SOIL SCIENCE SOCIETY OF AMERICA

PROCEEDINGSVOL. 38 NOVEMBER-DECEMBER 1974 No. 6

DIVISION S-l—SOIL PHYSICS

Diurnal Soil-Water Evaporation: Comparison of Measured and CalculatedSoil-Water Fluxes1

R. D. JACKSON, R. J. REGINATO, B. A. KIMBALL, AND F. S. NAKAYAMA2

ABSTRACT

The Philip and DeVries theory and the "isothermal" theorywere used to predict diurnal soil water fluxes near the soil sur-face. The predicted values were compared with those obtainedby measurements of soil-water content, soil temperature, andevaporation. Previously measured soil-water diffusivities wereused in the theoretical calculations. The thermal vapor diffu-sivities were calculated using both the "simple" and the "com-plete" theory of Philip and DeVries. Comparison of measuredand calculated fluxes indicated that the theory of Philip andDeVries predicts the measured values better at intermediatewater contents, but the "isothermal" theory predicts values bet-ter at high and very low water contents.

Additional Index Words: heat and water transfer, soil dry-ing, temperature gradients.

rr>HE PREDICTION of water movement in field soils hasJ. largely been based upon theories of isothermal water

movement which neglects water movement in response totemperature gradients. Philip and DeVries (1957) proposeda theory to predict water movement as a consequence oftemperature and water content gradients. They applied thistheory to experimental data obtained in the laboratory, andsubsequent tests have largely been with laboratory data.C. W. Rose (1968a, b) using a similar theory, was perhapsthe first to consider the effect of temperature gradients onwater movement in a field soil during drying. His analysiswas based on theoretical calculations of soil-water flux.Jackson et al. (1973) obtained soil-water flux data moredirectly by measuring flux at the soil surface with lysimetersand the water content changes with depth. These data alongwith much unpublished data from the same experiment,provided a basis for testing the theory of Philip and DeVries

1 Contribution from the Agricultural Research Service,USDA. Received 22 Apr. 1974. Approved 26 July 1974.

2 Research Physicist, Soil Scientists, and Research Chemist,respectively, US Water Conserv. Lab., 4331 East Broadway,Phoenix, Ariz. 85040.

for predicting soil-water flux under natural, diurnal fieldconditions.

BACKGROUND

The total soil-water flux (Jw) may be written (Philip andDeVries 1957, C. W. Rose 1968a) as

i-DT,VT

— DTvVT — Ki [cm sec-1], [1]

where Dei is the isothermal liquid diffusivity, Dga the isother-mal vapor diffusivity, DTl the thermal liquid diffusivity, DTl,the thermal vapor diffusivity, V0 and VT the gradients of watercontent and temperature respectively, K the hydraulic conduc-tivity, and i is the unit vector. The cgs system is used throughoutthis report.

The terms Dgl, Dgv, and K are functions of temperature,as well as water content, but are not multiplied by the tem-perature gradient and are referred to as "isothermal." Con-versely, the terms DTl and DTv are functions of water contentas well as temperature, but are not multiplied by the water con-tent gradient and are referred to as "thermal." In the followingsections, experimental data and theoretical calculations wereused to obtain our best estimates of the four diffusion coeffi-cients, as a function of water content and temperature. Usinghydraulic conductivity (K) data from Jackson (1973) the termKi was calculated and found negligible for all tests.

Isothermal Liquid Diffusivity—Experimentally determinedisothermal liquid diffusivities for Avondale clay loam—fine-loamy, mixed (calcareous), hyperthermic, Anthropic Torriflu-ventic, are shown in Fig. 1. In the figure, the solid circles andsolid triangles represent liquid flow measurements. Solid circlesrepresent soil-water diffusivities measured at several pressures(Jackson, 1965). Since water vapor flow is proportional to thereciprocal of pressure, a plot of soil water diffusivity versus thereciprocal of pressure resulted in a straight line, with the vapordiffusivity as the slope and the liquid diffusivity as the intercept.The solid triangles represent liquid diffusivity data obtained byplacing soil columns in a chamber held at 15 bars pressure dur-ing the diffusion time. The open triangles represent liquid andvapor diffusivity data taken at atmospheric pressure. The opencircles, taken from Jackson (1964a), represent both liquid andvapor flow, with the vapor predominating. The open squares

861

862 SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

10'

10

10

><n

10

10

Q

rrLJ

I

»'

10

I——0.5cmi—I cm

——i 0.5cmI——i Icm

i 0.5 cm \H I cm I25 Mat-

8 Mar.

Mar.

.05 .1 .15 .2 .25

GRAVIMETRIC WATER CONTENT.3

Fig. 1—Soil-water diffusivity vs. gravimetric water content at25C. Symbols are measured data. The solid line A was cal-culated from a polynomial equation fit to the data points.The broken lines are the sum of line A and the calculatedvapor diffusivities. Line B was assumed to improve the matchbetween calculated and measured fluxes.

represent desorption diffusivities for combined liquid and vaporflow (Jackson 1964b). The open diamonds represent data forpredominantly liquid flow at water contents near saturation(Jackson, 1963). The straight lines show the water contentranges (high, intermediate, low), for the date and depth indi-cated. The "high" water content range is high only in relationto the other two water content ranges and does not imply thatwater contents were near saturation.

To facilitate computations, equations were statistically fit tothe diffusivity data, and the results plotted as lines in Fig. 1.The solid line A represents the best fit line through the datapoints for liquid diffusivities. The solid line B was arbitrarilychosen for cases where the values from A resulted in considera-bly greater calculated fluxes than the measured values (see RE-SULTS AND DISCUSSION). The broken lines are the sum ofthe vapor and the liquid diffusivities.

The temperature dependence of the isothermal liquid diffu-sivity was assumed to be due to temperature effects on the sur-face tension (<r)-viscosity (17) ratio (Jackson, 1963). Tabularvalues alt), taken from Dorsey (1940), were used to develop

O.o0 O.I 0.2 0.3 0.4

WATER CONTENT(cm 3 cm 3 )

Fig. 3—Porosity factor, f ( e ) , versus the volumetric water con-tent.

0.05 0.10 0.15 0.20

GRAVIMETRIC WATER CONTENT

Fig. 2 — Thermal vapor diffusivities at 25C computed from"simple" theory (A), from "simple" theory times the temper-ature gradient correction (B), and the "complete" Philip andDeVries theory (C). The thermal liquid diffusivities are la-beled Dn and the horizontal broken line represents the watercontent range used in the calculation of soil-water flux.

a polynomial equation (referenced to 25C), which in turn wasused to calculate De! at various temperatures.

Isothermal Vapor Diffusivity— Isothermal vapor diffusivitiesfor sorption were calculated as the difference between measureddiffusion coefficients in relatively dry soils and the liquid dif-fusivities obtained by the pressure technique (Jackson, 1965).Coefficients for desorption were taken from Jackson (1964b),and the vapor diffusivities were computed in a similar manner,assuming that the desorption liquid diffusivities were the sameas those for sorption.

The components of Dev, which are temperature dependent,are the diffusion coefficient of water vapor in air (£>„) and thesaturated vapor density (p0). Data from Dorsey (1940) wereused to obtain polynomial equations for the temperature de-pendence of these factors, which allowed calculation of thevapor coefficients at various temperatures. The diffusivitiescalculated in this manner were in excellent agreement withmeasured data of Jackson (1965).

Thermal vapor diffusivity— The "simple" theory of vapordiffusion under thermal gradients gives the thermal vapor dif-fusion coefficient (DTv) as (Philip and DeVries, 1957),

[2]

where Da is the water vapor diffusion coefficient in air, f(e) is aporosity and tortuosity factor, h the relative vapor pressure, p(the density of water, and p0 the saturated vapor density. Therelative vapor pressure h was calculated using (Fink and Jackson,1973)

h = (!/[! + [3]

where a, B, and c are constants, and pb is the soil bulk density(g cm"3). The term dp0/dT was obtained by differentiating theequation used to describe p0 as a function of temperature. Valuesfor D7v calculated from Eq. [2] are shown as line A, Fig. 2.

Thermal Liquid Diffusivity-Philip and DeVries (1957) de-fined the thermal liquid diffusivity (DTl) as

D = [cm2 sec [4]

where K is the hydraulic conductivity (cm sec"1), \l/ the soil-water pressure head (cm), and 7 = (\/a)da/dT, the temperaturecoefficient of surface tension ("C"1)- Values of Z>Ti calculatedusing Eq. [4] were nearly zero for gravimetric water contentsless than 0.15 (Fig. 2),

JACKSON ET AL.: DIURNAL SOIL-WATER EVAPORATION 863

Eo 4

II-O- e

1.0 1.2 1.4 1.6

BULK DENSITY (gem"3)

Fig. 4—Bulk density—depth relationship.

The Porosity and Tortuosity Factor—The porosity and tor-tuosity factor may be written as (C. W. Rose, 1968a)

= D/D0 = (D/DS) (DS/D0) [5]

where D is the diffusion coefficient for an insoluble gas in amoist soil, and D0 and Ds are coefficients for the same gas inair and in dry soil respectively. The ratio D/DS accounts for thereduction of diffusion by the presence of water, and DS/D0accounts for the tortuosity or the presence of the soil matrix.Using information from Currie (1960, 1961) and C. W. Rose(1968a) and porosity data for Avondale clay loam, we deriveda relationship between f(e) and 9 (solid line, Fig. 3). In some cal-culations a linear porosity [/(e) = 0.66 e], was used (broken lineFig. 3).

The Temperature-Induced Enhancement Factor—Philip andDeVries argued that the temperature gradient that causes vaporflow is not the measurable macroscopic gradient, but the micro-scopic gradient acting across a pore. Denoting the averagemicroscopic temperature gradient across the pores as (V7\,they defined an enhancement factor

£ = (V7V VT = (VT)a/[e(VT)n

+ 8(VT)W + (1-e-fl) (VDJ [6]

where the subscripts a, w, and .s refer to the air, water, and solidphases, respectively. The meaning of (VJ")a is described morefully by DeVries (1963), and C. W. Rose (1966, 1968a).

The Water-Induced Enhancement Factor—Philip and DeVriesalso proposed a water induced enhancement factor. They arguedthat with or without a temperature gradient, vapor transferthrough soil containing air spaces is aided by liquid regions orislands. These islands of liquid are paths of low resistance andrapid transit to vapor, which can condense upstream and re-evaporate downstream from the island. Experimental data ofD. A. Rose (1963a, b) support this concept. Philip and DeVriesincluded this factor in their theory by taking the total crosssection available for transfer as equal to that occupied by airand water. So in Eq. [2], /(e) is replaced by et, the total porosity,for water contents up to the point where liquid continuity be-gins. At higher water contents they suggest replacing et withe/efc, where efc is the porosity at which liquid continuity begins.With this porosity correction and f multiplying the "simple" DTvyields the "complete" DTv, as given by Philip and DeVries (lineC, Fig. 2).

Water Content and Bulk Density—Figures 1 and 2 have gravi-metric water content as the abscissa instead of the volumetricbasis used in subsequent graphs. Volumetric water contents areobtained by multiplying the gravimetric water contents by thebulk density. Figure 4 shows the bulk density as a function ofsoil depth as used in these experiments.

0 06 1812 18 24 06 12

TIME OF DAY (hours)Fig. 5—Temperature (T), temperature gradient (VT) , volu-

metric water content (0») and water content gradient ( V f i u )as a function of time of day for 2 depths and 3 days ofmeasurement.

The diffusivity relations (Fig. 1) were obtained at one bulkdensity. Using them at other bulk densities requires referencingthe diffusivities to the gravimetric basis, while using water con-tents and water content gradients on the volumetric basis. Thediffusivity change for a given bulk density change accounted forin this manner was in reasonable agreement with the changesshown by Jackson (1962). The alternative to this approximationwould be to measure the diffusivities in situ, which is tech-nologically difficult at present.

EXPERIMENT AND CALCULATIONS

Experimental details are presented in Jackson et al.(1973) and Jackson (1973). In those reports, water con-tents were smoothed by running averages. Since then, datahave been smoothed using a Fourier transform technique(Kimball, 1974). Interpolations of water content at variousdepths and water content gradients were obtained using theparabolic spline technique reported by DuChateau et al.(1972). Temperatures and temperature gradients were ob-tained for specific times using the parabolic spline tech-nique on the temperature-depth data. Figure 5 shows thesmoothed temperatures, temperature gradients, water con-tents and water content gradients.

Measured soil-water flux data were obtained as outlinedin Jackson et al. (1973). Briefly, the flux (f0) at the soilsurface was measured with weighing lysimeters and the wa-ter content-depth relationship determined by sampling athalf-hour intervals. The water content differences for half-hour intervals were calculated for each depth and each timeperiod. The flux at the surface minus the integral of thewater content change with time between the surface anddepth z yielded the flux f(z) at depth z,

864 SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

Table I—Various combinations of diffusivities and terms within diffusivities used for the 9 sets of calculations

SetI

IIIII

IV

V.VI

VII

VIII

DC

D« (Fig- 1)line A

line BHne B

line B

(see remarks)

(see remarks)

(see remarks)

(see remarks)

D^Fig. 1)Sorption and desorp-tioo using simplehysteresis loop.

"

"

"(see remarks)

"

desorption only

sorption only

Dj^CFig. 2)line C"complete" theory

"complete" theoryline A"simple" theoryline B"simple" theorytimes tempera-ture enhancementfactor.= 0= 0

line C"complete" theory"complete" theory

"complete" theory

DXlEq. [4]

Eq. 14]Eq. [4]

Eq. [4]

= 0= 0

Eq. [4]

Eq. 14]

Eq.M

Remarks

No temperature de-pendeilce of D^, andD0V. Temperatureassumed as constant25C.Porosity functiontaken as D/DQ=0. 66eCompare with Set IIIn Table 2.Compare with Set IIin Table 2.

/» = /„— (' (ae/et)dz.•'n

The flux at other depths were obtained similarly. The "cal-culated" fluxes were obtained using Eq. [1]. Data for 3 days(8, 12, 25 March 1971) and two depths (0.5 and 1 cm)were used. The ranges of gravimetric water contents at thetwo depths for the 3 days are also given in Fig. 1.

Nine different sets of calculations summarized in Table1 were made.

RESULTS AND DISCUSSION

Soil-water fluxes were calculated for Sets I through VIfor 8 March 1971, all nine sets for 12 March 1971, and allsets but VII for 25 March 1971. These dates were selectedto show data for relatively high, intermediate, and air-drywater contents. The values were compared with the meas-ured flux data. As an index of how well the calculatedvalues matched the measured data, the sums of the absolutevalues of the differences for 24-hour periods were computedand are shown in Table 2. A perfect match would have anindex of zero.

For the data of 8 March the indices were largest for Set Iat both the 0.5- and 1-cm depths. Set I utilized completePhilip and DeVries DTv and the available experimental datafor Del (line A, Fig. 1). Figure 6 (lower half) shows themeasured (Jm) and the calculated (Jc) soil-water fluxes forthe 0.5- and 1-cm depths. For both depths, the calculatedvalues were greater than the measured until near noon, andthen were lower than the measured. The components of thecalculated flux Jgl, Jgv, JTv, and JTl are shown in the top half

Table 2—Sums of the absolute value of differences betweenmeasured and calculated soil-water flux for 48 time

periods (cm sec"1)

Set

I11

IIIIVV

VIVII

VIIIDC

8 March 710. 5cm1.370.800.600.660.590.83

_--

1 cm1.590.850.840.830.851.09

_--

12 March 710.5cm

0. 190.170.500.330.720.360.170.270.63

1cm0.270.260.350.210.490.360.260.310.66

25 March 710.5cm

0.540.540.270.390.130.39

-0.540.43

1cm0.470.470.210.310.200.35

-0.490.45

of Fig. 6. At the 0.5-cm depth the flux due to "isothermal"liquid flow,Jn , was the largest component, except for a fewafternoon hours. The vapor components, Jgv and JTv wereopposite in sign during the day, and at the 0.5-cm depthnearly cancelled each other. At 1-cm, Jev was very small,while JTv was significant.

The data in Fig. 6 indicated that the values of Del, shownin line A, Fig. 1, were too large. The lower diffusivities,shown in curve B were therefore assumed, and were used incalculations for subsequent sets. The indices for Set 11(Table 2) show that diffusivities using curve B improved theoverall match between measured and calculated fluxes, butdid not improve the match during the afternoon (line Jc II,Fig. 6).

The lowest index for 8 March data was obtained for SetV, i.e. DTv — DTl = 0. These results are shown as the lineJCV (lower half, Fig. 6). For the 0.5-cm depth the calculatedfluxes (/CV) came reasonably close to predicting the meas-ured (Jm). For the 1-cm depth, however, the measured andcalculated fluxes were out of phase for both Sets II and V.

The indices shown in Table 1 for the data of 12 Marchindicate that Sets I and II both predicted the measured fluxquite well. Little difference exists between the two sets be-cause diffusivity curve B joins curve A at these water con-tents (Fig. 1). The lowest index for the 1-cm depth is forSet IV. Figure 7 shows the calculated, measured, and com-ponent fluxes for Set II, 12 March. For the 1-cm depth thecalculated flux and the thermal vapor flux are also given forSet IV.

The difference between Sets II and IV is that the water-induced enhancement factor of Philip and DeVries was notused in calculating DTv in Set IV. This reduced DTv by afactor of about 2. The lines in Fig. 7 labeled /c II and JcIV show that the calculated and measured fluxes are morenearly in phase for Set IV, and more closely approximatethe measured flux.

The data for 25 March (Fig. 8) are for water contentsthat were essentially air dry. As expected, the indices inTable 2 show essentially no difference between Sets I andII. The lowest index is for Set V (DTv = DTl = 0). Fig-ure 8 shows the calculated (/c), measured (/„,), and com-ponent fluxes for Set II and the calculated fluxes for Set VC/CV). The calculated fluxes for Set II, (7CII) are out of

JACKSON ET AL.: DIURNAL SOIL-WATER EVAPORATION 865

06 18 2412 18 24 06 12TIME OF DAY ( h o u r s )

Fig. 6—Measured soil-water flux (}m), calculated flux (Jc) ,"isothermal" liquid flux (78[), "isothermal" vapor flux (]ov),thermal vapor flux ( /TU), and the thermal liquid flux On)for 2 depths, data of 8 March 1971. Roman numerals follow-ing flux symbols indicate the calculation set.

phase and sometimes opposite in sign to the measuredfluxes. The line for JCV very nearly predicts the measuredflux.

Reconciliation of Observations—Considering the datafor relatively high water contents (Fig. 6) one might con-clude that the theory underlying Eq. [1] does not predictmeasured fluxes with sufficient accuracy. The terms per-taining to the thermally driven components do not appearneeded, and the "isothermal" components yield values notin phase with the measured fluxes. The data for air drywater contents (Fig. 8) partially support this observation,because the thermally driven components were not needed,whereas the "isothermal" components provided calculatedfluxes of acceptable accuracy. Results for intermediate wa-ter contents (Fig. 7), however, indicate that the theory ofPhilip and DeVries can quite adequately predict measuredfluxes. These results agree qualitatively with laboratory re-sults reported previously. Philip and DeVries (p. 222)stated that "many investigators have observed that moisturetransfer under temperature gradients is negligibly small bothin very dry and in very wet media, but attains a fairly well-defined maximum at an intermediate moisture contentwhich appears to depend both on the soil-water tension andon air-filled porespace."

The above observations should be tempered by a consid-eration of the sensitivity of flux calculations to the value ofthe diffusion coefficients. A prime example of this is shownin Fig. 7 for the 1-cm depth. DTv for Sets II and IV are dif-ferent by about a factor of 2. Thus a change in DTv by afactor of 2 shifted the calculated line from out of phase toa reasonably good match with the measured values. Thus,

X3

cru

oCO

06 12 18 24 06 12 18 24

T I M E OF DAY ( h o u r s )Fig. 7—Same as Fig. 6 except for 12 March 1971.

one must know values of the diffusion coefficients quite ac-curately in order to use the theory to predict measured val-ues. The coefficients used for the 8 March and 25 Marchdata may not be sufficiently accurate. In the case of the"isothermal" coefficients, inaccuracy may have resulted

0.5

0.4

0.3 -

0.2

O.I

0

-O.I

-0.2

0.5

0.4

0.3

0.2

0.1

0

-O.I

-0.2

Fig,

0.5cm

- 0.5 cm

I cm

. . Icm

06 12 18 24 06 12 18 24

T I M E O F D A Y ( h o u r s )8 — Same as Fig. 6 except for 25 March 1971.

866 SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

from errors in measurement, errors in interpolation at wa-ter contents where data were unavailable, and from themethod used to obtain the temperature dependence of thecoefficients.

The thermal liquid and vapor coefficients were calculatedusing the theory of Philip and DeVries (1957). In derivingtheir theory for the calculation of DTv, they resorted to sim-plifying assumptions and emphasized that ". . . this treat-ment must be regarded as no more than a first approxima-tion . . ." We conclude that it is a good approximation, butit may need to be improved as better data become available.

At the higher water contents, hysteresis of the diffusioncoefficient-water content relation was ignored because oflack of data. Conceivably this factor may be responsible,at least in part, for the poor prediction of the 8 March data.

Additional Observations—Additional observations canbe made from examination of the indices given in Table 1.Comparison of the indices for 12 March, Sets II and VIIshow that the functional porosity and linear porosity, asshown in Fig. 3, yielded equivalent values for the calculatedflux. This was supported by graphs (not shown) similar toFig. 7. Thus, the flux calculation does not appear very sen-sitive to the porosity function.

As discussed earlier, a simple hysteresis loop was as-sumed for the "isothermal" vapor diffusivities. A compari-son of indices for 12 March, Sets II, VIII and IX shows thatthe set with the hysteresis loops (Set II) had the lowest in-dices of the three. The indices for the sorption vapor dif-fusivities were considerably larger than the desorption dif-fusivities.

Indices for Set VI for all three measurement days indicatethat the "isothermal" diffusion coefficients must be knownas a function of temperature, to obtain accurate predictions.Graphs of these values (not shown) demonstrate the needfor temperature-dependent diffusivities.

Of all nine sets of calculations, Set IV had the lowestoverall index (0.455, the average for the six measurementsin Table 2). This indicates that the "isothermal" theory un-derestimates and the "complete" theory overestimates themeasured fluxes. More accurate values of the diffusion co-efficients could well resolve the discrepancies.

CONCLUSIONS1) Our results indicate that the theory of Philip and

DeVries best predicts soil-water flux under diurnal fieldconditions at intermediate water contents. In general,the "complete" theory overestimates and the "isother-mal" theory underestimates the measured fluxes.

2) Diffusion coefficients used to calculate soil-water fluxunder field conditions must be accurately known, topredict reasonable flux values.

3) The temperature dependence of the "isothermal" dif-

fusion coefficients must be known for accurate predic-tions.

4) The calculated soil-water flux seems insensitive to theform of the porosity function.

5) Hysteresis in the "isothermal" vapor diffusion coeffi-cients should be accounted for.