Model for Estimating Soil Water, Plant, and Atmospheric Interrelations: I.Description and Sensitivity1

M. N. NlMAH AND R. J. HANKS2

ABSTRACT

A model and its numerical solution were developed to pre-dict water content profiles, evapotranspiration, water flow fromor to the water table, root extraction, and root water potentialunder transient field conditions. Soil properties needed arehydraulic conductivity and soil water potential as functions ofwater content. Plant properties needed are rooting depth andlimiting root water potential. Climatic properties needed arepotential evaporation and potential transpiration.

The model predicted significant changes in root extraction,evapotranspiration, and drainage due to the variations inpressure head-water content relations and root depth. Varia-tions in the limiting root water potential had a small influenceon estimated evapotranspiration, drainage, and root extraction.

Additional Index Words: soil water content profiles, evapo-ration, transpiration, evapotranspiration, root extraction, drain-age, soil water flow.

THE SOIL, plant, and atmospheric continuum is a verycomplicated system which has been studied in the past

by numerous investigators (Philip, 1966). With the avail-ability of computers, very complex models have been de-veloped to consider this system. Hanks, Klute, and Bresler(1969) and many others, as summarized by Freeze (1969),have considered soil water flow without plants.

Soil water flow to plant roots has been studied by a num-ber of investigators. Microscopic studies of Philip (1957)and Gardner (1960) considered the radial flow of water toa single root. Macroscopic studies (Ogata, Richards, andGardner, 1960; Gardner, 1964; Whisler, Klute, and Milling-ton, 1968; Molz and Remson, 1970, 1971; and Molz, 1971)dealt with the removal of moisture by the root zone as awhole without considering explicitly the effect of individualroots.

Few of the previous models have been tested under fieldconditions. The objective of this investigation was to de-velop a macroscopic model and numerical solution whichwould consider the many important variables as they existunder field conditions. This would allow the estimating ofinfiltration, soil water uptake by roots, soil water contentprofiles, drainage, and evapotranspiration. The present pa-per considers the model description and test of sensitivityfor limited field data for oats in 1970. Part II of this seriesgives a description of the field layout and a detailed fieldevaluation of the model.

1 Contribution from the Dep. of Soil Science & Biometeorol-ogy, Utah State Univ., Logan, Utah 84322. This work was sup-ported by Utah Agr. Exp. Sta. at Utah State Univ. and the Dep.of Interior, US Environmental Protection Agency under theirgrants WP-01492-0l(n)l and 13030FDJ. Utah Agr. Exp. Sta.Journal Paper no. 1302. Received Oct. 9, 1972. ApprovedMarch 6, 1973.2 Post Doctoral Fellow, Utah Water Research Lab., formerlyResearch Assistant; and Professor of Soil Physics, Dep. of SoilScience & Biometeorology, Utah State Univ., respectively.

MODEL DESCRIPTION

The general flow equation for one dimension without rootextraction is (Hanks et al. 1969)

——dt

——dz

——3z

[1]

where 9 is volumetric water content, t is time, z is depth, K ishydraulic conductivity, and H is hydraulic head (sum of pres-sure head, h, and gravity head). To modify the above equationalong the lines of Whisler et al. (1968) and Molz and Remson(1970), a plant root extraction term, A(z, t), is included giving

[2]

A (z, t) is the root extraction term defined as

A(z, t)- h(z, t) - s(z, Q] • RDF(z) K(9)

where Hroot is an effective water potential in the root at thesoil surface where z is considered zero and RRES is a rootresistance term equal to 1 + Re. Re is a flow coefficient in theplant root system assumed to be 0.05. When RRES is multipliedby z, the product will account for the gravity term and frictionloss in the root water potential. Thus, the root water potentialat depth z is higher than the root water potential at the surface(Hroot) by a gravity term and friction loss term (assuming thatthe friction loss in the root is independent of flow). h(z. t) isthe soil pressure head, s(z, t) is the salt (osmotic) potential (inequivalent head units), RDF(z) is the proportion of total activeroots in depth increment Az, K ( 9 ) is the hydraulic conductivityat depth z, and Ax is the distance between the plant roots at thepoint in the soil where h(z, t) and s(z, t) are measured. Ax isarbitrarily assumed to be one. Dividing by Az converts thetranspiration flux into change of water content per unit time.

As presently used, the model does not consider hysteresis orlayered soil although both of these have been considered earlier(Hanks et al., 1969; Bresler and Hanks, 1969). Further assump-tions are made that the soil properties, primarily the hydraulicconductivity and the pressure head-water content relation, donot change with time (there is no change in soil structure).RDF(z) will, in general, depend upon time as well as depth.However, for the 9-day interval considered here, RDF(z) wasassumed not to change with time. In 1971, when the completeseason was considered (Nimah and Hanks, 1973), measure-ments for alfalfa roots showed no significant change in rootdensity with time.

The value of the Hroot term is dependent on plant, climatic,and soil conditions. The value of Hroot will depend (i) on plantconditions since they govern the root distribution function,RDF(z); (ii) on climatic conditions since they define potentialtranspiration; and (iii) on soil conditions since h(z, t), s(z, t)are soil properties (which will vary greatly from wet to drysoil). In the model, a value of Hroot is "solved" to make theplant root extraction over the total profile equal to potentialtranspiration provided the value of Hroot is higher than thepotential below which wilting will occur (HW-M). Thus in themodel, Hroot is bounded on the wet end by (Hroot — 0.0) andthe dry end by (Hroot = Hwilt).

522

NIMAH & HANKS: SOIL WATER, PLANT, AND ATMOSPHERIC INTERRELATIONS: I. 523

The basic input data needed for the solution of the modelare:

1) Hydraulic conductivity-water content and pressure head-water content data covering the range of water content tobe encountered during the period of interest (soil prop-erty).

2) Air dry and saturated soil water contents (soil property).3) Root water potential below which the root will not go

(presumably the plant wilts) and the actual transpirationwill be less than potential transpiration (plant property).

4) Root distribution function RDF(z) for the period ofstudy. At present the model has no provisions for chang-ing this with time (plant property).

5) Water content-depth data at the beginning (initial condi-tions).

6) Potential transpiration and potential evaporation rate orpotential irrigation or rainfall as a function of time forthe whole period of run (boundary condition). At presentthe partition of evaporation and transpiration is donerather crudely based on an estimate of percent of plantcover. Potential evaporation was assumed to be 10% ofthe potential evapotranspiration. Potential evapotranspi-ration was defined here as equal to the potential evapora-tion from a free water surface multiplied by a crop factor.The potential evaporation from a free water surface wascalculated using the Penman equation as described byJensen (1966).

7) Osmotic potential of the irrigation water (boundary con-ditions) and osmotic potential-soil depth data (initialconditions).

8) Presence or absence of a water table at the bottom of thesoil profile (boundary condition).

The output data that the model will give is almost unlimited.The output data normally selected are the following:

1) Cumulative evapotranspiration, transpiration, and evapo-ration as functions of time

2) Volumetric soil water content, 9, and soil pressure head,h, as functions of time, and depth

3) Cumulative water flow (upward or downward) throughthe lower boundary as a function of time

4) The value of Hroot as a function of time.Equations [2] and [3] were approximated numerically in a

way similar to Hanks and Bowers (1962) except for a modifi-cation to account for the root extraction term and nonuniformdepth increments.

The finite difference for the root extraction term used on theright hand side of equation [2] is:

where

=

hp? = H' + RRES X zj, and hs\ = hj+ S? [5]

where the subscript i represents the depth of a node and thesuperscript / represents time.

The working numerical equation then becomes:

• j»

j— l

Az3

2Az2K•i—to

where cj~y* is as defined by Hanks and Bowers (1962), andAZJ, Az2, Az3, are variable depth increments and are defined by:

— zt_1)/2.

A surface pressure head is computed to give the estimatedwater flux at the surface (evaporation or infiltration but nottranspiration) in conformance with the boundary conditionsapplying at the time using the following equation:

ER= (-

[6]

[7]

where ER is the flux at the surface, htf and V"1 are tne Pres-sure heads at the surface at the end and beginning of the timeinterval, respectively, h^ and /i^'"1 are the pressure heads atdepth Zi at the end and the beginning of the time interval,respectively, and K^i-M is the hydraulic conductivity apply-ing between the surface and z — z\. The surface pressure headis allowed to vary only between limits (i.e., saturation or airdry). Thus the computed flux may be less than or equal to thepotential flux. An iterative procedure is used to hunt for thecorrect value of hj and K^'~^ (since they are both un-known) to give the correct flux provided the surface pressurevalue is within limits. When conditions are such that the surfacepressure head is between saturation and air dry, the computedwater flux at the surface will equal the potential flux.

However, when conditions are such that soil properties donot allow for the potential flux to occur, the computed flux willbe less than the potential. For example, after the initiation ofrainfall or irrigation, the soil water content may be fairly lowwhich will allow for infiltration to equal rainfall or irrigation.If the rainfall or irrigation rate is sufficiently high, the soilwater content near the surface will increase until it reachessaturation. After the soil surface is saturated, the infiltration ratewill be less than the rainfall or irrigation rate and ponding orrunoff will occur. A similar but reverse phenomenon will oc-cur at the initiation of evaporation (say after a rain); the soilwater content will be high which will allow for the evaporationrate to be governed by the potential rate. Subsequently, the soilwater at the surface decreases until it reaches the air dry stateat which time computed evaporation is less than the potentialrate. If the rainfall or evaporation rate decreases after the limit-ing surface water contents had been reached, the computed andpotential rate could again become equal.

COMPUTATIONAL PROCEDURE

The general computational procedure involves several stepsas outlined below:

1) Read input data. The input data include tables of conduc-tivity and soil water pressure head as functions of water content(Fig. 1 and 2) and root distribution as a function of depth(Table 1). The potential infiltration and salt content of theirrigation water at the soil surface as well as potential evapora-tion and evapotranspiration as functions of time (Fig. 3) arealso input data as are the maximum and minimum plant waterpotential. Other input information are the initial time incre-ment, A', to be used; the upper and lower limits on pressurehead and water content (that is, saturation and air dry); thelength of time the computation is to run; and the condition ofthe lower boundary (either a constant pressure head or no flux).

2) From the initial water content as a function of depth, asmeasured by the neutron probe, values of hydraulic conductiv-ity as a function of depth are computed by the procedure out-lined by Hanks and Bowers (1962). Values of specific watercapacity (C = A0/A&) as a function of depth are computedfrom the water content and pressure head-water content rela-tions.

3) The surface pressure head is deduced to conform to thesurface flux conditions using the iterative procedure describedearlier.

524 SOIL SCI. SOC. AMER. PROC., VOL. 37, 1973

H0.5

0,4

Condition B

Condition A

0.23

O.I

ICD

-1C* -10-ro5 -io4 -:o3 -lo2

Pressure potential -cm (log scale)

Fig. 1—Pressure potential vs. water content for Vernal sandyloam soil. Condition A was used in the computation. Condi-tion B was used to test the sensitivity of the model.

.5 _

10' 10 10 10K-cm/hour (log scale)

10" 10"

Fig. 2—Water content vs. hydraulic conductivity for Vernalsandy loam soil.

005

I

0.5

1.0

24 96 120Time-hours

168 192 216

I-IrrigationR=Rah

Fig. 3—Water flux at the surface (evapotranspiration and precipitation cm/hour vs. time for oats in the 9-day interval in 1970.

4) A value of Hroot is hunted to satisfy the potential tran-spiration conditions as described earlier. If precipitation is tak-ing place, root extraction is assumed to be zero.

5) The tridiagonal matrix made up of the series of linearequations for each depth is solved for the pressure head at the

Table 1—Root distribution (RDF) assumed for oats in 1970,initial electrical conductivity (SE), salt and initial water

content (0o) vs. depth used for the computation made.The value given for RDF applies for the depth

interval given by the depth indicated andthe depth immediately above

Soil depthcm

01358

1216202530354045557085

100115135155165

HDFOats1970

0.0000.0360.0730.0730.1090.1450.1450.1460.1820.0910.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000

SE

- mmhos0.00.50.50.50.50.51.01.41.82.22.63.03.53.12.72.42.11.81.41.00.6

9.volume fraction

0.0400.0410.0450.0500.0550.0640.0720.0800.0900.1000. 1430.1900.2380.2450.2560.2700.2850.3150.3700.4330.463

end of the time interval at each depth increment as describedby Hanks and Bowers (1962). Another computation is made ofthe water content at each depth increment from a knowledgeof pressure head at each depth increment and water capacityas function of depth and pressure head-water content relationsusing the following formula:

[8]

6) The program tests the total absolute change in water con-tent. If it is greater than a given value, the time increment isreduced by half and the program goes back to step 5. Other-wise, it will continue.

7) The desired output information is printed, a new A/ ischosen, and the values of hf and 6f are taken as the newinitial conditions hf~l, $£~l. The cumulative time is checkedand an adjustment to the potential boundary conditions at thesurface is made if necessary.

RESULTS AND DISCUSSION

The model was tested for a 9-day irrigation interval be-ginning with and following an irrigation in 1970. The sensi-tivity of the model to the variations in the various inputparameters was also checked. Figure 4 shows the actualand computed water content at four times after irrigation.The agreement between measured and computed soil water

NIMAH & HANKS: SOIL WATER, PLANT, AND ATMOSPHERIC INTERRELATIONS: I. 525

Water Content (6).2 .3 .4 5 0 .\ .2 3 A 5~-TI—i——i r r\ i i i r

20 .

60

180

Fig. 4—Comparison of the water content profiles as predictedand measured for oats in 1970 at various times after irriga-tion, (o) 24 hours, ( f o ) 72 hours, (c) 120 hours, and (d) 216hours.

content was good except for 24 hours after irrigation wherethere was considerable difference above 45 cm.

Figure 5 shows a comparison of the cumulative com-puted ET, measured ET, and potential ET. The daily read-ing of measured ET was obtained from two lysimetersinstalled in the field. The potential E was calculated usingJensen's modified Penman equation (1966). The crop fac-tor was assumed to be one. The computed ET at 216 hourswas 4.9 cm, which was 0.4 cm less than the actual ET. Thismight be due to the assumption of uniform root distributionper unit depth in the top 30 cm in the soil, which may notbe true. The Hroot value computed reached the minimumallowable (—15 bars) after 155 hours, (see Fig. 8) wherethe cumulative computed ET became less than the cumu-lative actual ET as shown in Fig. 5.

For the 9-day interval in 1970, the model predicted up-ward flow from the water table. This was not anticipated.However, upon checking, the measured data, showed thesame results. The upward or downward flow from the watertable was obtained from

Water Table Flow = AMo — / + ET [9]

where AMo is the total change in the water content profileoutside the lysimeters, / is irrigation or precipitation duringthe period, and ET is evapotranspiration as measured in thelysimeters. The equation assumes that ET inside the lysime-ters was the same as outside the lysimeters. Figure 6 showsthat the computed cumulative upward flow was very closeto the measured flow.

No actual physical measurements of root extraction were

14

32

-•Predicted evapotranspiration-•Actual evapotranspirarion-"Potential evapotranspiration

ISO ZOO100Time - hours

Fig. 5—Comparison of actual, predicted, and potential evapo-transpiration during the 10-day period with that predictedfor oats in 1970.

i62

lo200

|4Lo

Fig. 6—Comparison of actual (dots) and predicted (solidline) upward flow from the water table for oats during the9-day interval in 1970.

Water content - 6

20

40

60

80

0.1 0.2 0.3 04. 0.5

*——• measured•——'30cm root depth'——°45cm root deptht—t-»60cm root depth

loo

120

140

160

Fig. 7—Comparison of measured and predicted water contentprofiles at the end of a 9-day period in 1970 assuming rootdistribution 30, 45, and 70 cm.

made for oats (Avena saliva L.) in 1970. The 30-cm depthpattern with uniform extraction was estimated from meas-ured water content changes. To test the sensitivity of themodel to the root density function, 45- and 70-cm root ex-

526 SOIL SCI. SOC. AMER. PROC., VOL. 37, 1973

48 96Time - hours

144

Relative Root Absorption

192 240

60cm

45cm

-2

-4

-6

tosI -8

I-10

-12

-14.

-16 -

Fig. 8—Hroot during the 9-day period as predicted from 30,45, and 70 cm root distribution.

traction depths were used. The root density functions for45 and 70 cm were similar in shape but extended to differ-ent depths. Figure 7 shows the measured and computedwater content profiles for 30-, 45-, and 70-cm root extrac-tion depths at the end of the 9-day interval. The data showthe best fit for the 30-cm root extraction. Table 2 shows acomparison of computed evapotranspiration, upward flow,and Hroot using the three root depths. Evapotranspirationand upward flow for the 45- and 70-cm root depth extrac-tion from the water table were higher than the measuredand the 30-cm root extraction. Figure 8 shows a compari-son of the predicted value of //root for the three rootextraction patterns used. Only the 30-cm depth functionshowed that //root reached the minimum value allowed(—15 bars). Thus, although Hroot was different for the45- and 70-cm root depths, the predicted T was the samebecause the lowest value of Hmot allowed was not reached.The results of this test indicate that the model is quite sen-sitive to the root extraction function assumed. Note theincrease in the value of ffrooi in response to rains.

The relative root extraction at any depth (Fig. 9) wascalculated by multiplying the A(z, t) term by the thicknessof the soil layer and dividing by the total transpiration rate.Directly after irrigation, all the water was extracted fromthe top 5-cm soil layer. Later in the irrigation interval, the

Table 2—Comparison of evapotranspiration, upward flow ofwater from the water table, and Hroot at the end of the 90-

day interval in 1970 for three different root densityfunctions (root distribution is the same per unit

depth of soil in each case)Rootdepth

304570

Actu

Evapo-transplratlon

4.95.85.8

al 5.3

Upwardwater flow

2.22.32.72.1

Hroot(end of Interval)

bars-15-11- 2

—

&

.Q2 O3 Q4 Q5 .Q6 07 O^

10

20

E 30uI 40-

.c50

60

70-

80

10-

2D

3&Ef 40-

I 503

60

70-

8O

90

—30 cm Root-45 cm "

70cm " 20

Dot, one hour after irrigation

Q4 0,5 .Q6

70cm

90-Oat, 115 hours

Oat, 67 hours

.Ql .02 .Q3 Q4 O5 O6 .07 J08

45cm

-70 cm

Oat, 177 hours

Fig. 9—Relative root extraction of oats in 1970, as influencedby root depth function.

relative root extraction increased with depth. After 115hours, maximum extraction was at the bottom of the rootzone. At 67 hours there were two peaks in extraction, onenear the surface and one near the bottom. The surface peakwas due to the relatively high water content due to irriga-tion and the lower peak due to high water content main-tained by upward flow from the water table. Root extrac-tion, as predicted by the model, was not uniform but was acomplex function of soil water content, hydraulic conduc-tivity, depth, and time. This is in qualitative agreement withthe measurements of Reicosky et al. (1972) and Molz andRemson (1971).

A test of sensitivity of the model was made using twosoil conditions—A and B shown in Fig. 1. Condition Acaused evapotranspiration and transpiration to be greaterthan condition B, while the reverse was true for evapora-tion and upward water flow, as shown in Table 3. The wa-ter content profiles were quite different in the active rootextraction zone. At the end of the 9-day period, the watercontent at the 30-cm depth was about 0.10 for condition Aand about 0.12 for condition B as shown in Fig. 10. In this

Table 3—Comparison of predicted evapotranspiration, evapo-ration, transpiration, and water flow as influenced by dif-

ferent soil properties for a 9-day period starting July28, 1970 at Vernal, Utah (no precipitation)

Condition A,45-cm root depth

Condition B,45-cm root depth

Evapotranspiration 6.01Evaporation 0.64Transpiration 5.37Water flow from the water table 2.44

5.790.775.022.58

NIMAH & HANKS: SOIL WATER, PLANT, AND ATMOSPHERIC INTERRELATIONS: I. 527

0.1Woter content - 90.2 0.3 0.4T"

0——° Condition A0——° Measured

•——• Condition B

-V

20|

40

60e 80

* 100'6"120

140

160

ISO

Fig. 10—Comparison of measured water content profiles forsoil condition A and B at the end of the 9-day period in1970 oats.

test, Hroot fell to —15 bars 1 day earlier for condition Bthan for condition A. Thus it appears that computationsmade by the model are quite sensitive to the soil propertiesused.

Another test of sensitivity of the model was made usingdata from a different soil where the lower limit belowwhich HTooi was not allowed to go (Hwilt) was varied. Theoriginal soil water content was high simulating spring con-ditions. Figure 11 shows the computed cumulative ETwhere the lower limit was allowed to drop to —20 or —40bars as well as cumulative potential ET. The data show thatcumulative ET at 48 days was 7.6 cm for the —20 bar limitcompared to 8.3 for the —40 bar limit. The lower limit ofHroot reached —20 bars at 24 days and —40 bars at 30days. Thus, for the soil tested, the model indicates that thevalue chosen for the lower limit for the Wroot was not verycritical.

While not shown here, the model predicts the variationof Hrooi during the day in responding to changes in thepotential transpiration. For the purpose of this study, bound-ary conditions averaged for 1 complete day (Fig. 3) wereused except for irrigation or rain where the exact timeperiod was used.

i 14

>

|,o

J 8

6

4

2

'Potential ET' -20 Bar min allowable for Hroot1 -40 Bar min allowable for Hroot

30 35 40 45 50Time- days

Fig. 11—Cumulative evapotranspiration vs. time comparedwith predicted evapotranspiration where the lower limit ofHroot was — 40 bars and — 20 bars (data for desert soil fromCurlew Valley, Utah).