influence of different diffusivity-water content relations on evaporation of water from soils1

4
Influence of Different Diffusivity-Water Content Relations on Evaporation of Water from Soils 1 R. J. HANKS AND H. R. GARDNER L ' ABSTRACT Estimates of the influence of variations in the diffusivity- water content relation on evaporation showed that where the diffusivity was changed at high water contents, evaporation was changed accordingly. However, variations in the D(0) rela- tion at lower water contents showed little influence on cumu- lative evaporation. Estimates were also made for layered soils with the D(0) relation for one layer 10 times that of the other layer. Cumula- tive evaporation increased significantly, if the diffusivity of the bottom layer was greater, as the depth of the surface layer decreased, provided the surface layer depth was less than half the total depth of the soil. Where the diffusivity was greater for the surface soil, cumulative evaporation increased as the depth of the surface layer increased. All of the estimates were made based on the assumption that a diffusion-type equation is valid for evaporation. E VAPORATION of water directly from the soil is the process by which most of the precipitation received in many parts of the world is returned to the atmosphere. In moisture-deficient areas such as the Great Plains, it is estimated that about 60% of the precipitation is evapo- rated back to the atmosphere. Because of this large loss, much work has been done to develop methods for increas- ing the water available for plant growth, or other pur- poses, by decreasing evaporation from the soil. Mulches of various types have been advocated as management prac- tices for decreasing evaporation. However, as discussed by Army et al. (1), mulches effectively reduce evaporation only during the early stages of drying when evaporation is determined primarily by climatic factors. During the falling rate stage of drying, evaporation is determined largely by the water flow properties of the soil. During this stage, the soil surface dries and forms a "self-mulch" which is a more effective mulch than gravel or straw (4). In regions of limited rainfall, evaporation occurs most of the time in the falling rate stage. Thus, it would appear that any process that would substantially decrease evapora- tion would necessarily have to decrease evaporation during the falling rate stage of drying. The one-dimensional flow equation where O is the volumetric water content, D($) is the soil- water diffusivity (includes both liquid and vapor flow), x is distance, and t is time, has been used by many workers to describe the evaporation of water from soil (2, 3 and 9). While the flow equation is based on the assumption of isothermal conditions and it is well known that evaporation cannot be an isothermal process, Gardner (2), Gardner 1 Contribution from the Northern Plains Branch, Soil and Water Conservation Research Division, ARS, USDA. Received Feb. 10, 1965. Approved May 10, 1965. s Research Soil Scientists, USDA. Fort Collins, Colorado. and Hillel (3), and Philip (8) have indicated that the diffusion equation describes the major features of evapora- tion. Jackson (7) has shown that the flow equation holds for the three soils studied for desorption of relatively dry soil under isothermal conditions. The investigation re- ported herein was conducted to determine how certain changes in D(#) relation influenced evaporation. Both a uniform and two-layered soil were studied. While only one soil type was studied, the properties are similar to many others, as discussed elsewhere (6), and the results should be generally applicable. PROCEDURE The general procedure was to calculate evaporation from a "soil" having a finite depth (L) initially at a uniform water content. The soil used as a model was Fort Collins silty clay loam at an initial volumetric water content of 0.32 (% bar per- centage). Calculations were made for various D(0) relations. For the "soils" studied, these conditions allowed a constant evapora- tion rate for a relatively short time so that the results apply pri- marily to the falling rate stage of drying. The flow equation was solved using a modification of the numerical method described by Hanks and Bowers (5, 6) for infiltration. While the numerical method allowed for the solution of the more general flow equation where the gravity component is included, preliminary computations showed the gravity com- ponent to have no significant influence on the results. The numerical procedure required only a slight modification for solution of the evaporation problem. The modification involved the surface boundary condition, ER < A, where ER is evapora- tion rate and A is a constant rate (approximating constant evapo- rative demand). For the infiltration problem, the surface boundary condition was 8 = 0 s < sa turatto,,> for all t > 0. An additional stipulation on the surface boundary condition for evaporation was 0< S urfnce> = 0. <air dry> when ER < A. Thus, the surface water content varied from the initial water content, 0i, down to 0 a dur- ing the time that ER ,= A. The relations used are listed as follows: 1) Diffusivity related to water content by the relation DO in cmVday = 9.53 X 10" 2 exp!7.83 9. ABC of Fig. 1. 2) D(0) larger than ABC on dry end. Curve DEC of Fig. 1. 3) D(0) smaller than ABC on dry end. Curve EBC on Fig. 1. 4) D(0) larger than ABC on wet end. Curve AEF of Fig. 1. 5) D(0) smaller than ABC on wet end. Curve ABG of Fig. 1. 6) Layered soil with D(0) = 9.53 X 10~ 2 expl7.83 S (ABC) above a soil with D(0) = 9.53 X ICT 1 exp!7.830 (10 ABC). 7) Layered soil with D(0) = 9.53 X KT 1 expl7.83 B (10 ABC overlying a soil having a D(0) := 9.53 X 10~ 2 exp!7.83 6 (ABC). The "standard" relation used (ABC of Fig. 1) was taken from some crude measurements made by the authors for Fort Collins silty clay loam. While the data are undoubtedly subject to large absolute errors, they are realistic enough for purposes of this investigation. The computations were made using the following initial and boundary conditions: ER < 0.75 cm/day; 0i = 0.32, t := O for all x; = 0.01, t > O at x = O when ER < A; and column length "L" ,= 30 cm with 20 increments (Ax ,= 1.5 cm). The initial water content, 0i, was chosen as being just slightly wetter than required to maintain the ER at 0.75 cm/day at the 495

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Page 1: Influence of Different Diffusivity-Water Content Relations on Evaporation of Water from Soils1

Influence of Different Diffusivity-Water Content Relations on Evaporationof Water from Soils1

R. J. HANKS AND H. R. GARDNERL'

ABSTRACTEstimates of the influence of variations in the diffusivity-

water content relation on evaporation showed that where thediffusivity was changed at high water contents, evaporationwas changed accordingly. However, variations in the D(0) rela-tion at lower water contents showed little influence on cumu-lative evaporation.

Estimates were also made for layered soils with the D(0)relation for one layer 10 times that of the other layer. Cumula-tive evaporation increased significantly, if the diffusivity of thebottom layer was greater, as the depth of the surface layerdecreased, provided the surface layer depth was less than halfthe total depth of the soil. Where the diffusivity was greaterfor the surface soil, cumulative evaporation increased as thedepth of the surface layer increased.

All of the estimates were made based on the assumption thata diffusion-type equation is valid for evaporation.

E VAPORATION of water directly from the soil is theprocess by which most of the precipitation received in

many parts of the world is returned to the atmosphere.In moisture-deficient areas such as the Great Plains, it isestimated that about 60% of the precipitation is evapo-rated back to the atmosphere. Because of this large loss,much work has been done to develop methods for increas-ing the water available for plant growth, or other pur-poses, by decreasing evaporation from the soil. Mulchesof various types have been advocated as management prac-tices for decreasing evaporation. However, as discussed byArmy et al. (1), mulches effectively reduce evaporationonly during the early stages of drying when evaporationis determined primarily by climatic factors. During thefalling rate stage of drying, evaporation is determinedlargely by the water flow properties of the soil. During thisstage, the soil surface dries and forms a "self-mulch"which is a more effective mulch than gravel or straw (4).In regions of limited rainfall, evaporation occurs most ofthe time in the falling rate stage. Thus, it would appearthat any process that would substantially decrease evapora-tion would necessarily have to decrease evaporation duringthe falling rate stage of drying.

The one-dimensional flow equation

where O is the volumetric water content, D($) is the soil-water diffusivity (includes both liquid and vapor flow), x isdistance, and t is time, has been used by many workers todescribe the evaporation of water from soil (2, 3 and 9).While the flow equation is based on the assumption ofisothermal conditions and it is well known that evaporationcannot be an isothermal process, Gardner (2) , Gardner

1 Contribution from the Northern Plains Branch, Soil and WaterConservation Research Division, ARS, USDA. Received Feb. 10,1965. Approved May 10, 1965.

s Research Soil Scientists, USDA. Fort Collins, Colorado.

and Hillel (3), and Philip (8) have indicated that thediffusion equation describes the major features of evapora-tion. Jackson (7) has shown that the flow equation holdsfor the three soils studied for desorption of relativelydry soil under isothermal conditions. The investigation re-ported herein was conducted to determine how certainchanges in D(#) relation influenced evaporation. Both auniform and two-layered soil were studied. While only onesoil type was studied, the properties are similar to manyothers, as discussed elsewhere (6), and the results shouldbe generally applicable.

PROCEDUREThe general procedure was to calculate evaporation from a

"soil" having a finite depth (L) initially at a uniform watercontent. The soil used as a model was Fort Collins silty clayloam at an initial volumetric water content of 0.32 (% bar per-centage). Calculations were made for various D(0) relations. Forthe "soils" studied, these conditions allowed a constant evapora-tion rate for a relatively short time so that the results apply pri-marily to the falling rate stage of drying.

The flow e q u a t i o n was solved us ing a modification of thenumerical method described by Hanks and Bowers (5, 6) forinfiltration. While the numerical method allowed for the solutionof the more general flow equation where the gravity componentis included, preliminary computations showed the gravity com-ponent to have no significant influence on the results.

The numerical procedure required only a slight modificationfor solution of the evaporation problem. The modification involvedthe surface boundary condition, ER < A, where ER is evapora-tion rate and A is a constant rate (approximating constant evapo-rative demand). For the infiltration problem, the surface boundarycondition was 8 = 0 s < s a t u r a t t o , , > for all t > 0. An additionalstipulation on the surface boundary condition for evaporation was0 < S u r f n c e > = 0. < a i r dry> when ER < A. Thus, the surface watercontent varied from the initial water content, 0i, down to 0a dur-ing the time that ER ,= A.

The relations used are listed as follows:1) Diffusivity related to water content by the relation DO in

cmVday = 9.53 X 10"2 exp!7.83 9. ABC of Fig. 1.2) D(0) larger than ABC on dry end. Curve DEC of Fig. 1.3) D(0) smaller than ABC on dry end. Curve EBC on Fig. 1.4) D(0) larger than ABC on wet end. Curve AEF of Fig. 1.5) D(0) smaller than ABC on wet end. Curve ABG of Fig. 1.6) Layered soil with D(0) = 9.53 X 10~2 expl7.83 S (ABC)

above a soil with D(0) = 9.53 X ICT1 exp!7.830 (10 ABC).7) Layered soil with D(0) = 9.53 X KT1 expl7.83 B (10

ABC overlying a soil having a D(0) := 9.53 X 10~2 exp!7.836 (ABC).

The "standard" relation used (ABC of Fig. 1) was taken fromsome crude measurements made by the authors for Fort Collinssilty clay loam. While the data are undoubtedly subject to largeabsolute errors, they are realistic enough for purposes of thisinvestigation.

The computations were made using the following initial andboundary conditions:

ER < 0.75 cm/day;0i = 0.32, t := O for all x;0» = 0.01, t > O at x = O when ER < A;

andcolumn length "L" ,= 30 cm w i t h 20 i n c r e m e n t s (Ax ,=

1.5 cm) .

The initial water content, 0i, was chosen as being just slightlywetter than required to maintain the ER at 0.75 cm/day at the

495

Page 2: Influence of Different Diffusivity-Water Content Relations on Evaporation of Water from Soils1

496 SOIL SCIENCE SOCIETY PROCEEDINGS 1965

start of evaporation for relation ABC. Thus, computations weremostly confined to the conditions where soil properties had a largeinfluence on evaporation (the falling rate stage of evaporation).

RESULTS AND DISCUSSIONThe cumulative evaporation as a function of time is

shown for several of the treatments in Fig. 2. The datashow that variations in the D($) relation on the dry endof the scale, relations DEC and EEC, caused no significantchange in cumulative evaporation even after 20 days.

However, where the D($) relation was changed on thewet end, the cumulative evaporation was markedly influ-enced. For example, after 8 days, the cumulative evapora-tion for relation AEF was 2.2 cm compared to 1.3 cm forABC and 0.83 cm for ASG. Thus, it is evident thatchanges in the D(#) relation on the wet end have a largeinfluence on evaporation, but variations on the dry endhave little influence on evaporation. This is also in quali-tative agreement with the analysis of Gardner and Hillel(3).

These results seem reasonable from consideration of the#(x) profiles for the different variations. The numericalmethod used also gave Q as a function of x and t. Figure 3shows these data for the different treatments where thecumulative evaporation was about 1.7 cm. The $(x) curvesare all essentially the same for curves ABC, EEC, andDEC as would be expected from the data of Fig. 2. CurveAEF shows a smaller 6 at the bottom of the column anda larger 6 at the top of the column than ABC. AEG hasa larger B at the bottom of the column and a smaller O atthe top of the column than ABC. The average water con-tent for all curves is the same. The water contents for allcurves were greater than 0.10 for all depths except the

100 -

surface. Therefore, variations in the D(0) relation belowB =. 0.10 were important only very near the surface. TheD($) relation was the same for curves ABC, DEC, andEEC for all depths below 1.5 cm which was the bulk ofthe soil. Likewise, the water contents below 1.5 cm forcurves ABC, AEF, and AEG were all above 0.10 and theD (#) curves, where water movement was taking place, wereall different. This resulted in a different evaporation. Thetime at which the cumulative evaporation reached 1.7 cmwas 4.6, 12.6 and 28.4 days for curves AEF, ABC, andABG, respectively.

While the equation of Gardner and Hillel (3) appearsto describe evaporation qualitatively, the equation appearsto be quantitatively inadequate.3 Table 1 shows a com-parison of the evaporation rate for

3The Gardner and Hillel (3) equation isER .= D(ft,vE) WT2/4 La

where W is the water content of a column of soil of length, L,expressed as an equivalent depth of water (i.e., cm), D(0»vB) isthe diffusivity corresponding to the average water content of thecolumn (e.vB) = W/1).

2.2 -

TIME IN DAYS

Fig. 2—Cumulative evaporation as a function of time for theseveral variations in the D(0) relation studied.

W A T E R C O N T E N T - V O L U M E.10_______J5______.20 .30

.05 .10 .15 .20 .25WATER CONTENT-VOLUME

Fig. 1—Diffusivity-water content relations studied.

.30 Fig. 3—Water content vs. depth for the several D(ff) relationsstudied. The cumulative evaporation for all curves was about1.7 cm which was reached at 4.6, 12.6, and 28.4 days forcurves ABF, ABC, and ABG, respectively.

Page 3: Influence of Different Diffusivity-Water Content Relations on Evaporation of Water from Soils1

HANKS AND GARDNER: INFLUENCE OF DIFFERENT DIFFUSIVITY-WATER CONTENTS ON EVAPORATION 497

Table 1—Comparison of evaporation rate computed by themethod of Gardner and Hillel (G & H) and that computed

by the numerical method (NM); @,Tg is the averagevolumetric water content for a 30-cm soil column

eavg G & H

cm/day

NM

cm/day

Ratio eavg

Curve ABC0.300.290.280.270.260.250. 24

0.300.290.280.270.260.25

0.490.410.330.260.210.170.14

Can0.1140.1010.0850.07.10. 0670.060

0.330.190.130.100. 0800.0670. 055

re ASG0.1420.0850.0610.0460. 0370.033

1.52.12.52.62.72.52.5

0.290.280.270.260.250.240.24

G & H

cm/dayCurve

1.671.250.960.690.510.390.35

NM

cm/dayAEF

0.590.380.280.210.170.130.12

Ratio

2.83.33.43.33.03.02.9

Curve DEC

0.81.21.41.41.81.8

0.300.290.280.270.260.250.24

0.490.410. 330.260.210.170.14

0.340.200.140.110.0870.0720.060

1.42.02.42.42.42.42.3

various water contents computede by the Gardner—Hillelequation with the resul ts computed by the numericalmethod. The data show the Gardner—Hillel equation to yieldresults that are too high by a factor as high as 3.4. Theresults also indicate the Gardner and Hillel equation tobe more in error for the D(#) relations where D variedgreatest from dry to wet soil. Moreover, the Gardner andHillel equation was not in error by a constant factor forany one D(0) relation. The error increased to a maximumand then decreased as moisture content decreased. This,unfortunately, would indicate that a simple correction ofthe Gardner and Hillel formula is not apparent.

Figure 4 shows the cumulative evaporation vs. time com-puted for cases where the surface soil had a diffusivityrelation that was 0.1 that of the underlying soil. The dataindicate that cumulative evaporation increased as the depthof the surface soil decreased. The depth of the surface

ZA-

16 20TIME IN DAYS

Fig. 5—Cumulative evaporation as a function of time for thelayered soil. D surface = 10 D(ABC), D underlying = ABCrelation. The thickness of the surface layer is given.

22.

TIME IN DAYS

Fig. 4—Cumulative evaporation as a function of time for thelayered soil. D surface = ABC relation, D underlying =10 D(ABC). The thickness of the surface layer is given.

layer had little effect on cumulative evaporation for depthsgreater than 0.525 L.

Figure 5 shows that the topsoil has large influence onevaporation where the surface soil had a D($) relation 10times greater than the underlying soil. Evaporation in-creased as the thickness of the top layer increased. With atop layer of only 0.075 L cm thick, evaporation was con-siderably greater than with no surface layer (uniform soil).

Figure 6 shows the water content-depth curves for sev-eral of the layered cases where the cumulative evaporation(average moisture content of the entire column) was ap-proximately the same. There is a large difference in themoisture profiles due to the characteristics of the layer.Where the diffusivity was smaller in the top layer, com-pared to the bottom layer, the moisture content from thesurface to the boundary changed rapidly. Below the bound-ary, the moisture content increased only slightly with in-

WATER CONTENT-VOLUME

.10 .15 .20

. ABC (UNIFORM SOIL) |- D(UNOERLYING SOIU= 10 D(ABC) I•D(SURFACE SOIL)=IOD(ABC)

Fig. 6—Water content vs. depth for the layered soils. The cumu-lative evaporation for all curves was about 1.5 cm which wasreached at 2.4, 7.6, and 10.2 days for curves D (surface) =10 D(ABC), D(underlying soil) = 10 D(ABC) and ABC,respectively. The boundary of the layered soils was at 0.225 L.

Page 4: Influence of Different Diffusivity-Water Content Relations on Evaporation of Water from Soils1

498 SOIL SCIENCE SOCIETY PROCEEDINGS 1965

creasing depth. Where the diffusivity was greater for thetop layer compared to the bottom layer, there was a smallerchange in water content with depth above the boundary buta greater change with depth below the boundary.

The information contained in Fig. 6 is of interest pri-marily to show the general shape of the $(x) curves. Theabsolute magnitudes of the curves are somewhat unrealis-tic because two soils having different D(#) relations wouldbe expected to have different water content-suction relationsas well. Therefore, the water contents at the boundarywould not match and some displacement would be ex-pected. (This type computation was made for infiltrationby Hanks and Bowers (5) and could be made for evapora-tion as well.) The data of Fig. 6 apply strictly to the caseof two soils with similar water content-suction relations butwater content-conductivity relations that are different by afactor of 10.

These data strictly apply only to the initial and bound-ary conditions as well as the soil properties in the computa-tion. However, as discussed by Hanks and Bowers (6),the soil properties used are similar to many other soils.Since the initial and boundary conditions, for an evapora-tion problem of the type studied, are similar to theconditions for many other soils, there is no reason to ex-pect that results obtained are not generally applicable.

The data imply that, if evaporation during the fallingrate stage is to be altered, the D(0) relation should bechanged on the wet end of the scale. However, the wet endof the scale is represented by soils below the surface. Thisunderlying soil would be very difficult to treat, by chemicalor physical means, to change the D(0) relation. Changingthe D(#) properties of the surface soil by some meanssuch as mulches, etc., would appear to have little effectduring the falling rate portion of the evaporation process.

However, it should be remembered that the mulches areeffective in decreasing the evaporation during the constantrate portion of the evaporation process.

The data further imply the region of the D(#) relationwhere accurate measurements of D are needed as well asthe region where less accurate data are sufficient. If D($)data are to be used to estimate evaporation, accurate meas-urements must be made at the highest moisture contents.However, little accuracy is needed at the lower moisturecontent.