastic, glass was selected owingto its impenetrability to water vapor. Willis et al. (1965)measured the diffusivity coefficients for water throughacrylic plastic to be approximately 8 x 10~6 cm2/sec.The use of plastic or many other materials over a periodof several months may give rise to an appreciable loss ofwater from the soil. Gee6 attributed significant losses ofwater (about 10% of the total flow over a period of 8days) from his soil columns to water vapor loss throughplastic end-plates. Gamma-radiation attenuation measure-ments of soil-water-content distributions in Columns A andB showed no measurable net water loss for the durationof the experiment.

Future laboratory investigations of soil-water movementin response to temperature gradients need to be conductedusing soil columns in which water is allowed to transferacross both end boundaries. By maintaining a water vaporsource and sink at desired temperatures at each boundary

6 Ibid.


a steady, net flux of water through the column would beestablished and easily measured without resorting to transi-ent soil-water-content distributions to estimate values ofq in equations [2] or [4]. Owing to the imperfect natureof insulating materials, attempts to analyze the radial heatand water transfer within cylindrical soil columns shouldbe made using two-dimensional models solved with theaid of numerical computers.

APPENDIX

Values of the four water diffusivity functions appearing inequation [2] were calculated from the following relations:

1) The isothermal vapor diffusivity

Paagp dd,D =————-——— (cm2/ day)

evap (p-p)pwR'T 98

4) The thermal liquid diffusivity

DTliq = Kyj, (cnvVday °C)

whereDatm = the molecular diffusivity of water vapor in

(cmVday)P = total air pressure (mm Hg)p = partial pressure of water vapor (mm Hg)a = a tortuosity factor taken as 0.66a = volumetric air content (cmVcm3)g = gravitational acceleration (cm/sec2)p = density of water vapor (g/cm3)i// = soil water pressure head (cm)pw = density of liquid water (g/cm3)R' = gas constant of water vapor (ergs/g K)T' — temperature (degrees Kelvin)8 = soil-water content (cm3/cm3)

2) T lie isothermal liquid diffusivity

dtLD

6l* = K-^(™2'W

whereK = unsaturated hydraulic conductivity (cm/day)

3) The thermal vapor diffusivity

[a + f ( a ) • 6] £>atm P h dPo

°~- M"-rt——^("™ay°c>where

air

= value of a at which liquid continuity in pores nolonger exists

= a/ak for O < a < ak= 1 for a > ak= saturated vapor pressure of air (mm Hg)= p/'p0, the relative vapor pressure of air= density of saturated water vapor (g/cm3)= temperature (°C)= (VDa/(VD7

(VT)a = mean temperature gradient in air-filled pores(°C/cm)

(V7") = overall temperature gradient (°C/cm)

Ok

Ha)

7 de Vries (1952a,b) has worked out values of £ for sandand common soil minerals for different combinations of 6 anda + e. Philip and de Vries (1957) state that these J values areacceptable for use in the Philip-de Vries theory in the 10-30Crange.

where

y = I/y da/dT = temperature coefficient of the sur-face tension of water ("O1)

a = surface tension of water (dynes/cm)

soil-water movement in response to imposed temperature gradients1

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