simulation of the water balance of soil columns and fallow soils

SIMULATION OF THE WATER BALANCE OF SOIL COLUMNS AND FALLOW SOILS

H. R. ROWSE (Natwnul Vegetable Research Station, Wellesboume, Wmeuick)

Summary A model is described that predicts the evaporation of water from, and the distri-

bution of water in, a soil column evaporating into a constant environment. It is based on a numerical solution of the flow equation and requires only the initial water distribution in the column, the equilibrium (air-dry) water content at the soil surface and the relationship between volumetric water content and diffusivity. The model predictions show good agreement with a published analytical solution and with experimental results.

Modifications to the model that allow for rewetting of the sod by rainfall, and changes in atmospheric conditions above the soil, enable predictions to be made of the water balance of a fallow field. In general, good agreement was obtained with the measured distribution of water deficits in the soil profile, although the predicted water content of the surface 2.5 c m of soil showed systematic differences from the measured values. The reasons for this are discussed.

Introduth Two recently ublished mathematical models have shown that it is possible to pre 8 ict the movement of nitrate in a fallow soil (Burns, 1974) and the degradation of certain soil ap lied herbicides (Walker, 1973,

meteorological measurements. Both models involve calculating the distribution of water in the soil, and the evaporative loss from the soil surface. An empirical method for calculatin water loss (Stanhill, 1958)

measurements, and it does not provide any way of allowingfor differences in soil texture. An alternative approach is to attempt to apply the flow equation for water movement in soil.

Both analytical and numerical methods have been used to solve the flow e uation (Equation I below). Analytical methods have the advan- tage o 9 arithmetic simplicity but are not so generally ap licable as numerical ones. Gardner and Gardner (1969) and Gardner (973) have used an analysis based on the flow equation to calculate the evaporative loss of water from soil columns watered at various intervals. This treatment has recently been extended to field soils (Gardner, 1974). Black et al. (1969) have also used an analysis based on the flow e uation

of water to be calculated in addition to the evaporation, but it is limited to semi-infinite soil columns drying from a uniform water distribution into a constant environment.

The disadvanta e of numerical solutions is that they require a computer. Many of t a em are concerned with phenomena associated with

1974) from a knowledge of the soil p E ysical properties and standard

has, on occasion, given results which di if ered markedly from field

to calculate the evaporation and drainage from an uncrop ed fie 9 d soil. The analytical solution of Rose (1968) enables the vertica P distribution

Journal of Soil Science, VoL 26, No. 4.1975 6113.4 A a

338 H. R. ROWSE infiltration, such as the movement of solutes in the soil or surface run-off and do not consider evaporation from the soil surface. These include the works of van Keulen and van Beek (1971), Stroosnijder et al. (1972), Beek and Frissel 1973), Ferrari and Cuperus (I 73), and de Wit and

(19%) and A d j e t al. (1969) do consider evaporation from the soil surface but only in laborato conditions. None of the above treatments

The object of the current work was to develop a mode (hereafter called thefild model) based on a numerical solution to the flow equation, in an attempt to provide a simple means of calculating the loss of water and its distribution in the field soil profile, from a knowledge of the soil hydraulic properties and daily measurements of rainfall and evaporation from an o en water surface. To this end a more simple model

column evaporating into a constant environment was first written and tested.

van Keulen 19 2 . The numerical solutions o P Hanks and Gardner

attempt to predict the distri T ution of water in the field soil rofile. P

(hereafter calle c f the column model) describing the water balance of a soil

The column model The models

For soils drier than field ca acity the effect of gravity on water flow may be ifflored (Rose, 1968). &der these conditions the difisivity form of the c assical flow equation for the vertical isothermal flow of water in a soil may be written nn

F = soil water flux, positive direction upwards (cm day-l) D = soil water difisivity (cma day-1) 8 = volumetric water content (cm3 cm-3) z = depth (cm)

D is a function of 8 and is known to exhibit hysteresis de endin on whether the soil is wetting or drying. However, for a soil c$ing kom field capacity D may be considered to be a unique function o 8.

The model considers the soil as a vertical stack of layers each of thickness hz. If the relationship between D and 8 is known an estimate of the flux at depth x between two adjoining layers is given from equation - l.... 1 uy.

The value of D6 is taken as the value of the difisivity at a water content of

For a vertical column closed at its lower end the flow across the bottom layer of the stack is zero.

When water evaporates from a soil initially wet to near saturation into a constant environment, two distinct stages in the eva orative rocess can usually be detected (Keen, 1914; Philip, 1957; H&l, 1971r Initially the rate of evaporahon remains constant at a rate dependmg on the evaporative demand of the atmosphere and which is approximately

0.5{8b-@SaP) +8(6+c66&))1.

SIMULATION OF WATER BALANCE 339 equal to the rate of loss from an open water surface. This is known as the constant rate, or energy limiting hase, of evaporation. During this

the resistance to water movement in this zone increases. Eventually, the resistance to water movement becomes so large that the evaporative demand of the atmosphere can no longer be met and the evaporation rate starts to fall. This is known as the falling rate, or soil limiting phase, of evaporation when the soil water content at the soil surface is effec- tively at the air-dry value. For the present only the soil limiting stage of evaporation is considered. In the model this rate is calculated as

phase the water content near the so1 s surface falls, with the result that

where 8, is the air-dry volumetric water content at the soil surface. As before, the value of D8 is taken as the value of the diffusivity at a water content of:

Equation 3 is directly analogous to Equation 2 in that it involves the same ap roximations but strictly it estimates the flux at a depth of

associated with this are considered later. The model was programmed in C.S.M.P. (I.B.M. 1967) and run on

the Rothamsted com uter (ICL 4/70). For each run of the model the

0'5{e8+s(&5&))'

0-25A.z P rom the surface, rather than at the surface itself. The errors

following were spec if! ed: i the size and number of layers, ii the air-dry water content at the soil surface, iii the initial water content of each layer, iv the relationship between the water content 8 and the diffusivity

D which was assumed to be the same at all depths. In the programme equations 2 and 3 are used to calculate the flow of water between each layer and across the soil surface during a small time increment. From these values the new water contents of each layer at the end of that time increment are calculated. The process is then re- peated for the next time increment and so on, so that the series of small time-steps ap roximates to the continuous process that occurs in reality.

ThJieM model The model used for fallow soils in the field was similar to the column

model. It had sixteen layers with the following layer thicknesses numbered from the surface downwards; 1-5, I cm; 61, 2- cm; 8-11, -0 cm; 12-16, 10.0 cm. The boundary condition at the owest layer 70-80 cm) was changed from one of zero flux to one of zero change in

water content (i.e. 8 was held at field capacity This had the effect of allowing drainage (when the layer above the ottom layer was wetter than field ca acity), and upward movement (when this layer was

ii The length o P the time-steps was generally between 0.01 and 0.001 day.

6 i drier than fie P d capacity). However, the amount of upward movement

340 H. R. ROWSE from the bottom la er was never large because the gradients in water

The rate of evaporation from the soil surface was considered to be equal to the smaller of either a soil supply rate (i.e. the soil limiting rate) calculated from equation 3, or an atmospheric demand rate (i.e. the energy limiting rate) equal to the rate of loss from an o en water

from an eva oration tank. The open-water loss rate was considered to

In early runs of the model the rain was considered to start and finish at the times indicated by a recording rain gauge, and to fall at a rate e ual

subsequent runs, including those shown in this paper, the rain was arbitrarily considered to fall between noon and midnight. The two methods gave very similar results for the two soil parameters considered in this paper, i.e. the ogoo G.M.T. soil surface water content and the total soil water deficit.

During rainfall, each soil layer is successively returned to field capacity starting from the soil surface. Any additional rainfall after the whole profile has reached field capacity is considered to cause drainage. This simple way of treating infiltration was used by Bums (197 ) and it ap roximates well to the water distribution during infiltration. IfSypically in K ltrating water moves through a zone of almost constant water content (the transmission zone) which is clearly separated from the dry soil by a sharp wetting front (Hillel, 1971). However, there are also two practical reasons for treating rainfall 111 this way. First it means that it is only necessary to know the diffusivity relationship for water contents between air-dryness and field capacity. It appears that at least for the sandy-

content at this dept K were always small.

surface. In equation , 6, was taken as a fixed value equal to t R e labora- tory determined air- d ry value, and the open water loss was estimated

follow a ha 8 sine wave during the day and was zero at night.

to the daily total divided b the duration of rainfall. However, stan 8 ard agricultural meteorologica Y records do not include these data and in

loam soil used in the experiments described below, a single eva method is adequate, but diffusivity measurements between city and saturation would require an additional method. even if the complete diffusivity relationship is known, a more rigorous treatment of the infiltration process would be ex ensive in computer time. It has been shown (van Keulen and van geek, 1971) that the maximum permissible time-step that can be used without oscillations occurring in the model is a function of the diffusivity. The high values of diffusivity that are typically found between field capacity and saturation mean that very small time-ste s would have to be used, which in-

Experimental methods Column experiments

The column model was tested on triplicate columns of a coarse sand. The columns were 10 cm long and were constructed from aluminium rings I cm high and 3-16 cm I.D. taped together with waterproof tape. The bases of the columns were covered with a fine nylon cloth. They were carefully packed with the air-dry sand and saturated by a slowly

evitably means a large increase in t x e computer running time.

SIMULATION OF WATER BALANCE rising water table from the base of the column. After draining to a constant weight at a tension of 50 cm, the bases of the columns were sealed with plastic caps, and they were placed inside a closed evaporation chamber at a constant temperature (21 "C& I "C). The air inside the chamber was circulated over a large area of calcium chloride crystals so as to maintain a constant relative humidity of 0.3. The columns were removed from the chamber at various intervals for weighing.

34 I

1

0 0.05 0.1 0 0.1 5 Volumetric water content

FIG. I. The measured relationships between volumetric water content and difisivity for a coarse sand obtained from 2, 5, and 10 an soil columns.

The method of Gardner (1962) was used to calculate the relationship between volumetric water content and difisivity during the soil lirmting stage for the coarse sand. It involves measuring the rate of evaporation from a soil column and applying the equation

where L is the length of the column, de/dt is the instantaneous rate of water loss, 8 8,

is the instantaneous volumetric water content, is the final equilibrium water content of the column when evaporation has ceased.

Triplicate difisivity columns of various lengths (2, 5, and 10 cm) were prepared and allowed to evaporate as described above. The volumetric water content at 50 cm tension was 0.2 , but the rate of eva oration

columns respectively. The calculated relationship between volumetric water content and difisivity for each of the three column lengths is shown in Fig. I.

from the columns did not fall a preciab 4 y until their mean vo f umetric water contents were approximate P y 0-14,0.18, or 0-20 for 2,5, and 10 cm

342 H. R. ROWSE Field experiments

The experimental site was a small lot in a fallow sandy-loam soil in Big Cherry Ground at the National tegetable Research Station. The plot was cultivated in the spring of 1973 and thereafter kept free of weeds

0 0.1 0.2 (

._

Volumetric water content

FIG. 2. The measured relationship between volumetric water content and diffusivity for the sandy-loam field soil from the experimental site. The circles are the experi-

mental points and the lines ABC and AJ3D were used in the model (see text).

by spraying with paraquat. Every 2-3 days, four replicate soil samples were taken at ogoo G.M.T. from the top 2-5 cm of the experimental site and their moisture contents determined. Dihsivity measurements were made on six soil columns obtained cylinders (6-4 cm

flush with the soil and the excess

soil trimmed so that it was flush with the base of the column. The water content-difisivity relationshi was then determined by the method used above for the coarse sand. T R e results so obtained are shown in Fig. 2 where the points are the individual dihivities calculated from each column and the line is their geometric mean which was used in the model. The sharp discontinuity 111 the difisivity relationship at point

diam. x 10 cm long) into the soil surface. They were carefully

SIMULATION OF WATER BALANCE 343 B in Fig. 2 robably arises because the effect of gravity is not negligible

was extrapolated to D and both the relationships ABC and ABD were tested in the model as described below.

Results

at these hig R water contents. To examine this possibility the line at €3

Comparison with an analytical solution redictions were

of water in 30 cm-long soil columns at various times after the start of evaporation. The columns were brought to a tension of 15ocm and

To test the accuracy of the column model its compared with the data of Rose (1968). Rose measure : the distribution


FIG. 3. Comparison of the results from the column model (circles) with the data of Rose (lines). For explanation see text.

allowed to evaporate into an atmos here of constant temperature and

assume the columns to be sem-infinite, and calculated the relationship between soil water content and difisivity from the measured water distribution. The test of the present model involved the use of this relationshi to ether with the values for equilibrium surface water content (0, P f an the initial water content of the column, to calculate the final distribution of water in the column. A comparison of Rose’s experimental results and the model calculations for the six soils used, is shown in Fig. where the vertical axis is a transformed depth (Zt-*)

. He then used an analpica P solution to the flow equation which

where 2 is dept ?I (cm) and t is time since the start of evaporation (days).

344 H. R. ROWSE Differences between the measured water distribution and those calculated b the model include any errors associated with the calculation of the di sivity relationshi as well as those associated with the model. For this test a 30-layer mo el was used with the following layer thicknesses numbered from the surface downwards; I- , 0.2 cm; 6-15, 0-4 cm; 16-20, 1-0 cm; 21-30, 2.0 cm. The Runge-kutta variable step numerical integration method (I.B.M., 1967) was used with an error criterion sufficiently small to prevent oscillation.

B f i l

- --0 10 20 Time (days)

30

FIG. 4. Fall in the mean volumetric water content of a 10 cm column of coarse sand as determined from experiment (circles) and as calculated by the column model from the

diffusivity relationships shown in Fig. I .

Column experiments A similar model was used to predict the change in the mean water

content of ~ o c m columns of coarse sand. The model had 10 layers each I cm thick. Fig. 4 shows a comparison of the experimental data (mean of triplicate columns with the model results using the water contentdifisivity relations h, 'p obtained from the 2, 5 , and 10 cm columns. As the diffusivity was not known at high water contents, the model was started when the mean volumetric water content was 0.14. The water distribution in the columns at this mean water content was obtained from a separate ex eriment in which the columns were sectioned at various tunes and t R e water content of each ring determined. Other results from the sectioning ex eriment are shown in Fig. 5 where

are corn ared with those calculated from the model (2 cm dihsivity relations R ip).

Field studies The water content-difisivity relationship of the field soil (Fig. 2)

was tested by using it in the column model, and comparing the calculated changes in the water content of the diffusivity columns with the experimental values. Fig. 6 shows that both the relationships ABC and

the measured water distributions a P ter various periods of evaporation

SIMULATION OF WATER BALANCE 345 ABD gave excellent agreement. The relationship ABC was used in the field model.

Fig. 7 shows a comparison of the results from the field model with the measured ravimetric water contents of the surface 2-5 cm of the experimental p P ot. The model was also tested against the data of Bums

p, 7.0 days


FIG. 5. Comparison of measured (circles) and model calculated (lines) water distributions in a 10 cm column of coarse sand. The times shown indicate the times from the

start of the model run (at a mean volumetric water content of 14'4%).

NVRS Personal Communication) some of which has been published Bums 197 ) by comparin the measured and calculated soil water

field adjacent to, and of similar soil texture to that in which the field diffusivity measurements were made. Most of the measured deficits were around I cm due to the fact that the first centimetre of water was lost rapidly and any subsequent loss occurred at a relatively much slower rate. With the exception of the prediction for 13 July all

deficit to a t epth of 45 cm ( gr able I). These data were obtained from a

346 H. R. ROWSE differences between predicted and measured deficits can be accounted for by the variability in the field measurements.

Discussion Good agreement between observed results and those predicted by the

model can only be expected provided that the following conditions are satisfied

The numerical solution used is accurate. Numerically correct parameters are used in the model. The assumptions about the physical processes involved are valid.

FIG. 6. Fall in the water content of a 10 an column of sandy-loam field soil as measured (line) and calculated from the difFusivity curves ABC (circles) and ABD

(squares) shown in Fig. 2.

Fig. 3 shows that there are no large differences between the measured water distributions of Rose and those calculated by the model from his dfisivity relationships. It can therefore be concluded that condition (i) above is satisfied. No attem t was made to improve this agreement because these results suggest g a t any error introduced into the field model by the method of calculation is liable to be small compared with other sources of error.

water content and the field capacity water content (in the field modeyare easil measured and are not likely

parameter is undoubtedly the relationship between water content and

Parameters such as the air d

to produce major errors in the model pre J ictions. The most important

SIMULATION OF WATER BALANCE a

347

I

150 1 60 170 a

I

180 190 200 a

* . *

Time (days from 1 st Jan)

FIG.^. Comparison of the measured (circles) and calculated (line) gravimetric water contents in the surface 2.5 cm of a sandy-loam soil.

TABLE I Comparison of measured and calculated soil moisture dejicits in rg70

Measured S.M.D. 1-22 1-38 1-28 0.35 0.91 1.15

S.E.* 0.197 0.069 0.106 0.442 0.231 0.110

Calculated S.M.D. 1.39 1-32 1-77 0.72 0.70 1.07

Standard error for 3 D.F.

348 H. R. ROWSE diffusivity. The equation for calculating this relationship (Equation 4) proposed by Gardner (1962) requires the assumption of constant d i f i - sivit with depth in the column. In the coarse sand measurements this

It might therefore be expected that the agreement between the model and experiment would be best when the difisivity relationship determined from the 2cm columns was used. Fig. 4 confirms this. The failure to predict the distinct drying front that was found experimentally (Fig. 5 suggests that even the relationship from the 2 cm columns was

10 cm columns did not predict the water distributions as well as did that from the 2 cm column. It is concluded therefore that because the coarse sand exhibited a distinct drying front, the method used to determine the diffusivity relationship was the major source of disagreement between the model and the experiments.

In contrast to the results from the coarse sand there was excellent agreement between the column model and the rate of water loss from the 10 cm field column when the difisivity measurements from the 10 cm field columns were used. This is consistent with the fact that the sandy- loam soil did not show the distinct drying front of the coarse sand, and the assumption of a constant diffusivi with depth was more valid.

model, the predicted distribution of water deficits in the field agreed well with those measured by Burns (1974). It is considered therefore that the method used to measure the difisivity relationship in the field soil was adequate. To ensure that the diffusivity relationship used in the model was also representative of the experimental site the six samples for the diffusivity determination were taken from the 0-10 cm depth in the soil as the majority of water loss by evaporation occurs in this region.

The hysical princi les on which the column model is based are well

field model involves assumptions in addition to those in the column model. The assumption that rainfall successively wets each soil layer has been discussed previously and it is this that accounts for the slight underestimate of the water content in the surface 2- cm following rainfall (Fig. 7) when the soil surface is wetter than fie1 d capacity for a short time. There is also an implicit assumption of isothermal conditions in the field model and this robably results in the over-estimation of the

Durin days of high solar radiation the soil surface is warmer than the

tion from the soil and reduces the rate of water flow to the soil surface from below. The combined effect is that the soil surface dries more ra idly than it would do under isothermal conditions. A model that

on the thermal properties of the soil and would necessarily have been more complicated. The results described above (Fig. 7 and Table I) agree with the conclusions of Philip (1957) and Fritton et ul. (1970) that isothermal theory satisfactorily predicts the accumulated evaporation

con CT ition will be more nearly achieved with the shortest (2 cm) columns.

not su Bi ciently accurate. The difisivity relationships from the 5 and

Further, it was found that when this re 7 ationship was used in the field

establis R ed (Rose, 19 t? 8) and will not be discussed here. However, the

water content of the sur P ace 2-5 cm of soil several days after rainfall.

soil be K ow or the atmosphere above. This increases the rate of evapora-

ta H es into account the flow of heat as well as water would require data

SIMULATION OF WATER BALANCE 349 in non-isothermal conditions but does not predict the formation of a dry surface layer.

Ac?mmledgements The author wishes to thank J. Kerr and P. Pinkney for technical

assistance. REFERENCES

BEEK, J., and FRISSEL, M. J. 1973. Simulation of nitrogen behaviour in soils. Pudoc, Wageningen .

BLACK, T. A., GARDNER, W. R., and THURTELL, G. W. 1969. The prediction of evaporation, drainage, and soil water storage for a bare soil. Soil Sci. SOC. Am. Proc. 33,655-60.

BURNS, I. G. 1974. A model for predicting the redistribution of salts applied to fallow soils after excess rainfall or evaporation. J. Soil Sci. 25, 16598.

DE WIT, C. T., and VAN KEULEN, H. 1972. Simulation of transport processes in soils. Pudoc, Wageningen.

FERRARI, Th. J., and CUPERUS, J. L. 1973. Dynamic simulation of vertical non- absorbed anion transport. Plant and Soil 38,425-38.

FRITTON, D. D., KIRKHAM, D., and SHAW, R. H. 1970. Soil water evaporation, isothermal diffusion, and heat and water transfer. Soil Sci. SOC. h e r . Proc. 34, 183-9.

GARDNER, H. R. 1973. Prediction of evaporation from a homogeneous soil based on the flow equation. Ibid. 37,513-16. - 1974. Prediction of water loss from a fallow field soil based on soil water flow theory. Ibid. 38, 379-82. - and GARDNER, W. R. 1969. Relation of water application to evaporation and storage of soil water. Ibid. 33, 192-6.

GARDNER, W. R. 1962. Note on the separation and solution of diffusion type equations. Ibid. 26,404 (see erratum ibid. 26, 514).

HANKS, R. J., and GARDNER, H. R. 1965. Influence of different diffusivity-water content relations on evaporation of water from soils. Ibid. 29,495-8. - KLUTE, A., and BRESLER, E. 1969. A numeric method for estimating infiltration, redistribution, drainage and evaporation of water from a soil. Water Resour. Res. 5, 1064-9.

HILLEL, D. 1971. Soil and Water-Physical Principles and Processes. Academic Press, New York and London.

I.B.M. 1967. Continuous System Modelling Program (36oA-CX-I 6X), Users Manual. New York: I.B.M. publications.

KEEN, B. A. 1914. The evaporation of water from soil. J. agr. Sci., Camb. 6,456-75. PHILIP, J. R. 1957. Evaporation and moisture and heat fields in the soil. J. Met. 14,

ROSE, D. A. 1968. Water movement in porous materials. 111. Evaporation of water from a soil. Brit. J. Appl. Phys. (J. Phys. D) Ser. 2, 1, 1779-91.

STANHILL, G. 1958. The estimation of evapo-transpiration from meteorological data. Natn. Veg. Res. Stn. Unpublished monograph.

STROOSNIJDER, L., VAN KEULEN, H., and VACHAUD, G. 1972. Water movement in layered soils. 2. Experimental confirmation of a simulation model. Neth. J. agric. Sci. 20,6772.

VAN KRULEN, H., and VAN BEEK, C. G. E. M. 1971. Water movement in layered soils- A simulation model. Neth. J. agric. Sci. 19, 138-58.

WALKER, A. 1973. Use of a simulation model to predict herbicide persistence in the field. Proc. Ear. Weed Res. Coun. Symp. Herbicides-Soil, 1973. 240-9. - 1975. A simulation model for the prediction of herbicide persistence. J. Environ. Quality 4, In press.

(Received 12 November I974)

354-66.

simulation of the water balance of soil columns and fallow soils

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