an analytical solution for one-dimensional water infiltration and redistribution in unsaturated soil

7/21/2019 An Analytical Solution for One-Dimensional Water Infiltration and Redistribution in Unsaturated Soil

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Pedosphere 19(1): 104110, 2009

ISSN 1002-0160/CN 32-1315/P

c 2009 Soil Science Society of China

Published by Elsevier Limited and Science Press

An Analytical Solution for One-Dimensional Water Infiltration

and Redistribution in Unsaturated Soil1

WANG Quan-Jiu1,2, R. HORTON3 and FAN Jun2

1Institute of Water Resources Research, Xian University of Technology, Xian 710048 (China). E-mail: [email protected] Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Institute of Soil and Water Conse-rvation, Chinese Academy of Sciences and Ministry of Water Resources, Yangling 712100 (China)3Department of Agronomy, Iowa State University, Ames, IA 50011 (USA)

(Received February 14, 2008; revised November 16, 2008)

ABSTRACT

Soil infiltration and redistribution are important processes in field water cycle, and it is necessary to develop a

simple model to describe the processes. In this study, an algebraic solution for one-dimensional water infiltration and

redistribution without evaporation in unsaturated soil was developed based on Richards equation. The algebraic solution

had three parameters, namely, the saturated water conductivity, the comprehensive shape coefficient of the soil water

content distribution, and the soil suction allocation coefficient. To analyze the physical features of these parameters,

a relationship between the Green-Ampt model and the algebraic solution was established. The three parameters were

estimated based on experimental observations, whereas the soil water content and the water infiltration duration were

calculated using the algebraic solution. The calculated soil water content and infiltration duration were compared with

the experimental observations, and the results indicated that the algebraic solution accurately described the unsaturated

soil water flow processes.

Key Words: algebraic solution, Green-Ampt model, soil water infiltration and redistribution, unsaturated soil

Citation: Wang, Q. J., Horton, R. and Fan, J. 2009. An analytical solution for one-dimensional water infiltration and

redistribution in unsaturated soil. Pedosphere. 19(1): 104110.

INTRODUCTION

Soil water flow is an important process in the global water cycle. The important tasks related to

the study of soil water movement are the determination of soil water content distribution, soil water

infiltration, soil storage capacity, and plant water uptake. Several soil water models describing soil

infiltration processes have been developed (Green and Ampt, 1911; Parlange, 1971, 1972; Parlange et

al., 1992, 1997; Basha, 1999). Wang et al. (2003) presented an algebraic solution for describing the

unsaturated soil water movement. The solution used the short time assumption of Parlange (1971) inwhich the change in water content with time was assumed to be small relative to the rate of flux change

with distance. Thus, the time-dependent changing flux rate was cancelled, and the water flux was

assumed to be uniform in the wetted soil profile.

Redistribution of soil water following an infiltration event is an important process in the field water

cycle. Knowledge of water redistribution is also required to determine whether water or solutes can

penetrate the root zone. Some models have been developed to describe soil water redistribution (Alway

and McDole, 1917; Youngs, 1958; Bresler et al., 1969; Staple, 1969; Gardner et al., 1970; Youngs and

Poulovassilis, 1976; Smith, 1999; Shao and Horton, 2000). It is necessary to develop simple models to

analyze water infiltration and redistribution processes. In this study, an analytical solution to describe

1Project supported by the Knowledge Innovation Program of the Chinese Academy of Sciences (No. KSCX2-YW-N-003),

the National Basic Research Program of China (No.2005CB121103), and the National Natural Science Foundation of

China (No.50879067).


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SOIL WATER INFILTRATION AND REDISTRIBUTION 105

the unsaturated soil water movement during infiltration and redistribution without evaporation was

developed based on the algebraic model of Wang et al. (2003); the relationship between the proposed

model and the Green-Ampt model (1911) was analyzed; a description of how to estimate the model

parameters was included; and the model was tested against experimental data.

MATERIALS AND METHODS

Theory

The soil water retention curve used in this study is described as follows (Brooks and Corey, 1964):

rs r

=hd

h

N(1)

where is the soil water content (cm3 cm3), r is the residual water content (cm3 cm3), s is the

saturated water content (cm3 cm3), hd is the air entry suction (cm), h is the soil water suction

corresponding to the water content, and Nis an empirical parameter.The unsaturated soil conductivity k (cm min1) is expressed as follows (Brooks and Corey, 1964):

k(h) =ks

hdh

M(2)

or

k() =ks

rs r

M/N(3)

whereks is the saturated conductivity (cm min1), and Mis an empirical parameter.

For one-dimensional, vertical soil water movement, the Richards equation is:

t

=z

D()

z

k()z

(4)

wheret is the time,D() is the soil water diffusivity, andz is the vertical coordinate, positive downward.

The initial and boundary conditions for infiltration are: (z, 0) =i, (0, t) =s, and (, t) =i,

wherei is initial water content (cm3 cm3). Setting the time at zero for the beginning of redistribution

just after infiltration ceases, the boundary conditions for redistribution are: (0, 0) =sand(, t) =i.

Infiltration model

When the initial soil water content is small and can be approximated as r = i, integrating Eq. 4

with the algebraic model from Wang et al. (2003) for soil water provides the following four equations:

=

1 zzf

(s i) +i z zf

= i z > zf

(5)

F = (s i)zf

1 + (6)

f= kszf

+ks (7)

t= (s i)

(1 +)ks

zf

ln(zf+ 1)

(8)

wherezfis the wetting front distance, referring to the distance from the soil surface to the bottom of the


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106 Q. J. WANG et al.

wetted soil layer; = N/M, the comprehensive shape coefficient of the soil water content distribution,

which determines the amount of water in a soil profile for a fixed wetting front distance; is a soil

suction allocation coefficient; Fis the cumulative infiltration; and f is the infiltration rate. =M /a,

wherea is a constant.

The Green-Ampt model (Green and Ampt, 1911) describing soil water infiltration can be written

as:

i= kshs

zs+ks (9)

I= (s i)zs (10)

t= s i

ks

zs hsln

1 +

zshs

(11)

where i is the infiltration rate, hs is the suction at the wetting front, Iis the cumulative infiltration,

and zs is the wetting front distance in the Green-Ampt model.Letting:

zs = zf1 +

(12)

hs = 1

(1 +) (13)

and substituting Eqs. 12 and 13 into Eqs. 68, the Green-Ampt model can be obtained.

Redistribution model

The redistribution of soil water is an internal drainage process. We consider the redistribution as a

continuous process of infiltration assuming that soil water hysteresis is negligible during this redistribu-tion. Infiltration and redistribution processes can easily be combined.

Following Wang et al. (2003), the Richards equation can be analyzed as:

1

hdz=

dk

M(q(t) k) (14)

M

hdz=

dk

q(t) k (15)

where q(t) is the internal drainage rate, and k is the unsaturated soil water conductivity. Integrating

Eq. 15 leads to:

z0

M

hdz =

kzk0

dk

q(t) k (16)

where k0 is the hydraulic conductivity at the soil surface, and kz is the hydraulic conductivity corre-

sponding to the distance z :

z = 1

M

10

ds

h

lnq(t) kzq(t) k0

(17)

Letting 10

ds

h =

1

ahd , Eq. 17 is converted to:


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z = 1

ln

q(t) kzq(t) k0

(18)

Whenz is at the wetting front, zf, and the initial water content is low, the corresponding hydraulic

conductivity is also small; thus, Eq. 18 reduces to:

q(t) = k0zf

+k0 (19)

The unsaturated conductivity at any distance z is:

k= q(t) (q(t) k0)expM z

ahd

(20)

Whenr = i, Eq. 20 is approximated by a Taylor series and combined with Eq. 19:

=

1 z

zf

(0 r) +r (21)

Integrating Eq. 21, the water amount (W) retained in the soil profile is expressed as:

W = 1

1 +(0 i)zf (22)

For soil moisture redistribution, the water retained in the soil profile is a constant. Thus, the water

content (0) at the soil surface changes as the wetting front advances. According to Eq. 22, the water

content (0) at the soil surface is a function of the wetting front distance and:

0(ti+1) =(0(ti) i)zf(ti)

zf(ti+1) +i (23)

where0(ti+1) and 0(ti) are the water contents at soil surface associated with wetting front distances

zf(ti+1) and zf(ti) at time i and time i + 1, respectively.

With the development of internal drainage processes, the soil water content of the upper part of the

profile decreased, and the soil water content of the lower part increased. For any wetting front distance

interval, zf(zf = zf(ti+1) zf(ti)), the soil water content distribution profile for the wetting front

distance, zf(ti+1), must intersect with the soil water content profile for zf(ti) at a certain soil depth.

According to Eq. 20, the intersect depth (zs) may be expressed as:

zs =

1 0(ti) i

0(ti+1) i

1/1

zf(ti+1)

1

zf(ti) 0(ti) i

0(ti+1) i1/

(24)

Thus, the internal drainage associated with the wetting front distance interval zfcan be calculated.

According to Eq. 21, the internal drainage amount (Wid) is:

Wid = 1

1 +

zf(ti)(0(ti) i)

1

1

zszf(ti)

1+ zf(ti+1)(0(ti+1) i)

1

1

zszf(ti+1)

1+ (25)

The time interval to complete the internal drainage process is calculated from the internal drainage

rate,q(t), and the internal drainage amount (Wid):


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t= Wid

q (26)

Materials

To evaluate the redistribution model, the experimental data reported by Gardner et al. (1970) who

performed a series of infiltration-redistribution trials on vertical cylindrical columns of Gilat loess, a fine

sandy loam, was used. The air-dried soil was passed through a-2 mm screen and packed mechanically (to

a bulk density of 1.47 g cm3) into lucite tubes with 5 cm inner diameter and 160 cm length. A depth

of 5 cm of water was applied to each soil column and water content profiles were monitored repeatedly

by means of a gamma ray method during water redistribution. The detailed experimental description

can be found in Gardner et al. (1970).

RESULTS AND DISCUSSION

The infiltration model and the Green-Ampt model

The algebraic infiltration model was converted into the Green-Ampt model with the parameters

and using Eqs. 12 and 13, which indicated that the Green-Ampt model could also be developed

based on the assumption and theory that the algebraic model was established and the parameters in

the infiltration model could be estimated from each other. Thus, when the Green-Ampt model was

used to calculate the infiltration rate or cumulative infiltration, the soil water distribution could also be

calculated using Eq. 5. In this way, the shortage that the Green-Ampt model can not describe the soil

water content profile was overcome.

Evaluating the infiltration and redistribution models

There were three parameters in the infiltration and redistribution models. Whether the infiltrationand redistribution models described the soil water movement well during infiltration and post-infiltration

depended on the parameters, , , and ks. Wang et al. (2003) analyzed the influences of the three

parameters on the infiltration properties. For a given soil, the constant, ks, reflected the capacity of

the soil pores to transmit water. The comprehensive shape coefficient of the soil water distribution, ,

which determined the amount of water in a soil profile for a fixed wetting front distance, was a function

ofN and M. Eq. 22 indicated that the amount of infiltrated water in a soil profile (W) decreased as

increased. Fig. 1 shows the influences of on the amount of retained water in the soil profile when sandi were assumed to be 0.45 and 0.04 cm3 cm

3, respectively. The results indicated that the amount

Fig. 1 Influences of the comprehensive shape coefficient of the soil water content distribution () on the amount ofretained water in the soil profile. zf is the wetting front distance.

Fig. 2 Soil water content distribution during infiltration and redistribution. zf is the wetting front distance.


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of stored water decreased as increased. Also, as seen in Eq. 19, , a soil suction allocation coefficient,

influenced the internal drainage rate, q(t), which decreased as increased. Fig. 2 shows the moisture

redistribution withs, i, and assumed to be 0.45 cm3 cm3, 0.04 cm3 cm3, and 0.103, respectively.

The results indicated that the soil water content increased with the increase in the wetting front distanceand decreased in the upper part of the soil profile.

To further evaluate the redistribution model, the experimental data reported by Gardner et al.

(1970) was adopted. The tested soil water retention curve was:

h= 0.634.3 (27)

The unsaturated conductivity of the tested soil was:

k= 160 00010.6 (29)

where k is in cm d1. Also, in order to obtain the related parameters in Eq. 2, Eq. 29 was converted

into Eq. 30:

k= 9.68 0.04

s 0.04

9.3= 9.68

37h

2.79(30)

To analyze the moisture distribution, the wetting front distance at the beginning of the redistribution

was calculated based on the infiltrated water amount (W) from Eq. 22. Next, the soil water content at

the soil surface for a given wetting front distance, 0(ti+1), was determined from Eq. 23. Subsequently,

Eq. 21 was used to calculate the soil water content distribution () corresponding to a given wetting front

distance zf(ti+1). After that, according to the calculated soil water content distribution, the intersect

depth (zs) for the two soil water content profiles was calculated using Eq. 24, and the internal drainage

(Wid) corresponding to the two wetting front distances was calculated using Eq. 25. The calculated

internal drainage (Wid) was divided by the internal drainage rate, q(t), to obtain the drainage time

(Eq. 26). According to the hydraulic parameters given by Eqs. 28 and 30, the soil water content profilescorresponding to drainage time were calculated (Fig. 3) with results indicating that the model can be

used to predict soil moisture redistribution.

Fig. 3 Soil moisture redistribution. The points represent the measured data and the lines are the calculated values.

CONCLUSIONS

It was important to predict field unsaturated soil water flow precisely and quickly. When Richards

equation was used to describe soil water movement, numerical methods must be used, and the rele-

vant parameters were difficult to estimate accurately. The algebraic model presented in this study was

convenient and simple to use for predicting soil water movement, and the model parameters were eas-

ily estimated. The model could reflect the comprehensive features of unsaturated soil water movement

without evaporation.


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an analytical solution for one-dimensional water infiltration and redistribution in unsaturated soil

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