Infiltration Prediction based on Estimation of Green-Ampt Wetting Front Pressure Headfrom Measurements of Soil Water Redistribution1

S. K. CHONG, R. E. GREEN, AND L. R. AmjJA2

ABSTRACTA simple, physically based algebraic equation was derived to cal-

culate the wetting front matric pressure head, Hr, in the Green-Amptequation of infiltration. The derivation of the equation involved com-bining the simple hydraulic conductivity function with the wettingfront matric pressure head function proposed by R. G. Mein and C.L. Larson (1971). The parameters in the derived equation were de-termined from fitting a power function to the experimental data ofsoil water content and soil water pressure head vs. elapsed timeobtained during the post-infiltration redistribution period. In thederived equation, Hf is expressed as a function of antecedent soilwater content. At low initial water contents, the equation shows thatHf is essentially a constant. Evaluation of the calculated Hf values wasaccomplished by utilizing Hf in the Green-Ampt equation to predictinfiltration at seven field sites for soils at initial water contents belowfield capacity. Predicted cumulative infiltration was compared withinfiltration measured with a double-ring infiltrometer. The resultingGreen-Ampt predictions were accurate for initially dry soils, witha correlation value of r = 0.94 and a relative error of 12.1%.

Additional Index Words: wetting front matric pressure head, soilwater redistribution, infiltration prediction, Green-Ampt equation.

Chong, S. K., R. E. Green, and L. R. Ahuja. 1982. Infiltrationprediction based on estimation of Green-Ampt wetting front pres-sure head from measurements of soil water redistribution. Soil Sci.Soc. Am. J. 46:235-239.

K,t = / - A0(//0 - Hf)\n{l , [1]

RECENTLY in watershed infiltration analysis, muchinterest has been directed toward using the Green

and Ampt (1911) approach because of its simplicityand encouraging results (e.g., Mein and Larson, 1971;Swartzendruber and Hillel, 1975; Dangler et al., 1976).Moreover, this simple and empirical approach (Childs,1967, 1969) can be extended beyond the case of asingle-layered, uniform soil to consider infiltration intolayered soils (e.g., Childs and Bybordi, 1969; Bouwer,1969; Hillel and Gardner, 1970). The Green-Amptapproach is obtained by applying Darcy's equation toa wetting soil profile with assumptions of vertical flowand a transmission zone with both uniform water con-tent and uniform hydraulic conductivity with depth.Furthermore, it is assumed that in the wetting soilprofile there exists a distinct and precisely definablewetting front; the matric pressure head at the wettingfront is assumed to be constant regardless of time andposition during infiltration. With consideration of thegravity effect, the Green-Ampt approach gives thefollowing simple infiltration equation (Childs, 1969):

' Contribution from the Water Resources Research Center andHawaii Institute of Tropical Agriculture and Human Resources,Univ. of Hawaii at Manoa. Received 13 July 1981. Approved 17Nov. 1981.2 Research Assistant. Now Assistant Professor of Forest Hy-drology, Dep. of Forestry, Southern Illinois Univ. at Carbondale,Carbondale, IL 62901; Professor of Soil Science, Dep. of Agronomyand Soil Science, Univ. of Hawaii at Manoa, Honolulu; and As-sociate professor of Soil Science, Univ. of Hawaii at Hilo. NowSoil Scientist (Physics), USDA-ARS, Southern Plains Watershedand Water Quality Laboratory, Durant, Okla.

where Ks is the hydraulic conductivity in the trans-mission zone (cm/min); t is time (min); A0 is the dif-ference between field-satiated and antecedent soilwater content (cm3/cm3); H0 is the pressure head atthe water entry surface (cm of water); ///is the matricpressure head at the wetting front (cm of water); and/ is cumulative infiltration (cm).

Equation [1] is valid only for a soil profile withuniform physical properties. A major obstacle in usingEq. [1] is the difficulty of estimating the parameterHf. Especially in the field, the wetting front matricpressure head is difficult to define when the wettingfront is diffuse as a result of nonuniform antecedentwater content or variation of physical properties withdepth.

Bouwer (1964) suggested that /// can be calculatedby:

where Kr is the relative hydraulic conductivity (di-mensionless); ft is the matric pressure head (cm ofwater); and ft, is the matric pressure head at the an-tecedent water content (cm; of water).

Equation [2] has been djerived theoretically fromDarcy's law by Morel-Seytbux and Khanji (1974) fortwo-phase flow, and by Neuinan (1976) for water flow,neglecting air movement.

Several ways of estimating ///have been proposed.Bouwer (1966) developed an apparatus for in situmeasurement of the air-entry value which he used toestimate ///; Mein and Parrel (1974) determined wet-ting front matric pressure head by theoretical justifi-cations; Panikar and Nanjap'pa (1977) redefined ///bymultiplying Kr in Eq. [2] by relative soil water content;Brakensiek (1977) related ///to the bubbling pressurehead which can be obtained from a soil-water char-acteristic curve. However, most of the methods re-quire a knowledge of either the soil-water character-istic or the hydraulic conductivity-water content,K(6), relationship. Moreover, most of the requiredinformation they used was obtained in the laboratory.

In this study, a simple algebraic equation was de-rived for calculating /// based on in situ soil waterredistribution measurements. There is no attempt tocompare the calculated /// with values obtained byother methods. Instead, the calculated /// is used topredict infiltration using Eq. [1], and the predictedinfiltration is compared with the field-measured results.

THEORYInstead of using Eq. [2] for calculating Hf, Mein and Lar-

son (1971) proposed as an alternative, Eq. [3]:Hf = [/£${ hdK(0)V[K(0,) - K(0j\, [3]

where K(0) is the hydraulic conductivity (cm/min); 9S is thesatiated soil water content (cm3/cm3); and 0, the antecedentsoil water content (cmVcm3). Equations [2] and [3] give the

235

236 SOIL SCI. SOC. AM. J . , VOL. 46, 1982

same result (Swartzendruber, 1974). We used Eq. [3] in ourwork, even though we could have used Bouwer's Eq. [2]just as well.

During the post-infiltration redistribution period, both soilwater content and soil water pressure head in the profilecan be expressed as functions of time as shown by Chonget al. (1981) such that:

= atb; t > 0, [4]and

h = mf, [5]in which 0 is the average water content in the soil layer(cmVcm3); h is soil water pressure head (cm of water), t isthe redistribution elapsed time (min); and a, b, m, and nare constants.

With the assumption of a unit gradient [i.e., d6/dt =— KIL; L is the depth of the profile (cm)], the rate of changeof soil water content in the profile during redistribution canbe used to calculate hydraulic conductivity as shown byChong et al. (1981) such that

K(h) = C,hC2 for h 4= 0 or h = [K(h)ICtflc\ [6]where

C, = -Labm'c\ andC2 = (b - l)/n.

Substituting Eq. [6] into Eq. [3], we have:TJ _ \\itis i c ^ i r f^* (is if* \^d-A jcn mHf = [l/(Aj — A,J [JK: (A/C|J flAJ. L / J

Integrating Eq. [7] with the assumption that A"(0) = A"(0S)when h = /zs- (5 denotes a satiated condition), and alsousing the relation of Eq. [4] and Eq. [6], ///can be expressedas a function of water content, that is,

Hf = m(eja)"/b[(b - \)l(b + n - 1)] - (0jef+-™~\i - (0/0,)"-"* J' [8]

Equation [8] is a simple algebraic equation for calculatingwetting front matric pressure head as a function of soil watercontent. The specific advantages of Eq. [8] are (i) it is analgebraic equation in which 0, is the only independent vari-able, and (ii) /// can be obtained at any antecedent watercontent, if the parameters a, b, m, n, and 6S can be deter-mined. Equations [6] and [8] are developed based on thewater movement during the desorption process in the soil.Therefore, strictly speaking, we cannot use Eq. [8] for pre-dicting infiltration because of the well-known hysteresis ef-fect. However, it is hard to measure K(h) during the ab-sorption process in the field. Since the soils involved in thisstudy were satiated to 85% of the total porosity (Chong etal., 1981), the hysteresis should be less. Hence, we assumedthat the effect of hysteresis was small compared with otherfactors that influenced the results.

EXPERIMENTAL PROCEDURESField Measurements

In order to test Eq. [8] and Eq. [1] for calculating wettingfront matric pressure head and infiltration, the field exper-iments were conducted on soils of Molokai (Typic Torrox)and Lahaina (Tropeptic Haplustox) series in sugarcane fieldswith tilled Ap horizons 30 to 40 cm in depth on the WahiawaPlateau, Island of Oahu, Hawaii.

The infiltration measurements were conducted with dou-ble-ring infiltrometers, wherein a constant 2-cm depth ofponded water was maintained by controlling the water inlet.Water application was continued approximately 1 hour be-yond the time when an apparent steady infiltration rate wasobserved.

Initial wetting of the profile was accomplished with aninfiltration run on dry soil. After a redistribution period of1 day, a multiple tensiometer was installed at the center ofthe inner ring. The redistribution of soil water was allowedto proceed for another 2 days following the infiltration runon moist soil. The experimental redistribution measurementswere made immediately after this second infiltration run.Hydraulic conductivity at field satiation was calculated fromthe measured steady flux and pressure head gradient at theend of the infiltration run.

The starting time of redistribution was defined as thattime when applied water was absorbed and just disappearedfrom the soil surface. The soil water content during theredistribution period were obtained gravimetrically from soilsamples obtained between the inner and outer rings. Thesoil water content was measured seven times over a 14-dayperiod at the Hawaiian Sugar Planters' Association (HSPA)sites and four or five times over a 5- to 8-day period at theother sites. The earliest time for soil water content mea-surements varied between 1 hour (HSPA sites) and 24 hours(other sites). Volumetric water contents were calculatedusing bulk density data obtained from soil cores taken fromthe experimental site after redistribution measurements wereterminated. Water pressure heads during redistribution wereobtained from tensiometer measurements at increasing timeintervals throughout the drainage period, from every fewminutes initially to daily near the end of the several daysof measurements.

The soil surface inside the rings was covered with a plasticsheet during redistribution to prevent evaporation. A 2-cmthick Styrofoam sheet was placed on the plastic sheet toreduce extreme changes in soil temperature. A canopy wasinstalled above the experimental setup to prevent rainfallfrom entering the rings. More details of site preparation andinfiltration and redistribution measurements are given else-where (Chong et al., 1981).

Determination of Parameters in Eq. [8] andInfiltration Prediction

Parameters for Eq. [8] were determined using Eq. [4] andEq. [5] by regression with experimental data of soil watercontent and soil water pressure head vs. time measuredduring the redistribution period. In this study, 85% of totalporosity was used as the field-satiated soil water content,0S. This is mainly because the soil was not fully saturateddue to air entrapment, even though the infiltration was takingplace under ponded water. Details of the determination ofparameters in Eq. [8] and the reasons for using 85% of totalporosity as the field-satiated condition are given in Chonget al. (1981).

Equation [1] cannot be solved explicitly for / for givent. It was solved, instead, for t at given values of /. Valuesof / for given values of t were then obtained graphicallyfrom the t, I curve.

As mentioned before, Eq. [1] is valid only for a soil profilewith uniform physical properties. In our study the field pro-files were not considered uniform because of distinct Apand B horizons. In order to satisfy the boundary conditionsimposed on Eq. [1], we can deal only with the relativelyuniform Ap horizon; this means that Eq. [1] is valid onlyfor values of t less than the time required for the wettingfront to reach the interface of the Ap and B horizons.

The total amount of water infiltrated into the Ap horizon,/„ (cm), can be calculated as:

Im = A0 LAp, [9]where LAp is the depth of Ap horizon (cm), and A0 is thewater-fillable porosity, which for our purpose is assumedto be the difference between 85% of total porosity and theantecedent soil water content (cmVcm3).

CHONG ET AL.: INFILTRATION PREDICTION BASED ON ESTIMATION OF GREEN-AMPT PRESSURE HEAD 237

Table 1—Parameters from Eq. [4] & Eq. [5] determined by regression with experimental data, and initial dry run and field-satiatedwater contents for calculated wetting front metric pressure head from Eq. [10) for 7 experimental sites.

Parameters!

Site

HSPAAHSPABHSPACOP410 EOP410W

OP221 EOP221 W

a

0.60790.66020.60710.70580.6110

0.58950.7132

b

-0.0595-0.0633-0.0611-0.0797-0.0601

-0.0601-0.0769

m

-8.5570-1.1095-4.8220-5.8426

-12.1452

-6.6110-8.1103

n

Molokai soil0.32590.55050.38070.37180.2761

Lahaina soil0.35550.3446

Soil water content

Antecedent fl,-

0.280.360.330.220.23

0.240.23

Satiated 0S

0.5040.5200.4820.5320.536

0.5300.522

Matric pressureheadHf

cm of water

-34.50-18.33-31.67-33.02-29.97

-18.66-48.29

t Equations [4] and [5].

Since /„, can be determined from Eq. [9], the approximatetime that is required by the wetting front to reach depth LApin the profile can be obtained from the experimental cu-mulative infiltration results.

RESULTS AND DISCUSSIONCalculation of Wetting Front Matric Pressure HeadThe constants obtained from the regression of Eq.

[4] and Eq. [5] on experimental data for each exper-imental site are tabulated in Table 1. For all experi-mental sites, the absolute value of the correlation coef-ficient, r, between regressed and measured results for9 vs. t exceeded 0.95 and for h vs. t exceeded 0.98(Chong et al., 1981).

Examples of calculated K(6) (from Chong et al.,1981) and calculated ///from Eq. [8] (dashed line) vs.6 are shown in Fig. 1. The hydraulic conductivity isvery small and approaches zero as the water contentdecreases to 0.35 cmVcm3. On the other hand, when0,- = 6S, based on the L'Hospital rule, /// in Eq. [8]becomes m(6ja)'b and is equal to -23.9 cm of water.The ///values increase as 0, decreases and ///reachesa maximum of - 34.5 cm as the hydraulic conductivityapproaches zero.

In Eq. [8], the parameter n is always positive but<1.0, due to the increasing negative value of waterpressure head vs. time during drainage, i.e., h be-comes more negative with time in Eq. [5]. Also, b isalways negative because of decreasing water contentin the soil profile. Therefore, the power terms for (0,-/6S) in Eq. [8] are always positive and generally >5.0.Since (0,/0J is always much less than 1.0 for small 0,,the term (0,/0s) in Eq. [8] can be eliminated with littleerror for low antecedent water contents. This impliesthat /// was essentially constant for antecedent watercontents below the in situ field capacity (the soil watercontent at 48 hours after initiation of redistributionwas 0.36 to 0.39 cm3/cm3 for all of the experimentalsoils in this study). Equation [8] explains why /// isvirtually constant over a wide range of antecedent soilwater contents (Mein and Farrel, 1974). Hence, forantecedent water contents less than field capacity, theterm (OJ6S) in Eq. [8] can be neglected, and Eq. [8]becomes:

/// = m(esla)(nlb\(b - \)l(b + n - I)]. [10]

51 3

HO"u o 2

oX

Field Measured Satiated oHydraulic Conductivity

Calculated from Eq. (6)

-Eq. (10)

Eq. (8)

00.10 0.20 0.35 0.40 0.45 0.50ANTECEDENT SOU WATER CONTENT, 9, (cm3/cm3)

Fig. \—The relationships of hydraulic conductivity and wetting frontmatric pressure head, HSPA, Site A.

An example of the calculated /// using Eq. [10] isshown by the solid line in the lower graph of Fig. 1.The antecedent soil water contents in Table 1 (withthe exception of those for HSPA sites B and C) wereconsiderably below field estimates of field capacity.Thus, Eq. [10] is expected to provide essentially thesame result as Eq. [8], with only a small error beingintroduced when the initial water content is in thevicinity of field capacity. The differences in resultsfrom Eq. [8] and Eq. [10] for the data in Table 1(expressed as a percentage of the result from Eq. [8])ranged from 0 to 4.9%, with differences greater than0.06% being associated only with data for HSPA Band C of 4.9 and 1.3%, respectively. The /// valuescalculated with Eq. [10] for each experimental site areshown in the last column of Table 1.

Predicted InfiltrationThe Green-Ampt approach is more satisfactory for

calculation of water infiltration in dry soil than in wetsoil (Hillel and Gardner, 1970). This may be attributed

238 SOIL SCI. SOC. AM. J . , VOL. 46, 1982

Table 2—Parameters for Green-Ampt Eq. [1] used to calculatecumulative infiltration and corresponding measured

________infiltration for 7 experimental sites.

Hydraulic Soil Cumulative infiltration}conduc- 'horizon Fillable ———————————————tivityt depth porosity Calculated Measured

Site Ks LAn A9 /,. /m

HSPAAHSPABHSPACOP410EOP410W

cm/rain

0.04110.02170.00830.09330.1251

cm cm'/cm3

Molokai soil4040403040

0.2240.1600.1520.3120.306

5.214.411.878.009.80

7.104.702.017.67

10.57Lahaina soil

OP221 EOP221 W

0.14510.1010

5050

0.2900.292

10.6215.30

11.6712.30

t Field-satiated conditions.i Dry antecedent conditions.

to the diffuse boundary of the wetting front in a wetsoil; i.e., the movement of the wetting front essentiallydoes not conform to the Green-Ampt piston-type flowassumption when the soil is too wet.

In this study we are interested in calculating wettingfront matric pressure heads for various soils and theassociated cumulative infiltration with time. Only thedry antecedent condition of each soil will be used inthe evaluation of this prediction equation, since theGreen-Ampt equation is less likely to be valid for wetsoils. Predictions for initially wet soils were made onlyto examine the magnitude of prediction error whenthe assumptions inherent in the Green-Ampt equationare not satisfied.

The parameters required in Eq. [1] for calculatinginfiltration for all seven experimental sites are shownin Table 2 (Hf is given in Table 1). Ks is the field-satiated hydraulic conductivity (field-measured steadyflux divided by the appropriate measured gradient),which is required by the Green-Ampt equation. The

16

§ u

I12IE 10zui

| 8

1 6uQ

O

< 2u *

fillable porosity, A0, is the difference between ante-cedent soil water content and 85% of total porosity,which is assumed to be the field-satiated water contentin our study (see Chong et al., 1981). The antecedentsoil water content was obtained gravimetrically in thefield just before infiltration measurements.

Both the measured cumulative infiltration, Im, andthe calculated Green-Ampt cumulative infiltration, 7C,in Table 2 correspond to the infiltration period from5 minutes of elapsed time to the time, t. The first 5min of infiltration is neglected for the purpose of thesecomparisons because of uncertainties in the measuredinflow rates soon after initiation of infiltration. Thecalculated upper limit of time, t, for each site is basedon the estimated water storage available in the Aphorizon.

Im and Ic for dry antecedent conditions (Table 2)are plotted in Fig. 2 in relation to a 1:1 line to indicatethe accuracy of the Green-Ampt prediction of infil-tration for the seven sites. The average deviation ofpredicted cumulative infiltration from that measuredat each site is expressed as an average percentageerror, e (Topping, 1966):

e = (l//i)[2|/m - /C|//J • 100 [11]

160 2 4 6 8 10 12 14

MEASURED CUMULATIVE INFILTRATION (cm)

Fig. 2—Comparison of measured and calculated (by Green-Amptapproach) infiltration in relatively dry soils.

in which n is the number of sites.For the data in Fig. 2, e = 12.1%. The correspond-

ing correlation coefficient, r, for Im and Ic is 0.94.These results are promising in view of a likely incon-sistency between the Green-Ampt assumption of auniform soil profile and the actual variation of physicalproperties with depth in the Ap horizon of the fieldsoils of this study. As expected, the predicted infil-tration for wet antecedent conditions was less accu-rate, e = 55.4%, confirming that the Green-Amptapproach is usually not appropriate for wet soils.

CONCLUSIONEquation [8] is a simple algebraic equation in which

the wetting front matric pressure head, Hf, is givenas a function of antecedent water content. The re-quired parameters can be obtained from in situ mea-surements of water content and soil-water pressurehead during redistribution, assuming that the effectsof hysteresis are small. The values of ///thus obtainedwere used in the Green-Ampt equation to predictinfiltration. The prediction error was small for initiallydry soil, suggesting that this method has merit for soilswith low antecedent water contents.

The Green-Ampt equation was tested in this workonly for topsoils whose depths ranged between 30 to40 cm. For the dry antecedent conditions, the differentsoils required between 2 to 13 cm of infiltration waterto be satiated to this depth. This means that the top-soils (Ap horizon) would be the controlling horizonfor infiltration of rainfall, especially in areas whererainstorms are short and intense (e.g., Hawaii).Nevertheless, for long-duration rainfall we need toextend the application of the Green-Ampt equationto the subsoil B horizons of generally smaller hy-draulic conductivity and porosity, and a different soilmoisture condition. Further work is needed to eval-uate these problems under field conditions.

UEBLER & SWARTZENDRUBER: FLOW OF KAOLINITE AND SEWAGE SUSPENSIONS IN SAND AND SAND-SILT: i 239

ACKNOWLEDGMENTSThe work upon which this publication is based was sup-

ported in part by funds provided by the Office of WaterResearch and Technology (B-054-HI), U.S. Dep. of the In-terior, Washington, DC., as authorized by the Water Re-search and Development Act of 1978; and by the WaterResources Research Center and Hawaii Institute of TropicalAgriculture and Human Resources, Univ. of Hawaii atManoa. This is WRRC Contribution no. 122 and HITAHRJournal Series no. 2581.

Contents of this article do not necessarily reflect the viewsand policies of the Office of Water Research and Technol-ogy, U.S. Department of the Interior, nor does mention oftrade names or commercial products constitute their en-dorsement or recommendation for use by the U.S.Government.