J. Hydrol. Hydromech., 63, 2015, 2, 93–101
DOI: 10.1515/johh-2015-0012

Optimal parameters for the Green-Ampt infiltration model
under rainfall conditions
Li Chen1, 2*, Long Xiang1, Michael H. Young3, Jun Yin4, Zhongbo Yu1, 5, Martinus Th. van Genuchten6, 7
1State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China.
Division of Hydrologic Sciences, Desert Research Institute, 755 E. Flamingo Rd., Las Vegas, NV 89119, USA.
3
Bureau of Economic Geology, University of Texas at Austin, Austin, TX, USA.
4
Ministry of Forests, Land & Natural Resource Operation, Prince George, BC, Canada.
5
Department of Geoscience, University of Nevada, Las Vegas, NV, USA.
6
Department of Mechanical Engineering, Federal University of Rio de Janeiro, Brazil.
7
Department of Earth Sciences, Utrecht University, Utrecht, Netherlands.
*
Corresponding author. E-mail: li.chen@dri.edu
2

Abstract: The Green-Ampt (GA) model is widely used in hydrologic studies as a simple, physically-based method toestimate infiltration processes. The accuracy of the model for applications under rainfall conditions (as opposed toinitially ponded situations) has not been studied extensively. We compared calculated rainfall infiltration results forvarious soils obtained using existing GA parameterizations with those obtained by solving the Richards equation forvariably saturated flow. Results provided an overview of GA model performance evaluated by means of a root-meansquare-error-based objective function across a large region in GA parameter space as compared to the Richards equation,
which showed a need for seeking optimal GA parameters. Subsequent analysis enabled the identification of optimal GA
parameters that provided a close fit with the Richards equation. The optimal parameters were found to substantiallyoutperform the standard theoretical parameters, thus improving the utility and accuracy of the GA model for infiltration
simulations under rainfall conditions. A sensitivity analyses indicated that the optimal parameters may change for somerainfall scenarios, but are relatively stable for high-intensity rainfall events.
Keywords: Green-Ampt; Infiltration; Rainfall; Optimal parameters.
INTRODUCTION
The Green-Ampt (GA) infiltration model (Green and Ampt,
1911), and various modifications and extensions thereof, areused widely for analyzing relationships between rainfall, infiltration and runoff for a range of conditions and from different
theoretical and experimental perspectives (Ahuja and Ross,
1973; Barry et al., 2005; Basha, 2011; Bouwer, 1978; Chen andYoung, 2006; Gowdish and Muñoz-Carpena, 2009; Hilpert andGlantz, 2013; Hogarth et al., 2013; Mein and Larson, 1973;
Parlange et al., 1982; Voller, 2011; among many others). Because of its simplicity and physical basis, the model has been
implemented also in several codes addressing larger-scale hydrologic processes (e.g. Arnold et al., 2012; Flanagan and Nearing, 1995; O’Brien et al., 2009; Rossman, 2010).
As with most models, a critical issue in applying the GA parameterization is accurate estimation of the GA parameters. In
current practice, GA model parameters are determined using
both theoretical and empirical approaches (Neuman, 1976;
Rawls et al., 1983; Risse et al., 1994; Saxton and Rawls, 2006).
A number of studies have compared the accuracy of alternative
parameterizations against numerical solutions of the Richards(1931) equation for unsaturated or variably saturated flow,
mostly for ponded infiltration scenarios (e.g. Barry et al., 1993;
Hsu et al., 2002; Salvucci and Entekhabi, 1994). By comparison, the accuracy of the parameterizations has not been examined extensively for infiltration under rainfall conditions. Suchstudies are important since GA errors in estimating infiltration
can magnify considerably when runoff estimates are needed,since runoff in many situations is a small fraction of rainfall. As
compared to ponded infiltration, rainfall adds additional complexity to the accuracy of parameter estimation due to the transition from flux-controlled infiltration to profile-controlled

infiltration. The objective of this study was to evaluate the performance of the GA model in comparison with the Richards
equation, and to determine optimal GA parameters that would
improve the utility of the model for simulating infiltration underrainfall conditions. We further studied the sensitivity of the
optimal parameters to various rainfall intensities and durations.
THE GA INFILTRATION MODEL AND ITSPARAMETERS
Formulation of the GA model
The GA model assumes piston-type soil water movement
and a step-function for the water content in the soil profile involving two water contents, the initial water content θi and thesaturated water content θs near the soil surface as shown in Fig. 1.
Using Darcy’s law for these conditions, the infiltration rate iunder ponded conditions is given by (Green and Ampt, 1911):i=dI= Ks 1 + (θ s − θi ) S I 
dt

(1)

where t is time, I is the cumulative infiltration depth, Ks is thesaturated soil hydraulic conductivity, and S is the capillarypressure at the wetting front, sometimes referred to as the capillary drive (e.g. Morel-Seytoux and Khanji, 1974) or macroscopic capillary length (Philip, 1985). In some studies (Rawls et al.,1983) an effective hydraulic conductivity (Ke) is used rather
than Ks, to account for hysteresis effects. This difference ininterpretation will not affect the theoretical analysis in ourwork. Most studies directly use Ks in Eq. (6) for ponded infiltration studies (Mein and Larson, 1973; Morel-Seytoux andKhanji, 1974; Neuman, 1976). Our study also ignores the pond-93

Li Chen, Long Xiang, Michael H. Young, Jun Yin, Zhongbo Yu, Martinus Th. van Genuchten
θiθsSoil depth

where ψ is the capillary pressure under unsaturated conditions.
Eq. (6) assumes that the effect of the hydraulic conductivity Ki
at the initial capillary pressure head (ψi) or water content (θi) isvery small and can be ignored. For relatively wet initial conditions, Ks in Eq. (6) formally should be replaced by (Ks – Ki),
with the integration being carried out between ψi and 0 (e.g.
Smith et al., 2002).
Equation (6) requires integration of the hydraulic conductivity K as a function of the capillary pressure ψ. A number ofanalytical expressions for K(ψ) may be used for this purpose(e.g. Brooks and Corey, 1964; Kosugi, 1996; van Genuchten,1980). Because of their complexity, most of these functions
require numerical integration. An exception is the power law
function of Brook and Corey (1964), which has been used in a
number of GA applications, notably by Rawls and coworkers(Rawls et al., 1989, 1983). In our study we used the van
Genuchten (1980) function for K(ψ) and integrated Eq. (6)numerically. Numerical integration may be too cumbersome formany practical studies. A useful approximation of S based on
Eq. (6) for the K(ψ) function of van Genuchten (1980) is given
by Morel-Seytoux et al. (1996).

Water content

Wetting front

Fig. 1. Schematic soil water content profile in the GA model.

ing depth since its effect is usually very small. If large, theponding depth can be immediately added to the right-hand sideof Eq. (1).
An implicit function for I can be derived by integrating Eq.
(1) to obtain:I
K st = I − S (θ s − θi ) ln 1 +
 S (θ − θ ) 
s
i 


(2)

For infiltration under constant rainfall intensity, Eq. (2) canbe modified as (Chu, 1978; Mein and Larson, 1973):IK s t − t p − ts  = I − S (θ s − θi ) ln 1 +
 S (θ − θ ) 
s
i 


(

)

(3)

where tp is the time to ponding given by

tp =

Ip
p

=

(θ s − θi ) S t
p2 Ks − p(4)

in which p is the rainfall intensity, and Ip the infiltration depth attime tp. The variable ts in Eq. (3) is a virtual time defined by:

ts =

Ip
Ks

−

S (θ s − θi )
Ks



Ip
ln 1 +
 S (θ s − θi ) 

(5)This model was extended further by Chu (1978) to unsteady
rainfall events.
Model parameters and parameterization approaches
The GA model parameters include the saturated soil
hydraulic conductivity Ks, the capillary pressure at the wetting
front S, as well as the initial (θi) and saturated (θs) watercontents. Of these, Ks is a relatively standard hydraulicparameter that can be measured in the laboratory or field using avariety of approaches. For ponded infiltration, S is given by(Morel-Seytoux and Khanji, 1974; Neuman, 1976):

S=

94

1 0
K (ψ ) dψK s −∞(6)

S=

0.046m + 2.07 m 2 + 19.5m3(1 + 4.7 m + 16m 2 )α(7)where α and m = 1–1/n are parameters in the original van
Genuchten-Mualem formulation (van Genuchten, 1980).
The use of analytical functions for K(ψ) has the advantagethat many of the parameters in these functions are often immediately available, including through the use of pedotransfer
functions which estimate the parameters from soil texture andrelated data (Rawls et al., 1983; Schaap et al., 2001; Vereecken
et al., 2010). Alternatively, GA parameters could be estimatedusing field-based approaches to better account for site-specificconditions. Some studies aimed at deriving GA model parameters from field experiments, mainly focusing on the hydraulicconductivity. Single- and double-ring infiltrometer experiments(Nimmo et al., 2009; e.g. Stahr et al., 2004) and rainfall simulation (e.g. Risse et al., 1994) have been used for this purpose to
back-calculate Ks from continuous measurements of the infiltration depth. Unfortunately, S cannot be determined easily using
these approaches.
Most GA applications in the literature consider ponded infiltration, whereas many practical applications involve conditions
in which the soil surface is initially unsaturated. Mein and Larson (1973) modified the model to approximate rainfallinfiltration cases using the ponded GA model equation before
the onset of ponding. The appropriateness of parameters determined for ponded infiltration conditions may be reduced further
due to this approximation. In general, little attention has focusedon the performance of the GA model when theoretical parameters are employed. Only Mein and Larson (1973) conducted
limited tests to compare the theoretically parameterized GA
model to the Richards equation.
An issue with most or all existing approaches is that model
performance using estimated GA parameters is not quantified
by comparisons to actual infiltration data or numerical predictions, or evaluated relative to other combinations of parametervalues, especially under rainfall conditions. Therefore, infiltration simulation using the resulting parameters may not ensure
optimal results. Prior work seeking optimal GA infiltration
parameters is limited. To our knowledge, only Hsu et al. (2002)
addressed this issue through comparisons with infiltration rates
based on the Richards equation. However, as acknowledged inOptimal parameters for the Green-Ampt infiltration model under rainfall conditions

their paper, the approach was “less rigorous” in that GA parameters were determined by fitting to only three points rather thanthe entire infiltration curve. In addition, an artificial time shiftparameter was used by Hsu et al. (2002). According to Eqs (3–5), this parameter is not independent since it can be calculated
from other parameters.
The brief summary above of the GA model its parameters
shows a need to systematically investigate performance of themodel using existing parameterizations, as well a need for practical guidance in seeking optimal GA model parameters, especially for infiltration under rainfall conditions.

patterns, the infiltration curves obtained with the two parameterizations differed depending upon soil type.
For the sand soil, the GA model agreed closely with theRichards equation throughout the period of investigation. For
the sandy clay loam soil, the GA model over-predicted theinfiltration depth by up to 11% relative to the Richard equation.
For the clay loam soil, the theoretical GA model parameters
over-predicted the infiltration depth even more, this time by up
to 25%. These results suggest that a better GA parameterization
100000PERFORMANCE OF THE GA INFILTRATION MODELψ (mm)

1000

100

sand
sandy clay loam
clay loam

10

1

0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

100

500

θ
(a)
10000010000

ψ (mm)

1000

100

sand
sandy clay loam
clay loam

10

1

0.1
0.001

0.01

0.1

1

K (mm hr-1)

10(b)Fig. 2. van Genuchten (1980) soil hydraulic functions for the threetest soils: (a) Soil water retention curves; b) Unsaturated soil
hydraulic conductivity curves.
1

Normalized cumulative infiltration

In this study we combined the variables S and (θs–θi) in Eqs(1–5) into a single parameter, denoted as MS (i.e., MS = S(θs–θi)). This combination is preferred since the effect of the initialsoil moisture condition is then automatically taken into accountfor any θi value. Hereafter, we report on the two parameters Ks
and MS for the GA model.
We used the established approach, first suggested by Meinand Larson (1973), for comparing GA simulation results withnumerical solutions of the Richards equation to evaluate the use
and accuracy of the GA parameterization. The Richards equation was solved using the van Genuchten (1980) soil hydraulicproperty relationships for θ(ψ) and K(ψ). For the evaluation wenumerically integrated Eq. (6) very carefully using Romberg
integration, and multiplied the resulting S values with (θs–θi) to
obtain MS.
Table 1 provides van Genuchten soil hydraulic parameters
for the sand, sandy clay loam, and clay loam soils (Carsel andParrish, 1988) used in the simulations. These soils were chosen
to generate simulated infiltration data across a two-order-ofmagnitude range in Ks. Fig. 2 shows the van Genuchten (1980)
θ(ψ) and K(ψ) relationships for these soils. Table 1 also shows
the rainfall rates used in the calculations. We assumed constantrainfall intensities p, each lasting 10 hrs. To ensure that both
infiltration and runoff covered a substantial part of total rainfall,
rainfall intensities were set at approximately twice Ks for all
soils. The soil and rainfall data were used to compare GA infiltration results obtained by numerically solving Eqs (3–5) with
Hydrus-1D (Šimůnek et al., 2008) results based on the Richards
equation. Only relatively simple (one-dimensional, uniform)
soil profiles were considered, while ignoring such complexities
as soil layering, hysteresis, hydrophobicity, preferential flow
paths, and sloping soil surfaces. Hydrus-1D simulations werecarried out using a relatively deep soil profile with a free drainage lower boundary condition to avoid the impact of the lower
boundary, and a very fine grid size (1 mm) with small time
steps (adaptive with the smallest value being 10–5 hr) to ensurehigh numerical accuracy. Soil profile thickness and a lower
boundary do not need to be specified for the GA model.
Cumulative infiltration curves (normalized by total rainfall)obtained with both the GA model and the Richards equation for
the three soils are presented in Fig. 3. Before ponding commenced, the curves overlapped along a straight line representing
the cumulative rainfall rate. Ponding times differed for each soil
depending upon soil texture and rainfall intensity. After ponding
commenced, as a result of the reduced infiltration rate, the cumulative infiltration curves were situated below the straight line
representing cumulative rainfall. Although cumulative infiltration continued to increase over time, the rate of increase (i.e.,
the infiltration rate) gradually declined and ultimately approached a constant value equal to Ks in each case. While theRichards equation and GA model showed the same general

10000

The Richards Eq. vG Parameter Sand
GA model theoretical parameter Sand
The Richards Eq. vG Parameter Sandy Clay LoamGA model theoretical parameter Sandy Clay Loam
The Richards Eq. vG Parameter Clay LoamGA model theoretical parameter Clay Loamrainfall

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized time t*

Fig. 3. Comparison of infiltration curves (normalized by total
rainfall) obtained with the Richards equation and the GA model.

95

Li Chen, Long Xiang, Michael H. Young, Jun Yin, Zhongbo Yu, Martinus Th. van Genuchten
Table 1. Soil and rainfall parameters used for the numerical comparison.
vG parameters for the Richards equationαTheoretical GA parameters
MS(mm)

Rainfall intensityp(mm hr–1)

Sand

297

(mm–1)
1.45×10–2

2.68

Ko(mm hr–1)
297

14.8

500

Sandy clay loam

13.1

5.90×10–3

1.48

13.1

10.4

25

Clay loam

2.6

1.90×10–3

1.31

2.6

17.3

5

Ks
(mm hr–1)

is needed, or that parameters in the GA model should be adjusted (optimized), to provide improved infiltration predictions. The
latter option is addressed next.
OPTIMAL GA PARAMETERS
We next seek to determine optimal GA parameters through aleast square analysis. A straightforward approach to optimize
the GA parameters is to calculate the overall error between the
GA infiltration curve and the Richards equation for all possible
Ks and MS parameter combinations, and then to identify thosecombinations corresponding to a minimum root mean squared
error (RMSE). A similar approach was followed by Yin (2008).
The RMSE was used as the objective function (F) to be minimized:

F=

2
1 k
 yi − f ( ti , K s , MS ) 
k i =1(8)

where yi represents cumulative infiltration depths corresponding
to times ti obtained with the Richards equation, f(ti, Ks, MS) are
GA cumulative infiltration depths (i.e., I in Eqs (1) through (3))
for parameters Ks and MS, and k is the number of samples foreach infiltration curve. The objective function thus serves as anindicator of the overall accuracy of the GA model compared toinfiltration behavior identified using Hydrus-1D. The minimum
RMSE of F corresponds to optimal Ks and MS values. Ten-hour
rainfall durations were used in all simulations, considering thatrainfall duration is independent of soil type. This approachprovided performance of the infiltration parameters across awide range of GA parameter values.
To show results of different conditions on the same scale, weused normalized dimensionless variables. As dimensionless
objective function we used F* = F/IF, where IF is the final cumulative infiltration rate. We further used dimensionless (scaled)values K* = Ko/Ks and MS*=MSo/MS (=So/S), where Ko and MSo
are the adjusted (or scaled) values of the original (theoretical)
GA parameters Ks and MS.
Plots of the dimensionless objective function F* for the sand,sandy clay loam, and clay loam soils are shown in Figs 4–6,
respectively. F* represents the average relative error between
the GA model and the actual infiltration curve obtained withsolutions of the Richards equation. Note that errors in the final
infiltration amount may be significantly higher than the average
error given by F*. Substantial computational effort was required
to obtain each of these maps since hundreds of thousands of GA
model solutions were generated for each case. To show a wide
range of F* values on the plots in Figs 4–6, the upper bounds on
F* were set to 0.1 (10%), which is higher than acceptable errors
in most applications. Also shown in the figures are comparisonsof infiltration curves obtained with solutions of the Richards
equation and the GA model using the optimal parameters.

96n

The results in Figs 4–6 reveal several interesting features ofthe objective function. Most importantly, F* for all soils
converged to single minima that can be located on the plots.
This indicates that optimal GA parameters exist and can beobtained using the optimization analysis. The GA simulation
results with optimal parameters were notably better than thosegenerated using the theoretical parameters for especially themedium-and fine-textured soils (Fig. 3). Therefore, for practicalapplications, the standard (theoretical) parameterization can be
replaced with an optimization approach. In addition, the resultsin Figs 4–6 show that F* is more sensitive to K* than to MS*,
thus indicating that Ks is a more important parameter than S in
GA model applications (especially for coarser soils).
Searching for optimal parameters may be straightforward forcoarse-textured soils but difficult for fine-textured soils. For the
sandy soil, the elliptical shape of F* contoured a large area of
K*-MS* space, thus ensuring that the optimization algorithm
could reach the minimum. As soil texture becomes finer,
however, the difficulty of locating optimal parameters increased
since the shape of the F* contours tended to become morestretched so that the direction of local gradients started todeviate from the minimum. For such cases one may obtain
multiple acceptable solutions with the optimization although theobjective function still has only one global minimum.
Another significant aspect of our results relates to thedifference between the theoretical and optimal values of Ks and
MS. The dimensionless optimal parameters for the three test
cases are included in Table 2. The data indicate that for thecoarse- and medium-textured soils (the sand and sandy clayloam) in our study, the optimal hydraulic conductivity (Ko) was
relatively close to Ks, such that K* approached unity, while theoptimal MSo was substantially smaller than the theoretical MS
value (i.e., MS* <<1). For the fine-textured (clay loam) soil, theoptimal Ko was about 40% of Ks, while MSo was larger than thetheoretical value of MS.
From the F* plots in Figs 4–6 one can deduce that the overall
infiltration errors (i.e., F* values) using theoretical GA
parameters were several orders of magnitude higher than thosefor the optimal parameters. These results can be explained byconsidering the different infiltration mechanisms for the threesoils. The infiltration rate for the coarse- and medium-textured
soils approximated Ks most of the time since gravitational
forces for these soils should dominate the capillary pressure
gradients. The optimal Ko for those soils is hence expected
converge quickly to Ks. By comparison, the effects of capillarity
on infiltration remain important for a much longer period oftime. As such, the oversimplified moisture profile (Fig. 1) in the
GA model inaccurately reflects the effects of capillary forces on
infiltration of especially fine-textured soils. Still, the use of
optimal parameters (Ko and MSo )greatly improved the GA
predictions, with especially Ko becoming differently from Ks
when fine-textured soils are considered, as shown by the entriesof K* in Table 2.

Optimal parameters for the Green-Ampt infiltration model under rainfall conditions
Table 2. Sensitivity of optimal parameters Ko and MSo to rainfall intensity p and duration tr.

Case

Soil
textureRainfallintensityp(mm hr–1)

1
300

3
Sand

5

MS*

60.203

0.997

0.136

100.338

0.998

0.094

10

200.676

0.999

0.066

3

60.203

0.985

0.536

500

100.338

0.991

0.473

1.684

5
10
3

15

1.010

5

7

†

5

1.145

200.676

0.995

0.399

3.779

0.938

0.286

6.298

0.992

0.159

9

10

12.596

1.008

0.117

10

3

3.779

0.855

0.881

6.298

0.908

0.714

11
12

Sandy
clay loam

25

1.908

5
10

13

†

3

14

50

3.817

5

12.596

0.954

0.556

3.779

0.741

1.522

6.298

0.840

1.122

15

10

12.596

0.924

0.798

16

3

0.451

0.296

3.524

0.751

0.385

2.546

1.503

0.419

2.254

0.451

0.481

1.876

0.751

0.439

2.112

1.503

0.439

2.105

17
18
19

5

21

5
10

Clay
loam

20

‡

tr*
K*

6

8

†

p*

1.923
†

3
10

5

Theoreticalparameters

Optimal parameters

‡

3

2

4

Duration
tr
(hr)

3.846

10

K*

MS*

1

1

1

1

1

1

Cases corresponding to Figs 4, 5, and 6.
Subscript * denotes non-dimensional parameters, p* = p/Ks, tr* = tr/tc, K* = Ko/Ks and MS*=MSo/MS.

In view of the above results, we also studied the time scalethat characterizes the driving forces in the system to better
explain their impact on the objective function and the optimized
values of Ks and MS. Following Smith (2002), a characteristic
time scale can be defined as:tc =(θ s − θ r ) S
Ks(9)where S is the capillary pressure of the wetting front as usedpreviously. The time scale characterizes the dominant force
driving infiltration. Smith (2002) showed that for ponded cases,
gravitational forces dominate infiltration and that the infiltration
rate approaches Ks for t values larger than about 10tc, whilegravitational forces are negligible when t is much smaller thantc. In the current study (Table 1), values of tc can be obtained
easily for the three soils: tc = 0.05, 0.79, and 6.6 hr for the sand,sandy clay loam, and clay loam soils, respectively. The twoorders of magnitude difference in tc reflects the wide range insoil hydraulic conditions for the three soils.
If we use tr for rainfall duration, then the ratios of tr/tc (i.e.,the dimensionless rainfall duration, denoted further as t*) for aten-hour event are 200, 12.7, and 1.5 for the three soils, respectively. This explains why the optimal value of Ks is close to thetheoretical Ks value for the two coarser-textured soils when therainfall event extends to 10 hrs. These results also indicate thatfor short-duration events (i.e., events with a t* << 10), the GA

parameters that best describe the infiltration depth are considerably different from the theoretical values.
Figure 6a also suggests that small t* values can result in anarrow valley in parameter space where the GA parameters
produce very similar values of the objective function, with allparameters being far superior than the theoretical values. This
result has practical meaning, especially for arid environments
where runoff generating rainfall events are often very intense
and of short duration, thus yielding small t* values. For such
short-duration cases, optimal parameter sets should perform far
better than the GA theoretical parameters, thus providing moreaccurate predictions of runoff.
Sensitivity of optimal parameters to rainfall intensity andduration
We next conducted a sensitivity test to evaluate changes in
the optimal parameters as a function of rainfall intensity andduration. Two or three rainfall intensities and three durations
were tested for each soil type (Table 2). Optimal parameters foreach case were identified directly from plots of the objectivefunction (F*). Our objective of these tests was to found out if
optimal parameters applicable for the various scenarios exist
and could be identified.
Results of the sensitivity analysis are summarized in Table 2
for a total of 21 cases we examined. The results and input
conditions are provided in dimensionless form. They show that97

Li Chen, Long Xiang, Michael H. Young, Jun Yin, Zhongbo Yu, Martinus Th. van Genuchten
8

5

MS*

4

3

2

1

0.1
0.096
0.092
0.088
0.084
0.08
0.076
0.072
0.068
0.064
0.06
0.056
0.052
0.048
0.044
0.04
0.036
0.032
0.028
0.024
0.02
0.016
0.012
0.008
0.004
0.001

7

6

5

MS*

0.1
0.096
0.092
0.088
0.084
0.08
0.076
0.072
0.068
0.064
0.06
0.056
0.052
0.048
0.044
0.04
0.036
0.032
0.028
0.024
0.02
0.016
0.012
0.008
0.004
0.001

6

4

3

2

1

0
0.6

0.7

0.8

0.9

1

1.1

1.2

0.2

0.4

0.6

0.8

Ke*

(a)

1.2

1.4

(a)

3000

160

cumulative infiltration depth (mm)cumulat ive in filtratio n depth (mm)1

Ke*

The Richards Equation van Genuc hten Parameters
G-A with optimized parameters

2500

2000

1500

1000

500

0
0

1

2

3

4

5

6

7

8

9

10

time (hr)The Richards Equation van Genuchten Parameters
G-A with optimized parameters

140
120
100
80
60
40
20
0
0

1

2

3

4

5

6

7

8

9

10

time (hr)

(b)

(b)

Fig. 4. (a) Distribution of the objective function F* for the sandy
soil; (b) Comparison of the optimal GA infiltration curve andsolution of the Richards equation for the sandy soil.

Fig. 5. (a) Distribution of the objective function F* for the sandy
clay loam soil; (b) Comparison of the optimal GA infiltration curveand solution of the Richards equation for the sandy clay loam soil.

rainfall intensity and duration both affected the values of the
optimal parameters. Optimal K* values were generally close to 1
(i.e., Ko ≈ Ks) for long-duration events (large t*), while optimal
Ko values were less than Ks, sometimes by up to 70%, for shortduration events (small t*). Optimal MS values were less than 1.0
for low rainfall intensities or longer durations, but larger than
1.0 for short duration events (t* < ~10). This again indicates thattheoretical GA parameters are generally not the best choice for
infiltration simulations, and hence that optimal parameters
should be used. When Ko was set equal to Ks as a constraint in
the optimization, acceptable MS* values were generally lessthan 1.
Optimal parameters as a function of rainfall duration and intensity showed some general trends. For example, K* generally
increased with rainfall duration (tr*) and decreased with rainfall
intensity (p*), except for small tr* values when K* increased with
p*. Conversely, MS* generally decreased with tr* and increased
with p*, except for small tr* values when MS* decreased with p*.
In addition, for long-duration rainfall events (tr* > ~10), K*
approached 1 and MS* < 1, while for short-duration events (tr* <~1.5), K* < 0.5 and MS >>1. These results show that the GAmodel using theoretical parameters overestimated gravitational

effects for small tr*, but underestimated these effects as tr* increased. The optimal GA parameters compensated for such
effects and could improve the simulations. In general, relatively
large tr* values affected the capillary component most, in whichcase that the optimal MS* changed more significantly as a function of rainfall intensity.
Figure 7 shows examples of F* distributions for different
rainfall intensities and durations (Table 2). Cases for large tr*values are not included since the optimal parameters did notchange significantly for different rainfall conditions. Maps for
medium and small tr* values show F* values below 0.025 (i.e.,
overall errors were less than 2.5% of cumulative infiltration),
which we consider acceptable for these type of simulations.
Results indicate that overall performance of the GA model wasaffected substantially by rainfall conditions at medium tr* values(Fig. 7a) where the F* distribution and optimal parameterschanged significantly with rainfall conditions. For practicalapplications, optimal parameters for high rainfall intensities
resulted in generally smaller errors for other rainfall conditions.
This implies that optimal parameters are still applicable for thevarious conditions. For small tr*, the performance of the GAmodel was not significantly affected by rainfall conditions (Fig. 7b),

98

Optimal parameters for the Green-Ampt infiltration model under rainfall conditions
10

5.5

4

Ms*

3

6

Ms*

2

4

1
0.3

0.4

0.5

Ke*

2

0.025
0.024
0.023
0.022
0.021
0.02
0.019
0.018
0.017
0.016
0.015
0.014
0.013
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0

5
4.5
4
3.5

MS*

0.05
0.044
0.038
0.032
0.026
0.02
0.014
0.008
0.004
0.001

8

0.1
0.096
0.092
0.088
0.084
0.08
0.076
0.072
0.068
0.064
0.06
0.056
0.052
0.048
0.044
0.04
0.036
0.032
0.028
0.024
0.02
0.016
0.012
0.008
0.004
0.001

3
2.5
2
1.5
1
0.5
0
0.2

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

1.2

K*

1.6

Ke*

(a)

(a)
cumulative infiltration depth (mm)35

The Richards Equation van Genuchten Parameters
G-A with optimized parameters

30

25

20

15

10

5

0
0

1

2

3

4

5

time (hr)

6

7

8

9

10

(b)
Fig. 6. (a) Distribution of the objective function F* for the clay
loam soil (including a close-up around the minimum; (b)Comparison of the optimal GA infiltration curve and solution of the
Richards equation for the clay loam soil.

with optimal parameters for one case being immediately applicable to other cases. Table 3 shows the performance of the
optimal parameters for clay loam soil cases included in Table 2.
Results indicate that, in general, optimal parameters for highintensity rainfall events are more applicable to various conditions. This finding provides useful guidance for practical applications in terms of determining the most suitable GA parameters.
In real applications with complex rainfall conditions, the optimal parameters may be determined using as the objectivefunction the average F* for all significant rainfall intensities,
weighted by rainfall depth. In our study, however, such an approach was not followed since the weighing factors for various
rainfall conditions cannot not be determined without knowing
specific rainfall characteristics. For this reason we provided
only general recommendations.
CONCLUDING REMARKSThis study examined the general performance of the GAmodel for simulating infiltration under rainfall conditions by comparing GA calculations with numerical solutions of the Richards(b)Fig. 7. Comparison of F* distributions for different rainfall
intensities and durations: (a) from bottom to top: cases 7, 11, and
15 in Table 2; (b) from bottom to top, cases 17, 19, and 21 in Table 2.

equation. Our results demonstrate that standard (theoretical)
values of the two GA parameters (i.e., Ks and MS) will introduce significant errors in rainfall-infiltration modeling for fineto medium-textured soils assuming rainfall events with realistic
durations. We showed that optimal parameters for the GA model do exist and can be used to greatly improve the accuracy of
GA simulations.
Our results further show that optimal parameters differ fromtheoretical values and may be affected by the rainfall characteristics. In general, both rainfall intensity and duration changed
the values of the optimal parameters, with duration having amore significant impact. For long-duration (large tr*) rainfall
events, the optimal saturated conductivity (Ko) was close to Ks(the theoretical input value), while the optimal MSo value wasmuch less than the theoretical MS. For short-duration (small tr*)rainfall events, optimal Ko values may become significantly
smaller than Ks, whereas optimal MSo will be much greater than
the theoretical MS. Whereas optimal parameters for one rainfall

99

Li Chen, Long Xiang, Michael H. Young, Jun Yin, Zhongbo Yu, Martinus Th. van Genuchten
Table 3. Normalized objective function F* for the clay loam soil assuming different parameters.
0.29620.3846

0.41920.48080.43850.4385MS*

3.52372.54572.25381.87572.11162.1052

Rainfall intensityp*Rainfall duration
tr*1.92320.451

1.9232

†

K*

0.751

F* (%)
0.04†

0.04

0.07

0.07

0.1

0.12

0.2

†

0.02

0.07

0.13

0.11

0.15

†

1.9232

1.503

0.98

0.18

0.07

0.62

0.11

0.09

3.8462

0.451

0.94

0.6

0.24

0.04†

0.1

0.13
†

3.8462

0.751

0.56

0.44

0.16

0.25

0.07

0.1

3.8462

1.503

1.04

0.29

0.14

0.67

0.12

0.09†

F* of the optimal parameter set for the corresponding rainfall condition.

condition may not perform satisfactorily for all other conditions,especially for medium tr* values, parameters feasible for various
rainfall conditions may still be identified by comparing errors
for the different cases. A weighted average of errors can be used
for complex rainfall conditions if rainfall characteristics andtheir frequencies are known. Optimal parameters for high intensity rainfalls were found to be generally more applicable todifferent rainfall conditions.
In practical applications, parameter values other than the optimal values still may be acceptable. For short-duration rainfall
events (with likely more effects when soils are fine-textured),
acceptable parameters were located in a narrow valley in Ks –MS parameter space, thus implying that these two parameters
should be determined concurrently to ensure satisfactory predictions of infiltration. Applying the GA model actually implies
using a GA soil to approximate the real soil. A common goal of
infiltration modeling is to accurately predict cumulative infiltration. If this is the sole criterion for modeling performance, a GA
soil may not necessarily bear the same physical properties as thereal soil (e.g., Ko ≠ Ks). For a specific application, different GA
parameters may be acceptable to approximate the behavior of
the real soil, although optimal parameters would still perform
slightly better than other sets. This equifinality of model parameters has been studied by Beven and Binley (1992) and Beven
and Freer (2001), among others.
Practitioners may need to confront two situations when parameterizing the GA model. When a set of soil hydraulic parameters (such as the van Genuchten parameters or of otherrepresentations) are known, optimal parameters can be obtained
simultaneously by fitting the GA model to the Richards equation (using the known soil hydraulic parameters). As a first
approximation, pedotransfer functions could be used for this.
However, if the soil hydraulic parameters are not known, andpedotransfer functions are judged to be too imprecise, infiltration experiments (preferably rainfall experiments) could be
conducted to produce infiltration curves which then would
replace the Richards equation in the optimization approach.
Finally, we note that the present analysis was carried out for
ideal conditions involving a homogeneous soil profile. It may
be interesting to conduct similar analyses for more realistic field
conditions such as for layered soils, soils with sloping surfaces,or for macroporous (structured) soils exhibiting preferential
flow paths. The approach followed in this paper is very muchapplicable also to those conditions. Our ultimate objective in all
this was to obtain better estimates of infiltration for larger-scalehydrologic applications, including runoff estimation, for which

100

numerical solutions of the Richards equation are still largely
impractical.

Acknowledgements. This study was supported in part by the
Grant of Natural Sciences Foundation of China (No. 51349015),
the U.S. Army Corps of Engineers through the Urban Flood
Demonstration Program under contract W912HZ-08-2-0021,
the Flood Control District of Maricopa County under contract
IGA FCD 2008A014, the Chinese Fundamental Research Funds
for Central Universities (No. B13020174), the Thousand Talents Program of China, and CAPES-Brazil.
REFERENCES
Ahuja, L.R., Ross, J.D., 1973. A new Green-Ampt type model
for infiltration through a surface seal permitting transient parameters below the seal. In: Proc. Nat. Conf. Advances in Infiltration, December 12–13, 1983, Hyatt Regency Illinois
Center, Chicago, Illinois, Am. Soc. Agric. Eng, St. Joseph,MI.
Arnold, J.G., Moriasi, D.N., Gassman, P.W., Abbaspour, K.C.,
White, M.J., Srinivasan, R., Santhi, C., Harmel, R.D., Van
Griensven, A., Van Liew, M.W., 2012. SWAT: Model use,
calibration, and validation. Trans. ASABE, 55, 1491–1508.
Barry, D.A., Parlange, J.-Y., Sander, G.C., Sivaplan, M., 1993.
A class of exact solutions for Richards’ equation. J. Hydrol.,142, 29–46. doi:10.1016/0022-1694(93)90003-R
Barry, D.A., Parlange, J.-Y., Li, L., Jeng, D.-S., Crapper, M.,
2005. Green–Ampt approximations. Adv. Water Resour., 28,
1003–1009. doi:10.1016/j.advwatres.2005.03.010
Basha, H.A., 2011. Infiltration models for semi-infinite soil
profiles.
Water
Resour.
Res.,
47,
doi:10.1029/2010WR010253
Beven, K., Binley, A., 1992. The future of distributed models:Model calibration and uncertainty prediction. Hydrol. Process., 6, 279–298. doi:10.1002/hyp.3360060305
Beven, K., Freer, J., 2001. Equifinality, data assimilation, anduncertainty estimation in mechanistic modelling of complex
environmental systems using the GLUE methodology. J.
Hydrol., 249, 11–29. doi:10.1016/S0022-1694(01)00421-8
Bouwer, H., 1978. Groundwater Hydrology. McGraw-Hill,
New York.
Brooks, R., Corey, A., 1964. Hydraulic Properties of PorousMedia. Hydrology Papers, Colorado State University, Ft.
Collins, CO.

Optimal parameters for the Green-Ampt infiltration model under rainfall conditions

Carsel, R.F., Parrish, R.S., 1988. Developing joint probability
distributions of soil water retention characteristics. Water
Resour. Res., 24, 755–769. doi:10.1029/WR024i005p00755
Chen, L., Young, M.H., 2006. Green-Ampt infiltration model
for sloping surfaces. Water Resour. Res., 42, W07420.
doi:10.1029/2005WR004468
Chu, S.T., 1978. Infiltration during an unsteady rain. Water
Resour. Res., 14, 461–466. doi:10.1029/WR014i003p00461
Flanagan, D.C., Nearing, M.A., 1995. USDA-Water Erosion
Prediction Project: Hillslope profile and watershed model
documentation. NSERL Rep 10.
Gowdish, L., Muñoz-Carpena, R., 2009. An improved Green–
Ampt infiltration and redistribution method for uneven multistorm series. Vadose Zone J., 8, 470. doi:
10.2136/vzj2008.0049
Green, W.H., Ampt, G.A., 1911. Studies on soil physics, 1. The
flow of air and water through soils. J. Agric. Sci., 4, 1–24.
Hilpert, M., Glantz, R., 2013. Exploring the parameter space of
the Green–Ampt model. Adv. Water Resour., 53, 225–230.
doi:10.1016/j.advwatres.2012.12.001
Hogarth, W.L., Lockington, D.A., Barry, D.A., Parlange, M.B.,
Haverkamp, R., Parlange, J.-Y., 2013. Infiltration in soils
with a saturated surface: Infiltration and water profiles. Water Resour. Res., 49, 2683–2688. doi:10.1002/wrcr.20227
Hsu, S., Ni, C., Hung, P., 2002. Assessment of three infiltration
formulas based on model fitting on Richards equation. J.
Hydrol. Eng., 7, 373–379. doi:10.1061/(ASCE)10840699(2002)7:5(373)
Kosugi, K., 1996. Lognormal distribution model for unsaturated
soil hydraulic properties. Water Resour. Res., 32, 2697–2703.
doi:10.1029/96WR01776
Mein, R.G., Larson, C.L., 1973. Modeling infiltration during a
steady rain. Water Resour. Res., 9, 384–394.
doi:10.1029/WR009i002p00384
Morel-Seytoux, H.J., Khanji, J., 1974. Derivation of an equationof infiltration. Water Resour. Res., 10, 795–800.
doi:10.1029/WR010i004p00795
Morel-Seytoux, H.J., Meyer, P.D., Nachabe, M., Tourna, J., vanGenuchten, M.T., Lenhard, R.J., 1996. Parameter equivalence for the Brooks-Corey and van Genuchten soil characteristics: Preserving the effective capillary drive. Water Resour. Res., 32, 1251–1258. doi:10.1029/96WR00069
Neuman, S.P., 1976. Wetting front pressure head in the infiltration model of Green and Ampt. Water Resour. Res., 12, 564–
566. doi:10.1029/WR012i003p00564
Nimmo, J.R., Schmidt, K.M., Perkins, K.S., Stock, J.D., 2009.
Rapid measurement of field-saturated hydraulic conductivity
for areal characterization. Vadose Zone J., 8, 142.
doi:10.2136/vzj2007.0159
O’Brien, J.S., Jorgensen, C., Garcia, R., 2009. FLO-2D Software Version 2009. FLO-2D Software, Inc. Nutrioso, AZ,
Nutrioso, AZ.
Parlange, J.Y., Lisle, I., Braddock, R.D., Smith, R.E., 1982. The
three-parameter infiltration equation. Soil Sci., 133, 337–341.
Philip, J.R., 1985. Reply to comments on "Steady infiltrationfrom spherical cavities”. Soil Sci. Soc. Am. J., 49, 788.
doi:10.2136/sssaj1985.03615995004900030055x

Rawls, W.J., Brakensiek, D.L., Miller, N., 1983. Green-Ampt
infiltration parameters from soils data. J. Hydraul. Eng., 109,62. doi:10.1061/(ASCE)0733-9429(1983)109:1(62)Rawls, W.J., Brakensiek, D.L., Savabi, M.R., 1989. Infiltration
parameters for rangeland soils. J. Range Manage., 42, 139.
doi:10.2307/3899311
Richards, L.A., 1931. Capillary conduction of liquids throughporous mediums. Physics, 1, 318. doi:10.1063/1.1745010
Risse, L.M., Nearing, M.A., Savabi, M.R., 1994. Determining
the Green-Ampt effective hydraulic conductivity from rainfall-runoff data for the WEPP model. Trans. ASAE, 37, 411–
418.
Rossman, L.A., 2010. Storm water management model user’s
manual, version 5.0. National Risk Management ResearchLaboratory, Office of Research and Development, US Environmental Protection Agency.
Salvucci, G.D., Entekhabi, D., 1994. Explicit expressions forGreen-Ampt (delta function diffusivity) infiltration rate andcumulative storage. Water Resour. Res., 30, 2661–2663.
doi:10.1029/94WR01494
Saxton, K.E., Rawls, W.J., 2006. Soil water characteristic estimates by texture and organic matter for hydrologic Solutions.
Soil Sci. Soc. Am. J., 70, 1569. doi:10.2136/sssaj2005.0117
Schaap, M.G., Leij, F.J., van Genuchten, M.T., 2001. Rosetta: acomputer program for estimating soil hydraulic parameterswith hierarchical pedotransfer functions. J. Hydrol., 251,
163–176. doi:10.1016/S0022-1694(01)00466-8
Šimůnek, J., van Genuchten, M.T., Šejna, M., 2008. Development and applications of the HYDRUS and STANMOD
software packages and related codes. Vadose Zone J., 7, 587.
doi:10.2136/vzj2007.0077
Smith, E., Smettem, R.J., Broadbridge, P., Woolhiser, D.A.,
2002. Infiltration Theory for Hydrologic Applications. Water
Resour. Monograph 15, Am. Geophys. Union, Washington,
DC.
Stahr, L.M., Eisenhauer, D.E., Helmers, M.J., Dosskey, M.,
Franti, T.G., 2004. Green-Ampt infiltration parameters in riparian buffers. USDA Forest Service/UNL Faculty Publication 30.
van Genuchten, M.Th., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil
Sci.
Soc.
Am.
J.,
44,
892–898.
doi:10.2136/sssaj1980.03615995004400050002x
Vereecken, H., Weynants, M., Javaux, M., Pachepsky, Y.,
Schaap, M.G., van Genuchten, M.Th., 2010. Using pedotransfer functions to estimate the van Genuchten–Mualem
soil hydraulic properties: A Review. Vadose Zone J., 9, 795.
doi:10.2136/vzj2010.0045
Voller, V.R., 2011. On a fractional derivative form of theGreen–Ampt infiltration model. Adv. Water Resour., 34,
257–262. doi:10.1016/j.advwatres.2010.11.012
Received 29 May 2014
Accepted 4 December 2014101