REPORT DOCUMENTATION FORMWATER RESOURCES RESEARCH CENTER

University of Hawaii at ManoalReport Technical MeroorandwnNumber Report No. 78

3Title

Two-Dimensional InfiltrationFquations: Soil-Box LaboratoryExperiments

.8Author(s)

Mr. Shan-hsin ChiangDr. Yu-Si Fok

2COWRRField-Group 2G, 3F

"ReportDate August 1985

5NO. ofPages viii + 33

6No. of 17N~. ofTables 12 FJ.gures 16

9Grant Agency

Hawaii State General Fund

10Grant No.Project No. S-0l8

lloescriptors: 'A'inf1ltrat10n rate, 1rr1gat10n, hydrau11c conduct1vity, SOl1.porosity, soil water, rooisture contentIdentifiers: *two-dimensional infiltration equations, *soil-rooisture move­ments, *cumulative infiltration, degree of saturation, Makiki clay-loam

12Abstract (Purpose, method, results, conclusions)

Infiltration equations can be used to describe furrow and drip irri­gation, groundwater recharge, and wastewater injection. Physical two­dimensional (2-D) infil~ration equations for drip irrigation expressed inexplicit, power algebraic forms were developed based on four consecutivetime intervals which were derived in a previous study. 'lWo-dimensionalinfiltration is asswned to be a composite of 2-D upward infiltration anddownward infiltration components. These components of infiltration arecOJIq:>uted on the assunption that the loci of the wetting pattern for 2-Dinfiltration are serniellipses. Because the developed equations are pre­sented in explicit algebraic power forms, the physical parameters of thesoil rredia, such as hydraulic conductivity, capillary potential, pressurehead, soil porosity, and moisture content, can be expressed as dependentparameters of drip irrigation. The validity of the developed equations wasexamined in five laboratory experiments at different moisture contents.The results of computed infiltration show good agreement.

2540 Dole Street • Honolulu, Hawaii 96822 • U.S.A.• (808) 948-]84]

AI11'fICBS:

Mr. Shan-hsin Chiang*Graduate StudentDepartment of Geographyuniversity of Hawaii at Manoa(808) 948-8664

Dr. Yu-Si FokProfessor of Civil Engin=eringResearcher, water Resources Research Centeruniversity of Hawaii at Manoa(808) 948-7298

*Ph.D. degree (Geography) to beconferred August 1986.

$3.00/copyMail to: University of Hawaii at Manoa

Water Resources Research Center2540 Dole St.Homlulu, Hawaii 96822

Tel.: (808) 948-7847 or -7848

TWO-DIMENSIONAL INFILTRATION EQUATIONS:

SOIL-BOX LABORATORY EXPERIMENTS

Shan-hsin Qliang

Yu-Si Fok

Technical Memorandwn Report No. 78

August 1985

Research Project Completion Report

for

Laboratory Investigation of TWo-Dimensional InfiltrationProject No. &-018

Principal Investigator: Yu-Si Fok

Project Period: 1 July 1984-30 June 1985

The work on which this report is based was supported in part by the Stateof Hawaii General Fund and the Office of the Director, water Resources Re­search Center, University of Hawaii at Manoa.

WATER R&SOORCES RESEARQI CENI'ERUniversity of Hawaii at Manoa

2540 Dole StreetHonolulu, Hawaii 96822

v

Infiltration equations can be used to describe furrow and drip irri­

gation, groundwater recharge, and wastewater injection. Physical two­

dimensional (2-D) infiltration equations for drip irrigation expressed in

explicit, power algebraic forms were develoPed based on four consecutive

time intervals which were derived in a previous study. Two-dimensional

infiltration is assumed to be a corrp:>site of 2-D UIMard infiltration and

downward infiltration corrp:>nents. These corrp:>nents of infiltration are

conp.1ted on the asst.nrption that the loci of the wetting pattern for 2-D

infiltration are semiellipses. Because the develoPed equations are pre­

sented in explicit algebraic power forms, the physical parameters of the

soil media, such as hydraulic conductivity, capillary potential, pressure

head, soil porosity, and moisture content, can be expressed as dependent

parameters of drip irrigation. The validity of the develoPed equations was

examined in five laboratory experiments at different moisture contents.

The results of cOnp.1ted infiltration show good agreement.

ABsmAcr•••

INmCDUCl'ION. . .. . . .

· . . . . . . . . . . . .· . . .

vii

v

1

SOIL-K:>ISIDRE MJ\1EMENT.

Horizontal Flow CoIIp>nent.

Vertical Downward Flow Conq;:lonent •

Vertical Upward Flow Conponent •

· . . . . . . . .· . . . . . . . .

· . . . . . .

2

2

4

5

INFILTRATION EQUATIONS••••••

~RY SOIL-BOX EXPERIMENT. •

MEASURID AND (BSERVED D.Z\TA.

RESULTS AND DISCUSSICN.

· . .· . . .· . . . . . . . . .

7

8

10

11

CCNCLUSIONS . . · . . · · · · · · · · 30

Aa<NCMLEIX;MENTS . · · · · · · · · · · 31

GLOOSARY OF SYMBOLS . · · · · · · • · · 32

REFERENCES CITED. . . · · . . · · · · · · · · · · · · · 33

Figures

l. Assumed Loci of Wetting Fronts • · · · · · · · · · 32. Relationship Between Two Dimensionless Parameters,

yt/h and Kt/(nsh) of Elapsed Infiltration Time, t. · · · · · 63. Experimental Awaratus for 2-D Infiltration. · · · · · 9

4. Mariotte Tube set-up • · · · · · · · · · 9

5. Falling Head Perrneameter • · · · · · · · · · · · 116.1. Experimental Wetting Pattern at Time 1 Interval. · · · · · 236.2. Experimental Wetting Pattern at Time 2 Interval. · · · · · 236.3. Experimental Wetting Pattern at Time 3 Interval. · · · · 246.4. Experimental Wetting Pattern at Time 4 Interval. · · · · · 247. Wetting Processes for Soil sample I-A. · · · · · · · · · 258. wetting Processes for Soil sample I-B. · · · · 259. Wetting Processes for Soil sample 2-A. · · · · · · · · 26

10. Wetting Processes for Soil sample 2-B. · · · · 26

viii

11.

12.

wetting Processes for Soil Semple 2-C. • • • •

Measured and Catplted Cumulative Infiltrationfor Soil Semple I-A. • • • • • • • • • • • • •

. . 27

28

13. Measured and Catplted Cumulative Infiltrationfor Soil Semple I-B. • • • • • • • • • • • • • · . . 28

14. Measured and Conputed Cumulative Infiltrationfor Soil Semple 2-A. • • • • • • • • • • • • • • · . . 29

15. Measured and Conputed Cumulative Infiltrationfor Soil Semple 2-B. • • • • • • • • • • • • • . . . . 29

16. Measured and Conputed Cumulative Infiltrationfor Soil Semple 2-C. • • • • • • • • • • • • • · . . 30

Tables

17

· · . . 18

19

20

· · 21

· · . . 22

9. Cbserved and Conputed Values of Cumulative Infiltrationfor Soil Semple I-B. • • • • • • • • • • • • • • • • • •

1. Parameter Values of Makiki Soil Sanples, A' abu, Hawai' i. . . . . . 11

2. Numerical 8;}Uations for Advance of Horizontal (x),Upward (yt) , and Downward (yi-) Soil-water Movements,and Time Intervals (It, Ii-). • • • • • • • • • • • • • • • • • 12

3. Cbserved and Computed Values of Horizontal, Downward,and Upward Movement for Soil Sample I-A. • • • • • • • • 13

4. Cbserved and Conputed Values of Horizontal, Downward,and Upward Movement for Soil Semple I-B. • • • • • • • • 14

5. Cbserved and Conputed Values of Horizontal, Downward,and Upward Movement for Soil Semple 2-A. • • • • • • • • • • • • • 15

6. Cbserved and Conputed Values of Horizontal, Downward,and Upward Movement for Soil Semple 2-B. • • • • • • • • 16

7. Cbserved and Computed Values of Horizontal, Downward,and Upward Movement for Soil Sample 2-C•••••••••

8. Cbserved and Computed Values of Cumulative Infiltrationfor Soil Sample I-A. • • • • • • • • • • • • • • • • • • •

10. Cbserved and Conputed Values of cumulative Infiltrationfor Soil Semple 2-A. • • • • • • • • • • • • • • • • • •

12. Cbserved and Conputed Values· of Cumulative Infiltrationfor Soil Sample 2-C. • • • • • • • • • • • • • • • • • • •

11. Cbserved and Computed Values of Cumulative Infiltrationfor Soil Semple 2-B. • • • • • • • • • • • • • • • • • • •

IN'DOXlCl'ION

Infiltration equations can be used to describe furrQ\' and drip irriga­

tion, groundwater recharge, and wastewater injection. The flow of water

fran a source, such as a drip irrigation emitter or an irrigation furrow,

into the soil profile is a two-dimensional (2-D) phenomenon. The charac­

teristics of 2-D infiltration of water into the soil profile has been

studied I:¥ Philip (1957), Toksoz, Kirkham, and Baumann (1965), Hillel

(1970), Fok (1970), Kirkham and Powers (1973), and Selim and Kirkham (1973,

1974) • Fok (1967) used four consecutive l-D explicit and power algebraic

functions in the first attenp: to relate the vertical dowr7Nard length of

wetting fran a water source on the soil surface to wetting front to total

head loss; the hydraulic conductivity of the transmission zone; the period

of infiltration; and soil p:lrameters, such as soil porosity and the net

increment of degree of water saturation. '!be total head loss is the sum of

the depth of water on the soil surface, capillary potential head at wetting

front, and the pressure potential loss in the wetting zone. Fok, Clung,

and Liu (1982) awlied successfully Fok' s early four algebraic forms of

time intervals to express the IiJysical 2-D infiltration equation fran a

point source on the soil surface. Fok and Cliang (1984) developed four 2-D

algebraic infiltration equations for furrow irrigation based on Fok' s four

time-interval equations.

Fok' s four consecutive time-interval, infiltration equations were

based on four assumptions: (1) the soil is homogeneous and isotropic,

(2) the soil structure does not change after wetting, (3) the wetting locus

of a wetting front fran a IX>int source is semielliptical, and (4) 2-D

infiltrationcomIX>nents, namely, horizontal and dowr7Nard, can be expressed

I:¥ one-dimensional infiltration equations.

'!be main objective of this research is to extend the 2-D infiltration

equations derived I:¥ Fok, O1ung, and Liu (1982) fran a IX>int water source

on the soil surface to a given depth beneath the soil surface. '!be 2-D

infiltration equations for a point water-source at a given depth beneath

the soil surface constitute the dowrItlard, horizontal, and lJl:Mard infiltra­

tions fran the IX>int water-source.

Three derived consecutive time-interval infiltration equations for

llpiard infiltration in this study were examined in laboratory expe~iments

2

by using a soil-box. Five experiments using Makiki clay-loam (Andie Ustic

Humitropepts) soil were conducted to observe the wetting patterns of the

soil profile at various time intervals during the infiltration processes.

The measured cmnulative infiltrations show good agreement with corres­

ponding computed cmnulative infiltration fran the developed 2-D infiltra­

tion equations.

OOIIr-K>IS'IURE ftD\7EMENT

This study was based on four assumptions: (1) the soil is hanoge­

neous, fine, mixed, and isohypothermic; (2) the soil structure does not

change after wetting; (3) the wetting locus of the wetting front fran a

point source of a drip irrigation emitter is a composite of two semi­

ellipses; and (4) the vertical downward and upward COIIlfX>nents and the

horizontal component of the wetting front can be described by a I-D infil­

tration equation. Figure 1 shows the assumed locus of the wetting fronts

in which x represents the horizontal wetting front, yt the vertically

upward flCM component of wetting front, y-l- the vertically dOWI'Mard flow

component of wetting, and t 1 , t 2 , and t 3 are the observed infiltration

times.

Horizontal Flow ComIx>nent

Infiltration in the horizontal direction is the simplest case of in­

filtration because the component of gravity is zero in the horizontal flow

of soil rooisture and the flow of soil rooisture in the horizontal direction

is drawn only by the matric suction force. Green and Ampt (1911) expressed

the lateral advance of the wetting front of horizontal infiltration as a

function of the square root of infiltration time. 'Ibksoz, Kirkham, and

Baumann (1965) also reported that the empirical lateral advance of the

wetting front for I-D horizontal infiltration is a function of the square

root of infiltration time. Fok, Chung, and Liu (982) derived one equation

which proved the lateral advance of the wetting front as a function of the

square root of infiltration time as follows:

(1)

3

wherex = horizontal advance of wetting front during infiltration

K = hydraulic conductivity in transmission zone

hx = horizontal total head loss in transmission zone

n = soil porosity

s = net incranent of degree of water saturation, which is eqUal toSl - So

Sl = degree of saturation after infiltration

So = degree of saturation before infiltration

t = ela~ed time during infiltration.

Soi 1 Surface

yt

-----..../

I

----...... ..........

'-"-

"-

""\\\

-. ............

"-"-

"-\

\

-- ......"-

"­\

------

..-­.... --,.-

//

/I

....-­/

/'/

//

/II

I II ,I I

/ // /

/ /t z/ //' /

,/ t 3 /.-/ /

...- /'/'

/....

II

//

t 1 //'

/---

-------

I\\,

\\

\

""-"­

'-

I\\,

\\

\

""-"­

"-'- ...... -.

x ....--+---,I----+--------:l.-------t---+-----lr----~xI Water /\ Source\

\"­

"- ......

y+

00l'E: t l = infiltration time observation period 1t z = infiltration time observation period 2t 3 = infiltration time observation period 3x = horizontal wetting front

yt = upward flow wetting fronty+ = downward flow wetting front.

Figure 1. AssLmled loci of wetting fronts

4

Vertical Downward Flow CcmIponent

According to Hansen (955) and Fok and Hansen (1966), the 2-D downward

water movement into the soil during infiltration can be expressed as

yi- _ In (1 +.¥f) Kt1* !lot = nsh

where

yi- = vertically OOwBiard canponent from water source to wetting front

hi- = total head loss in downward transmission zone

and K, N, s, and t are as defined previously.

Based on equation (2), Fok (967) developed four consecutive IX'Wer

functions related to y+ and t as follows:

For 0 ~ t < t 1, y1i- = 1.45 [~~tr~ (3)

For t 1 ~ t ~ t z, yzi- = 1.82 (Khi-~:1Str'5 5 (4)

For t z ~ t ~ t 3, Y3+ = 2.19 (Kh+~~7°tr'B 8 (5)

For t 3 ~ t ~ t" y,i- = 1.83 (Kh+~~77tJ0,8 5 (6)

The end times, t 1, t z, t 3, and tIt' in equations (3) to (6) can be ex­

pressed in terms of the soils properties as

For y+/h+ = 0.1, t1i- = 0.00476 n:hi- (7)

For Yi-/hi- = 1.0, tzi- = 0.316 n:hi- (8)

For y+/hi- = 5.0, t 3i- = 3.26 n:hi- (9)

For yi-/h+ = 30.0, t,,+ = 26.86 nshi- • UO)K

S

Vertical Upward Flow Cooponent

The l-D upward water lOOVement into the soil profile during infiltra­

tion can be derived using Darcy's law and the continuity equation.

The total hydraulic head loss of upward flow in a vertical column of

soil should be h - Lg, in which h, is the constant pressure head loss in

the transmission zone, and Lg is gravity head. If h is expressed in terms

of energy per unit weight, the total hydraulic head and the gravity head

can be expressed by unit of length. Therefore, Lg should be equal to y,

which is the advance of the wetting front in a vertical direction. Darcy's

law is

Q = KA h - Y Ul)y

in which Q is the upward flow rate, A is the cross-sectional area of flow,

and K, h, and y are as defined previously.

For upward flow, the continuity equation is expressed as

Q = ns A ~ • (12)

Since we have indicated the downward movement as fOsitive in direc­

tion, the upward movement should be negative, as

KA h + Y - A d (-y) or_y - ns dt '

h-y_ 9YKA Y -nsA dt •

Equation (13) can also be written as

h - yt _ MKA yt - ns dt •

(13)

(4)

US)

By rearranging and integrating equation (14) with time = 0 to t, and y = 0

to y, equation US) can be ootained as follows:

Yt _ In (1 + Yi) = Kth h nsh •

&;Iuation (IS) shows that yt is an i.Irl'licit function of the elapsed

infiltration time, t. To find the explicit relationship between yt and t,

a graphic approximation is obtained by plotting the two dimensionless

parameters, yt/h and Ktlnsh of equation US) on log-log paper (Fig. 2).

6

r 0.01 Eo; yt/h < 0.3 +- 0.03.0; yt/h < 0.8 +-0.8 Eo; yt/h--

~---- yt/h., 1.41(Kt/nsh)o.s ------<+yt/h., 0.9(Kt/nsh)O~yt/h"0.86(Kt/nsh)O.lS

1.0

~

./1-0"

/

--". yt/h - In (1 + yt/h) ., Kt/nsh

./~

V

.s::~ 0.1>-

0.010.0001 0.001 001

Kt/nsh0.1 1.0 10

Figure 2. Relationship between two dimensionless p:irameters,yt/h and Kt/nsh, of elapsed infiltration time, t

curvilinear relationship between the two p:irameters, yt/h and Kt/nsh, can

be represented approximately by several straight lines, such as the three

consecutive lines in Figure 2. Three IXMer equations showing yt/h as a

IXMer function of Kt/nsh were obtained fran an evaluation of the three

lines in Figure 2 for different ranges of yt/h as follows:

for yt/h < 0.3, y1t/h = 1.40 (Kt/nsh)D.5

for 0.3 ~ yt/h < 0.8, Y2t/h = 0.90 (Kt/nsh)~a,

for 0.8 ~ yt/h < 1.0, yat/h = 0.86 (Kt/nsh) 0.15 •

(16)

(17)

(18)

The typical end times t 1t , t 2t , and tat for equations (16) to (18)

that can be derived in terms of soil properties are

for yt/h = 0.3, t1t = 0.0567

for yt/h = 0.8, t 2t = 0.809 nsh/K·

for yt/h = 0.99, tat = 3.615 nsh/K •

(19)

(20)

(21)

Thus, the IXMer function relationships between yt and t can be 0b­

tained by evaluating yt in equations (16) to (18) according to the corres­

ponding infiltration periods:

for 0 ~ t < tl' Yl t = 1.40 (Kht/ns) 0.5 (22)

for t 1 ~ t~ t 21 yz+ = 0.90 (Khl.'Ot/ns)o.H

for t z ~ t < t 3, Y3+ = 0.86 (Kh S.6 7t/ns) G.1S •

INFIL'IRATION EQUATIONS

7

(23)

(24)

Based on the assumption that the loci of wetting fronts fran a p:>int

are semiellifSes, the upward 2-D cumulative infiltration oornp:>nent can be

expressed as

1TIt = '2 xytbns (25)

in which It is llpiard cumulative infiltration, b is depth of the soil-box,

and x, yt, n, and s are as defined previously.

SUbstituting yt and x in equation (25) and the cumulative infiltration

It for various time intervals, t1t, tzt, and t 3t can be derived as follows:

The downward cumulative infiltration, Ih derived 1::¥ Fok, Chung, and

Liu 1982) for various time intervals, t 1 +, tz+r t 3h and t,+ can be derived

as follows:

for 0 ~ t < t 1, 11+ = 3 .22 bKh~'s hO.s t

for t 1 ~ t < t z, I z+ = 4.04 bK1.0 S hO.s hMS (ns) -0.0 S t1.0 Sx

for t z ~ t < t 3, 13+ = 4.85 bK1.18 hO.s hO.3 z (ns) -0.18 t 1•18x

for t 3 ~ t < t p I,+ = 4.07 bK 1.1 S hO.s ho.1s (ns) -0.3 S t 1.38 .X

(29)

(30)

(31)

(32)

The total cumulative infiltration [I should be the summation of the

upward cumulative infiltration It and the downward cumulative infiltration

1+, and can be expressed for the same time zone as

and the infiltration rate as

["I = It + 1+

. dIJ. = dt •

(33)

(34)

8

Thus, the infiltration rate for upward movement can be expressed as

follows:

for 0 ~ t < t u it = 3 .11 bKh~s hO.s (35)

for t l ~ t < t u izt = 1.68 bKo.u h~s hO.u (ns)O·18 t-o.18 (36)

The infiltration rates for downward movement were developed by Fok,

Chlmg, and Liu (1982) as follows:

for 0 ~ t < t l , i l + = 3.22 bKhl~ hO.s (38)

for t l ~ t < t u i z+= 4.24 bKl.O S h*s hO., S(ns) -0.0 S to.o S (39)

for t z ~ t < t 3, i 3+ =5.37 bK L18 hf(s hO.3S(ns) -0.18 t 0.18 (40)

for t 3 ~ t < t" id = 5.49 bK L3S h*s hD.lS (ns) -0.3 S to.36 . (41)

The total infiltration rate, H, is the summation of the upward infil­

tration rate and the downward infiltration, as

L:i = it + i+ •

l.AB(EAT()RY SOIL-BOK EXPERIMENT

(42)

An experimental soil box alIOOst similar to that used by Fok, Chung,

and Liu (1982) was designed to examine the validity of their derived equa­

tions. The box was constructed of plywood with inside dimensions of 1.01m<39.4 in.) wide x 12 m deep x 0.06\ m (2.5 in.) thick. Three sides and the

bottom of the box were of plywood. The front was sealed with a 12 m x 12 m

Plexiglas sheet so that the wetting front of the soil during infiltration

could be easily traced through the glass at selected time intervals. The

Plexiglas front also enabled the evaluation of oorizontal, vertically d~

ward and upward soil-IOOisture IOOvements (Fig. 3).

A Mariotte tube was used to· maintain a continuous infiltration water

sUWly and to provide visual readings of the volume of infiltration water

during selected time intervals (Fig. 4). A falling head permearneterde­

termined the hydraulic conductivity, K (Fig. 5).

Two samples of Andic Ustic Humitropepts from Maroa Valley, O' ahu were

used to conduct the experiments. The Inceptisols Order soil was compacted

Figure 3. Experimentalapparatus for2-D infiltration

9

Figure 4. Mariotte tubeset-up

10

Figure 5. Falling headpermeameter

evenly into the soil box with a wooden rod. '!be moisture content of the

soil before and .inmediately after wetting was determined 1::¥ the oven-dry

method. '!be porosity, n, and the net increment of degree of water satura­

tion, s, were canputed fran the meaSured data by the OITen-dry method.

Five experiments were conducted on soils of two different densities

with different initial water contents. The soil parameter values for these

five experimental soil ~les are listed in Table 1.

Based on the measured and computed soil parameters, the numerical

equations for the advance of horizontal, vertically upward and downward

soil-water roovements, the time intervals for upward and downward movement,

and the cumulative upward and downward infiltration are listed in Table 2.

The measured and conputed data of the horizontal, upward and downward

11

TABLE 1. PARAMETER VALUES CF MARIKI roIL SAMPLFS, O'ABU, HAWAI'I

(%)

16.27 63.07 57.13 32.5

22.96 65.47 32.26 34.3

25.63 60.58 32.75 39.1

19.86 59.08 19.84 19.2

22.20 60.32 44.36 34.1

2.81

2.81

2.47

2.47

2.47

DENSITY

Bulk True

1.04

0.98

0.97

1.02

0.98

4.08

3.29

9.69

1.01

2.55

HYDRAULICa:mUC­TIVITY

:m:R&=MENTALDEGREE

OFSATURA­

TIOO

PORCSlTY

INITIALVCLtrMETRICWATER,

CCNl'ENl'

I-A1-B

2-A

2-B

2-C

Wl'E: Arnic Ustic HlDRitropepts (Inceptisols) soils.

lOOVements for the five experiments are respectively listed in Tables 3

to 7.The total cwnulative infiltration for five soil samples are listed in

Tables 8 to 12.

The experimental wetting patterns for various times are shown in .

Figures 6.1 1;:0 6.4, and the measured wetting patterns of the total twcr

dimensional infiltration for five soil samples are presented in Figures 7

to 11.

RESULTS AND DIS<lJSSIOO

The five experiments show that the conputed and observed cumulative

infiltrations have a linear relationship with infiltration time on log-log

paper. The results also indicate that the difference between the COlrputed

and measured values is quite large at the beginning during a short time

interval and also after a relatively long time period (Figs. 12-16).

The COIIplted data were consistently less than the observed data in time

interval zone I-perhaps the result of the actual water supply at the

beginning of the experiment being more than that which was recorded. The

excess amount of water during the initial period of infiltration was caused

by some difficulty in maintaining a constant flow using the manually

12

TABLE 2. mMERICAL EOUAT1ONS FOR NNl>N:E OF HORIZCNl'AL (x),UIWARD (yt), AND~ (Y+) SOIL-WATER KNEMENTS,AND TIME INTERVALS (I t, 1+)

SAMPLEI-A I-B 2-A 2-B 2-C

x 0.86 t°oS 1.03 t°oS 1.88 t°oS 0.58 t 0.5 0.72 to.5

Yd 0.85 t°oS 1.02 t OoS 1.93 t°oS 0.57 to.5 0.71 t o.s

Yzt 1.86 to.H 2.15 t°.3' 3.44 t 0.3' 1.21 t 0.3' 1.58 to.H

Y3t 8.48 t 0.15 9.31 to.lS 12.36 tOol 5 5.20 tOol 5 7.20 t o.15

Yl+ 0.88 t°oS 1.06 t 0.5 1.93 t°.s 0.59 to.s 0.74 t 0.5

Yz+ 0.74 t°oS5 0.91 t o•55 1.80 to.55 0.50 t O.s 5 0.62 t 0.55

Y3+ 0.32 to.SI 0.40 to.u 0.91 to.s I 0.22 t o.s I 0.27 t o.s I

Y,+ 0.07 t°.B5 0.09 t°.B5 0.24 t 0.1 5 0.05 t°.B5 0.06 to.B 5

t 1t 163 min 125 min 46 min 127 min 162 min

tzt 2,323 min 1,782 min 647 min 1,812 min 2,300 min

t 3t 10,380 min 7,960 min 2,892 min 8,094 min 10,278 min

td 14 min 11 min 4 min 11 min 14 min

t z+ 908 min 696 min 253 min 708 min 899 min

t 3+ 9,361 min 7,179 min 2,608 min 7,299 min 9,269 min

td 82,117 min 59,144 min 21,487 min 60,136 min 76,364 min

I 1t 2.68 t 3.51 t 7.64 t 0.39 t 1.41 t

l zt 6.16 to.I , 5.03 t°.B' 14.31 t°.B' 0.87 t 0.1' 3.23 tOol'

1 3t 26.71 t o.s5 20.72 to.S5 48.82 t u.s 5 3.58 t o.S5 13.98 to.s 5

Id 2.78 t 3.63 t 7.91 t 0.41 t 1.46 t

l z+ 2.34 t 1.05 3.10 tl.05 7.11 tl.O 5 0.35 tl.O 5 1.23 t 1.05

1 3+ 1.00 t 1•1B 0.89 t 1•1B 3.59 t 1.1I 0.15 t 1.1B 0.52 t 1•1I

1,+ 0.65 t 1•35 0.63 t 1•35 2.44 t 1•35 0.32 t 1.35 0.43 t1.35

13

TABLE 3. <BSERVED AND OOMPUTED VALUES CF H<:RIZCNl'AL, ~,AND UIWARD~ FOR SOIL SAMPLE I-A

SOIIrWATER MJVElt1ENT

TIMEHorizontal (x) Downward (y-l) tp/ard (y+)

L(Obs.) * R(Obs.) t Corrp.· Cbs. Corrp. Cbs. Comp.(min) -(cm)-

3 1.6 1.8 1.5 1.6 1.5 1.6 1.5

6 2.1 2.3 2.1 2.1 2.2 2.1 2.1

12 3.2 2.9 3.0 2.9 3.0 2.8 2.9

20 3.8 3.5 3.8 3.4 3.8 3.5 3.8

30 4.5 4.4 4.7 4.2 4.8 4.2 4.6

40 5.2 5.0 5.4 4.8 5.6 4.8 5.4

60 6.6 6.4 6.6 6.2 7.0 5.9 6.6

80 7.4 7.4 7.7 6.9 8.2 6.8 7.6

100 8.2 8.3 8.6 8.1 9.2 7.6 8.5

120 9.2 9.0 9.4 8.9 10.2 8.4 9.3

150 10.5 10.4 10.5 10.1 11.5 9.5 10.4

180 11.6 11.4 11.5 11.2 12.9 10.4 10.9

230 13.1 13.0 13.0 12.8 14.7 11.8 11.8

280 14.5 14.5 14.4 14.2 16.4 12.9 12.6

335 15.8 15.6 15.7 15.8 18.1 13.7 13.4

395 17.3 17.3 17.1 17.7 19.9 14.9 14.2

450 18.4 18.4 18.2 18.9 21.1 15.9 14.8

*Left.tRight.

14

TABLE 4. (BSERVED AND <X>MPUTED VALUES CF HORIZCNl'AL,~,AND U1WARD MJIJD1ENT FOR SOIL SAMPLE 1-B

SOII.cWATER IOJFlt1ENTTIME Horizontal (x) Downward (yi-) . tplard (yt)

L(Obs.) * R(Obs.) f eonp. C1:>s. eonp. Cl:>s. CoIrp.(min) (em)

5 2.3 2.6 2.3 2.5 2.4 2.3 2.3

10 3.4 3.5 3.3 3.6 3.4 3.4 3.2

15 4.3 4.3 4.0 4.4 4.0 4.2 4.0

20 4.8 4.9 4.6 5.0 4.7 4.8 4.6

25 5.4 5.5 5.2 . 5.5 5.3 5.3 5.1

35 6.5 6.4 6.1 6.4 6.4 6.2 6.1

45 7.4 7.2 6.9 7.2 7.3 7.0 6.9

60 8.3 8.4 8.0 8.3 8.6 7.8 7.9

75 9.2 9.2 9.0 9.3 9.7 8.7 8.9

95 10.3 10.3 10.1 10.3 11.0 9.6 9.9

125 11.7 11.6 11.6 11.7 12.9 10.6 11.4

165 13.5 13.3 13.3 13.4 15.0 11.9 12.2

205 14.8 14.7 14.8 15.0 16.9 13.1 13.1

255 16.4 16.2 16.5 16.5 19.1 14.2 14.1

315 17.9 17.8 18.3 18.2 21.4 15.5 15.2

380 19.6 19.7 20.1 20.1 23.7 16.9 16.2

445 21.3 21.2 21.8 21.4 25.9 18.2 17.1

510 22.5 22.6 23.3 22.7 27.9 19.2 17.9

555 23.6 23.7 24;3 24.0 29.2 19.9 18.4

*Left.tRight.

15

TABLE 5. (BSERVED AND COMPUTED VALUES CF HORIZCNrAL, ~,AND umARD KJVEMENT FOR SOIL SAMPLE 2-A

SOIL-WATER MJIJEMENT

TIMEHorizontal (x) Downward (y-l-) Upward (yt)

LWbs.) * R«bs.) f Conp. Cbs. Conq;>. Cbs. Conp.-(min) (em>

10 6.6 7.0 6.0 6.5 6.4 6.3 6.1

15 8.2 8.5 7.3 8.4 8.0 7.5 7.5

20 9.3 9.7 8.4 9.3 9.4 8.5 8.6

25 10.4 10.8 9.4 10.3 10.6 9.4 9.7

30 11.4 11.9 10.3 11.1 11.7 10.2 10.6

40 13.2 13.6 11.9 13.5 13.7 11.8 12.2

50 14.5 15.4 13.3 15.3 15.5 12.8 13.0

60 15.6 16.8 14.6 16.4 17.1 13.7 13.8

80 18.0. 18.9 16.8 18.9 20.0 15.2 15.3

100 20.1 20.6 18.8 21.6 22.7 16.7 16.5

120 22.2 22.3 20.6 23.8 25.1 17.8 17.5

ISO 25.0 24.7 23.1 26.8 28.3 19.6 18.9

180 26.8 26.2 25.3 29.5 31.3 20.6 20.1

225 29.7 29.2 28.2 ·33.5 35.4 21.7 21.7

285 32.8 32.8 31.8 37.6 42.3 23.8 23.5

345 35.5 35.2 35.0 41.7 48.2 24.8 25.1

420 38.2 38.3 38.6 46.2 55.1 26.6 26.8

500 41.3 41.4 42.1 50.5 . 62.0 27.8 28.5

*Left.fRight.

16

TABLE 6. CBSERVED AND roMPUTED VALUES CF HORIZCNmL, ~,AND UIWARD~ FOR SOIL SAMPLE 2-B

SOIL-WATER MJVEMENT

TIMEHorizontal (x) Downward (y+) Upward (yt)

L<Obs.) * R«bs.) f eonp. Cbs. Comp. Cbs. Comp.(min) (ern)

40 4.0 4.2 3.6 3.9 3.8 3.6 3.6

60 5.0 4.9 4.5 4.8 4.8 4.2 4.4

120 7.0 6.6 6.3 7.5 7.0 5.6 6.2

165 7.8 7.6 7.4 8.5 8.3 6.5 6.9

210 8.9 8.4 8.3 9.5 9.5 7.3 7.4

280 9.8 9.6 9.6 10.5 11.1 8.6 8.2

365 11.5 11.2 11.0 11.8 12.8 9.1 9.0

545 13.6 13 .4 13.4 14.5 16.0 11.0 10.3

790 16.4 16.5 16.2 17.0 20.5 14.1 11.7

1070 17.9 18.1 18.8 19.2 25.3 16.2 13.0

1325 21.5 21.1 20.9 21.5 29.2 17.8 13.1

1685 22.5 22.8 23.6 25.0 34.4 19.4 15.1

2150 25.8 26.0 26.7 28.0 40.6 22.0 16.4

2740 28.5 28.0 30.1 32.2 47.9 24.0 17.0

3155 30.4 30.1 32.3 36.0 52.7 25.1 17.4

3655 32.0 31.9 34.8 38.5 58.2 26.2 17.8

4235 34.3 34.5 37.1 41.1 64.4 27.5 18.2

*Left.fRight.

17

TABLE 7. (BSERVED AND COMPUTED VALUES <F HORIZCNl'AL, ~,AND UPWARD ~1T FOR SOIL SAMPLE 2-C

SOIL-WATER MJVEMENT

TIME Horizontal (x) Downward (Y-t) Upolard (y+)L(Obs.) * R«l)s.) T Corrp. Cbs •. Corrp. Cbs. Corrp.

(min) (em)

20 2.9 3.2 3.2 3.0 3.2 3.0 3.2

40 4.8 5.0 4.6 4.7 4.7 4.8 4.5

120 8.0 8.4 7.9 8.1 8.7 8.0 7.8

165 10.5 10.8 9.2 9.7 10.3 10.3 9.2

280 12.8 13.1 12.0 13.3 13.8 12.3 10.7

400 15.2 15.4 14.4 14.8 16.8 15.0 12.1

580 18.3 18.2 17.3 18.0 20.7 17.0 13.7

1020 23.5 23.6 23.0 22.0 29.6 19.8 16.7

1365 26.8· 26.9 26.6 26.0 36.0 22.2 18.4

1715 30.6 30.8 29.8 32.4 42.2 25.2 19.9

1995 34.0 33.9 32.2 36.7 46.8 27.2 20.9

2465 38.6 38.1 35.7 43.6 54.0 29.3 23.2

2835 40.8 40.3 38.3 46.4 59.5 30.7 23.7

3155 42.9 42.7 40.4 50.2 63.9 32.3 24.1

3375 46.8 46.2 41.8 53.0 67.0 33.3 24.4

3765 47.4 47.1 44.2 57.0 72.1 35.0 24.8

*Left.fRight.

18

TABLE 8. OOSERVID AND cntPUTED VALUES CF aJMULATIVE INFIL'mATIONFOR roIL SAMPLE I-A

INFIL'lRATION

TIME Upward (It) Downward (I ~) Total cumulative, HConp.lted Conplted Cbserved Conplted

(min> (cm3 )

3 8.1 8.3 21.1 16.4

6 16.1 16.7 41.1 32.8

12 32.2 33.3 76.2 65.5

20 53.6 54.3 113.3 107.9

30 80.4 83.2 166.5 163.6

40 107.2 112.5 225.6 219.7

60 160.9 172.2 340.9 333.1

80 214.5 232.9 471.2 447.4

100 268.1 294.5 580.5 562.6

120 321.7 356.6 680.8 678.3

ISO 402.1 450.8 855.3 852.9

180 483.2 545.8 996.8 1 029.1

230 593.7 706.1 1 241.6 1 299.8

280 700.4 868.1 1 496.3 1 568.5

335 814.3 1 047.9 1 751.1 1 862.2

395 935.1 1 299.8 2 141.9 2 234.9

450 1 043.3 1 428.6 2 280.9 2 471.9

TABLE 9. CJ3SERVED AND CXJo1PUTED VALUES CF QJMULATIVE INFILTRATIONFOR SOIL SAMPLE I-B

INFIL'lRATION

TIMEUpward (It) Downward (I -1-) Total cumulative, H

Conplted Corrplted <l:>served Corrplted(min> (cm3 )

5 17.5 18.2 48 .3 35.7

10 35.1 36.3 81.5 71.4

15 52.6 53.3 129.7 105.9

20 70.2 72.0 165.9 142.2

25 87.8 91.0 202.1 178.8

35 122.9 129.6 268.5 252.5

45 157.9 168.7 358.9 326.6

60 210.6 228.3 473.5 438.9

75 263.3 288.5 594.1 551.8

95 333.5 369.8 732.8 703.3

125 421.2 472.6 962.0 893.8

165 366.7 660.3 1 257.7 1 027.0

205 440.0 829.3 1 510.8 1 269.3

255 528.5 1 042.9 1 812.4 1 571.4

315 631.2 1 301.9 2 246.6 1 933.1

380 738.9 1 585.4 2 614.5 2 324.3

445 843.7 1 871.3 2 970.4 2 715.0

510 946.1 2 159.3 3 338.2 3 105.4

555 1 015.7 2 359.8 3 615.7 3 375.5

19

20

TABLE 10. OOSERVED AND OOMPUTED VALUES CF ClJMULATIVE INFIL'IRATIOOFOR SOIL SAMPLE 2-A

INFIL'IRATIOO

TIMEUpward (It) Downward (I -1-) Total CUmulative, rI~ted Computed Cl:>served Conplted

(min) (cm3 )

10 76.4 79.7 238.9 156.1

15 114.6 122.0 352.4 236.6

20 152.8 165.1 447.9 317.9

25 191.0 208.6 543.5 399.6

30 229.2 252.7 645.0 481.9

40 305.6 348.1 824.1 647.4

50 382.7 432.0 997.3 814.7

60 445.9 523.1 1 158.6 969.1

80 567.9 707.6 1 505.0 1 275.2

100 684.9 894.5 1 839.4 1 579.4

120 798.3 1 083.2 2 138.0 1 881.4

150 962.8 1 369.2 2 603.8 2 332.0

180 1 122.2 1 658.1 3 004.0 2 780.3

225 1 353.5 2 095.8 3 637.0 3 449.3

285 1 650.9 2 686.3 4 383.5 4 337.2

345 1 938.2 3 283.0 5 136.0 5 221.2

420 2 286.5 4 036.2 5 954.2 6 332.7

500 2 647.1 4 847.1 6 694.7 7 494.2

TABLE 11. OOSERVID AND <ntPUTED VALUES CF OJMULATIVE INFILTRATIONFOR SOn. SAMPLE 2-B

INFIL'IRATION

TIMEUpward (It) Downward (I",) Total cumulative, L:I

Conplted Computed Cl:>served Conputed(min) (crn3 )

40 15.7 16.7 44.1 32.4

60 23.5 65.2 49.0

120 47.1 52.7 127.1 99.8

165 63.1 73.7 166.0 136.8

210 77.2 94.9 211.2 172.1

280 98.3 128.4 278.6 226.7

365 122.8 169.6 357.4 292.4

545 172.0 258.4 510.4 430.4

790 234.9 400.7 705.1 635.6

1070 303.2 574.6 900.7 877 .8

1325 362.8 739.4 1 150.0 1 102.2

1685 444.0 981.9 1 359.2 1 425.9

2150 524.1 1 309.1 1 729.8 1 833.2

2740 614.4 1 742.8 2 179.8 2 357.2

3155 673.3 2 058.3 2 312.4 2 731.6

3655 740 .9 2 448.5 2 801.5 3 189.4

4235 815.3 2 913.3 3 060.1 3 728.6

21

22

'mBLE 12. CBSERVED AND CXJttPUTED VALUES CF QJMULATIVE INFIL'1RATIONFOR roIL SAMPLE 2-C

INFILTRATION

TIME UJ;Mard <It) Downward <I+) Total Cumulative, 1:1Carplted Conplted Cbserved Conputed

(min> (cm3 )

20 28.2 28.6 68.0 56.8

40 56.3 59.2 142.4 115.5

120 168.9 187.5 393.2 356.4

165 235.5 262.0 513.6 497.5

280 367.1 456.5 825.1 823.6

400 495.4 663.8 1 140.3 1 159.2

580 676.8 979.5 1 535.4 1 656.3

1020 1 087.5 1 845.7 2 658.6 2 933.2

1365 1 389.0 2 602.9 3 321.4 3 991.9

1715 1 682.6 3 407.5 4 254.2 5 090.1

1995 1 910.5 4 073.2 4 908.6 5 983.7

2465 2 240.1 5 228.1 6 235.0 7 648.2

2835 2 452.8 6 166.2 7 355.1 8 619.0

3155 2 629.4 6 995.6 7 685.3 9 625.0

3375 2 747.2 7 640.9 8 483.2 10 388.1

3765 2 949.5 8 618.0 9 187.6 11 567.5

23

Figure 6.1. Experimental wettingpattern at time 1interval

Figure 6.2. Experimental wettingpattern at time 2interval

24

Figure 6.3. Experimental wettingpattern at time 3interval

Figure 6.4. Experimental wettingpattern at time 4interval

DISTANCE (em)18

16

18

20

Figure 7. Wetting processes for soil sample I-A

DISTANCE (em)22

~

c

..§. 24t-+t+H-ff.-H*"-tt-tt+1H+1-t1-#-'H-t--D--t.-+t+H+-t-+-t--+1f--Hft-tf-rt-+t+t+-tl

UJ:E

I-

Figure 8. wetting processes for soil sample I-B

25

26

DISTANCE (em)30

45

UJ:z:I-

~

c

..§. 45t--1H-H-+f-f+-I-+-f-+tl-HtlI++--()~t-H-tttt-1I"tH--tt+-Ht-+++-;t--I-tt----i

50

55

Figure 9. Wetting processes for soil sample 2-A

DISTANCE (em)30

~

c

..§. 35UJ t+14-t+--++-+-tH-tlI+f-+Ht++H-(J--I*+t+-t-+-\-+;H-+J4~H-+t+-tl:z:I-

40

45

Figure 10. Wetting processes for soil sClI\l>le 2-B

27

DISTANCE (on)40

60

Figure 11. wetting processes for soil sample 2-C

by sane difficulty in maintaining a constant flow using the manually

operated flow regulator. The measured data, however, were less than the

canputed data after a relatively long period of infiltration, usually in

time zone 3. This difference might be based on the fact that soil param­

eters, such as moisture content, hydraulic conductivity, and total head

loss, carmot be considered as constants. Although the theoretical assUffiIF­

tions and the experiments may have various possible sources of errors, such

as fran soil sample canpaction, measuranent, reading, recording, and opera­

tion of the manual flow regulator, the computed total cumulative infiltra­

tions are closely similar to the measured data in these five experiments

within time interval zone 3, especially within 1,000 min. '!he results of

the I-D movements, horizontal, vertically upward and downward, of wetting

fronts are almost equal between the canputed and the measured data within

500 min. After 500 min, the computed downward movement is greater than the

observed data, and the computed upward movement is less than the observed

data. These facts may constitute the basis for further IOOdifications of

assumptions.

~Q)

TIME (min)400 60040 60 80 100 20020

TIME (min)2 3 4 5 67 8910

. ---- Measured data------ Computed data

Ul:> 100- 80I-~..... 60:::>:E:::> 40(,.)

20

101

4000, Iii iii iii iii iii Iii i' i i ~

2000

­'"E 1000~ 800

z 600o~ 400a:1-......i: 200Z

40040 60 80 100 200202 3 4 5 67 8910

---- Measured data

------ Computed data

3000

2000

- 1000e 800U- 600Z0

400

~a:I-

200.....i:z

100Ul80:>

~ 60.....:::> 40:E:::>(,.)

20

101

Figure 12. Measured and computed cumulativeinfiltration for soil sample 1-A

Figure 13. Measured and computed cumulativeinfiltration for soil sample 1-B

TIME (min)

---- Measured data----- Computed data Measured data

Computed data

20

40

400

10080

60

200

w>

~....I::>~::>u

zo

2000

5000. I I [i iii Iii I iii [ I tit I I i4000

10' ! I ! I , I I !, !!" ! ! ! 'I ,,' ,

10 20 406080100 200 400600800100020004000

TIME (min)

~a:I­....I

U.Z

--­'"EU 1000

- 800

600

40 60 80 100 200 400 600202 3 4 5 678910

100008000

6000

4000

--- 2000'"Eu-Z 10000 800-I- 600~a:I- 400....I-U.Z

200w>-~ 100....I 80::>~ 60::>u

40

20

101

Figure 14. Measured and corrputed cumulativeinfiltration for soil sample 2-A

Figure 15. Measured and COItq?Uted cumulativeinfiltration for soil sample 2-B

I\)\D

~·lj •

30

10000 ---- Measured data8000

6000- Computed data

4000

.... 2000EU-z

10000

~800

'" 600I-....I- 400LA..Z

LoJ 200>l-e:(....I::::> 100~

80::::>u

60

40

20

1°1~0-~2::1;:O-L.....:J40:--'-6~0""":8~0":-'10:"::0--:-20~O---l~4~00:-":-:60.l.::0~80~0...J.l0:"::0"':"'0-:2-:-100~0--L4-:-1000

TIME (min)

Figure 16. Measured and cOJ1P1ted cumulativeinfiltration for soil sample 2-C

CCHLUSIONS

Experimental laboratory results showed that the two-dimensional in­

filtration };i1enomenon can be expressed by the explicit, };X)Wer, and alge­

braic equations derived fran one-dimensional infiltration. Soil par~

eters, such as hydraulic conductivity, capillary potential, total head

loss, soil porosity, rooisture content, and infiltration time, are expressed

explicitly as dependent variables of the cumulative infiltration. These

explicit, algebraic infiltration equations in };X)Wer forms with their appro­

priate application time zones p~ovide drip irrigation system designers an

easy to use mathematical· guide to evaluate the feasibility and performance

of their design.

laboratory experiments on five Makiki clay-loam soil s~les showed

that the measured, total cumulative infiltrations were quite similar to the

31

computed total cumulative infiltrations within 1,000 min, despite the many

assumptions made for soil parameters, such as constants for porosity, net

incranental degree of saturation, hydraulic conductivity, and total head

loss, as well as the various possible sources of experimental errors, such

as soil canpaction, measuranent, reading, recording, and operation of the

manual flow regulator.

The authors are indebted to Dr. L. stephen Lau, Director, water

Resources Research Center, for his interest in and encouragement of this

research project. We also wish to thank Andr~ H. Oshita, Technician,

Department of Civil Engineering, for the construction of the laboratory

soil-box.

32

A cross-sectional area of flow

b thickness of soil sanple in soil box

h total constant pressure head loss in vertical direction in trans­mission zone

hx horizontal constant pressure head loss in horizontal direction intransmission zone

It upward cumulative infiltration

I + downward cumulative infiltration

it llplard infiltration rate from point source in soil nedium

i + downward infiltration rate from point source in soil nedium

K hydraulic conductivity in transmission zone

Lg gravitational head of infiltration flow

n constant porosity of soil nedium

o upward flow rate in soil nedium

s net incranental degree of saturation in transmission zone, i.e.,Sl - So

Sl constant degree of saturation in transmission zone after infiltration

So constant degree of saturation of soil nedium before infiltration

ns incranental· soil water content by volume per unit volume of bulksoil = ~e

t time of infiltration

x distance of wetting in horizontal direction fran given point source towetting front

yt vertical llplard length of wetting from given point source to wettingfront

y+ vertical downward length of wetting fran given point source to wettingfront

SUbscripts

1-4 typical time for separation of infiltration period

t 'upward movement or length

+ downward movement or length

33

Fok, Y.S. 1967. Infiltration equations in exponential forms. J. Irrig.Drain. Div., Am. SOC. Civ. Eng. 93 <IR4) :125-35, Proc. Paper 5686.

1970. A study of two-dimensional infiltration. Trans. Am. SOC.Agr. Engr. 13(5):676-81.

___, and Hansen, V.E. 1966. Q1e-dimensional infiltration into hano­geneous soil. J. Irrig. Drain. Div., Am. SOC. Civ. Eng. 92<IR3):35­48, Proc. Paper 4912.

__~,; Chung, S.O.; am Liu, C.C.K. 1982. 'IWo-dirnensional exponentialinfiltration equations. J. Irrig. Drain. Div., Am. SOC. Civ. Eng.108(IR4):231-41, Proc. Paper 17565.

___, and Chiang, S.-h. 1984. 2-D infiltration equations for furrowirrigation. J. Irrig. Drain. Div., Am. SOC. Civ. Eng. 110(2):208-17,Proc. Paper 18947.

Green, W.H., and Ampt, G.A. 1911. Studies on soil physics. Part I-'Iheflow of air and water through soils. J. Agric. SCi. (G.B.) 4:1-24.

Hansen, V.E. 1955. Infiltration and soil water movement during irriga­tion. Soil SCi. 79(2):93-105.

Hillel, D. 1970. Soil and water. New York: Academic Press.

Kirkham, D., and Powers, W.L. 1973. Advanced soil physics. New York:Wiley-Interscience. 242 pp.

Philip, J.R. 1957. MJrnerical solution of equations of the diffusion typewith diffusivity concentration-dependent II. Aust. J. Phys. 10:2~42.

Selirn, H.M., and Kirkham, D. 1973. Unsteady two dimensional flow of waterin unsaturated soils above an i.npervious barrier. Proc. Soil SCi.SOC. Am. 37:489-95.

___, and Kirkham, D. 1974. Unsteady state two-dimensional watercontent distribution and wetting fronts in soil. Geodermall:25~74.

Toksoz, S.; Kirkham, D.; and BalDIlann, E.R. 1965. 'IWo-dirnensional infil­tration and wetting fronts. J. Irrig. Drain. Div., Am. SOC. Civ. Eng.9l<IR3) :65-79.