the effect of initial moisture content and rain (sprinkler) intensity on wetting front advance...

CATENA vol. 15, p. 491-505 Braunschweig 1988

THE EFFECT OF INITIAL MOISTURE CONTENT AND RAIN (SPRINKLER) INTENSITY

ON WETTING FRONT ADVANCE DURING RAIN (SPRINKLER) INFILTRATION

Y.Z. E l -Sha fe i , A l e x a n d r i a

Summary

A laboratory experiment was carried out on a 5 cm inside diameter by 70 cm long column of silty clay soil under five initial moisture contents, and five sprinkler intensities using a sprinkler simulator. A general equation was derived taking into account the initial moisture content and water application rate, to compute the wetting front advance during sprinkler infiltration. The experimental results strongly supported the derived equation, and the wetting front advance could be predicted under any combination of initial moisture content and sprinkler intensity. The study also revealed that the advance of wetting front increases either by increasing the sprinkler intensity or initial moisture content.

The numerical solution of the partial differential equation of water flow of- fered by HANKS et al. (1969) was applied to compute the wetting front advance during the sprinkler infiltration. A good agreement was found between the two methods of computations; the derived equation and the differential equation. However, the solution of the partial

ISSN 0341-8162 (~)1988 by CATENA VERLAG, D-3302 Cremlingen-Destedt, W. Germany 0341 8162/88/5011851/US$ 2.00 + 0.25

differential equation needs very accurate estimation of K -- 19 and o2 - 19 relationships to achieve reliable results.

1 Introduction

The rate at which water can be applied to the soil without causing runoff is an important factor in irrigation and hydrology. The term rain or sprinkler infiltration as used here refers to a down- ward entry of water into the soil at a rate sufficing to prevent the formation of an enduring water cover (surface ponding).

ABRAMOV (1954) reported that for each kind of soil, there exists a certain relationship between intensity of sprin- kling and duration of irrigation until the moment when runoff appears above the surface level. He also stated that the rain (sprinkler) infiltration is greatly in- fluenced by the state of moisture in the upper layer before irrigation. HANSEN (1955) utilized Darcy's law and intro- duced an equation to determine the wetting front advance during infiltration, depending mainly on the characteristics of the transmission zone. RUBIN & S T E I N H A R D T (1963) solved numerically the partial differential equation (1) for the rain infiltration under the initial and boundary conditions of interest.

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492 EI-Shafei

O0 8 (K Olp) aK -Si - + T £

( 1 )

in which ® = the moisture content on volume basis.

= the matric or capillary potential. K = the soil capillary conductivity. L = the depth of wetting front.

They also neglected the effect of seal- ing due to raindrop impact and stated that the rain infiltration may continue indefinitely into a deep profile without causing ponding if the rain intensity is equal or less than the saturated conductivity of the soil. RUBIN et al. (1964) presented a linear equation to predict the advance of wetting front taking into account the initial moisture content (Oi) and rain intensity (I). However, their equation required the values of the lim- iting transmission zone moisture content and the soil capillary conductivity at the initial moisture content. HANKS et al. (1969) described a general numerical method for equation (1) from which they could compute the wetting front advance and the moisture profile under constant rain (sprinkler) intensity. However, the solution of equation (1) is required input data of ~ c - ® relationship and K - ® relationship in addition to the values of ®i and I.

PARLANGE (1972) presented a numerical integration of the infiltration equation (of PHILIP 1969) under constant imposed flux to calculate the moisture distribution. PHILIP (1973) presented an analysis for equation (1) by introducing a parameter (F) called flux- concentration relation, which is the ratio of the flux at any point to the flux at the soil surface. An approximation for F as:

F = ( O - - - -

where Os is the value at soil surface, was given by WHITE et al. (1979),

CLOTHIER et al. (1981), and PER- ROUX et al. (1981). BOULIER et al. (1987) solved equation (1) analytically and numerically by introducing the parameter F to predict the ponding time. They mentioned that F have a complex behavior.; the time dependence of F is hard to estimate and is usually ignored. They also confirmed the inaccuracy of numerical solution and concluded that the analytical solution is less accurate but easier to apply.

It is the purpose of this paper to de- velop a mathematical equation to predict the rate of the wetting front advance during rain (sprinkler) infiltration, taking into account the effect of initial soil moisture (®i) and rain (sprinkler) intensity (I).

The following assumptions were made in developing the theory dealing with rain (sprinkler) infiltration into a vertical soil column:

1. The soil is regarded as a semi-infinite, homogeneous, isotropic body, the bulk density is uniform through the profile and remained constant during water- ing.

2. One dimensional flow is assumed in the system.

3. The initial moisture content is uniform throughout the profile.

4. The water application rate is considered to be constant throughout the wa- tering.

5. The kinetic energy of the falling rain drops is so small that surface disturbance will be negligible.

The one-dimensional flow of water in unsaturated soil can obey Darcy's law:

v = - K (2) OL

CATENA--An Interdisciplinary Journal of SOIL S C I E N C E - - H Y D R O L O G Y ~ E O M O R P H O L O G Y

Effect Initial Mois ture Content, Infiltration 493

in which q~ = soil water potential = tp + L V = water velocity in the soil.

Unde r the fulfillment o f the previous assumption, a quasi-state flow is taking place in the vertical direction and V is almost constant. Then, one can write the following equat ion:

dL v = ( ~ ) ( o r - o i ) (3)

in which dL (~-) = the advance rate of the wetting front

under rain (sprinkler) infiltration. Oi = the initial moisture content, on volume

basis. O T = the moisture content in the

transmission zone. Equat ion (2) can be written as:

V = K r ~ (4)

or

hr + L v = K r - - (5) L

in which hT = d~v, head loss in transmission zone. KT = capillary conductivity in transmission

zone. Substituting equat ion (5) into equat ion

(3) one obtains:

dL KT ,hT + L ,

(-~-) = ( O r -- 0 3 ( ~ ) (6)

By integrating the equat ion with initial condit ion L = 0, t = 0, and assuming the capillary potential 0P) is constant along the transmission zone (hT = 0), and letting K r = I.

dL I

(-~) - o r - o ~ (7)

foo L ~ ' I oi) dt dL = (OT --

I t L - (8)

( O r - Oi)

Equat ion (8), so far, has been derived on the assumpt ion o f unit hydraulic gradient between any two points in the transmission zone which is followed by a sharply defined wetting front, with zero slope, assuming the depth o f wetting zone to be zero. In other words, equat ion (8) has been derived by using piston flow characteristics. But in real- ity, especially with a soil having appre- ciable amounts o f clay, that is not of- ten the case. Usually, the moisture content o f the transmission zone decreases slightly with depth and there exists a zone with decreasing moisture content between the transmission zone and wetting front. Also, ( O r - - O i ) is not quite in- dependent o f L and t, since the moisture content o f the transmission zone slightly decreases with depth and increases with time ( Y O U N G S 1960).

2 Mathematical Development

I f one takes all the above considerat ion into account, equat ion (8) can be written:

I t L = ( O r - Oi )n (9)

and letting the value

1 (O--TT -- Oi )n = C = constant. (10)

Since, O r actually takes different values depending on Oi and I, one can consider C as a constant mainly dependent on rain (sprinkler) intensity, I , and initial moisture content in the soil (03 , dimensionally in consistent. Equat ion (9) can be rewritten as:

L = Ct n ( l l )

CATENA An Interdisciplinary Journal of SOIL SCIENCE- HYDROLOGY GEOMORPHOLOGY

494 EI-Shafei

Rotating shaft

Motor (1 r.p.m.)

5 cm

From water reservoir

i[ ° ° ° ~

I

Water pump

Capillary tubes

Lid

Drops

Micro-tensiometer

Manometer _ ~,, soil column

Slot (3 mm)

Base plate

Fig. 1: Schematic diagram of the experimental set up.

in which n = a constant having values between 0.8

and 1.0 dependent on the soil type.

The value of n, more likely, will be closer to 1.0 as the soil possesses a higher degree of uniformity bo th in bulk density and mois ture content .

If L is plot ted as a funct ion of t on log-log paper, one expects to ob ta in a series of parallel lines for different values of I. On the other hand, plot t ing C at a uni t time as a funct ion of I, a series of parallel lines for different values of ®i are expected. Mathemat ical ly , this may be expressed as:

C = B(Oi)I" (12)

in which m = the slope of parallel lines as a result of

plotting C as a function of I, dimensionless.

B(Oi) = a constant dependent of the initial moisture content, dimensionally inconsistent

The cons tan t B(®i) can be obta ined from the following relat ionship:

B ( O i ) I ( O i ) 1 - - = - - ( 1 3 )

B(®i)2 (0/)2

Equa t ion (11) can be writ ten in a general form:

CATENA An Interdisciplinary Journal of SOIL SCIENCE--HYDROLOGY~EOMORPHOLOGY

Effect Initial Moisture Con ten t, Inliltra tion 495

I 0 9

E 8 U

i 7 ..J

" 6

c 5 0 > "0 o 4

C "2 2

n = 0 . 8 7

¢o ¢9"

l ( c m / h r . )

q,.~ ~. ,,.~ Ix" / /

e . = 6 . 6 % ,6

IO 2 0 50 4 0 50 IOO 2 0 0

T i m e _ m i n u t e s

Fig. 2: Wetting front advance for different sprinkler intensities, ®i = 6.6%; comparing measured and computed values.

L = B(®~)Imt n (14)

and the advance rate of the wetting front can be obtained by:

dL (~[) = B(®i)nlmt "-] (15)

If we restrict our attention to rain infiltration as defined previously and neglect- ing the small surface disturbance due to the small Kinetic energy of the falling drops, equation (15) can predict the advance rate of the wetting front for any combination between I and ®~.

3 Experimental procedures

A laboratory experiment was carried out on 5 cm (diameter) by 70 cm long soil

columns under the possible combination of five sprinkler intensities (1.50, 2.50, 3.50, 4.50 and 5.5+0.05 cm/hr) and five initial moisture contents (6,6, 12, 18.5, 23.8 and 27.7 percent by volume). The experiment was always terminated at the beginning of surface ponding and before the wetting front reached the bottom of the column by at least 7 cm depth.

The soil used was silty clay (43%, 52% silt and 5% sand) sampled from the 30 cm top layer at Kafr E1-Zaiat province, Egypt. The soil was air-dried and then sieved through 2 mm-screen. The soil samples were brought uniformly to the desired initial moisture (®i) by applying the ice moisturizing technique. It involves the addition of water as ice scorings to the dry soil at -8°C. The soil

CATENA An Interdisciplinary Journal of SOIL SCIENCE HYDROLOGY--~EOMORPHOLOGY

496 E1-Shafci

15 n = O . B 7 ~¢0 / ' b g / ~ \"

E• I 0 , 9

8 • 7 u c 6 0

0

~ 4 0

e 3

2

~ - = 12° /o

I0 20 50 40 50 I00 200 3C0

Time . minutes

Fig. 3: Wetting front advance for different sprinkler intensities, Oi = 12% ; comparing measured and computed values.

was taken to a deep-freeze room and spread out on paper so that it was cool- ing rapidly. The required amount of ice, equivalent to the desired initial moisture content, was passed through a 2 mm- screen. The ice-soil mixture was then carefully rolled until it was uniform, left a few hours in the deep-freeze room, transferred to a tight container, returned to the laboratory, and kept overnight to warm at room temperature. This technique produced a uniform soil moisture with maximum spread equal to 0.25 percent.

The soil column was packed in section- ized lucite column by 5 cm increments, weighing out enough soil to give a bulk density of 1.20 g/cm 3. The water application apparatus (rain simulator) used in the experiment was designed to uniformly sprinkle the soil surface at any constant application rate (EL-SHAFEI 1974). Essentially, it was constructed by water supply reservoir, rotating lid feeded by 16 polyethylene capillary tubes (0.28 mm i.d.), and piston pump (fig.l).

The average drop size produced by the rain simulator was 2.2 mm. The falling

CATENA An Interdisciplinary Journal of SOIL S C I E N C E - - H Y D R O L O G Y ~ E O M O R P H O L O G Y

Effect Initial Moisture Content, Infiltration 497

0 I

,.,I

Q) (,,) r -

0

" 0 0

o

O~ ,r.

,e- G)

4 0

30

20

I 0 9 8 7 6

n = 0 . 8 7

(0 I (Cm / hr. ) ~ \ .

%.= ,8.5 %

I i i i , , , , I , l P l I I0 20 50 40 50 I O0 200 300 5 0 0

T i m e _ m inu tes

Fig. 4: Wetting front advance for different sprinkler intensities, Oi = 18.5%; comparing measured and computed values.

distance was 5 cm (fig.l). The kinetic energy = 23 erg/drop and the normal energy expended per cm rain on each m 2 of soil surface = 4129 x 104 erg (4.13 Joules) which is considered very small accoording to S C H L E U S N E R & KID- D E R (1960) and E L - S H A F E I (1976).

The wetting front is the foremost point of advance of the moving water was

recorded from visual observation and for the micro-tensiometers (as sensors) which were installed along the soil column (fig.l). The K (®) relationship was obtained by applying a series of rain (sprinkler) intensities to the long vertical columns of soil. Rain infiltration was maintained until almost unit potential gradient (as indicated by micro-

CATENA An Interdisciplinary Journal of SOIL SCIENCE HYDROLOGY~EOMORPHOLOGY

498 E1-Shafei

50

40

50

• 2 0 E U

I

- 1 1 5

g C

"o 9 O

8

6

.c_ 5

3 4

n : 0 . 8 7

l(Cmlhr.) 'b ~ b q,.

zr / z3.8 %

I , I [ I r , , h I I I I I I I0 20 30 40 50 I00 200 300 500

T i m e . m i n u t e s

Fig. 5: Wetting front advance for different sprinkler intensities, 19i = 23.8% ; comparing measured and computed values.

tensiometers) had been established in the top 20 cm of the column. This procedure also gave the moisture-tension relationship W (®) for the soil (YOUNGS 1960, EL-SHAFEI & FAHMY 1975, PER- ROUX et al. 1981). This procedure provides a good facility to obtain the

(19) relationship during the wetting cy-

cle (rain infiltration). Either the K (®) or ~o (O) relationship was plotted on semi- log scale and one smooth curve was obtained as unique characteristics for the soil. The moisture content was deter- mined gravimetrically (on weight basis) for each section (5 cm height) of the soil column, then be converted to O (volume


EFfect Initial Moisture Content, Infiltration 499

6 0

5 0

40

5O

E 0

I

j 20

U C

0

o 9 8

o, 7 .c_ ~- 6

~ 5

n = 0 . 8 7

3

4

2

I ~ I ' I0 2 0

I (Cm/hr . ) ,~.¢0 / ¢ b " / \ ' ~

¢ / /

I ~ ~ f I i i I , I I I I 30 40 50 IOO 200 300 500

T i m e _ m inu tes

Fig. 6: Wetting front advance for different sprinkler intensities, Oi = 27.7%; comparing measured and computed values.

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500 EI-Shafei

1.0

0.6

0.5

0.4

0.3

O 0.2

OlO

0.0~

0.06

0.05

O~ ---27.7*/ .

- m= 0.75 ~ e~. - - -25 .8% / - / /

e z ~ 2 . 1 .

eg 6 .6"1 .

I I I I h L I I I I i I I 2 3 4 5 6 I0

I ( C m / h r. )

Fig. 7: The constant (C) as a function of the sprinkler intensity ( I ) at different initial moisture content (®i).

basis) by using the bulk density. Each experimental value used in the results was an average of two determinations.

4 Results and discussion

Figures 2, 3, 4, 5 and 6 present the wetting front advance, L, as a function of time t, during rain (sprinkler) infiltration for different rain intensities at different initial moisture contents. The figures show how the advance of the wet-

ting front increases as the initial moisture content increases. This is in accor- dance with HANSEN (1955) who ob- served that the rate of entry in moist soils was less than in dryer soils, but the wetting front advanced more rapidly when the soil is wet. This result is attributed to the decrease of moisture deficit ( O r - ® i ) as ®i increases (equations (7) and (8)). It may also be explained by the fact that less energy is required, at the wetting front to expand the meniscuses to al- low water movement into the zone, as

CATENA--An Interdisciplinary Journal of SOIL SCIENCE--HYDROLOGY-~3EOMORPHOLOGY

2 . 0

f 1.0

¢-r

E w.

> , ,

U - i ' ID ,..

0 U

lO-I e~

o U


I O - ~ I ~ 1 I I O 10 2 0 50 4 0 50

Moisture content~ 0 (%)

60 Fig. 8: Capillary conductivity of silty clay soil as a function of moisture content.

®i increases. There is less energy loss, since the initial flow occurs in large water channels; hence, less energy expen- diture would result in moist soil, even though the total available head may be less than in a dry soil. Figures 2 to 6 also show the excellent agrrement between computed and measured values. The results revealed that the exponent (n), in equations (14) and (15) has a value 0.87 for this silty clay soil at a bulk density 1.2 g/cm 3.

Tab.1 gives values of the constant C (in equation (11)) which are dependent

on the rain (sprinkler) intensity (I) and initial moisture content of the soil (®i).

Fig.7 presents the constant C as a function of I at different values of ®i. The constant C can also be obtained under any condition of rain (sprinkler) intensity and initial moisture content by using equations (12) and (13). The value of the constant B (®3 in equation (12) can also be obtained from fig.7 by taking the value of C at a unit rain intensity.

Consider the following example. Sup- pose one had conducted a quick experiment on a homogeneous soil which had

CATENA An Interdisciplinary Journal of SOIL SCIENCE H Y D R O L O G Y ~ E O M O R P H O L O G Y

502 El-Shafei

200C

I OOC

500

400 300

200

E U

I 0 0

g 0

"~ 5 0 C

~ 4 0

3O

~ 2 0

0

I 0

5

I I I f I 0 tO 2 0 3 0 4 0 5 0 6 0 Fig. 9" Moisture tension of silty

clay soil as a function of moisture Moisture con.tent~ O ( % ) content.

I O i (%) (cm/hr) 6.6 12 18.5 23.8 27.7

1.5 0.068 0 .122 0 .190 0 .245 0.286 2.5 0.100 0.180 0 .280 0 .360 0.420 3.5 0.128 0.230 0 .358 0 .461 0.538 4.5 0.154 0 .277 0 .431 0 .554 0.647 5.5 0.180 0 .324 0 .504 0 .648 0.756

Tab. 1- The values of C in equation ( I I ) .

CATENA--An Interdisciplinary Journal of SOIL SCIENCE HYDROLOGY~GEOMORPHOLOGY

Effect Initial Moisture Content, Intiltration 503

20

I0

e ¢ ' = 1 2 %

I ( C m / h r . )

e o el,'

E U

t

6

5

4

3

E q u o t i o n (14)

, x o H o n k s ~ e t o l .

I L / I I I I I I I I0 20 30 40 50 I00 200 500

T i m e _ m i n u t e s

Fig. 10: Wetting front advance for different rain (sprinkler) intensities, ®i = 12%; comparing computed values from the derived equation (14) with HANKS et al. solution.

®i = 6.6% and Db = 1.2gcm 3, to determine L under three or four values of I (cm/hr). By the end of this experiment, he would be able to calculate the values of the constants, n,m and B(6.6). From figures 2 and 7, the values are 0.87, 0.75 and 0.05, respectively. Now suppose it is desired to predict theoretically the L- equation for the same soil under the condition of ®i = 18.5% and I = 3.5 cm/hr.

1. Using equation (13) :

B(6.6) 0.05 6.6

B(18.5) B(18.5) 18.5

which gives B(18.5) = 0.14

2. Using equation (12):

C = 0.14(3.5) 0.75 = 0.358

(which coincides with the value of C in tab.1 and fig.7)

3. The computed L-equation is:

L = 0.358t °87

which is in excellent agreement with the measured one:

L = 0.36t °'88

Fig.8 shows the measured values for capillary conductivity (K) plotted as a function of moisture content (O). Fig.9 shows the measured values for soil moisture tension (02) plotted as a function of soil moisture content (®). The data of K - - O relationship and 02--0 relationship were taken from the smoothed curves in

CATENA An Interdisciplinary Journal of SOIL SCIENCE - H Y D R O L O G Y ~3EOMORPHOLOGY

504 EI-Shafei

4O

3o[

2 0 0 £ = 2 3 . 8 %

I(C m/hr. )

q," e o "b" / \ .

IO E 1.}

!

. - I

6

5

/

1 IO

Equat ion (14)

• x o H a n k s ~ e t o l .

f , I r , ~ r l l ....... ! I ; 20 30 40 50 I00 200 ~00

T ime_minutes

Fig. 11 : Wetting front advance for different rain (sprinkler) intensities, 19i = 23.8%; comparing values from the derived equation (14) with HANKS et al. solution.

figures 8 and 9 respectively which show a good fit. These data were used in the numerical solution (HANKS et al. 1969) of equation (1), by the computer program. Figures 10 and 11 show a good agreement for the calculated wetting front advance by two different computations; the derived equation (14) and the partial differential equation (1). The good agreement is attributed to accurate measure- ment of K - ® and ~p- O relationships

which were taken under rain (sprinkler) infiltration process (wetting cycle) using the same soil column in the experiment. The rough (approximated) estimation of K - O or ~ p - O relationship will def- initely lead to disagreement if equation (1) is used. Practically, one of the main reasons which usually hinder the use of equation (1) is the problem of getting reliable values for K - O and ~p - O relationships.



Acknowledgement

The author gratefully acknowledge the help of Food Science Dept., Faculty of Agriculture, Alexandria University, for allowing the use of the deep freeze room.

References

ABRAMOV, A. (1954): Sprinkler irrigation ver- sus intake rate of soils. Pochvovedenie, No. 11. (Russian).

BOULIER, J.F., PARLANGE, J.-Y., VAUC- LIN, M., LOCKINGTON, D.A. & HAVER- KAMP, R. (1987): Upper and lower bounds of the ponding time for near constant surface flux. Soil Sci. Soc. Am. J., 51, 1425-1428.

CLOTHIER, B.E., KNIGHT, J.H. & WHITE, L. (1981): Burger's equation: Application to field constant - Flux infiltration. Soil Sci., 132, 255-261.

EL-SHAFEI, Y.Z. (1974): Surface ponding in relation to rain, sprinkler intensities and infiltration capacity of soils. Z. f. Kulturtechnik u. Flurberein., 15, 43 52.

EL-SHAFEI, Y. (1976): Soil and water conserva- tion under sprinkler irrigation. Z. f. Kulturtech- nik u. Flurberein., 17, 15-24.

EL-SHAFEI, Y.Z. & FAHMY, I.M. (1975): Flow of water in unsaturated uniform and layered porpus materials. Alex. J. Agric. Res., 23, 173-184.

EL-SHAFEI, Y.Z. & EL-NAGGAR, I.M. (1981): Change of soil-moisture tensions during water capillary rise and evaporation processes. Z. f. Kulturtechnik u. fturberein., 22, 13-20.

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

HANKS, R . J , KLUTE, A. & BRESLER, E. (1969): A numerical method for estimating infiltration, redistribution, drainage and evaporation of water from soil. Water Res. Research, 5, 1064-1069.

PARLANGE, J.-Y. (1972): Theory of water movement in soils: 8. one diemsnional infiltration with constant flux at the surface. Soil Sci., 114, 1-4.

PERROUX, D.E., SMILES, D.E. & WHITE, L (1981): Water movement in uniform soils during constant - flux infiltration. Soil Sci. Soc. Am. J., 45, 237-240.

PHILIP, J.R. (1969): Theory of infiltration. Adv. in Hydrosci., 5, 215-305.

PHILIP, J.R. (1973): On solving the unsaturated flow equation. 1 The flux-concentration relation. Soil Sci., 116, 328-335.

RUBIN, J. & STEINHARDT, R. (1963): Soil water relations during rain infiltration: 1 - The- ory. Soil Sci. Soc. Am. Proc., 27, 246~251.

RUBIN, J., STEINHARDT, R. & REINIGER, P. (1964): Soil water relation during rain infiltration. II - Moisture content profiles during rains of low intensities. Soil Sci. Soc. Am. Proc., 28,1 5.

SCHLEUSENER, P.E. & KIDDER, E.H. (1960): Energy of failing drops from medium pressure irrigation sprinkler. Agric. Eng., 41, 100-103.

WHITE, I., SMILES, D.E. & PERROUX, K.M. (1979): Absorption of water by soil: The constant flux boundary condition. Soil Sci. Soc. Am. J., 43, 659-664.

YOUNGS, E.G. (1960): The hysteresis effect in soil moisture studies. 7 th Int. Congress Soil Sci., 1.4, 107 112.

Address of author: Prof. Dr. Yehia Z. EI-Shafei Soil and Water Science Dept., Faculty of Agriculture University of Alexandria Alexandria, Egypt

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the effect of initial moisture content and rain (sprinkler) intensity on wetting front advance...

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