SCIENCE SOCIETY OF AMERICA

PROCEEDINGSVOL. 38 MAY-JUNE 1974 No. 3

DIVISION S-l—SOIL PHYSICS

Prediction of Water Loss from a Fallow Field Soil Based on Soil Water Flow Theory1

H. R. GARDNER2

ABSTRACT

A simple technique based on soil water flow theory to predictevaporation from nonhomogeneous field soils is presented.From cumulative evaporation measurements on undisturbedcores of soil, a dimensionless curve is drawn relating fractionalwater loss to the square root of time divided by the amount ofwater available for evaporation. A procedure is described toaccount for the residual water left in the soil from one rainfallevent to the next. Predictions of cumulative evaporation madeby the use of the dimensionless curve were compared with 4years' of lysimeter data on Rago silt loam. The system wouldwork best in an area with high average potential evaporationand low rainfall such as the Great Plains.

Additional Index Words: evaporation, diffusivity, soil mois-ture.

rr\HE THEORY applied by Gardner and Gardner (1969) forJ_ describing water loss by evaporation was based on the

diffusivity equation and involved a solution requiring aconstant water content at the soil surface and a finite depthof wetting. Plots of 0/0; vs (D0t/L2)V> were presented inFig. 5 and 6 of Gardner and Gardner (1969) where 6 wasthe amount of water lost by evaporation; 6{, the initialamount of water present for evaporation; t, the time; L, thedepth of wetting; and D0, the diffusivity of air dry soil.Their solution to the diffusivity equation used an exponen-tial relationship between the diffusivity and the water con-tent.

A previous paper by Gardner (1973) described the fol-lowing technique for estimating evaporation based on theabove theory. A known amount of water was added to eachof a series of packed homogeneous soil columns. The depthof wetting and the cumulative amount of water evaporationversus time were measured. These data were used to con-

1 Contribution from ARS, USDA, in cooperation with Colo-rado State Univ. Exp. Sta. Scientific Journal Series 1867. Re-ceived 16 July 1973. Approved 15 Feb. 1974.2 Soil Scientist, USDA, Fort Collins, Colorado.

struct a plot of 0/0{ vs. (D0t/L?)V2. This plot was com-pared with the theoretical plot described above to obtainvalues of Di/D0 and D0. On the basis of these values, theamount of evaporation from this soil at a given time couldthen be predicted from the amount of water added and thedepth of wetting.

For use in a field soil, the above technique presentsseveral complications. The depth of wetting cannot be con-veniently measured. The water distribution within the soilafter wetting often is not uniform. The potential evapora-tion rate is not constant. The cumulative evaporation-timecurve that is produced from the evaporation data from fieldsoil is different from that curve produced from soil pre-pared in the laboratory because the field soil is not uniformwith depth in either texture or density. The field structureand texture tend to reduce the cumulative evaporation sig-nificantly below that of homogeneous soil. This paper pre-sents the modifications and assumptions necessary to adaptthe methods suggested by Gardner (1973) for use in thefield and compares the calculated seasonal cumulative evap-oration with evaporation from bare weighing lysimeters.

PROCEDURE

Four acrylic plastic columns 30 cm in length and 10.5 cm indiameter were pushed vertically into the soil surface on an areathat had been treated normally for a fallow soil with no me-chanical disturbance for about 30 days before sampling. Thesoil was Rago silt loam and had been rained on several timesbefore the sampling which left the surface in a rather puddledcondition. The columns were removed and the soil protrudingfrom the bottom smoothed off and bottom plates installed onthe columns. The columns were oven dried at 105C andweighed daily until constant weight was obtained.

The columns were then placed on a 1-m-diameter turntablewith a rotation of 2 rpm. Heat lamps were placed over the circleof rotation to achieve approximately 0.75 cm/day evaporationfrom a free water surface in a column of the same diameter andheight as the soil columns.

The treatments in duplicate were 1 and 3 cm of water addedto the soil from the top. The columns were weighed daily atfirst, then less often after the evaporation rate had slowed. Theweights were converted to centimeters of water lost.

379

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

WATER CONTENT —»

J

Fig. 1—Hypothetical water content distribution after tworains. A is the distribution several days after a large rain.B is the distribution after a small rain. C is the assumed dis-tribution after A and B are added together.

For simplification, the depth of welting, L, is replaced by theamount of water present to be evaporated since the depth ofwetting is quite closely related to the amount of water added tothe soil. MI is substituted for 6-t from Gardner and Gardner(1969) since H usually stands for volumetric water content.No attempt is made to match a theoretical dimensionless curveto evaporation from undisturbed test columns of field soil forIhe reasons noted earlier. The assumption that D0 = 1 is madeand M/M, vs. (t/MfY^ is plotted for each treatment (Fig. 2).

The field data used for comparison with the predictions weretaken from four fallow lysimeters on Rago silt loam in the samegeneral area from which the undisturbed columns were taken.The lysimeters were 1 m2 and 0.9 m deep. They were includedin a crop rotation experiment and each had been bare fromOctober of the year prior to taking the evaporation data. (Theyears included are 1965, 1966, 1967, and 1968.) Winter wheat(Triticum ae.itivum L.) was planted approximately 15 Septem-ber at the end of the fallow year. Since the lysimeters were ina rotation the data for each year is from a different lysimeter.Thus, the evaporation predictions include the variability amonglysimeters.

After each rain and its subsequent evaporation period theremay be unevaporated water remaining in the soil. This watercannot be neglected since it makes a contribution to the storedwater and should be taken into account with the next rain andevaporation period (Gardner, 1973). The water in the soilimmediately after a rain at the beginning of an evaporationperiod was assumed to be uniformly distributed with depth.The degree to which this matches reality depends on how muchof the water from the last addition was lost and how much wasadded in the next rainfall. If the water added did not return thewater content to the previous amount in the profile, then the dis-

PACKED COLUMNS

.UNDISTURBED COLUMNS

• DATA POINTS

O I 2 3 4 5 6(t/M;2)"2

Fig. 2—Fractional water lost compared to square root of timeof evaporation in dimensionless terms.

tribution of water within the profile might look something likea combination of curves A and B in the hypothetical case pre-sented in Fig. 1 where A remains from the first rain and B isadded from the second rain. The assumption of a uniform dis-tribution of water similar to curve C in Fig. 1 at the beginningof each period would not hold. If the rain sequence for a seasonincluded a few large rains interspersed with many small rainsand the calculation of evaporation began again at the time ofeach rain, the calculated loss would be too large.

The technique that is used to adjust for the nonuniformityof water content in the profile is to compare the remaining waterin the soil plus the amount received in the next rainfall with theprevious rainfall. If the sum of the rainfall plus residual wateris more than the previous rainfall, then the assumption of uni-form distribution with depth is used and the evaporation isdetermined in the usual manner. If the sum is less than theprevious addition, then the water is assumed to be in a distribu-tion similar to that of curves A and B combined in Fig. 1. Theresidual water and the new added water in the profile are con-sidered as two independent quantities. The calculation of evapo-ration from the previous rainfall is extended for this additionaltime period, the new water is treated as new rain, and its cumu-lative evaporation is calculated alone. The two resultantamounts are then summed. The process is then repeated forthe next rain. Any residual water left from the second calcula-tion is added to the residual of the first so that only one residualamount is carried forward.

RESULTS AND DISCUSSION

The cumulative evaporation data from the undisturbedcores were plotted in Fig. 2 as previously noted. The curvelabeled "Packed columns" in Fig. 2 was a theoretical curveobtained from Gardner (1973). This curve was matched todata obtained using columns of Rago soil that had beensieved through a 2-mm sieve and packed in the columns.This curve is presented for comparison with the curve fromthe undisturbed soil. The maximum amount of water thatcan be applied to these columns of soil without violating

Table 1—Table of values used in a sample calculation of cumulative evaporation for Bago silt loam

Line

122a34

1

Time

11

16

2Cumulative

time

2

7

3

Rain

6.6040.33

4.340.203

4Rain +

residual

7.483(7.10)

,(10.79),10. 873

(10. 282)

5

(t/Mj) V2

0.1333.0300.1890.091

12. 0660.243

6

M/M.

0.0950.7510.1380.0730.9560.172

7

M

0.7100.2481.0320.7930.1941.870

8Cumulative

M

0.710

1.2802.074

3.344

9

Residual

6.7720.0826.450

10.0790.0099.003

10Cumulative

residual

7.100.082

0.0090.009

GARDNER: PREDICTION OF WATER LOSS FROM A FALLOW FIELD SOIL 381

the lower boundary condition is that amount that will wetthe soil just to the bottom of the column. This limit is nec-essary since the depth of wetting is assumed to be directlyrelated to the water content and the water added is beingused in the place of the depth of wetting in determining thecurve.

An example of the procedure for predicting cumulativeevaporation is presented here by the use of a series of fourrains and their respective evaporation times from rainfalldata of 1965. The necessary numerical quantities areshown in Table 1.

Line 1—First Rain

1) Calculate (tlM^j* from time and rain plus residualwater. The residual comes from a previous rain.(0.134)

2) Determine MIMi from the value on curve. (0.095)3) Calculate M from M/Mt. (0.711)4) Find residual water from M{-M. (6.773)

Line 2, Second Rain

1) Add second rain (0.33) to residual from Line 1(6.773) and enter in column 4, rain plus residual(7.10). This residual quantity is less than the rainplus residual of Line 1; therefore, rain 2 will be con-sidered separate and rain 1 will be recalculated.

2) Calculate (t/M?)* for rain 2 alone. (3.03)3) Determine MJMi from the graph. (0.751)4) Determine M from M/Mt. (0.248)5) Determine the residual from MCM. (0.082) Enter

this in column 10.

Line 2a

1) Recalculation of evaporation from rain 1. Use thecumulated time from rains 1 and 2 entered in column2 (2 days) and the original rain plus residual fromrain 1 (7.48) to determine (t/Mf)^. (0.189)

2) Find M/Mi from the graph. (0.138)3) Determine M. (1.032)4) Determine the cumulative M by subtracting the pre-

vious M from rain 1, line 1 (0.711) and adding in theloss M from the recalculation of rain 1 (1.032) andalso the loss from rain 2 (0.248) to get a total of(1.28).

5) Determine residual for line 2a by subtracting cumu-lative M for 2a from the rain plus residual for rain 1(line 1). (6.45)

Line 3—Third Rain

1) Add the residual from line 2a, column 9, to the rainfor rain 3 (6.45 + 4.34 = 10.79). This value isgreater than rain 1 plus its residual so the processwill begin again.

2) Cumulative residual from line 3 is added to the10.79. (10.87)

3) (t/M?)* is calculated. (0.092)4) M/Mi is found from the graph (Fig. 2). (0.073)5) M is calculated. (0.793)

30

Ul

5O0

1965

DATA —

PREDICTION

120 140 160 180 200DAY OF YEAR

220 240 260

Fig. 3—Predicted and measured cumulative evaporation froma Rago soil lysimeter for 1965.

30

1966

- DATA

PREDICTED

o-20

>tu

5

120 140 160 180 200DAY OF YEAR

220 240 260

Fig. 4—Predicted and measured cumulative evaporation froma Rago soil lysimeter for 1966.

6) M is accumulated in column 8. (2.074)7) Residual is found by subtracting M from Mf. (10.079)

Line 4—Fourth Rain1) Residual (10.079) plus rain (0.203) is calculated

(10.282). This is less than the rain plus residual forrain 3 so this calculation will be the same as for lines2 and 2a.

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

30

o

z'20o4ecoo_<I>LJ

UJ

*»<_i3

1967

- DATA

PREDICTED

120 140 160 180 200DAY OF YEAR

220 240 260

Fig. 5—Predicted and measured cumulative evaporation froma Rago soil lysimeter for 1967.

The procedure for predicting cumulative evaporationpresented above was applied to the rainfall data for 1965,1966, 1967, and 1968 and the predicted cumulative evapo-ration curves are presented in Fig. 3, 4, 5, and 6, respec-tively. The measured losses as determined from the lysime-ter weights are also included. The predicted evaporationagrees well with the measured evaporation when the timeconsidered is extended. A short time comparison of a fewdays shows more deviations. The prediction was made onlyfor the time period for which lysimeter data were available.The lysimeters are not reliable during the winter since snowcaught in the lysimeters may differ from that caught in arain gauge.

The prediction of cumulative evaporation assumes nowater loss by movement downward out of the zone of con-sideration. In this case the predicted losses by evaporationwere compared with water lost from lysimeters which hadno drainage. A prediction of loss from the field, if checkedby soil water measurements, could appear to be too low de-pending on how much water moved downward. The drain-age could be a significant amount if the initial water contentin the soil, was high and the time period considered waslong.

Comparison of the shape of the curve presented in Fig.1 with that used by Black et al. (1969) showed that, forhis situation, the loss per cycle did not depart from thesquare root of time curve. In our case, the cumulative evap-oration after almost every small rain departed from thesquare root of time, so the determination of a square rootof time curve alone was not sufficient. The portion of thecurve that departed from the square root of time couldbe determined only from undisturbed core data. No datapoints in Fig. 2 exceeded a fractional loss of 0.73. The re-mainder of the curve was extrapolated. Points beyond 0.73could have been obtained if the evaporation had continued

5og20

<a:oa.<>UJ

UJ

pio<t_i3

UO

1968

DATA

PREDICTED

120 140 160 180 200DAY OF YEAR

220 240 260

Fig. 6—Predicted and measured cumulative evaporation froma Rago soil lysimeter for 1968.

for a longer time, but the position of this part of the curveis not critical since this part is not reached except for verysmall rains.

The potential evaporation rate in the field is variable butthe only time that the external environment has any signifi-cant effect on the evaporation rate is during the time whenthe water supply is sufficient to allow the external energysupply to be limiting (Gardner and Hillel, 1962). A largereduction in evaporation rate during the energy limitingphase is necessary before the cumulative loss is affected sig-nificantly, as pointed out in Gardner and Gardner (1969).In a humid region, the effect of the potential evaporationrate on the cumulative evaporation would be greater thanin a drier region because of the greater proportion of thetotal evaporation that takes place in the energy limitingphase. In the Central Great Plains, because of the low rain-fall and high potential evaporation throughout the growingseason, the energy supply to the soil surface can be neg-lected as a variable in the evaporation when the entiregrowing season is considered.

In areas of limited rainfall such as the Great Plains, theuse of the technique to estimate evaporation from bare soilworks well for time periods of a few weeks or more. Shorttime comparisons of a few days would show more deviationsince the potential evaporation rate would have more sig-nificance. A different plot of M/Mt vs. (t/Mf)^ wouldhave to be obtained for each soil since it is now known howthe curves for field soils of similar texture would compare.