Green-Ampt Infiltration Models for Varied FieldConditions: A Revisit

Ravindra V. Kale & Bhabagrahi Sahoo

Received: 5 May 2010 /Accepted: 16 June 2011 /Published online: 12 July 2011# Springer Science+Business Media B.V. 2011

Abstract The Green-Ampt (GA) infiltration model is a simplified version of the physicallybased full hydrodynamic model, known as the Richards equation. The simplicity andaccuracy of this model facilitates for its use in many field problems, such as, infiltrationcomputation in rainfall-runoff modelling, effluent transport in groundwater modellingstudies, irrigation management studies including drainage systems etc. The numerousinfiltration models based on the Green-Ampt approach have been widely investigated fortheir applicability in various scenarios of homogeneous soils. However, recent advances inphysically based distributed rainfall-runoff modeling demands for the use of improvedinfiltration models for layered soils with non-uniform initial moisture conditions undervarying rainfall patterns to capture the actual infiltration process that exists in nature. Thedifficulty that modelers are facing now-a-days includes the estimation of time of pondingand the application of the infiltration model to unsteady rainfall events occurring inheterogeneous soil conditions. The investigation in this direction exhibits that only fewinfiltration models can handle these situations. Hence, it is of vital importance to analyzethe usefulness of different variants of the Green-Ampt infiltration models in terms of theirdegree of accuracy, complexity and applicability limits. This paper provides a brief reviewof these infiltration models to bring out their usefulness in the rainfall-runoff and irrigationmodeling studies as well as the drawbacks associated with these models.

Keywords Green-Ampt . Infiltration . Layered soil profile . Rainfall-runoff modeling

Water Resour Manage (2011) 25:3505–3536DOI 10.1007/s11269-011-9868-0

R. V. Kale (*)Scientist B, National Institute of Hydrology Roorkee, Jal Vigyan Bhawan, Roorkee 247667, Indiae-mail: ravikale2610@rediffmail.com

B. SahooScientist, Soil and Water Conservation Engineering, ICAR Research Complex for NEH Region,Nagaland Centre, Jharnapani, Medziphema 797 106, Nagaland, Indiae-mail: bsahoo2003@yahoo.com

1 Introduction

The importance of the infiltration process in pedology, hydrology, and environmentalsciences had led to considerable literature dealing with experimental observations,theoretical, analytical, numerical and empirical modeling of infiltration. When water isapplied to the soil, it completely infiltrates into the soil column until the rate of itsapplication exceeds the infiltration rate into the soil profile without surface ponding.Consequently, when the rate of water application exceeds the infiltration rate, storage ofwater builds up on the soil surface in the form of rainfall excess that leads to surface runoff,resulting in the initiation of soil erosion process. R. E. Horton [as quoted by Beven andRobert (2004)] is the best known originator of this concept for storm hydrograph analysisand prediction. The excerpt from the selected research papers on infiltration phenomenonby Horton can be found in Beven and Robert (2004). The accurate estimation of pondingtime and time distribution of runoff is of specific interest to agricultural and civil engineersfor proper design of irrigation and drainage systems, rain water harvesting reservoirs, andhydraulic structures at the watershed scale. Among the various infiltration models found inthe literature, the Green-Ampt (GA) model (Green and Ampt, 1911) is widely used inpractice, e.g., in the Best Approximation of Surface Exchanges (BASE) (Desborough andPitman 1998) and Soil Water-Atmosphere-Plants (SWAP) (Gusev and Nasonova 1998) landsurface models.

The main aim of this paper is to provide an insight to the various infiltration studiescarried out by different researchers using the Green and Ampt approach to accomplish thesimulation of infiltration under variable conditions of soil profile and rainfall. Thesubsequent sections highlight on the factors influencing infiltration process in watershedmodeling; framework and performance of the Green-Ampt model for homogeneous andheterogeneous/layered soils with and without cracks, macro-pores, and surface seal;variants of this model based on the solution scheme and accounting for rainfall pattern;usefulness of the Green-Ampt equation in modeling studies, challenges in using the Green-Ampt model for hydrology research, and advantages and shortcomings of the modelvariants based on the past research work.

2 Factors Influencing Infiltration Process in Watershed Modeling

Infiltration is a complex phenomenon controlled by a number of soil andclimatological variables. In a field, there would typically be cracks and macrosporesat the soil surface which are responsible for increasing the infiltration rate into the soil.At the watershed scale, the layered soils are encountered which affect the infiltrationprocess depending on the pedo-hydrological properties of each soil layer. The factorssuch as actual hydraulic properties of the vertical soil profile, water contentdistribution along the depth and rainfall intensity affect the infiltration process atlocal scale. These basic factors are properly accounted for when one extends theanalysis to infiltration in locally non-uniform soil profiles, and spatially andtemporally varying systems at the watershed scale. Nielsen et al. (1973) conductedcomprehensive field experiments for studying infiltration and redistribution of soil-waterfollowing irrigation at 20 sites over a 150 hectare of agricultural field in the CentralValley of California. In their experiment, water was ponded at 20 random locations onthe plots of 6.5 m2 each for measurement of steady infiltration rate, and the monitoringof soil matric potential at six depths were carried out while allowing the plot to drain out.

3506 R.V. Kale, B. Sahoo

Statistical analyses of variations of soil-water properties with soil depth throughout thefield established the extensive spatial variation in transport and retention propertiesamong the 20 sites, and at different depths between 30 and 180 cm in the same site,while the static soil properties such as the bulk density and porosity varied little amongthe replicated measurements. Gusev (1993) pointed out that according to theobservations, the saturated hydraulic conductivity Ksat may differ within the order ofseveral meters which envisages that the spatial variability of Ksat is high. Therefore, evenfor a small area, the spatial variability of Ksat should be taken into account whilecalculating the infiltration of water into soil. Accordingly, the spatial variability forheterogeneous areas can be accounted by using the approximations that a) the spatialvariability of Ksat can be characterized by its standard deviation σK from its mean valueKsat and b) Ksat is uniformly distributed on some interval of a to b. Use of theseassumptions helps in derivation of the analytical expression for the areally averagedinfiltration rate given by (Gusev and Nasonova 1998)

f ¼2A13 K3=2

sat

���K»

sat

aþ A2

2 K2sat

��K»

sat

aþrA2KsatjbK»

sat; if a < K

»sat � b

r if K»sat � a

2A13 K3=2

sat

���baþ A2

2 K2sat

��ba; if K

»sat > b

8>>><>>>:

ð1Þ

A1 ¼0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hsrwΔW t�1

pr

q2ffiffiffi3

psK

; A2 ¼ 1

2ffiffiffi3

psK

; a ¼ �ffiffiffi3

psK ; b ¼ Ksat þ

ffiffiffi3

psK ð2Þ

where K»sat = saturated hydraulic conductivity at which the infiltration rate is equal to the

precipitation rate; ρw = water density; ΔW = difference between soil porosity and fieldcapacity and tpr = duration of the period which is used in the infiltration curve andcalculated from the beginning of rainfall. Similarly, to identify the grid size resolutionwhich captures the essential variability at the watershed scale, Farajalla and Vieux(1995) advocated the entropy or information content of soil map as measure of thespatial variability of infiltration parameters in the Green-Ampt infiltration model. Therate of change of entropy with different resolutions is the Hurst coefficient, which is themeasure of spatial variability that describes the persistence of floods and droughts. Theentropy curve is useful for selection of critical cell resolution which captures theessential variability of distributed hydrological model parameters. Recently, Craig et al.(2010) developed a set of variants of the up-scaled GA model for calculating regionallyaveraged infiltration rates into heterogeneous soils which require specification of thespatial distribution of saturated hydraulic conductivity and/or initial soil water deficit inthe sub-basin. A large body of literature established that the spatial variability of soilproperties affect the infiltration at field scale (Smith and Hebbert 1979; Warrick andNielsen 1980; Russo and Bresler 1982; Greminger et al. 1985; Sivapalan and Wood1986; Hawkins and Cundy 1987; Milly and Eagleson 1988; Binley et al. 1989;Saghafian et al. 1995; Logsdon and Jaynes 1996; Sullivan et al. 1996; Woolhiser et al.1996; Merz and Plate 1997; Shin et al. 1998; Smith and Goodrich 2000; Assouline andMualem 2006). However, till now, the use of these techniques in practical applications islimited. Moreover, in the hydrological predictions, the uncertainty involved in capturingthe essential spatial variability of soil properties needs to be studied in more detail.

Green-Ampt Infiltration Models for Varied Field Conditions 3507

Further, the formation of a surface seal has significant influence both on the infiltrationand runoff processes (Duely 1939; McIntire 1958). Surface seal of thickness 1–5 mm(Tackett and Pearson 1965) generally develops during the rainfall impact (McIntire1958), soil erosion, artificial seal resulting from application of polymeric substances, anddeposition of fine soil particles due to overland flow. It can reduce the soil hydraulicconductivity by 2000 folds (McIntire 1958) resulting in a reduction of infiltration rate byabout 50% (Eisenhauer 1984). However, a significant reduction in the infiltration rateoccurs after development of the temporally and spatially varying surface seal which aredifficult to implement accurately in a simulation model. Subsequently, Assouline andMualem (2002) studied the combined effect of soil spatial variability and profileheterogeneity on the infiltration process. The main inference of this study suggests thatthe soil surface seal has a greater effect on the infiltration process than the spatialvariability of soil.

3 Green-Ampt Model for Homogeneous Soils

Green and Ampt (1911) derived an approximate mechanistic model for infiltration underponded condition into a deep homogeneous soil with uniform initial moisture content whichassumes a piston-type water content profile with a well defined wetting front. This model isbased on the hypothesis that there is the existence of sharp wetting front having a constantmatric potential and the wetting zone is uniformly wetted with a constant hydraulicconductivity (Fig. 1). These assumptions were based on the approximations of real soil

Fig. 1 Infiltration profile for the Green-Ampt model

3508 R.V. Kale, B. Sahoo

behavior obtained from experience, so that the model they developed still has instructivevalue today. This model can be derived by combining the Darcy’s law with the continuityprinciple. The resulting Green-Ampt equations is given as

f ðtÞ ¼ K 1þ ho þ hsL

� �ð3Þ

FðtÞ � h0 � hsð Þ$q ln 1þ FðtÞh0 � hsð Þ$q

� �¼ Kt ð4Þ

where f(t) = dF(t)/dt = infiltration rate; K = effective hydraulic conductivity; ho = depth ofponding water over the soil surface; hs = capillary suction head at the wetting front; L =depth of wetting front below the bottom of pond; F(t) = cumulative infiltration depth;and Δθ = fe−θi = soil moisture deficit; fe = effective porosity; and θi = initial(antecedent) moisture content. The cumulative infiltration depth in Eq. 4 can beobtained implicitly by using the numerical iterative methods or explicitly by using theapproximation which is discussed in detail in the subsequent sections. The predictedcumulative depth at any desired time interval can be used to estimate the infiltration rateby using Eq. 3.

From Eq. 3, it may be noted that the Green-Ampt infiltration equation differs fromthe Darcy’s law, in which the depth of wetting front is necessary in the calculationsinstead of the hydraulic gradient. The assumption of sharp wetting front may bereasonable in fine grained soil, as shown by dye studies in an experimental earthen liner(Albrecht and Cartwright 1989). The Green-Ampt equation can also be obtained as anexact solution of the Richards equation considering the diffusivity function D(θ) as aDirac-Delta-type function δ(θ1−θ) with a non-zero value only at the saturated watercontent (Philip 1957); where θ1 is the moisture content imposed at the boundary. Milly(1985) demonstrated that the delta-function diffusivity is not a sufficient condition forexistence of stable Green-Ampt profile in the delta-function soil. They argued that forstability of the Green-Ampt profile, the second derivatives of hydraulic conductivitywith respect to moisture content (d2K/dθ2) should not be negative for all the values ofmoisture content. Usually, the coarse-textured, well-sorted soil such as sands, sandfraction and glass bead media most closely exhibit the Green-Ampt profile as these soilssatisfy the condition d2K/dθ2>0. Further, after comparing the Green-Ampt equation withvarious methods for calculating the water infiltration into soil, Whisler and Bouwer(1970) concluded that the numerical analysis of diffusion models gave the bestagreement with the observed data, however, require considerable amount of input dataand tedious calculation procedure; whereas, the Green-Ampt model was easier to useand provided reasonably accurate estimate of infiltration which are sufficient for most ofthe field problems. Further, this model can give reasonable estimates of the depth ofwetting, infiltration capacity, and cumulative depth of infiltration with readily availableinput data, but may not be able to accurately reproduce the actual water content/pressureprofile as a function of time or space. Gill (1978) proposed a modification in the Green-Ampt model of one-dimensional vertical infiltration into homogeneous soil in which thesaturated and unsaturated zones are regarded as two homogeneous stratifications withdifferent values of hydraulic conductivity. Hence, infiltration is conceived to beoccurring in a medium of two homogeneous layers with the same porosity but differenthydraulic conductivities. The study on the appropriateness of the Green-Ampt model for

Green-Ampt Infiltration Models for Varied Field Conditions 3509

predicting water movement in homogeneous unsaturated soil by Smiles et al. (1981),Dagan and Bresler (1983) and Govindaraju et al. (1992) shows that soil water storage ispredicted accurately by the Green-Ampt model under flux-dependent conditions at thesoil surface. The Green-Ampt model for homogeneous unsaturated soil can be used as analternative to the Richards equation (Govindaraju et al. 1996). This model, althoughapproximate, provides analytical solutions to the flow field. It was found that the flowquantities required for the analytical solution of convective transport (specifically thesoil water storage) were predicted accurately by the Green-Ampt model. Thecomparisons of numerical and analytical solutions of solute concentration profilessuggested that the Green-Ampt water flow model was adequate for convective solutetransport predictions, and that the flow field need not be computed by the morenumerically expensive Richards equation for this problem. For modeling horizontalinfiltration, Kargas and Kerkides (2011) computed a Green-Ampt model type infiltrationprofile by using the codes of HYDRAUS-1D (Simunek et al. 1998).

Moreover, Eq. 3 is applicable only when water starts ponding on the soil surface fromthe beginning of the rainfall event. Therefore, to use this equation considering the conditionof ponding some time after the start of rainfall, Mein and Larson (1971, 1973) developed atwo-stage model for infiltration (under a constant-intensity rainfall) into a homogeneoussoil with uniform initial moisture content. The first stage predicted the volume ofinfiltration at the moment when surface ponding begins. The second stage, which is usedfor prediction of infiltration after the occurrence of the ponding, described the subsequentinfiltration behavior, wherein they provided the equations for the calculation of pondingtime and accumulated infiltration depth. This variant of the Green-Ampt model of Mein andLarson (1973) can also be found in the referred textbooks (Chow et al. 1988; Rawls et al.1992). Furthermore, Swartzendruber (1974) used the Green-Ampt approach to describeponded condition whose analysis is similar to that of Mein and Larson (1973) for thecomputational time before ponding occurs (t ≤ tp), where tp is the time of ponding. Fordescribing the ponding characteristics after the time of ponding, Swartzendruber (1974)presented two explicit forms of equations together with an interesting analysis of the errorsinvolved in the process of transformation from implicit to explicit. Both of these models areapplicable only for constant rainfall occurring over a homogeneous soil. In these models,the cumulative depth of infiltration is obtained mainly by using the Newton-Raphsonmethod. In case of overland flow, the ponded depth is negligible as compared to thecapillary suction and the wetting front depth (Chow et al. 1988). Although thisassumption is acceptable in many overland flow studies, however, it is only a crudeassumption. Argyrokastritis et al. (2009) showed that after ponding and the transforma-tion of the boundary conditions from flux-controlled to flooding, there is a closecoincidence of the moisture profiles for the case of sand mixture, retaining always theGreen–Ampt mode of wetting front advancement. Freyberg et al. (1980) investigated theperformance of the Green-Ampt model for addressing the infiltration problem with timevarying water depth. They suggested that the effective suction head must be considered asa function of surface water depth, initial moisture content and soil type. After examiningthe different definitions of effective suction head for using in the infiltration model, theysuggested that the choice should be made based on a particular problem, as the differencesbetween various effective suction heads by various definitions is marginal. The Green-Ampt model was further modified to account for the effect of entrapped air and crustformation (Ahuja 1974; Morel-Seytoux and Khanji 1974; Risse et al. 1995). Morel-Seytoux and Khanji (1974) used two-flow infiltration theory equations of water and airanalytically to clarify the physical meaning of the terms in the Green-Ampt infiltration

3510 R.V. Kale, B. Sahoo

equation. The parameter which Morel-Seytoux and Khanji obtained is the capillarypressure head at the wetting front and is defined as

hs ¼Z0

hci

fwdhc ð5Þ

where hc = capillary pressure head; hci = capillary pressure head at initial saturation; fw =fractional flow function mathematically expressed as

fw ¼ 1þ Kra

ma

mw

Krw

� ��1 Krw

mwð6Þ

where Kra and Krw = relative permeability to air or to water; and μa and μw = dynamicviscosity of air or water.

However, it is somewhat difficult to understand the meaning of the term fw, whereas theterm hs can be estimated theoretically. The required relative permeability to air (Kra) in theevaluation of fw is seldom measured, although, the flow function is relatively insensitive toKra. Morel-Seytoux and Khanji (1974) argued that the resistance due to fluid viscosityretards the flow of water into the soil pores.

3.1 Performance of Green-Ampt Model with Cracks and Macro-pores

The presence of surface connected macroscopic channels such as worm holes, root holesand cracks in the soil can enhance the vertical transfer of water and solute (Edwards et al.1979; Beven and Germann 1981). Water running into the channel infiltrates laterallythrough the wall into the surrounding soil medium at a depth. To model infiltration underthese conditions, Davidson (1984) pointed out that GA model despite requiring less soilinput data was capable to generate the results which were easier to interpret in two-dimensional systems than those based on the Richards equations. Therefore, he proposed atwo-dimensional Green-Ampt model for infiltration into soil containing regularly spaced,water filled, vertical cracks when the soil surface is saturated (the crack-filling stage isignored). Subsequently, he assumed a two-dimensional soil containing regularly spacedvertical cracks which are open to surface at all times and having a length of a and spacing2b, while width of each crack is ignored. Further, soil between the cracks is taken ashomogeneous and non-swelling. A Green’s function which satisfies the homogeneousboundary conditions was used to derive the integral equations to determine the velocity atwetting front. The results of this study shows that the cumulative infiltration was greater inthe presence of cracks (infinite b/a) than it is in their absence (infinite a/b), and it increaseswith the decreasing value of b/a and increasing value of -hs/a. The similar behavior of theinfiltration rate can be expected except when b/a=0.5. Note that, when b/a=0.5, thenumerical method breakdowns before the wetted front expand out horizontally wherein theadjacent cracks can interact. He reported that the results arrived at by using the GA modelcoincide with those arrived at by using the Richards equation (Edwards et al. 1979; Beven andGermann 1981). Although, this model seems to be appropriate when assumption of abruptwetting front is well satisfied, but it should be used cautiously when the conditions are notmet especially in fine-textured soils likely to more susceptible to cracks. Subsequently, anasymptotic GA model of infiltration into non-swelling saturated soil with the presence ofpermeable cracks and holes (Davidson 1985) and for soil whose surface are impermeable

Green-Ampt Infiltration Models for Varied Field Conditions 3511

except for cracks and holes (Davidson 1987) has been advocated. However, with a very largecrack spacing or column radius, the validity of the GA model is questionable.

3.2 Variants of Green-Ampt Model Based on the Explicit Solution Scheme

Li et al. (1976) developed a quadratic approximation of the Green-Ampt equation based onthe power series expansion of the logarithmic term in Eq. 4. The explicit solution of thisassumption gives a maximum error of 8%, when the approximation is performed first onthe accumulated infiltration volume (F), and not on the infiltration rate (f). The implicitsolution of Li et al. (1976) by refining the explicit solution using the second order Newtonmethod has a resulting error of 0.0003%. Schmid (1990) proposed a number of explicitequations for time-dependent infiltration rate and cumulative infiltration based on theextension of Mein and Larson (1973) infiltration model. Stone et al. (1994) derived theirapproximation based on two first terms in a Taylor series expansion. The proposedapproximation can be used for any event of constant rainfall and variable time of ponding.The investigation of the appropriateness of the approximation shows good fit with amaximum error of 3.5%, and a better fit to the Green-Ampt infiltration depth as comparedto the quadratic approximation. Salvucci and Entekhabi (1994) introduced the explicitGreen-Ampt model, whose four term expression yields an error of less than 2% in theestimation of infiltration for any given time period. The model of Salvucci and Entekhabi(1994) is based on rapidly conversing power series which is valid for a wide range of soiland hydraulic parameters. Serrano (2003) presented the improved decomposition solutionof the Green-Ampt equation which is an improvement over its explicit solution advocatedby him (Serrano 2001). Further, Barry et al. (2005) derived the approximations for theGreen-Ampt infiltration equation from the approximations of the Lambert W−1 function andvice versa (see Table 1).

Most of the above mentioned forms of the explicit solutions of the Green-Ampt equation arederived from an implicit solution. Mailapalli et al. (2009) proposed a nonlinear single stepmethod known as Ramos’ nonstandard explicit integration algorithm (EIA) for solving theGreen-Ampt equation using small step sizes. Conventionally, the multistep methods e.g.,Runga-Kutta and Newton-Raphson have been employed to solve the Green-Ampt equation.For the sake of brevity, a simple algorithm proposed by Mailapalli et al. (2009) is given here.

Step 1: Input the required value of Ke, hs, θs, θi, Δt, Tmax (Maximum simulation time) andn (=0).

Step 2: Initialize F (t) (= 0.0001).Step 3: Estimate value of F (t + Δt) at time t (=(n+1)Δt) using equation

F t þ $tð Þ ¼ FðtÞ þ2$tKe

1þhs qs�qið ÞFðtÞ

h i2� $t � Kshs qs�qið Þ

FðtÞ2� � ð7Þ

Step 4: If t ≤ Tmax, then n=n+1, go to step 3, else print results and terminate the program.

The Green-Ampt solution of Mailapalli et al. (2009) with the EIA technique has a betterperformance in terms of accuracy and run-time efficiency as compared to the conventionalsolution technique such as the Newton-Raphson algorithm and the explicit method of Barryet al. (1995) and Salvucci and Entekhabi (1994). The only disadvantage with this techniquelies in the requirement of small step sizes.

3512 R.V. Kale, B. Sahoo

Table 1 Variants of the explicit Green-Ampt model

Li et al. (1976)

et

al.

et

al.

Green-Ampt Infiltration Models for Varied Field Conditions 3513

Table 1 (continued)

Modified Li et al. (1976) equation Stone et al.

approximation (Li et al. 1976),

et al.

3514 R.V. Kale, B. Sahoo

3.3 Variants of Green-Ampt Model Accounting for Rainfall Pattern

Since rainfall intensity is a temporally varying entity, James and Larson (1976) extended theMein and Larson model (Mein and Larson 1973) to represent the infiltration andredistribution of soil water during intermittent water applications. However, this modelover-predicted the infiltration rate when the application rate exceeded the infiltrationcapacity of the soil. Similarly, Chu (1978) modified the Green-Ampt equation for unsteadyrainfall falling on the homogeneous soil surface. The time shift scale, termed as “pseudotime”, was used as a correction for considering the cumulative infiltrated depth of water atthe time of ponding during an unsteady rainfall event. The expressions are given for the twosurface condition indicators to examine whether the surface is ponded or not ponded duringthe rainfall event. Chu (1978) has also presented the methodology to predict the infiltration

Table 1 (continued)

Barry et al.

Green-Ampt Infiltration Models for Varied Field Conditions 3515

rate, time to ponding and cumulative infiltration depth under variable rainfall conditions. Asimilar algorithm is also presented by Chow et al. (1988) to separate the infiltration andrunoff during the occurrence of varying rainfall. Hachum and Alfaro (1978) also extendedthe Green-Ampt approach for analyzing the infiltration under unsteady rainfall patterns.Further, Chu (1987) extended the Mein and Larson model (Mein and Larson 1971) toanalyze the infiltration process under a variable rain on a covered natural soil. The Green-Ampt model was further modified to account for the effect of time varying ponding(Warrick et al. 2005; Hugo and Huang 2007).

The Green-Ampt infiltration model has its important application in irrigation systemmanagement. In the design and management of sprinkler and drip irrigation system, theknowledge of the advance rate, volume of wetting pattern to determine emitter or sprinklerhead discharge, time of application, irrigation period, rate of solute transport in case ofsoluble fertilizer applications and leaching requirement to treat the salt affected soil are ofvital importance. For this purpose a thorough understanding of the time relationshipbetween wetting pattern of dripping water from the ground surface and soil-profilecharacteristics is of prime importance. Although, numerous studies based on the Richardsequation are available in the literature, the requirement of soil water retention and soilcapillary conductivity curves as the input information are seldom available for most of thesoil that limits their application. In such a circumstance, the Green-Ampt model would bean appropriate choice due to the readily available physically based parameters. Chu (1994)has proposed a three dimensional GA model to describe the wetting pattern of surfaceemitter for drip irrigation system by assuming movement of water in radial direction fromthe dripping source in which the wetting pattern assumed to form wedge-shaped volumeelement. According to this development, the equations for infiltration rate and applicationtime of irrigation can be expressed, respectively, as

f ¼ Ksatr0R H þ R sinBð ÞR� r0

r�2 ð8Þ

1

3R30 � r30

� � r02

R20 � r20

� ¼ KsatHr0M

t ð9Þ

where R = wetted radius (i.e., the radial distance from the base of element to the sourcecentre); r = radial distance from a point in the element to the source center; H = GApotential parameter; B = angle between ground surface and volume element; R0 = wettedradius on ground surface; r0 = dripping source radius; Ksat = saturated hydraulicconductivity parameter; and M = GA water content parameter. The emitter discharge canbe estimated by the infiltration-capacity curve that represents the time distribution ofwetting-pattern volume of a water source with unlimited supply. A significant feature of theproposed infiltration capacity curve is that the infiltration-capacity rate increases with timeduring the extended infiltration period which is in contrast with deceasing rate ofinfiltration-capacity behavior exhibited by the one-dimensional GA model. Although, thisconclusion was arrived at theoretically, however it is not properly tested under differentfield and soil conditions. The implications of the phenomenon such as crust formation duewater drop impact, redistribution of soil moisture and non-uniformity of soil profile werenot taken into account while developing the infiltration-capacity curve. Therefore, thesecurves should be applied in drip irrigation design with caution and if possible withmodifications.

3516 R.V. Kale, B. Sahoo

4 Infiltration Process in Layered Soils

Under the natural condition, the soil matrix is seldom homogenous or isotropic. The topsoil layer usually has a loose texture which overlays the bottom soil layers withrelatively lower hydraulic conductivity. However, the identification of distinguishingboundary between two soil layers seems to be tough task. Generally, it is assumed thatthe soil matrix is composed of different soil layers and there exists a clear boundary.The layered soil profile consists of a soil mixture with varied texture, permeability andstructure, randomly encountered in different patterns one above another from thesurface. Such an arrangement of soil layers in different patterns reduces the infiltrationrate irrespective of the composition of underlying soil layer as finer or coarser than thesurface soil layer. In the fine textured soil, reduction in infiltration rate is attributed tothe high resistance of soil pores for an easy movement of water. Although, the coarsetextured soil has a higher hydraulic conductivity than the underlying fine textured soillayer, the lower matric potential at the wetting front causes partial saturation of thecoarse textured soil (Tagski 1960; Bouwer 1976). The resulting actual unsaturatedhydraulic conductivity is lower than the above wetter finer soil layer causing furtherreduction in the infiltration rate. However, in case of the dry soil, layered as given above,water is not able to enter into the coarse textured soil layer until sufficient pressure hasbuilt up to wet the large pores which may result in the occurrence of persistent narrowflow channels (fingers) throughout the coarse textured zone (Baker and Hillel 1990; Juryand Horton 2004).

When the material of high hydraulic conductivity overlies one with a lower value,positive soil water potential develops in layered soils during infiltration. Thisphenomenon occurs in the soil profile that consists of coarse textured soil (sand)overlaying a fine textured soil (clay). When the infiltrating water from the sand layerreaches the clay, it begins to accumulate at the layer interface due to the differences inhydraulic conductivities of the two materials. Initially, the underlying fine soil layerhaving higher suction head as compared to the overlain coarse layer has higherattraction for water which causes slight increase in the infiltration rate as the thin layergets wetted and, subsequently, the higher resistance of the finer soil to water movementcauses a remarkable reduction in water flow (Miller and Gardner 1962). In other words,the sand layer with higher hydraulic conductivity than that the clay will deliver water at arate greater than the clay can accept. Initially this results in an increasing of water contentin the soil above the interface. Subsequently, development of subsurface layer ponding,free water, and positive soil water potential occurs. This effect has been observed tohappen in fields with similar soil profiles. The occurrence of free water and positive soilwater potential is documented in the literature dealing with hillslope stability (Sidle et al.1985) and the GA model can be used in integration with the slope stability model for theanalysis of shallow type slope failure such as land slides (e.g., Muntohar and Liao 2010).Further, Hill (1992) and Deliman (1994) observed through laboratory experiments thatinfiltration into the second layer of a two layered soil does not begin until the top layer iscompletely saturated.

When the wetting front passes through the interface of the layered soils, some internaladjustment of water flow condition takes place due to abrupt changes in soil properties oftwo distinct layers so as to reach a new internal equilibrium. In such a case, the actualwetting front may not be a sharply defined one and, hence, the adjustment process start alittle bit earlier. As a result, the transition from high to low infiltration rates can be smootherwithin a small time period (Chu and Marino 2005).

Green-Ampt Infiltration Models for Varied Field Conditions 3517

5 Green-Ampt Model for Heterogeneous/Layered Soils

Over the last five decades, several studies have focused on infiltration into layered soils.The first experimental study on infiltration into layered soils was attempted by Colman andBodman (1945). Whisler and Klute (1966) attempted to treat the infiltration intomultilayered soil. An application of the Green-Ampt model to layered soil profiles wassupported by some limited experiments by Childs (1967). It was Childs and Bybordi (1969)who extended the Green-Ampt analysis to model the infiltration process into layered soilswith a decreasing hydraulic conductivity from the soil surface. Equation 3 was modified byChilds and Bybordi (1969) to estimate the infiltration rate and time at which the wettingfront reaches some location in the nth layer which is expressed as

f ¼ Kn Lþ h0 þ hsnð ÞLþPn

j¼0zj

KnKj� 1

� � ð10Þ

t ¼ tn�1 þ qsn � q0nKn

L�Xn�1

j¼0

zj þXn�1

j¼0

zjKn

Kj� 1

� �� h0 þ hsnð Þ

" #ln

Lþ hsn þ h0

h0 þ hsn þPn�1

j¼0zj

0BBB@

1CCCA

8>>><>>>:

9>>>=>>>;

ð11Þ

where n = total number of soil layers; hsn = capillary pressure head at the nth layer; zj = depth ofjth layer; Kn = saturated hydraulic conductivity of nth layer; Kj = saturated hydraulicconductivity of jth layer; t = time at which the wetting front is observed in nth layer; tn−1 = timeat which the wetting front reaches the boundary between (n−1)th layer and nth layer; θsn =saturated water content of nth layer; and θ0n = initial water content of nth layer.

The parameters in the above equations were determined empirically by experiment and itwas found that the computed infiltration rate was in good agreement. Therefore, Childs andBybordi (1969) argued that the Green-Ampt infiltration equation is valid in multilayeredsoil. It is noted that in their investigation, the pressure at the boundary between two layersare assumed to be continuous. Further, using the concept of Morel-Seytoux (1974), Sonu(1986) arrived at the same equations as Eqs. 10 and 11, except that it includes the effects ofviscosity of fluid on the infiltration rate expressed as

f ¼ Kn Lþ h0 þ hsnð Þ

bn L�Pn�1

j¼1zj

!þ Pn�1

j¼1

KnKj

� �zj

ð12Þ

where

bn ¼ qsn � q0nð ÞZqsnq0n

�f000wn

mwVrndq ð13Þ

t ¼ tn�1 þ bnKn

qsn � q0nð Þ L�Xn�1

j¼1

zj h0 þ hsn þXn�1

j¼1

zj � Kn

bn

Xn�1

j¼1

zjKj

lnLþ hsn þ h0

h0 þ hsn þPn�1

j¼1zj

0BBB@

1CCCA

8>>><>>>:

9>>>=>>>;

ð14Þ

3518 R.V. Kale, B. Sahoo

where βn = viscous correction resistance, the value which need not to be evaluated at everylayer of the soil; fwn = fractional flow function evaluated for the nth layer; and Vrn = relativeviscous resistance.

Generally, the less permeable soil layer governs the infiltration process regardless ofwhether this layer lies above or below the more permeable layers. Therefore, the lesspermeable layer is the main layer to control the entire infiltration process. Hence, the effectof layer interface has to be considered in the infiltration models (Rao et al. 2006). Further,the hydraulic conductivity of the porous material decreases with the decrease in soilmoisture content. Therefore, in order to transmit the water flowing at relatively low ratefrom upper to lower soil layer, the lower layer should have the moisture content less thanthe saturation moisture content. This situation is in contradiction to the concept of theGreen-Ampt model; hence, it is not applicable to model this situation. Conversely, Bouwer(1969, 1976) demonstrated the applicability of the Green-Ampt model for both decreasingand increasing conductivity profiles under ponded water condition. In most of multi-layeredmodels (e.g. Chu and Marino 2005), the analytical solution, implicit in terms of zw havebeen obtained for multi-layer soils, for which layers have constant value of Green-Amptparameters. However, Kacimov et al. (2010) used a 1-D Green Ampt model in which theCauchy problem for a nonlinear ordinary differential equation describing the wetting frontpropagation in the soil profile is solved by computer algebra routines. Particularly, theyhave addressed a special kind of heterogeneity common in gravelly beds of reservoirs ofrecharge dams in Oman in which saturated hydraulic conductivity and suction pressurehead at wetting front subjected to abrupt increase or decrease with depth of heterogeneitysoil having constant ponding or infiltration-evaporation depleted ponding on the surface,The impact of variation in ponding depth h0 due to operation of sluice gate on the variationof Green-Ampt conductivity-capillarity constituting parameters and, thus, resulting positionof wetting front at any target time have been assessed. This study has also analyzed thestrange F(t) curves (concave-convex or concave-convex-concave) due to acceleration anddeceleration of Lagrangian water particle traveling from soil surface.

Similarly, Childs and Bybordi (1969), and Hachum and Alfaro (1980) detailed the use ofthe Green and Ampt equations for modeling infiltration into layered soils with decreasinghydraulic conductivity. In both of these models, the monitoring of the wetting front is to bedone along the soil column. As long as the wetting front remains in the top soil layer, thegeneral Green-Ampt equation can be used; whereas, when the wetting front enters into thesecond soil layer, the hydraulic conductivity value changes to the harmonic mean value ofthe two soil layers for the wetted depth, and the capillary head is set equal to the suctionhead of the second soil layer. The same methodology is applied for the successive soillayers with the harmonic mean of the next two soil layers. However, this approach is notapplicable for the case wherein the hydraulic conductivity increases with the depth of thesoil profile. Thus, there is a need to use a different approach.

Subsequently, Moore and Eigel (1981) extended the Green-Ampt model of Mein andLarson (1973) to predict the infiltration rate and time of ponding in a two layered soilprofile. In their study, Moore and Eigel (1981) investigated the effect of the layer sequencein a two layered soil profile having significant different properties and rainfall intensity onthe infiltration process. The model of Moore and Eigel (1981) estimates the hydraulicproperties of the second soil layer, when the fine textured soil layer lies over the coarsetextured soil layer. The comparative study of this model with the numerical solution of theRichards equation shows that for coarse-over-fine layer arrangement, the time to pondingincreases with the decrease in water application rate and increase in layer thickness ofcoarse soil as estimated by both the models as well as by the observations. However, when

Green-Ampt Infiltration Models for Varied Field Conditions 3519

the fine-over-coarse soil layer arrangement is used, the ponding time estimated by theGreen-Ampt model is independent of the surface layer thickness. However, this is notthe case for the numerical model; particularly, for low intensity rainfall. Further, theproposed model by Moore and Eigel (1981) under-predicts the infiltration volume ascompared to the numerical solution and observed values. The range of prediction error ofthe infiltration volume was −2.9 to −5.9% and −4.7 to +8.3% for the coarse-over-fine andfine-over- coarse layer arrangements, respectively. These results show the fairly goodagreement of the Green-Ampt model with the numerical model. It was surmised from thestudy by Moore and Eigel (1981) that the infiltration through the higher hydraulicconductivity soil layer can be governed by the harmonic mean of the upper soil layer.Moreover, the two models proposed by Moore and Eigel (1981) are useful only under theinitial ponding condition.

Flerchinger et al. (1988) developed an explicit equation for the accumulatedinfiltration over a time for layered soil by using the approximation to a logarithmic term.This is basically the extension of infiltration rate equation given by Fok (1970), whenwetting front passes through a number of soil layers. Further, this model was extended forprediction of infiltration volume during discrete time steps. Similarly, Wang et al. (1999)used the original Green-Ampt equation for a two-layer soil profile. The differencebetween the models of Wang et al. (1999) and Flerchinger et al. (1988) lies in theprediction of the average suction head and the wetting front depth. The model thusdeveloped was only applicable for the constant rate of rainfall intensity. Leconte andBrissette (2001) developed a GA model for two layered soil profile assuming one-dimensional vertical unsaturated flow, where the upper soil layer has higher saturatedhydraulic conductivity than the soil below it. The proposed GA equations (Eqs. 13 and 18in original manuscript) are solved with classical forth-order Runga-Kutta method. Whiletesting this model on many soils, it was found that the proposed model can track averageupper soil moisture accurately for complex rainfall sequence with significant periods ofredistribution. This model was capable to perform well against the numerical solution ofRichard’s equation to represent local infiltration at 8–10 times faster rate. Deliman (1994)provided the algorithm to cope with the situation of positive soil water potential occurringin the coarse-over-fine soil profile. Beven (1984) and Selker et al. (1999) derived simpleexpressions for the rate of infiltration using the Green and Ampt approach for soils withpermeability that decreases with depth following linear, power law and exponentialrelationships. These analytical expressions for small catchment were developed based onsoil description and assuming initial ponding at the start of rainfall. Subsequently, Allaireet al. (2010) cautioned that it is not sufficient to describe the subsurface soil heterogeneityby accounting only for soil layers, but also additional features such as cracks in soil,regions of hydrophobicity and bioturbation need to be considered properly. However, suchheterogeneity feature exhibit a range of length scales which may be multifractural natureof soils, can be expected to be power-law distributed (Voller 2011). Hence, Voller (2011)have obtained a fractional GA infiltration model which is based on the postulate that if thelength scales of heterogeneity can be assumed to be power law distributed, then it may beappropriate to model the infiltration in heterogenous medium in terms of fractionalderivatives. Therefore, a Caputo fractional derivative (of order 0<α≤1) to expresshydraulic flux has been used in the development of this model. Although, the fractionaldiffusion model (Voller 2011) was found to be appropriate for infiltration intoheterogeneous soils, there is need for building a compelling physical link or analogybetween the nature of heterogeneity and mathematics and statistics of the fractionalcalculus to understand the nature of fractional derivatives.

3520 R.V. Kale, B. Sahoo

5.1 Variants of Green-Ampt Model Accounting for Surface Seal

The Green-Ampt equation with the parameter altered to account for the effect ofsurface seal was applicable for homogeneous soil topped with crust (Moore 1981a,b;Powell and Steichen 1982; Brakensiek and Rawls 1983; Rawls et al. 1990). Later, Chu(1985), Chu et al. (1986) and Wolfe et al. (1988) extended the technique to three layersoil profile including crust, tillage and subsoil layer. Subsequently, this model wasupdated to include the effect of initial moisture content on the parameter estimation in ahomogeneous soil layer. Moreover, Kim and Chung (1994) evaluated the model of Chu(1985) for the three layered soil by taking into account the temporally varying physicalproperties of the soil. Skonard and Martin (2002) developed a two-dimensionalphysically based infiltration model for furrow irrigation using the Green-Amptinfiltration approach. Simulation tests of this model showed that the two-dimensionalmodel is capable of estimating the cumulative infiltration volume within 8% of errorlevel as compared to the simulated infiltration using the finite element HYDRUS-2Dmodel. Application of the two-dimensional model in a surface irrigation advance schemeallows the irrigation performance parameters to be predicted without extensive soilexperiments. All these models are applicable for the surface irrigation modelling. Amethod based on the Green-Ampt approach was also proposed by Paschepsky andTimlin (1996) to model the infiltration into layered soil (especially three layers, viz.,crust/seal, soil and subsoil layers) profiles topped with crust deposition in the croppedfield. This method uses harmonic mean hydraulic conductivity in the wetting zone,assumes the pressure head in the wetting zone to be close to pressure head under theseal, and treats the wetting front pressure as a value which provides for the conservationof the integral mean hydraulic conductivity of the wetting front. The validation of thismethod with the numerical solution of the Richards equation suggests that, when thehydraulic conductivity decreases with depth, the difference between the cumulativeinfiltration and the infiltration rate was 2-5%; whereas it was 10–15%, when thehydraulic conductivity increases with depth. Hence, this model is applicable for thesituation where the hydraulic conductivity decreases with depth. Enciso-Medina et al.(1998) also presented the one-dimensional (1D) Green and Ampt approach to model theinfiltration into the seal-formed soil, wherein the Runge-Kutta method was used to solvethe infiltration function. The algorithm presented by them involves the tedious algebraicmanipulation which results in excessive execution time and error in computation. Toovercome these difficulties, Rao et al. (2006) extended the model of Moore and Eigel(1981) in the form of 1D GA model for modeling the infiltration in uniform and layeredsoils considering three cases, viz., seal-free uniform, seal formed uniform and sealformed layered soils. On the basis of the model evaluation with the numerical solution ofthe 1D Richards equation and explicit solution of the Green-Ampt model for theaforementioned three cases, it was concluded that, for an accurate simulation of theinfiltration process on tilled soils, surface seal formation has to be considered and to beincorporated in the infiltration models. The reduction in infiltration due to seal formationin uniform soils is 10 and 48% for sandy loam and silty-clay-loam soil, respectively.While, in case of three layer (seal-tillage-subsoil) soil profile, the reduction in infiltrationis 4% and 18% for sandy loam and silty-clay-loam soil, respectively. This model isapplicable only for the irrigation modeling with the inclusion of surface seal, tillage andsubsoil layers. The important point to be noted herein is that, when the coarse soil layerlies over the fine soil layer, the parameter, such as, the satiated hydraulic conductivityand moisture content of the lower layer remains unchanged. However, when the upper

Green-Ampt Infiltration Models for Varied Field Conditions 3521

soil layer is less permeable than the lower soil layer, the satiated hydraulic conductivityand moisture content of the lower soil layer get modified based on the seal condition.The satiated hydraulic conductivity and moisture difference of subsoils are the mostsensitive parameters which vary with the surface seal condition, but not with the soiltexture.

According to Rao et al. (2006), the position of the wetting front, zw can be given by

f ðzwÞ ¼ ðA� BhÞ lnzw þ h

h

� � �þ Bzw � t ð15Þ

When the wetting front is in the tillage layer (second layer below the seal layer), then theconstants A and B in Eq. 15 are given as

A ¼ zs=Ksð Þ$qs � zs=Ktð Þ$qt; B ¼ $qt=Ktð Þ for zs < zw � zs þ ztð Þ ð16Þ

Similarly, when the wetting front is in the subsoil layer below the tillage layer,

A ¼ zs=Ksð Þ$qs þ zt=Ktð Þ$qt � zs þ ztð Þ=Kb½ �$qb; B ¼ $qb=Kbð Þ for zw > zs þ ztð Þ ð17Þ

where zw = position of the wetting front below the soil surface; h = total head includingdepth of ponding water above the ground surface; zs and zt = thickness of seal and tillagelayer, respectively; Ks, Kt and Kb = satiated hydraulic conductivity of seal, tillage andsubsoil layers, respectively; and Δθs, Δθt and Δθb = change in moisture content in seal,tillage and subsoil layers, respectively.

The cumulative infiltration corresponding to the period (t) in tillage layer Ft(t) andsubsoil layer Fb(t) is given by (Rao et al. 2006)

FtðtÞ ¼ zs$qs þ zw � zsð Þ$qt for zs < zw � zs þ ztð Þ ð18Þ

FbðtÞ ¼ zs$qs þ zt$qt þ zw � ðzs þ ztÞ½ �$qb for zw > zs þ ztð Þ ð19Þ

If the seal is more permeable than the tillage layer, there is no need to account for thesealing effect and Eqs. 18 and 19 can be used directly.

Further, Jia and Tamai (1997) presented a generalized GA model to simulate theinfiltration into a multi-layered soil during unsteady rain. When surface ponding isoccurring from the beginning of a rain event and is continuing, then the cumulativeinfiltration (F) and infiltration rate (f) can be expressed as

f ¼ Km 1þ Am�1Bm�1þF

� �F � Fm�1 ¼ Km t � tm�1ð Þ þ Am�1 ln

Am�1þBm�1þFAm�1þBm�1þFm�1

� � ð20Þ

With

Am�1 ¼Pm�1

iLi �

Pm�1

iLiKm=Ki þ hsm

� �$qm

Bm�1 ¼Pm�1

iLiKm=Ki

� �$qm � Pm�1

iLi$qi

Fm�1 ¼Pm�1

iLi$qi

ð21Þ

3522 R.V. Kale, B. Sahoo

When surface ponding occurs while the wetting front penetrates the mth soil layer

f ¼ Km 1þ Am�1Bm�1þF

� �F � Fp ¼ Km t � tp

� þ Am�1 lnAm�1þBm�1þFAm�1þBm�1þFp

� � ð22Þ

with

Fp ¼ Am�1rp=Km�1 � Bm�1; tp ¼ tm�1 þ Fp�Fm�1

rpð23Þ

Where tm−1 = time when the wetting front reached the interface of mth and (m−1)th soillayers and it can be solved successively from t1, t2, ...... to tm−1 with t1 by Eqs. 1 and 2; L =the depth of wetting front; Li = the thickness of ith soil layer. Numerical test of GA modelby Jia and Tamai (1997) under unsteady rain showed that this model behaves similarly tothat of the the Richards equations solution.

In a subsequent work, Chu and Marino (2005) presented a modified Green-Ampt modelaccounting for two distinct periods, such as, pre-ponding and post-ponding that deals withthe infiltration into layered soils under unsteady rainfall condition. The generalized Chu andMarino infiltration model for the infiltration rate and cumulative infiltration can beexpressed, respectively, as

iðzÞ ¼ zþ hsnPn�1j¼1

zj�zj�1

Kjþ z�zn�1

Kn

ð24Þ

IðzÞ ¼ I zn�1ð Þ þ z� zn�1ð Þ qsn � q0nð Þ¼Pn�1j¼1 zj � zj�1

� qf j þ z� zn�1ð Þqfn ð25Þ

where i(z) = infiltration rate at the zth soil layer from the ground surface; z = location of thewetting front in the nth layer from the ground surface (zn−1<z<zn); hsn = suction pressurehead in the nth soil layer; Kj = hydraulic conductivity of the jth soil layer; I(z) = infiltrationrate at the zth soil layer from the ground surface; θ0n = initial volumetric water content ofthe nth soil layer; θsn = saturated volumetric water content of the nth soil layer; θf j =difference between the volumetric saturated water content and initial water content of thejth soil layer; and θfn = difference between the volumetric saturated water content and initialwater content of the nth soil layer.

The Chu and Marino infiltration model tracks the wetting front along the soil profile anddeals with the shift between the ponding and non-ponding periods. This model is also ableto handle the fully saturated flow condition. However, the time to ponding can only beidentified, if all the infiltration process is solved step-by-step and the cumulative infiltrationis computed according to the rainfall time discretization assuming the instantaneoushydraulic equilibrium at the interface of the layers. This model is tested for three cases,which includes unsteady rainfall over a uniform soil profile, steady rainfall over a layeredsoil and unsteady rainfall over a layered soil. Comparison of the simulated model resultswith the observed data, numerical and Chu (1978) model for the first case demonstrate theaccuracy and applicability of this model. This model was further extended to simulate theno rainfall periods by incorporating a compartmental model that accounts for the drainageand moisture redistribution in the soil profile. To take into account the variability of themodel parameters with the initial water content, Chu (1995) established a relationshipbetween the initial water content and the Green-Ampt parameters using the Brooks-Coreyrelationship (Brooks and Corey 1964) of soil water retention curve, which made this modelmore versatile. Based on these developments, WINDOWS based software “HYDROL-

Green-Ampt Infiltration Models for Varied Field Conditions 3523

INF” is also developed by Chu and Marino (2006). Further, the performance of theHYDROL-INF model can be improved by modifying the algorithm with modified Green-Ampt redistribution method (MGAR) proposed by Gowdish and Muñoz-Carpena (2009).Furthermore, Liu et al. (2008) extended the GA model of Jia and Tamai (1997) for thesimulation of infiltration into layered soil with non-uniform initial moisture content duringthe unsteady rainfall. The equation was developed to predict the average suction to includethe effect of initial moisture content utilizing the equation derived by Morel-Seytoux andKhanaji (1974). The comparison of this model with the Chu (1978) model for varyingrainfall over a homogeneous soil was in good agreement. However, application of thismodel to a multi-layered soil profile causes an abrupt instability in the calculation of theinfiltration capacity as well as the infiltration rate, especially, at the interface between thetwo soil layers. Further, this model is tested only with the limited observed data. Hence,there is a need to test this model extensively to confirm its use in modeling study.

In all of above described models for multi-layered soil profile, the hydraulic conductivityfor wetted zone lying above wetting front was considered as the saturated hydraulicconductivity, namely Ksat. However, Ma et al. (2010a) pointed out that because of entrappedair, the soil pores in saturated zone cannot be fully filled with water (see Bouwer 1966;Hammecker et al. 2003). Therefore, the actual saturated hydraulic conductivity of saturatedzone should be taken as the hydraulic conductivity at residual air saturation (K0) which issomewhat less than Ksat. Bouwer (1966) suggested K0=0.5Ksat. Ma et al. (2010a) attemptedto modify the multi-layered GA model by introducing the saturation coefficient todetermine the water content and hydraulic conductivity of the wetted zone. The saturationcoefficient was determined by the ratio between measured moisture volume and totalsaturated moisture volume of the wetted zone. In this study wetting front suction head wasdetermined by Bouwer (1966) and Neuman (1976) methods. To test the developed model, a300 cm long five layered soil column under controlled experimental condition was used.The comparison of the developed model results with those by traditional multi-layered GAmodel and HYDRUS one-dimensional model shows that the modified GA model withBouwer method could adequately capture the infiltration rate, the accumulated infiltration andthe movement process of wetting front in the large layered soil column. Therefore, theyargued that the presented approach was highly effective to simulate the infiltration in multi-layered soils. Note that, in this experiment, the obtained saturation coefficient throughmeasurement was 0.8. Subsequently, Ma et al. (2010b) proposed a modified Green-Amptmodel (MGAM) to simulate water infiltration into layered soil with consideration ofentrapped air instead of the complex two-phase (gaseous and liquid phase) flow model.Thus, they have attempted to modify the multi-layered GA model (Chu and Marino 2005)termed as traditional Green-Ampt model (TGAM) by replacing the parameters θs,n and Ks,n

with θa,n and Ka,n, respectively. Note that, in these equations θa,n and Ka,n are the actualwater content and hydraulic conductivity of the nth soil layer in wetted zone, respectively.Similarly, using Bouwer suggestion the modified GA model (Ma et al. 2010a) termed asBouwer Green-Ampt model (BGAM) was also tested in this study. In MGAM model, asaturation coefficient (Sa) to account for the resistance effect of air entrapment oninfiltration is given by expression

Sa ¼ 1� qra þ qrwqs

ð26Þ

where θra and θrw are the residual air and water content, respectively. However, these twoparameters are difficult to measure as they require the complex laboratory experiments and

3524 R.V. Kale, B. Sahoo

precise set-up. To overcome this problem, Ma et al. (2010b) used the parameters of the soilwater retention curve model (Brooks and Corey 1964) to determine θra and θrw.Subsequently, they have obtained saturation coefficient (Sa) approximately by usingexpression

Sa ¼ 1� qrqs

ð27Þ

Note that for fully saturated soil lying above wetting front Sa=1. All these threemodels were tested using 300 cm long five layer soil column and a 280 cm deep eightlayered field soil profile. They have shown from the simulated results that TGAMoverestimate the infiltration rate and cumulative infiltration whereas BGAM underesti-mate the infiltration rate and cumulative infiltration. However, the MGAM satisfactorilysimulated the infiltration process in both the laboratory soil column and the field soilprofile. However, it can be noted that the performance of the MGAM is largely dependson the accuracy in determination of the saturation coefficient (Sa) and the suction head(hs) at the wetting front.

6 Usefulness of Green-Ampt Infiltration Equation in Modeling Studies

The Green-Ampt model is the most suited model only when the cumulative infiltration isassessed (Schmid 1990; Qi 2008). Chen and Young (2006) have extended the Green-Amptinfiltration model to account for the effect of land slope on the infiltration and runoffgeneration processes which is essentially a modification of the Green-Ampt equationspresented by Chu (1978). Following the development by Chen and Young (2006), forponding condition, the infiltration rate can be expressed as

f ¼ fc ¼ KeL cos g þ hs þ H

Lð28Þ

where fc = infiltrability equal to infiltration rate f under ponding condition; Ke = effectivesaturated hydraulic conductivity; L = depth of wetting front in the direction normal to theland surface; H = h0cosg, the depth over the sloping ground surface; and Lcosg = gravityhead in the wetting front.

The GA equation for cumulative infiltration is given as (Chen and Young 2006)

Ket cos g ¼ FðtÞ � H þ hsð Þ$qcos g

ln 1þ FðtÞ cos gH þ hsð Þ$q

�ð29Þ

Equation 29 is a nonlinear equation which may be solved by using an iterative methodsuch as the Newton-Raphson method. Expanding the second term on the right hand-side ofEq. 29, it can be shown that for small time step

Ket � 1

2

F2h cos

2 gH þ hsð ÞΔq

ð30Þ

where Fh ¼ FðtÞ= cos g, indicating that sloping ground surface increases the infiltration bya factor (1/cosg). However, when t→∞, Ket≈Fh, which indicates that the slope effectdecreases with time and for large t, it gets completely vanished.

Green-Ampt Infiltration Models for Varied Field Conditions 3525

For steady rainfall case, the infiltration rate before ponding is given by

f ðtÞ ¼ r cos g; t � tp ð31ÞWhen f(t) = r, the ponding occurs over the ground surface and the cumulative infiltration

is given by

Fp ¼ H þ hsð Þ$qr cos g=Ke � cos g

ð32Þ

From this Eq. 32, the ponding time can be calculated as

tp ¼ Fp=r cos g ð33Þ

At post-ponding condition (t > tp), the infiltration rate is given by Eq. 28 aftercomputation of F(t) by Eq. 29. However, ponding does not occur initially at t=0, and hence,the revised equation to account for the time lag between the actual ponding time tp and theponding time ts at which ponding occurs, when the infiltration takes place at a potential ratefollowing the ponding from the start of the rainfall can be expressed as (Chen and Young2006)

Ke t � tp � ts�

cos g� ¼ FðtÞ � H þ hsð Þ$q

cos gln 1þ FðtÞ cos g

H þ hsð Þ$q �

ð34Þ

Note that with g=00, all the Eqs. 28–34 reduce to that derived by Chu (1978). Thecumulative infiltration depth and time to ponding on a sloping surface with slope angle gversus that expected on the horizontal plane under the same steady rainfall event can begiven as

FhpðgÞ ¼ Fhpð0Þ=cos2g ð35aÞ

tpðgÞ ¼ tpð0Þ=cos2g ð35bÞwhere Fhp = vertical infiltration depth when ponding occurs.

In this study, Chen and Young (2006) attempted to address a long term controvercialissue about the slope angle impact on the runoff generation process. On the effect of theslope angle on the runoff generation process, a group of researchers argue that runoffdecreases with the increase in slope angle due to a) thinning of soil crust or increased rillerosion and b) effect of negative slope on differential soil cracking. Another group ofresearchers attribute the negative slope angle effect for the decrease in depression surfacestorage and ponding depth and, based on this assumption, argue that there is an increase insurface runoff with the increase in slope angle. While a third group of researchers foundthat there is no significant effect of the slope angle on runoff. However, Chen and Young(2006) have numerically and theoretically shown that the infiltration increases with increasein the slope angle—“For the cases with ponded infiltration, the slope effect was generallynot significant for mild to moderate slope (g<100), but the slope effect becomes moreimportant for low intensity and short duration rainfall events, especially as it delayed thetime for ponding”. Further, the non-vertical rainfall which deflects in large angle to upslopecould also result in increased runoff which may be the possible explanation to theconflicting arguments by various researchers. The Green-Ampt model proposed by Chenand Young (2006) performed well as compared to the Richards equation which is supposed

3526 R.V. Kale, B. Sahoo

to be applicable for isotropic and anisotropic soils except for some small-scale topographicelements. However, for highly anisotropic soil, the lateral flux gradient may becomesignificant at large scale; hence, this model required to be used with caution.

Recently, most of the investigators opt for the Green-Ampt infiltration model in theoverland flow modeling study mainly due to its simplicity and robustness (e.g., Fiedlerand Ramirez 2000; Estéves et al. 2000; Castillo et al. 2003; Liu and Singh 2004). Inmodeling study dealing with overland flow, the main concern is for cumulativeinfiltration; hence, most of the overland flow simulation packages give priority to thisapproximate physically based infiltration model. The other important point which is ofinterest to the modeler is the simplicity of this model and the ease with which thenecessary input data can be obtained from soil physical properties. In this respect, the useof finite difference solution of the Richards equation is generally considered to be morerigorous for general purpose models of soil-water dynamics, due to the problem ofnumerical stability and un-assured guarantee of convergence by the iterative procedurewhich also results in excessive computer execution times. However, in such a situation,higher computational speed and greater model simplicity with less rigorousness can beachieved with the Green-Ampt model (Short et al. 1995). The accuracy of this model hasalready been demonstrated by comparing these results with the experimental data (Chuand Marino 2005; Hillel and Gardner 1970; Idike et al. 1980; Moore and Eigel 1981),solution of the Philips equation (Swartzendruber and Youngs 1974; Gill 1978; Qi 2008)and solution of the Richards equation (Freyberg et al. 1980; Moore 1981a, b; Ahuja 1983;Chu and Marino 2005; Qi 2008). Therefore, not surprisingly, the Green-Ampt equationhas been the choice model of infiltration estimation in many physically-based hydrologicmodels (Freyberg et al. 1980). Further, the United State Departments of Agriculture(USDA) and Agricultural Research Service (ARS) has done extensive work to developempirical relations for obtaining the Green-Ampt model parameters (e.g. Brakensiek andOnstad 1977; McCuen et al. 1981; Rawls and Brakensiek 1982; Springer and Cundy1987). Significantly, Rawls et al. (1983a) developed a procedure to determine the Green-Ampt infiltration parameters from data available from published soil survey andconstructed a chart for the determination of Green-Ampt parameters by knowing thesoil properties such as percentage of clay, percentage of sand, percentage of organicmatter, and bulk density. Ahuja et al. (1984, 1989) advocated a generalized Kozeny-Carman equation to determine the saturated hydraulic conductivity, Ksat as

Ksat ¼ Bfne ¼ 764:5f3:29e ð36Þwhere fe = effective porosity; and B and n are the constants varying with soil. Rawls et al.(1983a) has approximated fe as: fe=ft−θr , where ft is the total soil porosity and θr is theresidual soil moisture as a decimal or percentage. Using the same soil properties asRawls et al. (1983b), Alberts et al. (1989) arrived at an equation to estimate Green-Amptparameter as

ft ¼ 1� rb=2650ð Þ ð37Þwhere ρb = bulk density (kg/m3).

The residual soil moisture is calculated by

qr ¼ 0:000002þ 0:0001OM þ 0:00025Cl � CEC 0:45r

� rb ð38Þ

where OM = fraction of organic matter in soil; Cl = fraction of clay in soil and CECr =cation exchange capacity ratio.

Green-Ampt Infiltration Models for Varied Field Conditions 3527

The effective soil matrix potential is given by Ns ¼ hs fe � qið Þ, in which hs is related tothe soil properties as (Rawls and Brakensiek 1983)

hs ¼ eb ð39Þwhere the coefficient b is defined as

b ¼ 6:531� 7:33fe þ 15:8Cl2 þ 3:81f2e þ 3:40ClSa� 4:98Safe þ 16:1Sa2f2e

þ 16:0Cl2f2e � 14:0SaCl � 34:8Cl2fe � 8:0Sa2fe ð40Þ

where Sa = fraction of sand in the soil.Brakensiek et al. (1983) found that the predicted infiltration rate by using these estimates

amount to well within one standard deviation of the field measurements. Further, the waterretention characteristic of the soil is defined as the relationship between the soil water contentand the soil suction or matric potential. However, the soil water content and the soil matricpotential have a power function relationship which can be frequently presented by the wellknown relationship proposed by Brooks and Corey, Campbell and van Genuchten. Thedetails about these relationship and Green-Ampt parameters in tabulated form are available inthe well referred textbooks (e.g. Chow et al. 1988; Rawls et al. 1992; Williams et al. 1998). Astudy by Mohamoud (1991) advocating a two-stage method to determine the GA parametersunder different management practices of tillage, residue cover and crusting reveals that theraindrop impact can reduce the infiltration rate due to the formation of crust; however, themanagement practices which leaves behind near about 30% of residue cover protect the soilfrom the raindrop impact. Subsequently, the residue cover and crust have significant influenceon the infiltration process which was satisfactorily modeled by the GA infiltration model.

As highlighted in many study, the spatial variability of the soil properties has a significantimpact on infiltration process which results in scale problem when the point scale model isapplied on the macro-scale. Usually, there are three methodologies to comprise the currentinfiltration models for spatially heterogeneous soil (as reviewed by Hu et al. 2009): (1) Gridfining method (2) Parameter upscaling method and (3) Equation upscaling method. In gridfining method, the computational grid is required to be divided into finer grid to assure thehomogeneous assumption in final grid. However, this method is applicable for smallexperimental catchments due to limitations of data availability and computational capacity.In parameter upscaling method, the real heterogeneous soil is replaced by an equivalenthomogeneous soil with effective parameters estimated through upscaling methods such asmodel inversion, experimental relationship, analytical analysis and numerical experiments.This is widely used method because it maintains the original form of the model. In equationupscaling method, the spatial variability of the soil properties is described by probabilitydistribution function, and the point scale model is upscaled to macro scale using the stochastictheory. The resultant macro-scale model governs the ensemble infiltration processes on macroscale while retaining the physical interpretation of the point scale model and physical meaningof the parameters. The examples of this method are Monte Carlo simulation, analyticalapproximating method and statistical equation method. Recently, Hu et al. (2009) haveaccounted for heterogeneity of saturated hydraulic conductivity along the horizontal directionby adopting the probability distribution function, while layered soil profile have been used toaccount for heterogeneity of saturated hydraulic conductivity along the vertical direction. Thistask was accomplished by coupling the multilayered GA model proposed by Jia and Tamai(1997) with spatial averaging approach proposed by Chen et al. (1994a, b), the coupled modelwas termed as spatial averaging infiltration (SAI) model. The GA spatial averaging model by

3528 R.V. Kale, B. Sahoo

Chen et al. (1994a, b) is an example of equation upscaling method which uses a probabilitydistribution function. Hu et al. (2009) have argued that the negligence of the spatialheterogeneity in the horizontal direction results in the overestimation of the infiltration rate bythe GA model, whereas the negligence of spatial heterogeneity in vertical direction results inunderestimation of the infiltration rate. Thus, the application of the SAI model by these authorsin a catchment scale hydrological model has shown a very good agreement with the observeddata. However, the generalized use of this model requires the rigorous testing under differentfield conditions. Further, as stated earlier, to address the heterogeneity at the sub-basin scale,Craig et al. (2010) presented a direct method for upscaling the GA solution for laterallyheterogeneous soils based on the approximation of explicit GA formulation. For that purpose,they have introduced dimensionless hydraulic conductivity variable Ksat/r and dimensionlesstime X ¼ 1þ hs qs � qið Þ=rtð Þ½ ��1 as a mean of examining GA infiltration over a wide rangeof parameters space rather than the original four variables such as (Ksat, r, hsΔθ and t) tovisualize and analyze the problem for the entire range of plausible parameters. They found thatthe simulation results obtained by using the upscaled GA model were comparable with theMonte Carlo simulations of spatially random infiltration without run on with maximum errorof 3%. Subsequently, Liu et al. (2011) have utilized the proposed upscaled GA model by Craiget al. (2010) to investigate the feasibility and applicability of the GA model for the flow infinite soil bounded below by a shallow water table, frost table, or impermeable base. The studyby Craig et al. (2010) and Liu et al. (2011) emphasize the feasibility of GA model for regional-scale analysis. Therefore, the Green-Ampt model is capable of providing additional impetus forinclusion in many watershed models (Goldman 1989; Hu et al. 2009).

Furthermore, Wilcox et al. (1990) applied two models, namely the Soil ConservationService (SCS) curve number method and the Green-Ampt model to predict the runoff fromsix ungaged rural rangeland catchments located in Texas, Oklahoma, Arizona, Nebraskaand Idaho. The Green-Ampt model was parameterized based on the available soil andvegetation information and, exclusively, on individual catchment characteristics without themodel calibration. They concluded, based on the obtained results, that the physically basedGreen-Ampt model has the potential to be used as a management tool for predicting theland use impacts on runoff and infiltration. van Mullem (1991) has also found that theGreen-Ampt model is able to more accurately predict the measured runoff volumes andpeak discharges from storm with varying intensities than the SCS curve number methodwhile analyzing the data collected from 99 sites in Montana and Wyoming. Similarly,Madramootoo and Enright (1990) used the GA model to predict the surface runoff from asmall agricultural watershed in western Quebec, Canada. Recently, Turner (2006) verifiedvarious infiltration models by comparing the performance with the observed infiltrationcurve for Poplar Hill site on the Eastern Shore of Maryland and found that the Green-Amptand Philips equations are more versatile as compared to the other models, since they areapplicable both in the situations where rainfall rate is less than or greater than the actualinfiltration rate, and they do not rely on the earlier infiltration data obtained under differentfield conditions. The flexibility in consideration of variable rainfall intensities and soilprofile conditions make this model a handy tool in infiltration modeling.

7 Challenges in Using the Green-Ampt Infiltration Model

The application of the Green-Ampt infiltration equation in modeling infiltration duringsteady and unsteady rainfall events involves a number of restrictions on its use. Theassumption that the soil water movement is in the form of a wetting front and there is no

Green-Ampt Infiltration Models for Varied Field Conditions 3529

effect of diffusion is merely an approximation used for mathematical simplicity. In fact, thesoil profile is seldom homogeneous as assumed in most of the Green-Ampt infiltrationmodels. The soil water properties that greatly influence the infiltration rate includes thewater content, water retention characteristics, hydraulic conductivity, and hysteresis. Thevariation of antecedent soil moisture conditions within the soil profile is due to the soilphysical characteristic, such as, particle size, morphological and chemical properties, andbiological activities which makes the subsurface flow simulation process most complicated.The spatial distribution pattern of rainfall and topographical features, and soil types inelementary watershed are not identical during the various seasons of the year. Further, theinformation on ground surface water retention capacity and antecedent moisture capacityare seldom available. Despite of all these difficulties, with the advancement of technology,it is not difficult to accept this challenge of accurate estimation of infiltration. Further, withthe radar technology, it is possible to predict the areal extent as well as spatial distributionof rainfall (Sokol 2003). Various automatic rainguages with data loggers are helpful inaccurate measurement of rainfall intensities on monthly, daily, hourly basis or even at finertemporal scales from the remote area. Furthermore, the soil moisture measurements withvarious sensors are quite easier now-a-days. In such a scenario, the method described byKowalsky et al. (2005) may be helpful to capture the properties and system state ofheterogeneous soil. Further, the enhancement in efficiency of data storage devices makes itpossible to store a lot of information on rainfall as well as soil profile data. Therefore, useof physically based model for accounting infiltration process will be acceptable innumerous studies in watershed development and management.

8 Conclusions

The Green-Ampt infiltration model is a simplified physical model based on the Richardsequation which relates the rate of infiltration to measurable soil properties such as the porosity,hydraulic conductivity, and moisture content of a particular soil. This approach was developedfor three reasons: a) the solution of the Richards equation is difficult and not justified given thatthis equation is, at best, only a rough approximation of the actual field infiltration; b) asimplified solution still produces the exponentially decreasing relationship between infiltrationcapacity and cumulative infiltration; and c) the parameters of the methods can be related to soilproperties that can be measured in the laboratory, such as porosity and hydraulic conductivity.The main advantage of the Green-Ampt equation is that the analytical solution available for thecomputation of wetting front location and only two parameters required for characterizing thesoil properties, whereas the solution of the Richards equation requires more parameters. Thispaper deals with an extensive review of the Green-Ampt infiltration model for simulation of theinfiltration processes in the overland flow and surface irrigation modelling. The capabilities ofthe available models with their inabilities in handling various physical situations such as thelayered soil profile with varying initial moisture content and unsteady rainfall application arediscussed in detail. In this regard, it may be surmised that the Green-Ampt model is a widelytested infiltration model which is simple and self-sufficient to handle various field conditions.This model is sufficient to represent the soil spatial heterogeneity in a lumped manner ratherthan the use of more accurate Richards equation, which is prone to mismatching between thecomputational scales when it is coupled with the land-surface process components in overlandflow modelling studies, data intensive, and computationally expensive. Further it is morepractical to use the Green-Ampt model applicable for the multi-layered soil, when informationon the geological formation is available for the study area.

3530 R.V. Kale, B. Sahoo

Numerous modifications have been suggested in the literature in order to address thevariety of practical situations which falls outside the scope of the original GA model. Thesemodifications primarily focus on accounting of unsteady rainfall conditions, ponding ornon-ponding condition either constant or varying, non-uniform heterogeneous/multilayersoil profile, soil moisture redistribution, seal and crust formation impact, muddy water flow,sloping condition, rangelands, spatial soil heterogeneity in horizontal and vertical direction,two or three dimensional flow conditions, presence of regular cracks or holes, and presenceof shallow boundary conditions. However, during application of the GA model at regionalscales wherein the assumed idealized conditions seldom met, it is not clear that whatimplications these models have on the regional scale watershed modeling (e.g., Liu et al.2011). It could be expected that these model modifications would be plausible at theregional watershed modeling. Although, most of the modifications provide satisfactoryresults for the amended conditions and also in comparison with the dynamic Richardsequation, there is no study attempted to investigate the improvement achieved over theoriginal GA model. Further, the Green-Ampt model results are more likely to sensitivewhile using the saturated hydraulic conductivity, and its accurate accounting in the model issomewhat challenging task. The consideration of every phenomenon occurring at micro-scales does not necessarily result in improved prediction; rather it causes the burden ofincreased number of unknown parameters that yields into unwarranted risk of modeluncertainty. It seems that the multilayered model which accounts for the surface seal andcrust formation is appropriate in modeling the surface irrigation system, design of drip orsprinkler system or land slide estimation, and various applications having limited areal sizeand detailed geological information of the area under consideration. However, theapplication of the multi-layer GA model over a larger catchement area is doubtful and,thus need to be investigated in detail for an added improved prediction over the single layermodels. The study on the impacts of preferential flow using the GA model is another futureresearch area.

Acknowledgements The first author thanks Prof. M. Perumal, Department of Hydrology, Indian Institute ofTechnology Roorkee and Dr. V. C. Goyal, Scientist F and Head RCMU, National Institute of HydrologyRoorkee for encouraging working on this manuscript. Both the authors would like to thank the Editor-in-Chief Dr. G. Tsakiris and two anonymous reviewers of this article for many worthwhile and helpfulsuggestions which ultimately resulted in improvement of the previous contents of this manuscript.

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