effect of temperature-dependent hydraulic properties on soil water movement1

Effect of Temperature-dependent Hydraulic Properties on Soil Water Movement1

J. W. HOPMANS AND J. H. DANE2

ABSTRACTThermal effects on the soil water regime are often ignored. In

addition to the effects of temperature on the driving forces for flow,there are also direct thermal effects on the soil's hydraulic proper-ties. Accounting for this temperature dependency of the soil's hy-draulic properties, the pressure head form of the general flow equa-tion of water was solved by the predictor-corrector method for differenttemperature regimes and for a variety of initial and boundary con-ditions. Corrections of soil hydraulic properties due to temperaturechanges were either based on experimental data extracted from theliterature or on theoretical considerations. Temperature effects werefound to be much more pronounced for a pressure head than for aflax boundary condition at the soil surface. For temperature varyingwith both time and depth, the effect seemed to be minimal if thehydraulic properties were determined at the mean temperature ofthe profile. The latter suggests that soil hydraulic properties deter-mined in the laboratory, but to be used for field conditions, shouldprobably be determined at a temperature that approaches the meanfield soil temperature. The simulations also showed that temperatureeffects are more pronounced as the temperature coefficient of soil-water-pressure-head increases, but that the kind of effect dependson whether the existing temperature is above or below the referencetemperature (temperature at which the hydraulic properties weredetermined).

Additional Index Words: surface tension, viscosity, infiltration,redistribution, temperature coefficient.

Hopmans, J.W., and J.H. Dane. 1985. Effect of temperature-depen-dent hydraulic properties on soil water movement. Soil Sci. Soc.Am. J. 49:51-58.

ALTHOUGH the upper part of a soil profile is easilyrecognized to be a nonisothermal medium, the

thermal effects on the soil water regime in many soilwater movement studies have been tacitly ignored.The Darcy-based flow theory does not consider flowdue to forces that arise because of temperature gra-dients. In addition to the effects of temperature uponthe driving forces for flow, there are also direct ther-mal effects on the soil's hydraulic properties, viz., thewater retention and hydraulic conductivity functions.

If temperature gradients are present, water transportin unsaturated soils occurs in the liquid and vaporphase. Temperature differences induce vapor pressuregradients, the driving force for water vapor flow. To-gether with water vapor flow, heat is transported. Thelatent heat of vaporization is absorbed when watervaporizes and is liberated when the vapor condenses.The contribution of transport in the vapor phase can,however, be ignored without serious error (Philip andde Vries, 1957) if water flow in the liquid phase is ofconsiderable magnitude.

Temperature gradients may also cause liquid waterto flow. Although some researchers (Gary, 1965; Win-

1 Contribution from the Alabama Agric. Exp. Stn, AuburnUniv.,AL 36849. AAES Journal no. 3-84609. Received 16 April1984. Approved 29 August 1984.2 Graduate Research Assistant and Associate Professor of SoilPhysics, respectively, Dep. of Agronomy and Soils, Auburn Univ.,Auburn, AL 36849.

terkorn, 1947) found liquid film flow to be the primarymechanism for water flow under temperature gra-dients, it is generally believed that the bulk of liquidwater moves as capillary water due to differences insurface tension at different temperatures (Philip andde Vries, 1957).

Wilkinson and Klute (1962) derived an expressionfor the temperature effect on soil water pressure head(h) from the dependence of surface tension of wateron temperature. The effect of temperature on the pres-sure head-water content relationship was studied byGardner (1955), Haridasan and Jensen (1972), andYong et al. (1969), but none of them could explainthe observed effect of temperature on the pressurehead-water content relationship on the basis of changesin surface tension at the air-water interface alone. Ex-planations that have been offered for the larger thanexpected changes in pressure head due to temperaturechanges are based on the effect of temperature on en-trapped air and on the presence of surface active con-taminants at the air-water interface.

Observed increases in hydraulic conductivity (K) ata given water content (6) with increasing temperaturecould be attributed almost entirely to the decrease inviscosity of water (Haridasan and Jensen, 1972). Hy-draulic conductivity as a function of pressure head,however, did not reflect any temperature dependence.Two mutually counteracting effects of temperaturewere mentioned in this regard. As temperature in-creases, (i) hydraulic conductivity decreases becausethe amount of water held in the soil at a given pressurehead decreases, and (ii) hydraulic conductivity in-creases because the viscosity of water decreases. Re-sults reported by Constantz (1982) and Flocker et al.(1968), however, were less conclusive. These research-ers measured a temperature dependence of K(0) thatwas at least one order of magnitude larger than pre-dicted from the temperature dependence of the vis-cosity of water. Constantz noted that the viscosity ofsoil water may have a much greater temperature coef-ficient than free water and that both time and tem-perature may induce structural changes in the soil ma-trix.

Moore (1940) studied the effect of temperature oninfiltration and showed experimentally that the max-imum water content in the wetted layer decreased whilethe infiltration rate increased with increasing soil tem-perature. On raising the soil temperature from 0.3°Cto 25°C Klock (1972) measured an increase in drain-age that could be fully explained by the temperatureeffect on surface tension. Biggar and Taylor (1960)measured an increase in the quantities of water ab-sorbed by a dry soil as the temperature increased.

The objective of this study was to investigate theinfluence of soil temperature on soil water distributionduring and after infiltration, taking into account thetemperature dependency of the soil hydraulic prop-erties as calculated from theoretical considerations andbased on reported experimental measurements.

51

52 SOIL SCI. SOC. AM. J., VOL. 49, 1985

-0.018 -

-0.019 -

-0.027 -

-0.028 -

T, °C

Fig. 1—Temperature coefficient of surface tension of water (7) vs.temperature (T).

THEORYAssuming one-dimensional flow, the general flow equa-

tion in its pressure head form can be written as (Klute, 1969)C(h,T) (dh/dt) = d/dz { K(h,T) [dh/dz) +1]} [1]

where C is the specific water capacity, T is temperature, zis distance (0 at reference level and > 0 above it), t is time,and all other symbols were denned before. The Douglas-Jones approximation (implicit method with implicit linear-ization) was used to solve Eq. [1]. For details see Appendixand Hopmans and Dane (1984).

The computer model was adapted to account for temper-ature dependency of the hydraulic properties. According toPhilip and de Vries (1957) and Wilkinson and Klute (1962),the change of pressure head (h) with temperature (T) can bedescribed by

dh/dT = - hy(T) [2]where dh/dT is, the temperature coefficient of soil water pres-sure head (kPa/°C), a is the surface tension at the air-waterinterface (N/m), and y(T) is the temperature coefficient ofsurface tension of water (Fig. 1). Application of Eq. [2] andknowledge of a reference soil-water pressure head value (hnf)at a reference temperature (T^) allows the soil water pres-sure head value (hT) at any other temperature to be approx-imated byhT = hre{ - rref) - dh/dT = - rref)or

hre{ [3]provided | T — Tn{\ is small. The reference temperature isdefined as that temperature at which the hydraulic proper-

eFig. 2— Relation between soil water pressure head (h) and water

content (6) at reference temperature (!"„,) and a given temperatureT.

ties were determined. As 7 is temperature dependent andJ T - TKf is not always small, (T - Ta[)y(T) was approx-imated by summation over a finite number of temperaturesteps of 0.01 °C,i.e.

where Tt = TK( and Tk+{ = T, so that

, )y(Ti ) href = a( T)href . [5]*T = 1 + 2 ( T<+ 1I '=1

The coefficient a(7) is a function of both depth and time ifT changes with depth and time. An approximation for HT,as calculated from Eq. [5], can be obtained from Eq. [3] bysubstituting y(Tm) for y(T), Tm being (T + Tnf)/2.

The water capacity, C(h), can also be determined as afunction of temperature (Fig. 2):

W)- I6!

Figure 2 shows how for an arbitrary value of a, the pres-sure head at any temperature (hT) can be calculated fromthe water characteristic curve at the reference temperature(Tnf). The dashed line indicates how the water capacity attemperature T can be found from the water capacity at thereference temperature.

The hydraulic conductivity, K(6), at any temperature, T,can be calculated from

KT = [7]where 7?ref and »?r denote the viscosity of water (Ns/m) at thereference temperature and the soil temperature in question,respectively, and Knf is the hydraulic conductivity value atthe reference temperature. It is assumed that the changes inwater density with temperature are negligible. Although Kin Eq. [1] is presented as a function of h and T, K wasdetermined indirectly from 6(h,T) and K(6, T). The effect oftemperature-dependent hydraulic properties on soil water

HOPMANS & DANE: EFFECT OF TEMPERATURE-DEPENDENT HYDRAULIC PROPERTIES ON WATER MOVEMENT 53

0.10 0.20 0.30 0.40

6

6Fig. 3—Soil-water pressure head vs. water content and hydraulic

conductivity vs. water content of a porous medium at temperaturesof 20 and 50 °C (Solid lines: data from V. Novak, 1975; dashedlines: predicted from Eq. [5] and (7]).

movement is investigated using Eq. [5] and [7]. The trendspredicted by the model may, in certain cases, be underes-timated as experimental data have shown that the temper-ature effect on the hydraulic properties can be larger thanpredicted by Eq. [5] and [7].

SIMULATED EXPERIMENTSSimulation no. 1

Novak (1975) determined water retention curves and hy-draulic conductivity functions for a granular glass materialat temperatures of 20 and 50°C (Fig. 3, solid lines). A pro-cedure published by Van Genuchten (1978) was used to de-termine a functional relationship between 0 and h, while Kwas fitted by a third degree polynomial. These functionalrelationships were used to simulate infiltration at these twotemperatures for the following initial and boundary condi-tions:

6(z,0,T) = 0.2, -0.4 < z < 0 m0(Q,t,T) = 0.4, t>0

6(-QA,t,T) = 0.2, [8]To solve Eq. [1] these water content values were first con-verted to pressure head values at the two temperatures.

The dashed lines in Fig. 3 represent the hydraulic con-ductivity and water retention curve at 50°C as calculatedfrom Eq. [5] and [7] for a reference temperature of 20°C.

Simulation no. 2Using theoretical changes in the soil's hydraulic properties

with temperature as calculated from Eq. [5], [6] and [7],infiltration was simulated at temperatures of 10 and 40°Cfor both a pressure head and a flux surface boundary con-dition. Infiltration was also simulated for a linearly decreas-ing temperature with depth from 40 to 15°C. The hydraulicproperties of the sandy soil, experimentally determined byHaverkamp et al. (1977), were used and assumed to be validat 20°C. The initial and boundary conditions were:

h(z,0,T) = -6.15 kPa, -0.8 < z < 0 mh(0,t,T) = -3 kPa or q(Q,t,T)

= -0.1369 m h-1, t>0 [9]h(-0.8,t,T) = -6.15 kPa, t>0

where q is the flux density of water.

0.07

Fig. 4—Soil-water pressure head vs. water content for a sandy soilat temperatures of 10,20 and 40°C (the temperature coefficient ofsurface tension was multiplied by a factor three).

15 -

14 -

13 -

12 -

11 -

10 -

V« 9 -£

in" 8 '^X 7 -

6 -

5 -

4 -

3 -

2 -

1 -

0 -

O.O4 0.08 0.12 0.16

e0.20 0.24 0.28

Fig. 5—Hydraulic conductivity vs. water content for a sandy soil attemperatures of 10, 20 and 40°C, as calculated from Eq. [7].

The pressure head-water content (the temperature coef-ficient of surface tension was multiplied by a factor three)and the hydraulic conductivity-water content relationshipsat the different temperatures are displayed in Fig. 4 and 5.To obtain closer agreement between the theoretically cal-culated values and the experimentally determined values ofthe temperature dependency of pressure head, the infiltra-tion process was also simulated with temperature coeffi-


+ 18*10'

-0.02

-0.03Upper boundary condition

Fig. 6—Upper boundary condition for simulation no. 3.

cients that were arbitrarily selected to be three and five timesthe theoretical values.

Simulation no. 3To approach a more realistic situation, a long-term sim-

ulation was carried out, during which infiltration and evap-oration were alternated and varied in magnitude (Fig. 6).The flux (evaporation) changed according to a sine-function,except for the rainfall events. The bottom boundary con-dition was one of changing pressure head. The pressure headwas generated by the model in such a way that at any timestep the pressure head gradient approached zero. The hy-draulic properties and initial condition were the same as

20 C, reference———— 50°C, theory

50°C, experiment

those in the second simulation. In addition to these changingboundary conditions, a varying temperature was applied withregard to both time and position, viz.,T(z,t) = 20 + 10 exp(+z/0.226)

sin (0.2618? + z/0.226). [10]

Fig. 7—Water content profiles at temperatures of 20 (reference tem-perature) and 50 °C after 13.9 and 111.1 h of infiltration. Hy-draulic properties at these temperatures were either measured (solidand dotted lines) or calculated from theory (dashed lines).

Equation [10] indicates an average daily temperature (rav)of 20°C at any depth and a temperature amplitude at thesoil surface of 10°C. Although the damping depth is a func-tion of water content, it was chosen to be constant (0.226m) since the only purpose of Eq. [10] was to obtain a rea-sonable change of temperature with time (t in h) and depth.The constant 0.2618 is the angular frequency (h"1)- Tne re-sults of this flow problem and those of a similar flow prob-lem but with a Tm of 25 °C were compared with the resultsof the same flow problem subjected to a constant tempera-ture of 20°C with respect to both time and depth.

RESULTS AND DISCUSSIONSimulation no. 1

Results from the first simulation are shown in Fig.7. Infiltration was simulated at temperatures of 20(solid lines) and 50°C (dotted lines). The hydraulicproperties at these two temperatures, on which thesimulations were based, were experimentally deter-mined (Novak, 1975). The dashed lines also corre-spond to water content profiles at a soil temperatureof 50°C, but the hydraulic properties were calculatedfrom the temperature dependency of hydraulic prop-erties on surface tension and viscosity of water, withhydraulic properties at 20°C serving as reference val-ues.

Based on measured temperature dependency of thehydraulic properties, increasing the temperature from20 to 50°C increased the total amount of infiltratedwater after 111.1 h from 1.92 to 3.51 cm of water. Ifthe theoretical changes in 8(h) and K(ff) were consid-ered, the cumulative infiltration at 50°C was 2.63 cm.It was found that the change of hydraulic conductivitywith temperature affects the water-content distribu-tion more than the change in soil water pressure headwith temperature. This was concluded by assuming atemperature independent B-h relationship, while cor-recting K(0) for temperature (cumulative infiltration3.68 cm) and vice versa (cumulative infiltration 1.89cm). The results of this experiment indicate that the


e0.10 0.15 0.20 O25

e0.15 0.25 0.30

0.0

0.1-

0.2

0.3

0.5-

0.6-

0.7

0.8

Simulation no. 2Flux boundary condition

18min

48min

Fig. 8—Water content profiles at fixed temperatures of 10°C (solidlines), 40 °C (dashed lines) and at a linear decreasing temperaturewith depth (dotted lines) after 18 and 48 min of infiltration witha top flux boundary condition.

temperature effect on the soil's hydraulic propertiesinfluences water transport in soils. The increase in in-filtration rate with increasing temperature is in agree-ment with results published by Moore (1940) and Big-gar and Taylor (1960). The effect of temperature-dependent hydraulic properties on soil water move-ment is further substantiated by the results of the fol-lowing simulated experiments.

Simulation no. 2In this simulation, distinction was made between a

constant pressure head and a constant flux top bound-ary condition. Water content profiles for the fluxboundary condition are shown for 10 and 40 °C in Fig.8 after 18 and 48 min of infiltration. Changes in hy-draulic properties were based on theory (Tn{ = 20°C).

The higher temperature caused not only a decreasein the initial water content values in the profile (Eq.[9] and Fig. 4) but also a decrease in the water contentvalues in the transmission zone during infiltration. Astime proceeds, the influence of the gravitational headgradient increases relative to the pressure head gra-dient. The water content in the transmission zone willtherefore attain a value that sustains a hydraulic con-ductivity near or equal to the applied flux density atthe surface. This water content will decrease with in-creasing temperature (Fig. 5). Due to the low initialwater content, the water flux at the bottom of the pro-

Simulation no. 2Flux boundary condition

18 min10°C

48 min

Fig. 9—Comparison of wetting front depths for different values ofdh/dT after 18 min of infiltration at a soil temperature < 20°C(reference temperature) and after 48 min of infiltration at a soiltemperature > 20°C.

file was hardly affected by temperature changes. As theinfiltration rate at the top boundary remained con-stant, the total increase in the amount of water in thesoil profile at any given time after initiation of thesimulation was temperature independent. Conse-quently, the wetting front moved faster at the elevatedtemperature.

To attain closer agreement with experimentally de-termined values of dh/dT, the temperature coefficientof surface tension of water (7) was multiplied by threeand five. The subsequent effect on infiltration, espe-cially the wetting front advance, is demonstrated inFig. 9. For a temperature (T = 40°Q above the ref-erence temperature (Tref= 20°), this multiplication of7 caused the wetting front to move slower (dashed anddotted line) than for the theoretically calculated valueof dh/dT (solid line) while for a temperature (T =10°C) below the reference temperature, this multipli-cation of 7 caused the opposite to happen (dotted line).From Eq. [5] it follows that a(T) < 1 for T > TTe{(7<0). Therefore, the value of a(T) for T>TK{is re-duced when 7 is multiplied by a constant greater thanunity. Consequently, the smaller the value for a, thelarger the deviation between 6(h) at Tte{ and 0(h) at T(Fig. 2) and therefore the lower the water content atT for a given pressure head value. For a constant in-itial pressure head condition, the initial water contentvalues will therefore decrease as the multiplication


ao 0.05 0.10

e0.15 0.20 O25 0.0 0.05 0.10

60.15 0.20 0.25

0.0

0.1 •

0.2 -

0.3-

0)Q

0.5 •

0.6-

0.7 •

0.8-

Simulation no. 2Pressure head boundary

condition

Fig. 10—Water content profiles after 83.3 min of infiltration at tem-peratures of 20 and 40°C for different values of dh/dT.

factor of 7 is increased. From the mass balance prin-ciple and the constant flux boundary condition, it fol-lows that the wetting front advances slower the greaterthe multiplication factor. For T = 10°C < TKf =20 °C, the opposite happens.

Water content profiles resulting from a constantpressure head rather than a constant flux at the topboundary are shown in Fig. 10. The elevation in tem-perature from 20°C (dashed line) to 40°C (solid line)caused the amount of infiltrated water and the rate ofwetting front advance to increase if the theoreticalvalue for dh/dT was used. After 83 min of infiltration,water storage had increased 5.7 cm in the 20°C profileand 7.0 cm in the 40°C profile. Multiplying the tem-perature coefficient of surface tension of water with afactor three (Fig. 10, dotted line) or five (dashed-dot-ted line) had a significant effect on the water contentdistribution. Only a total of 4.8 and 3.2 cm of water,respectively, had infiltrated into the profile. The re-sults show that a pressure head upper boundary con-dition has a much more dramatic effect than havinga flux boundary condition.

Simulation no. 3The third simulation involved infiltration and evap-

oration (Fig. 6). The temperature changed according

0.1

0.3 -

0.5-

0.7

0.8

41.7H

2.8 h

Constant temp.20°C

Variable temp.Tav=2O°C

___ Variable temp.Tav=25°C

Fig. 11—Water content profiles after 2.8, 29.8, and 41.7 h of sim-ulation at a constant temperature of 20°C (reference temperature),a variable temperature with a mean temperature of 20 °C and avariable temperature with a mean temperature of 25°C.

to Eq. [10] with Tav either 20 or 25°C, or was a con-stant 20°C with respect to both time and depth. Thewater content distributions at specific times for thethree temperature regimes are shown in Fig. 11. Dur-ing the simulation, 2 h of infiltration were followedwith evaporation (Fig. 6) and a second 2-h applicationof water after 27.8 h. For the remaining period of time,the top boundary condition was again changed toevaporation. Figure 11 shows that temperature did notsignificantly affect the soil water content profile for avarying temperature with Tav = 20°C as compared toa constant temperature of 20°C. More distinct differ-ences occurred when the mean soil temperature (Tav= 25°C) differed from the temperature at which thehydraulic properties were measured (rref = 20°).

Hysteresis was not considered, but since it definitelyinfluences the water content distributions when wet-ting and drying cycles occur (Dane and Wierenga, 1975;Milly, 1982), the combined effects of temperature andhysteresis should be investigated.

SUMMARY AND CONCLUSIONSSoil water movement was simulated under different

temperature regimes for a variety of initial and bound-ary conditions. Corrections of soil hydraulic proper-


ties, due to temperature changes, were either based onexperimental data extracted from the literature or ontheoretical considerations. In the latter case, multiplesof the calculated temperature coefficient of surfacetension of water, which affect soil water pressure head,were also used to more nearly approximate experi-mentally determined values of dh/dT. The effect oftemperature on water movement was found to dependon the type of boundary condition, especially one atthe soil surface where changes were most drastic, andon the kind of temperature regime superimposed onit. Temperature effects were much more pronouncedfor a pressure head than for a flux boundary conditionat the soil surface. For temperature varying with bothtime and depth, the effects of temperature seemed tobe minimal if the hydraulic properties were deter-mined at the mean temperature of the profile. Thelatter suggests that those soil hydraulic propertieswhich are determined in the laboratory for applicationto field conditioins should probably be determined ata temperature that is near the mean soil temperatureunder field conditions. The results also showed thattemperature effects on soil water movement becomemore pronounced as the temperature coefficient of soilwater pressure head increases and the kind of effectdepends on whether the existing temperature is aboveor below the reference temperature (temperature atwhich the hydraulic properties were determined).

APPENDIXTo solve the flow equation, Eq. [1] is written in its quasi-

linear form:tfh9z2

[Al]

which can be written as a combination of two functions

0,/,(M,z)f+/4^,f). [A2]

The method that will be described to solve Eq. [Al] wasintroduced by Douglas and Jones (1963). The independentvariables t and z will be subscripted by j and /, respectively.

Let L be the depth of the profile under consideration and7"the total simulation time required, then for — L<z<0 and0<r<T, the gridpoints (z,, tj) are denned for i = 0, 1,...,N, andj =0,1,..., Msuch that z0 = 0, ZN = -L, t0 = 0and tff = T. The Douglas-Jones approximation uses twoequations, the predictor and the corrector. Each equationadvances the solution one-half time increment. The predic-tor is a modification of the implicit method, and calculatesthe unknowns (ht) at the (/ + 1/2) time level. The correctoris a modification of the Crank-Nicolson scheme and usesthe values of C(h) and K(h) calculated at the (/' + 1/2) timelevel to solve for hi at time (/+ !)• The grid spacing in thez-direction is fixed and is denoted by Az. The time incre-ment is variable: A(, = tj+l — t,.

Ifd2h/dz2 can be represented by two functions ft andf2 asin Eq. [A2], the predictor-corrector analog leads to linearalgebraic equations. The predictor is

C(h,T)K(h,T)

1K(h,T)

dh3r

1K(h,T)

dK(h,T)8z

dK(h,T)9z

9^3z

[

42(AV+1/2) = /i (*/. tj+1/2, h,j) f y y + l / 2 ~ hij

to/2

followed by the correctorhi,i+t ~hu1A2( V, + hl<f) = /,(z, ,tj+l/2,hiJ+l/2) -Z--

+/2(zt' {/+1/2' ̂ y+1/2 > Ây+1/2)[A4]

where

and

sh - h'+l-J h'-^6A> 2Az

-i hi+1 — 2hi, -t- h, iA 2 t _ '+ 'J___________' 'J

(Az)2

An advantage of the predictor-corrector method is that itgives rise to sets of linear equations with tri-diagonal coef-ficient matrices. The predictor-corrector method is uncon-ditionally stable and the truncation error is of the order (Az)2

+ (At)3/2. Using Eq. [A3] and [A4] the general flow equationin finite difference form yields for the predictor:

f y + l . y + l / 2 ~ 2^+1/2 + fy-1^+1/2

(W2

_ QJ fyy+i /2 ~ fyyKt, It/2

1 Ki+ \,j ~ Ki- \j \ hi+ \j ~ hi- ivK:i 2Az 2Az + 1

[A5]and for the corrector:

1 / hi+\J+ 1 ~ 2flij+l + hi-lj+l + hi+\,j ~ 2h',j + h'-l,j

2 \ (te)<

_ Cy+i/2 îy+1 ~ îy _ __1_K,, A?,.\y+i/2 -v

Kl+lj+\/2~ Ki-\J+\/2

2Azhi+\j+t/2 ~ hi-tj+\/2

2Az

K:•iy+i/2

+ 1 . [A6]

+f2(zi,tj^/2,hij,8zh,j) [A3]

for / = 1, ... , N-l.The mass balance equation was used to control the time

step size (Dane and Mathis, 1981). At time tj+i, the massbalance MBj+1 is defined as:

MBj+1 = | J_0L[0(z,tj+1)-0(z,tj)] dz - J^+1[v(0,t) -

v(-L,t)] dt| , [A7]

If MBj+1 was larger than a specific value e, the values of hij+1were rejected, the time step decreased, and new values forhij+1 were calculated. Very small values for MBj+1, e.g. O.le,resulted in an increase of the time step.


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Douglas, J., and B.F. Jones. 1963. On a predictor-corrector methodfor nonlinear parabolic differential equations. SIAM J. Appl. Math.Vol. 2, no. 1:195-204.

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Gardner, R. 1955. Relation of temperature to moisture tension ofsoil. Soil Sci. 79:257-265.

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effect of temperature-dependent hydraulic properties on soil water movement1

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