An Analytical Solution for One-Dimensional Water Infiltration and Redistribution in Unsaturated Soil

Pedosphere 19(1): 104-110, 2009ISN 1002-0160/CN 32-1315/PPEDOSPHERE⑥2009 Soil Science Society of ChinaPublished by Elsevier Limited and Science Presswww. elsevier com/locate/pedosphereAn Analytical Solution for One- :Dimensional Water Infiltrationand Redistribution in Unsaturated Soil*lWANG Quan-Jjiu1+2, R. HORTON3 and FAN Jun21Institute of Water Resources Research, Xi'an University of Technology, Xi'an 710048 (China). E-mail: wquanjiu@163.com2State Key Laboratory of Soil Erosion and Dryland Farmning on the Loess Plateau, Institute of Soil and Water Conse-rvation, Chinese Academy of Sciences and Ministry of Water Resources, Yangling 712100 (China)3 Department of Agronormy, lowa State University, Ames, IA 50011 (USA)(Received February 14, 2008; revised November 16, 2008)ABSTRACTSoil infiltration and redistribution are important processes in field water cycle, and it is necessary t切o develop asimple model to describe the processes. In this study, an algebraic solution for one-dimensional water infiltration andredistribution without evaporation in unsaturated soil was developed based on Richards equation. The algebraic solutionhad three parameters, namely, the saturated water conductivity, the comprehensive shape coefficient of the soil watercontent distribution, and the soil suction allocation coeficient. To analyze the physical features of these parameters,a relationship betwen the Green-Ampt model and the algebraic solution was establshed. The three parameters wereestimated based on experimental observations, whereas the soil water content and the water infitration duration werecalculated using the algebraic solution. The calculated soil water content and infltration duration were compared withthe experimental observations, and the results indicated tbat the algebraic solution accurately described the unsaturatedsoil water flow processes.Key Wornds:algebraic solution, Green- Ampt model, soil water iniltration and redistribution, unsaturated soilCitation: Wang, Q. J., Horton, R. and Fan, J. 2009. An analytical solution for one dimensional water infltration andredistribution in unsaturated soil. Pedosphere. 19(1): 104-110.INTRODUCTIONSoil water flow is an important process in the global water cycle. The important tasks related tothe study of soil water movement are the determination of soil water content distribution, soil waterinfiltration, soil storage capacity, and plant water uptake. Several soil water models describing soilinfiltration processes have been developed (Green and Ampt, 1911; Parlange, 1971, 1972; Parlange etal, 1992, 1997; Basha, 1999). Wang et al. (2003) presented an algebraic solution for describing theunsaturated 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 fAux changewith distance. Thus, the time dependent changing fux rate was cancelled, and the water fux wasassumed to be uniform in the wetted soil profile.Redistribution of soil water following an infiltration event is an important process in the field watercycle. Knowledge of water redistribution is also required to determine whether water or solutes canpenetrate the root zone. Some models have been developed to describe soil water redistribution (Alwayand McDole, 1917; Youngs, 1958; Bresler et al, 1969; St中国煤化工1970; Youngs andPoulovssilis, 1976; Smith, 1999; Shao and Horton, 2000)Y片CNMH P simple models toanalyze water infiltration and redistribution processes. In tnis stuay, an ana.ytlcal 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 ofChina (No. 50879067).SOIL WATER INFLTRATION AND REDISTRIBUTION105the unsaturated soil water movement during infiltration and redistribution without evaporation wasdeveloped based on the algebraic model of Wang et al. (2003); the relationship between the proposedmodel and the Green-Ampt model (1911) was analyzed; a description of how to estimate the modelparameters was included; and the model was tested against experimental data.MATERIALS AND METHODSTheoryThe soil water retention curve used in this study is described as follows (Brooks and Corey, 1964):θ- Or(1)0。-0:=(管)"where 0 is the soil water content (cm3 cm-3), 0r is the residual water content (cm3 cm- 3), 0g is thesaturated water content (cm3 cm-3), hd is the air entry suction (cm), h is the soil water suctioncorresponding to the water content, and N is an empirical parameter.The unsaturated soil conductivity k (cm min 1) is expressed as follows (Brooks and Corey, 1964):k(h)= kse(智)(2)or1θ_ θr、M/Nk(6)=的((0。-0,)(3)where ks is the saturated conductivity (cm min- 1), and M is an empirical parameter.For one dimensional, vertical soil water movement, the Richards equation is:8k()(4)0=最(0)-8zwhere t is the time, D(0) is the soil water diffusivity, and z is the vertical coordinate, positive downward.The initial and boundary conditions for infiltration are: 0(z,0)= 0, 0(0,t) = 0g, and 0(∞,t)= 0;,where 0; is initial water content (cm3 cm~ 3). Setting the time at zero for the beginning of redistributionjust after infiltration ceases, the boundary conditions for redistribution are: 0(0,0) = 0g and 0(∞,t) = 0.Infiltration modelWhen the initial soil water content is small and can be approximated as 0r = 0, integrating Eq. 4with the algebraic model from Wang et al. (2003) for soil water provides the following four equations:[o=(1-)°(0.-0)+0 z≤z(5)(θ=0qz> 4(0。一0)ZF=1 +o(6)f=βzt+kg中国煤化工(7)MHCNMHGt=(6。-0)(x-l(0z<+2)(8)(1+ a)的β)where z{ is the wetting front distance, referring to the distance from the soil surface to the bottom of the10Q. J. WANG et alwetted soil layer; a = 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; β is a soilsuction allocation coeficient; F is the cumulative infitration; and f is the infltration rate. β = M/a,where a is a constant.The Green- Ampt model (Green and Ampt, 1911) describing soil water infltration can be writteni= kh +h(9)I= (0。- 0})(10)0。-0;k。[x1.1(+剖(11)where i is the infltration rate, hg is the suction at the wetting front, I is the cumulative infiltration,and z is the wetting front distance in the Green- Ampt model.Letting:(12)1+a1hg=(1 +a)(13)and substituting Eqs. 12 and 13 into Eqs. 68, the Green- Ampt model can be obtained.Redistribution modelThe redistribution of soil water is an internal drainage process. We consider the redistribution as acontinuous 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 a:lkhd=- M(q()-百)(14)Ma.dkhdz= g()-k(15)where q(t) is the internal drainage rate, and k is the unsaturated soil water conductivity. IntegratingEq.15 leads to:(16)J。M-=-J" awhere ko is the hydraulic conductivity at the soil surface, and kz is the hydraulic conductivity corre-sponding to the distance z:。 q(t)-kzZ=中国煤化工(17)M|J。万ds" q(t)- koMYHCNMHGLettingf1 ds__iEq. 17 is converted to:Johaha’soIl WATER INFITRATION AND REDISTRIBUTION1079(t)-kz(18)q(t)- koWhen z is at the wetting front, zf, and the initial water content is low, the corresponding bhydraulicconductivity is also small; thus, Eq.18 reduces to:的q(t)= + ko(19)βzfThe unsaturated conductivity at any distance z is: .k= q(t)- (q(t)- ko)exp((器)(20)When 0; = 0;, Eq. 20 is approximated by a Taylor series and combined with Eq. 19:0=(1-三)° (0o-0,)+0.(21)Integrating Eq. 21, the water amount (W) retained in the soil profile is expressed as:1W=-(0o - ;})zq(22)For soil moisture redistribution, the water retained in the soil profle is a constant. Thus, the watercontent (0o) at the soil surface changes as the wetting font advances. According to Eq. 22, the watercontent (Oo) at the soil surface is a function of the wetting front distance and:0o(t:+1)=(0(t)一()(5) +0(23)z(i+1)where 0o(ti+1) and 0o(t:) are the water contents at soil surface associated with wetting front distanceszt(ti+1) and z(t;) at time i and time i + 1, respectively.With the development of internal drainage processes, the soil water content of the upper part of theprofle decreased, and the soil water content of the lower part increased. For any wetting front distanceinterval, Oz(Ozt = zr(ti+1) - z(i)), the soil water content distribution profile for the wetting frontdistance, z(ti+1), must intersect with the soil water content profile for z(t;) at a certain soil depth.According to Eq. 20, the intersect depth (2) may be expressed as:_ (. 0o(t:)-0: \1/a(0(+1)- 0;石=(24)11 (t)-0)1/azr(t:+1)~ z&(t) (0o(i:+1)-0)Thus, the internal drainage associated with the wetting front distance interval Ozq can be calculated.According to Eq.21, the internal drainage amount (Wia) is:Wid= i1 +a{ <(Q() 0)1-(1-商) - +(+-+1)[+(1-1中国煤化工2(4+))*}MHCNMHG(25)The time interval to complete the internal drainage proces is calculated ftom the internal drainagerate, q(t), and the internal drainage amount (W:a):108Q. J. WANG et alOt =Wid(26)qMaterialsTo evaluate the redistribution model, the experimental data reported by Gardner et al. (1970) whoperformed a series of infiltration-redistribution trials on vertical cylindrical columns of Gilat loess, a finesandy loam, was used. The air-dried soil was passed through a-2 mm screen and packed mechanically (toa bulk density of 1.47 g cm - 3) into lucite tubes with 5 cm inner diameter and 160 cm length. A depthof 5 cm of water was applied to each soil column and water content profiles were monitored repeatedlyby means of a gamma ray method during water redistribution. The detailed experimental descriptioncan be found in Gardner et al. (1970). .RESULTS AND DISCUSSIONThe infiltration model and the Green-Ampt modelThe algebraic infiltration model was converted into the Green- Ampt model with the parametersa and β using Eqs. 12 and 13, which indicated that the Green-Ampt model could also be developedbased on the assumption and theory that the algebraic model was established and the parameters inthe infitration model could be estimated from each other. Thus, when the Green-Ampt model wasused to calculate the infltration rate or cumulative infiltration, the soil water distribution could also becalculated using Eq. 5. In this way, the shortage that the Green Ampt model can not describe the soilwater content profile was overcome.Evaluating the infitration and redistribution modelsThere were three parameters in the infiltration and redistribution models. Whether the infltrationand redistribution models described the soil water movement well during infiltration and post-infiltrationdepended on the parameters, a, β, and kg. Wang et al. (2003) analyzed the infuences of the threeparameters on the infitration properties. For a given soil, the constant, kg, reflected the capacity ofthe soil pores to transmit water. The comprehensive shape coefficient of the soil water distribution, a,which determined the amount of water in a soil profile for a fixed wetting front distance, was a functionof N and M. Eq. 22 indicated that the amount of infiltrated water in a soil profile (W) decreased as aincreased. Fig. 1 shows the infuences of a on the amount of retained water in the soil profile when 0。and 0; were assumed to be 0.45 and 0.04 cm3 cm -3 , respectively. The results indicated that the amount160.5z,=10a=0.050.4言14。 12...... a=0.210. a=0.30.330=-== a=0.40.20.0.0~01(2034020253035中国煤化工z, (cm)E(cm)Fig. 1 Infuences of the comprehensive shape cofficient of the 8MH.CNMHG(a)ontheamontofretained water in the soil profle.矿is the wetting front distance.Fig.2 Soil water content distribution during infiltration and redistribution.社t is the wetting front distance.SOIL WATER INFILTRATION AND REDISTRIBUTION109of stored water decreased as increased. Also, as seen in Eq 19, β, a soil suction allocation cefficient,infuenced the internal drainage rate, q(t), which decreased as β increased. Fig. 2 shows the moistureredistribution with 0g, 0;, and a assumed to be 0.45 cm3 cm-3 , 0.04 cm3 cm-3, 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= 0.630-4.3(27)The unsaturated conductivity of the tested soil was:k = 160 00010.6(29)where k is in cmd-'. Also, in order to obtain the related parameters in Eq. 2, Eq. 29 was convertedinto Eq. 30:k= 9.681θ- 0.04 9.32.79(30)(8。-0.04) =9.68(元)“To analyze the moisture distribution, the wetting front distance at the beginning of the redistributionwas calculated based on the infltrated water amount (W) from Eq.22. Next, the soil water content atthe soil surface for a given wetting front distance, 0(t:+1), was determined from Eq 23. Subsequently,Eq.21 was used to calculate the soil water content distribution (6) corresponding to a given wetting frontdistance z(t:+1). After that, according to the calculated soil water content distribution, the intersectdepth (zx) for the two soil water content profiles was calculated using Eq. 24, and the internal drainage(Wia) corresponding t如o the two wetting front distances was calculated using Eq. 25. The calculatedinternal drainage (Wia) 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 proflescorresponding to drainage time were calculated (Fig. 3) with results indicating that the model can beused to predict soil moisture redistribution.0.50.4一-一t=0 min0.3全--. t= 1 min.--- t= 2 min-◆- t= 10 min-◆- t= 29 min0.5015 2025335Distance (cm)Fig. 3 Soil moisture redistribution. The points represent the measured data and the lines are the calculated values.CONCLUSIONSIt was important to predict feld unsaturated soil wat中国煤. When Richardsequation Wa8 used to describe soil water movement, nun化上sed, and the relevant parameters were difficult to estimate accurately. TheTYHCNMHGiinthisstudywasconvenient 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 movementwithout evaporation.11Q. J. WANG et al.REFERENCESAlway, F. J. and McDole, G. R. 1917. Relation of the water retaining capacity of a soil to its hydroscopic cofficient. J.Agr. Res. 9: 27-71.Basha, H. A.1999. Onedimensional nonlinear steady infltration. Water Resour. Res. 35: 1 697-1 704.Bresler, E., Kemper, W. D. and Hanlks, R. J. 1969. Infltration, redistribution and subsequent evaporation of water fromsoil as affected by wetting rate and hysteresis. Soil Sci. Soc. Am. J. 33: 832- -839.Brooks, R. H. and Corey, A. J. 1964. Hydraulic Properies of Porous Media. Hydrol. Paper 3. Colo. State Univ, FortCollins.Gardner, W. R, Hillel, D. and Benyamini, Y. 1970. Post irrigation movement of soil water: 1. Redistribution. WaterResour. Res. 6: 851-861.Green, W. H. and Ampt, G. A.1911. Studies on Boil pbyeices: 1. Flow of air and water through soils. J. Agric. Sci. 4(1):1-24.Parlange,J. Y. 1971. Theory of water movement in soils: 2. One dimensional infiltration. Soil Sci. 111: 170-174.Parlange, J. Y.1972. Theory of water movement in soils: 8. One dimensional infiltration with constant fux at the surface.Soil Sci. 114: 1-4.Parlange, J. Y., Barry, D. A., Parlange, M. B., Hogarth, W. L, Haverkamp, r, Ross, P. J., Ling, L. and Steenhuis, T.S.1997. New approximate analytical technique to solve Richards equation for arbitrary surface boundary conditions.Water Resour. Res. 33: 903- 906.Parlange, M. B., Prasad, S. N, Parlange, J. Y. and Romkens, M. J. 1992. Extension of the Heaslet Alksne technique toarbitrary soil water diffusivities. Water Resour. Res. 28: 2 793-2797.Shao, M. A. and Horton, R. 2000. Exact solution for horizontal water redistribution by general similarity. Soil Sci. Soc.Am. J. 62: 561-564.Smith, R. E. 1999. A conceptual model for infltration and redistribution in crusted soil. Water Resour. Res. 35:1385- 1 393.Staple, W. J. 1969. Comparison of computed and measured moisture redistribution following infiltration. Soil Sci. Soc.Am. Proc. 33: 840- 847.Wang, Q. J., Horton, R. and Shao, M. A.2003. Algebraic model for one dimensional infiltration and soil water distribution.Soil Sci. 168: 671- 676.Youngs, E. G.1958. Redistribution of moisture in porous materials after infltration. Soil Sci. 86: 117-125.Youngs, E. G. and Poulovassilis, A. 1976. The diferent forms of moisture profile development during the redistributionof soil water after infiltration. Water Resour. Res. 12: 1007-1 012.中国煤化工MYHCNMHG