Estimation of Infiltration

Rate in the Vadose Zone:

Application of Selected

Mathematical Models

Volume II

United States

Environmental Protection

Agency

National Risk Management

Research Laboratory

Ada, OK 74820

EPA/600/R-97/128b

February 1998
EPA/600/R-97/128b

February 1998

ESTIMATION OF INFILTRATION RATE IN VADOSE ZONE:

APPLICATION OF SELECTED MATHEMATICAL MODELS

Volume II

by

Joseph R. Williams

U.S. EPA

and

Ying Ouyang

ManTech Environmental Technologies, Inc. (METI)

and

Jin-Song Chen

Varadhan Ravi

Dynamac Corporation

ALM (METI) Contract No. 68-W5-0011 DO 022

Dynamac Corp. 68-C4-0031

Project Officer Delivery Order Project Officer

David S. Burden David G. Jewett

Subsurface Protection and Remediation Division

National Risk Management Research Laboratory

Ada, Oklahoma 74820

In cooperation with:

CENTER FOR REMEDIATION TECHNOLOGY AND TOOLS

OFFICE OF RADIATION AND INDOOR AIR

OFFICE OF AIR AND RADIATION

WASHINGTON, DC 20460

NATIONAL RISK MANAGEMENT RESEARCH LABORATORY

OFFICE OF RESEARCH AND DEVELOPMENT

U.S. ENVIRONMENTAL PROTECTION AGENCY

CINCINNATI, OH 45268
ii

NOTICE

The U.S. Environmental Protection Agency through its Office of Research and

Development partially funded and collaborated in the research described here as indicated through

the authors’ affiliations. Additional support for this project was provided by the Office of

Radiation and Indoor Air (ORIA) of the Office of Air and Radiation in collaboration with the

Office of Emergency and Remedial Response (OERR) of the Office of Solid Waste and

Emergency Response and the U.S. Department of Energy (EM-40). ManTech Environmental

Technologies, Inc. (METI) is an in-house support contractor to NRMRL/SPRD under 68-W5-

0011, Delivery Order 022 (Dr. David Jewett, Delivery Order Project Officer). Dynamac is an

off-site support contractor to NRMRL/SPRD under 68-C4-0031 (Dr. David S. Burden, Project

Officer). Ron Wilhelm and Robin Anderson (ORIA) provided valuable technical review and

administrative support to this project. The assistance of Sierra Howry, a student in the

Environmental Research Apprenticeship Program through East Central University in Ada,

Oklahoma, in the completion of this project is greatly appreciated. This report has been subjected

to the Agency’s peer and administrative review and has been approved for publication as an EPA

document.

Mention of trade names or commercial products does not constitute endorsement or

recommendation for use.

All research projects making conclusions or recommendations based on environmentally

related measurements and funded by the U.S. Environmental Protection Agency are required to

participate in the Agency Quality Assurance Program. This project did not involve environmental

related measurements and did not require a Quality Assurance Plan.
iii

FOREWORD

The U.S. Environmental Protection Agency is charged by Congress with protecting the Nation’s

land, air, and water resources. Under a mandate of national environmental laws, the Agency

strives to formulate and implement actions leading to a compatible balance between human

activities and the ability of natural systems to support and nurture life. To meet these mandates,

EPA’s research program is providing data and technical support for solving environmental

problems today and building a science knowledge base necessary to manage our ecological

resources wisely, understand how pollutants affect our health, and prevent or reduce

environmental risks in the future.

The National Risk Management Research Laboratory is the Agency’s center for investigation of

technological and management approaches for reducing risks from threats to human health and

the environment. The focus of the laboratory’s research program is on methods for the prevention

and control of pollution to air, land, water, and subsurface resources; protection of water quality

in public water systems; remediation of contaminated sites and ground water, and prevention and

control of indoor air pollution. The goal of this research effort is to catalyze development and

implementation of innovative, cost-effective environmental technologies; develop scientific and

engineering information needed by EPA to support regulatory and policy decisions; and provide

technical support and information transfer to ensure effective implementation of environmental

regulations and strategies.

Numerous infiltration estimation methods have been developed, and have become an integral part

of the assessment of contaminant transport and fate. This document selects six (6) methods

having utility under varying conceptualizations. The methods are described in detail and are

provided in electronic format for actual application.

Clinton W. Hall, Director

Subsurface Protection and Remediation Division

National Risk Management Research Laboratory
iv

ABSTRACT

Movement of water into and through the vadose zone is of great importance to the

assessment of contaminant fate and transport, agricultural management, and natural resource

protection. The process of water movement is very dynamic, changing dramatically over time and

space. Infiltration is defined as the initial process of water movement into the vadose zone

through the soil surface. Knowledge of the infiltration process is prerequisite for managing soil

water flux, and thus the transport of contaminants in the vadose zone.

Although a considerable amount of research has been devoted to the investigation of

water infiltration in unsaturated soils, the investigations have primarily focused on scientific

research aspects. An overall evaluation of infiltration models in terms of their application to

various climatic characteristics, soil physical and hydraulic properties, and geological conditions

has not been done. Specifically, documentation of these models has been limited and, to some

extent, non existent for the purpose of demonstrating appropriate site-specific application. This

document attempts to address this issue by providing a set of water infiltration models which have

the flexibility of handling a wide variety of hydrogeologic, soil, and climate scenarios. More

specifically, the purposes of this document are to: (1) categorize infiltration models presented

based on their intended use; (2) provide a conceptualized scenario for each infiltration model that

includes assumptions, limitations, boundary conditions, and application; (3) provide guidance for

model selection for site-specific scenarios; (4) provide a discussion of input parameter estimation;

(5) present example application scenarios for each model; and, (6) provide a demonstration of

sensitivity analysis for selected input parameters.

Six example scenarios were chosen as illustrations for applications of the infiltration

models with one scenario for each model. The intention of these scenarios is to provide

applications guidance to users for these models to various field conditions. Each model

application scenario includes the problem setup, conceptual model, input parameters, and results

and discussion.
v

CONTENTS

NOTICE ................................................................. ii

ABSTRACT ............................................................. . iv

LIST OF FIGURES ........................................................ . vii

LIST OF TABLES ........................................................ . ix

CHAPTER 1. INTRODUCTION ............................................. . 1

1.1 Phenomena of Water Infiltration in the Unsaturated Zone .................... . 1

1.2 Intended Use of this Document ....................................... . 3

CHAPTER 2. DESCRIPTIONS OF INFILTRATION MODELS ..................... . 5

2.1 Classification of Infiltration Models Chosen in the Document ................ . 5

2.1.1 Semi-Empirical Model ...................................... . 5

2.1.2 Homogeneous Model ........................................ . 5

2.1.3 Non-Homogeneous Model .................................... . 6

2.1.4 Infiltration Model for Ponding Conditions ........................ . 6

2.1.5 Infiltration Model for Non-Ponding Conditions .................... . 6

2.1.6 Infiltration/Exfiltration Model ................................. . 6

2.2 Infiltration Model Conceptualization ................................... . 6

2.3 Selected Models .................................................. . 9

2.3.1 SCS Model .............................................. . 9

2.3.2 Philip’s Two-Term Model ................................... . 9

2.3.3 Green-Ampt Model for Layered Systems ........................ . 10

2.3.4 Explicit Green-Ampt Model .................................. . 12

2.3.5 Constant Flux Green-Ampt Model ............................. . 13

2.3.6 Infiltation / Exfiltration Model ................................ . 14

CHAPTER 3. METHOD FOR SELECTION OF INFILTRATION MODEL ............ . 16

3.1 Model Selection Criteria ...................................... . 16

3.2 Example for Selecting the Water Infiltration Model .................. . 17

CHAPTER 4. INPUT PARAMETER ESTIMATIONS ............................ . 19

4.1 General Description of Parameters .................................... . 19

4.2 Estimation of the Fw in SCS Model .................................... . 23

4.3 Estimation of Green-Ampt Model Parameters ........................... . 24

4.4 Estimation of Philip’s Two-Term Model Parameters ....................... . 29

CHAPTER 5. EXAMPLE APPLICATION SCENARIOS .......................... . 30

5.1 Semi-Empirical Model (SCS) ........................................ . 30

5.2 Infiltration Model for Homogeneous Conditions (PHILIP2T) ................ . 32
vi

5.3 Infiltration Model for Non-homogeneous Conditions (GALAYER) ........... . 34

5.4 Infiltration Model for Ponding Conditions (GAEXP) ...................... . 36

5.5 Infiltration Model for Non-ponding Conditions (GACONST) ................ . 38

5.6 Infiltration and Exfiltration (INFEXF) ................................. . 40

REFERENCES ........................................................... . 42

APPENDIXES

Appendix A. User’s Instructions for Mathcad Worksheets ................... . A-1

Appendix B. Mathematical Models in Mathcad ........................... . B-1

1. SCS Model .............................................. . B1-1

2. Philip’s Two-Term Model ................................... . B2-1

3. Green-Ampt Model for layered systems ......................... . B3-1

4. Green-Ampt Model, Explicit Model ........................... . B4-1

5. Constant Flux Green-Ampt Model ............................. . B5-1

6. Infiltration/Exfiltration Model ................................ . B6-1

Appendix C. Sensitivity Analysis Discussion and Demonstration .............. . C-1
vii

LIST OF FIGURES

Number Page

1. Zones of the infiltration process for the water content profile under ponded

conditions (Adapted from Hillel, 1982) .................................... . 2

2. Infiltration rate will generally be high in the first stages, and will decrease with time

(Adapted from Hillel, 1982) ............................................. . 3

3. Conceptual model for simulating water infiltration using the SCS model ........... . 31

4. Surface infiltration rate as a function of daily rainfall rate ...................... . 31

5. Surface runoff rate as a function of daily rainfall rate ......................... . 31

6. Conceptual model for simulating water infiltration using the Philip’s Two Term

Model ............................................................ . 33

7. Water infiltration as a function of time .................................... . 33

8. Cumulative infiltration as a function of time ................................ . 33

9. Conceptual model for simulating water infiltration into a layered soil ............. . 35

10. Comparison of water infiltration between Scenarios 1 (solid line) and 2 (dash line) ... . 35

11. Conceptual model for simulating water infiltration using the Explicit Green-Ampt

model ............................................................. . 37

12. Water infiltration as a function of time .................................... . 37

13. Cumulative infiltration as a function of time ................................ . 37

 

14. Conceptual model for simulating water infiltration using the constant flux GreenAmpt model ........................................................ . 39

15. Water infiltration as a function of time .................................... . 39

16. Cumulative infiltration as a function of time ................................ . 39

 

17. Conceptual model for simulating water infiltration and exfiltration ............... . 41
viii

18. Water infiltration as a function of time .................................... . 41

19. Water exfiltration as a function of time ................................... . 41
ix

LIST OF TABLES

Number Page

1. Infiltration model classification ........................................... . 5

2. Summary of assumptions, limitations, and mathematical boundary conditions for

each model selected, where 2 is the volumetric water content. In this table,

features are identified as being present for the particular model (Y for YES), or not

considered (N for NO). The model names are defined in next section.. ............. . 8

3. Summary of common site conditions and the selected models capabilities of

simulating those conditions. Y indicates the model is designed for the specific site

condition and N indicates that it is not.. ................................... . 18

4. Listing of input parameters for running the infiltration models. .................. . 19

5. Curve numbers for hydrologic soil-cover complex for the average antecedent moisture

condition and Ia = 0.2Fw . Ia is initial water abstraction (USDA-SCS, 1972). ..... . 23

6. Description of hydrologic soil groups. .................................... . 24

7. Typical values of saturated volumetric water content (2s). ..................... . 25

8. Typical values of residual volumetric water content (2r). ...................... . 26

9. Typical values of air-entry head (hb).. ..................................... . 26

10. Typical values of pore size index (8). ..................................... . 27

11. Typical values for soil hydraulic property parameters (Rawls et al., 1992).. ........ . 28

12. Input parameter values for simulations with a sandy soil using the SCS model.. ..... . 30

13. Input parameter values for the simulations in a sandy soil using PHILIP2Tmodel. ... . 32

14. Input parameter values for the simulations using GALAYER model.. ............. . 34

15. Input parameters for the simulations in a clay soil using GAEXP.. ............... . 36

16. Input parameters for the simulations with r >Ks in a clay soil using GACONST.. .... . 38

17. Input parameters for the simulations using the INFEXF model. ................. . 40
1

CHAPTER 1. INTRODUCTION

1.1 Phenomena of Water Infiltration in the Unsaturated Zone

Water applied to the soil surface through rainfall and irrigation events subsequently enters the

soil through the process of infiltration. If the supply rate of water to the soil surface is greater

than the soil’s ability to allow the water to enter, excess water will either accumulate on the soil

surface or become runoff. The process by which water enter the soil and thus the vadose zone

through the soil-atmosphere interface is known as infiltration. Infiltrability is a term generally

used in the disciplines of soil physics and hydrology to define the maximum rate at which rain or

irrigation water can be absorbed by a soil under a given condition. Indirectly, infiltrability

determines how much of the water will flow over the ground surface into streams or rivers, and

how much will enter the soil, and thus assists in providing an estimate of water available for

downward percolation through drainage, or return to the atmosphere by the process of

evapotranspiration.

Understanding water movement into and through the unsaturated zone is of great importance

to both policy and engineering decision-makers for the assessment of contaminant fate and

transport, the management of agricultural lands, and natural resource protection. The process of

water movement is very dynamic, changing dramatically over time and space. Knowledge of the

infiltration process is a prerequisite for managing soil water flux, and thus the transport of

contaminants in the vadose zone.

The distribution of water during the infiltration process under ponded conditions is illustrated in

Figure 1. In this idealized profile for soil water distribution for a homogeneous soil, five zones

are illustrated for the infiltration process.

1) Saturated zone: The pore space in the saturated zone is filled with water, or saturated. 

Depending on the length of time elapsed from the initial application of the water, this zone

will generally extend only to a depth of a few millimeters.

2) Transition zone: This zone is characterized by a rapid decrease in water content with depth,

and will extend approximately a few centimeters.

3) Transmission zone: The transmission zone is characterized by a small change in water

content with depth. In general, the transmission zone is a lengthening unsaturated zone

with a uniform higher water content. The hydraulic gradient in this zone is primarily

driven by gravitational forces.

4) Wetting zone: In this zone, the water content sharply decreases with depth from the water

content of the transmission zone to near the initial water content of the soil.

5) Wetting front: This zone is characterized by a steep hydraulic gradient and forms a sharp

boundary between the wet and dry soil. The hydraulic gradient is driven primarily by

matric potentials.

Beyond the wetting front, there is no visible penetration of water. Comprehensive reviews of the

principles governing the infiltration process have been published by Philip (1969) and Hillel

(1982).
2

Figure 1. Zones of the infiltration process for the water content profile under ponded conditions 

 (Adapted from Hillel, 1982).

Soil water infiltration is controlled by the rate and duration of water application, soil physical

properties, slope, vegetation, and surface roughness. Generally, whenever water is ponded over

the soil surface, the rate of water application exceeds the soil infiltrability. On the other hand, if

water is applied slowly, the application rate may be smaller than the soil infiltrability. In other

words, water can infiltrate into the soil as quickly as it is applied, and the supply rate determines

the infiltration rate. This type infiltration process has been termed as supply controlled (Hillel,

1982). However, once the infiltration rate exceeds the soil infiltrability, it is the latter which

determines the actual infiltration rate, and thus the process becomes profile controlled. 

Generally, soil water infiltration has a high rate in the beginning, decreasing rapidly, and then

slowly decreasing until it approaches a constant rate. As shown in Figure 2, the infiltration rate

will eventually become steady and approach the value of the saturated hydraulic conductivity

(K ). s

The initial soil water content and saturated hydraulic conductivity of the soil media are the

primary factors affecting the soil water infiltration process. The wetter the soil initially, the

lower will be the initial infiltrability (due to a smaller suction gradient), and a constant

infiltration rate will be attained more quickly. In general, the higher the saturated hydraulic

conductivity of the soil, the higher the infiltrability.

Naturally formed soil profiles are rarely homogeneous with depth, rather they will contain

distinct layers, or horizons with specific hydraulic and physical characteristics. The presence of

these layers in the soil profile will generally retard water movement during infiltration. Clay

layers impede flow due to their lower saturated hydraulic conductivity; however, when these

layers are near the surface and initially very dry, the initial infiltration rate may be much higher

and then drop off rapidly. Sand layers will have a tendency to also retard the movement of the
The process of water movement through layers can be investigated through an interactive 1

software package entitled SUMATRA-1 developed by M. Th. van Genuchten (1978), USDA/

ARS, US Salinity Laboratory, Riverside, CA 92521.

3

Figure 2. Infiltration rate will generally be high

in the first stages, and will decrease with time

(Adapted from Hillel, 1982).

wetting front due to larger pore size and thus a

higher hydraulic gradient would be required for

flow into the layers. The surface crust will 1

also act as a hydraulic barrier to infiltration due

to the lower hydraulic conductivity near the

surface, reducing both the initial infiltrability

and the eventually attained steady infiltrability. 

As might be expected, the slope of the land

can also indirectly impact the infiltration rate. 

Steep slopes will result in runoff, which will

impact the amount of time the water will be

available for infiltration. In contrast, gentle

slopes will have less of an impact on the

infiltration process due to decreased runoff. 

When compared to the bare soil surface,

vegetation cover tends to increase infiltration by retarding surface flow, allowing time for water

infiltration. Plant roots may also increase infiltration by increasing the hydraulic conductivity of

the soil surface. Due to these effects, infiltration may vary widely under different types of

vegetation.

A number of mathematical models have been developed for water infiltration in unsaturated

zone (Philip, 1957; Bouwer, 1969; Fok 1970; Moore 1981; Ahuja and Ross, 1983; Parlange et

al., 1985; Haverkamp et al., 1990; Haverkamp et al., 1991). A thorough review of water

infiltration models used in the fields of soil physics and hydrology has been presented in a

previous volume prepared in conjunction with the current project (Ravi and Williams, 1998).

1.2 Intended Use of this Document

Soil water infiltration has been determined from experimental measurements (Parlange, et al.,

1985) and mathematical modeling (Philip, 1957; Bouwer, 1969; Fok 1970; Moore 1981; Ahuja

and Ross, 1983; Parlange et al., 1985; Haverkamp et al., 1990; Haverkamp et al., 19941). 

Although a considerable amount of research has been devoted to the investigation of water

infiltration in unsaturated soils, these investigations have primarily focused on scientific research

aspects. An overall evaluation of infiltration models in terms of their application to various

climatic characteristics, and soil physical and geological conditions has not been done. 

Specifically, documentation of these type models has been limited and, to some extent, nonexistent for the purpose of demonstrating appropriate site-specific application. This document

attempts to address this issue by providing a set of water infiltration models which have been

applied to a variety of hydrogeologic, soil, and climate scenarios. More specifically, the
4

purposes of this document are to: (1) categorize infiltration models presented based on their

intended use; (2) provide a conceptualized scenario for each infiltration model that includes

assumptions, limitations, mathematical boundary conditions, and application; (3) provide

guidance for model selection for site-specific scenarios; (4) provide a discussion of input

parameter estimation; (5) present example application scenarios for each model; and, (6)

provide a demonstration of sensitivity analysis for selected input parameters. This document does

not provide an in-depth sensitivity analysis for each model presented; however, a detailed

discussion of the process is provided in Appendix C.
5

CHAPTER 2. DESCRIPTIONS OF INFILTRATION MODELS

2.1 Classification of Infiltration Models

Six infiltration models (Table 1) were chosen for inclusion in this document based on several

considerations: (1) relatively simple approach, easy to use, and realistic in applications; (2) ability

to handle various field conditions including surface ponding, non-ponding, various rainfall rates,

surface runoff, and wetting and drying; and (3) application to both homogeneous or

nonhomogeneous soil profiles. The categories presented are not considered to be all-inclusive,

but do provide a wide range of model applications.

Infiltration models can be divided into various categories, depending on the purpose of the

model, boundary conditions, and the nature of subsurface systems. In this study, six categories

were identified for which the models chosen could be associated. The categories were chosen for

representativeness to site-specific conditions, which is discussed in the following.

Table 1. Infiltration model classification.

Category Model Selected Reference

Semi-Empirical SCS model USDA-SCS, 1972

Homogeneous Philip’s two-term model Philip, 1957

Nonhomogeneous Green-Ampt model for layered systems Flerchinger et al., 1988

Ponding Green-Ampt explicit model Salvucci and Entekhabi, 1994

Non-ponding Constant Flux Green-Ampt model Swartzendruber, 1974 

Wetting and Drying Infiltration/Exfiltration model Eagleson, 1978

2.1.1 Semi-Empirical Model: As indicated by the term empirical, these type infiltration

models are developed entirely from field data and have little or no physical basis. The empirical

approach to developing a field infiltration equation consists of first finding a mathematical

function whose shape, as a function of time, matches the observed features of the infiltration rate,

and then attempting a physical explanation of the process (Jury et al., 1991). Most physical

processes in semi-empirical models are represented by commonly accepted and simplistic

conceptual methods, rather than by equations derived from fundamental physical principles. The

commonly used semi-empirical infiltration models in the field of soil physics and hydrology are

Kostiakov’s model, Horton’s model, and the SCS (Soil Conservation Service) model (USDASCS, 1972; Hillel, 1982).

2.1.2 Homogeneous Model: Most infiltration models have been developed for application in

homogeneous porous media. These models are commonly derived from mathematical solutions

based on well-defined, physically based theories of infiltration (e.g., Richards equation). Since

this mechanistic infiltration model is derived from the water flow equation, considerable insight is

given to the physical and hydraulic processes governing infiltration. Infiltration models commonly
6

used for homogeneous soil profiles include the Green-Ampt infiltration model, the Philip

infiltration model, the Burger infiltration model, and the Parlange infiltration model. Descriptions

of these models are provided in documentation by Hillel (1982), Ghildal and Tripathi (1987), and

Jury et al. (1991).

2.1.3 Non-Homogeneous Model: Naturally developed soil profiles are seldom uniform with

depth, nor is the water content distribution uniform at the initiation of infiltration. Because of the

nonuniform soil profile, field infiltration measurement data frequently show different

characteristics than the models based on theoretical calculations for the uniform soil profile. For

most field observations of infiltration rates, the field observations would be less than what would

be predicted by models designed for homogeneous systems. In conceptualization of a nonhomogeneous soil profile, it is usually more convenient to divide the profile into layers or

horizons, each of which is assumed to be homogeneous (Childs and Bybordi, 1969; Hillel and

Gardner, 1970). The application of the Green-Ampt equation to calculate cumulative infiltration

into nonuniform soils has been studied by Bouwer (1969), Childs and Bybordi, (1969), Fok

(1970), Moore (1981), and Flerchinger et al. (1988).

2.1.4 Infiltration Model for Ponding Conditions: When the application rate of water to the

soil surface exceeds the rate of infiltration, free water (i.e., surface water excess) tends to

accumulate over the soil surface. This water collects in depressions, thus ponding on the soil

surface. Depending on the geometric irregularities of the surface and on the overall slope of the

land, some of the ponded water may become runoff, if the surface storage becomes filled. Under

ponded conditions, the cumulative infiltration is a function of soil properties, initial conditions,

and the ponding depth on the soil surface. Several infiltration models for ponding conditions have

been developed, including those by Parlange et al. (1985), Haverkamp et al. (1990, 1994), and

Salvucci and Entekhabi (1994).

2.1.5 Infiltration Model for Non-ponding Conditions: When the rate of water supply to the

soil surface does not exceed the soil infiltrability, all water can percolate into the soil and no

surface ponding of water occurs. This process depends on the rate of water supply, initial soil

water content, and the saturated hydraulic conductivity. Infiltration models for non-ponding

conditions have been developed by several researchers (Philip, 1957; Childs and Bybordi, 1969;

Hillel and Gardner, 1970). Probably the most common approach is an implementation of the

Green-Ampt model in an explicit approach.

 

2.1.6 Wetting and Drying Model: Alternate infiltration and exfiltration of water at the soil

surface will result in an unsteady diffusion of water into the soil. The presence of transpiring

vegetation adds another mechanism for moisture extraction distributed over a depth which is

related to root structure. Infiltration models for such conditions have been developed by

Eagleson (1978) and Corradini et al. (1994).

2.2 Infiltration Model Conceptualization

Water infiltration into unsaturated porous media for various climatic conditions, soil physical

and hydraulic properties, and geological conditions is very complex, resulting in a challenge to

mathematically simulate observed conditions. An understanding of the principles governing 
7

infiltration and factors affecting infiltration processes must be attained. Since a universal

mathematical model is not available to address water infiltration for all field conditions,

assumptions are commonly made, and limitations are identified during the model development and

application process. Assumptions used in developing water infiltration models commonly include

the following:

(1) Initial soil water content profile: Most infiltration models assume a constant and uniform

initial soil water content profile. However, under field conditions, the soil water content

profile is seldom constant and uniformly distributed.

(2) Soil profile: There are two types of soil profiles which exist under field conditions: (a)

homogeneous and (b) heterogeneous. Infiltration models developed for the homogeneous

soil profile cannot be used for heterogeneous soil profile without simplifying assumptions

concerning the heterogeneity.

(3) Constant and near saturated soil water content at the soil surface: This is a common

assumption for most infiltration models, which will allow for the ignoring of the initially

high hydraulic gradient across the soil’s surface. The length of time the surface is not near

saturation is very small when compared to the length of time associated with the

infiltration event.

(4) Duration of infiltration process: Some infiltration models are only valid for a short term

period of infiltration, which can limit their usefulness to field applications where infiltration

may last for longer time periods. A short term period might be representative of a rainfall

or irrigation event.

(5) Surface crust and sealing: None of the selected models can be used for the surface crusting

boundary condition. This boundary condition is complex and dynamic.

(6) Flat and smooth surface: These surface boundary conditions can be easily incorporated

into mathematical models, whereas an irregular surface or sloping surface can result in an

added dimension to the mathematical modeling.

Boundary conditions are those conditions defined in the modeling scenario to account for

observed conditions at the boundaries of the model domain. These conditions must be defined

mathematically, the most common of which are given as follows:

(1) Constant or specified flux at the surface: The rate of water applied to the soil surface can

be constant or time-varying. Because of its simplicity as compared to other boundary

conditions, a constant flux condition has been used in most infiltration models.

(2) Surface ponding condition. The surface ponding or non-ponding conditions are dependent

on the rate of water application as well as soil infiltrability. Whenever the rate of water

application to the soil exceeds the soil infiltrability, the surface ponding occurs. The

opposite case will result in surface non-ponding conditions.

(3) Finite column length at the lower boundary. Some infiltration models are limited to a finite

column length and others may allow for an infinite column length. Infiltration models

with the finite column length condition may limit their applications for deeper water

infiltration.

(4) Based on Richards equation. Several infiltration models (e.g., Philip’s model and

Eagleson’s model) were developed based on Richards equation. These models commonly

have well-defined physically based theories and give considerable insight into the

processes governing infiltration. These would require information related to the water

retention characteristics of the soil media. 
8

A summary of assumptions, limitations, and boundary conditions for each model chosen in this

document is given in Table 2. Detailed descriptions of the models are provided below.

Table 2. Summary of assumptions, limitations, and mathematical boundary conditions for each

model selected, where 2 is the volumetric water content. In this table, features are

identified as being present for the particular model (Y for YES), or not considered (N

for NO). The model names are defined in next section.

Model

name

Model Features

Assumptions and Limitations Boundary Conditions Constant and uniform initial 2 Homogeneous soil profile Valid for only short term infiltration Surface crust and sealing Constant water content at top boundary Constant flux at the top boundary Surface ponding condition Finite column length

Based on Richards equationBased on empirical equation

SCS N N N N N N N N N Y

Philip’s two

term Y Y Y N Y N N N Y N

Green-Ampt

model for

layered systems

Y Y N N Y N Y Y N N

Green-Ampt

explicit Y Y N N Y N Y Y N N

Constant flux

Green-Ampt Y Y N N Y Y N Y N N

Infiltration/Exfi

ltration Y Y N N Y N N N Y N
R'

(P&0.2Fw)

2

P% 0.8Fw

P> 0.2Fw

0, otherwise: P<_ 0.2 Fw

q ' P & R

The Mathcad worksheets provided in Appendix B are referenced in parentheses. These 2

are also used as the filenames.

9

(1)

(2)

2.3 Selected Models

2.3.1 SCS Model

Description: The empirical approach of developing a water infiltration equation consists of

first identifying a mathematical function whose shape, as a function of time, matches the observed

features of the infiltration rate, followed by an attempt at providing a physical explanation of the

process (Jury et al., 1991). For semi-empirical models, most physical processes are represented

by commonly accepted, and simplistic conceptual methods rather than by equations derived from

fundamental physical principles. The Soil Conservation Service (SCS) model is a commonly 2

used semi-empirical infiltration model in the field of soil physics and hydrology (USDA, 1972).

Equations: Mathematical functions for the SCS model are given as: 

and

where R is amount of runoff (inches), P is the daily rainfall amount (inches), Fw is a statistically

derived parameter (also called the retention parameter) with the units of inches, and q is the daily

infiltration amount (inches). Justification for the use of this model is based on a consideration that

the model is simple, it requires few input parameters, and has been widely used and understood in

the fields of soil physics and hydrology.

Assumptions and Limitations: A major limitation for applying the SCS model lies in that the

coefficients in Equation 1 must be evaluated with field data for each specific site. Since the model

is not derived from fundamental physical principles, it can only be used as a screening tool for

initial approximations.

2.3.2 Philip's Two-Term Model

Description: The Philip's two-term model (PHILIP2T) is a truncated form of the Taylor

power series solution presented by Philip (1957). During the initial stages of infiltration (i.e.,

when t is very small), the first term of Equation 3 dominates. In this stage, the vertical infiltration

proceeds at almost the same rate as absorption or horizontal infiltration due to the gravity
q(t)'

1

2

St &1⁄2%A

I(t)'St

1⁄2%At

10

(3)

(4)

component, represented by the second term, being negligible. As infiltration continues,

the second term becomes progressively more important until it dominates the infiltration process. 

Philips (1957) suggested the use of the two-term model in applied hydrology when t is not too

large. 

Equations: The Philip’s two-term model is represented by the following equations,

and

where q is infiltration rate (cm/h), t is time for infiltration (h), S is the sorptivity (cm/h ) and is a 1/2

function of the boundary, initial, and saturated water contents, A is a constant (cm/h) that

depends on soil properties and initial and saturated water contents, and I(t) is the cumulative

infiltration (cm) at any time, t.

Assumptions and Limitations: The three primary assumptions for this model are: (1)

homogeneous soil conditions and properties; (2) antecedent (i.e., conditions prior to infiltration)

water content distribution is uniform and constant (i.e., single-valued); and (3) water content at

the surface remains constant and near saturation. There are basically four limitations for this

model, which are as follows: (1) field soils are seldom homogeneous, and this model is not

designed for layered systems; (2) initial moisture content is seldom uniformly distributed

throughout the profile; (3) in most situations of rainfall or irrigation, the soil surface is rarely at a

constant water content; and (4) the approach is not valid for large times. Regarding the third

limitation identified above, if the rainfall or irrigation rate is smaller than the saturated hydraulic

conductivity, the soil may never be saturated at the surface, and no ponding will occur. Under

these conditions, the infiltration rate will be equal to the rainfall or irrigation rate. Even when the

rate is greater than saturated hydraulic conductivity, there will be some time lag between the

beginning of infiltration and the time of surface saturation and subsequent ponding. Under these

circumstances the constant flux Green-Ampt model presented by Swartzendruber (1974) may be

used. 

2.3.3. Green-Ampt Model for Layered Systems

Description: The Green-Ampt Model has been modified to calculate water infiltration into

nonuniform soils by several researchers (Bouwer, 1969; Childs and Bybordi, 1969; Fok, 1970;

Moore, 1981). Flerchinger et al. (1988) developed a model, based on the Green-Ampt model, for

calculating infiltration over time in vertically heterogeneous soils. This model is referred to as the

Green-Ampt model for layered systems (GALAYER).

Equations: Mathematical formulations for this model are given below: 
f ' f

(Kn

f

( '

F

( % 1

F

( % z(

F

( '

1

2

[t

( & 2z( % (t( & 2z(

)

2

% 8t

(

]

t

( '

Knt

)2(Hn% j

n&1

i'1

z

i

)

z

( '

Kn

(Hn% j

n&1

i'1

z

i

)

j

n&1

i'1

z

i

Ki

11

(5)

(6)

(7)

(8)

(9)

where f is infiltration rate (cm/h), f* is dimensionless infiltration, Kn is hydraulic conductivity

(cm/h) of the layer n containing wetting front, F* is the dimensionless accumulated infiltration in

layer n, z* is dimensionless depth accounting for thickness and conductivity of layers behind the

wetting front, t is time (h), )2 is change in volumetric water content (cm /cm ) as the wetting 3 3

front passes layer n, H is potential head while the wetting front passes through layer n, z is n i

thickness (cm) of layer i, and K is hydraulic conductivity of layer i. As can be seen, accounting for i

layers significantly increases the complexity of this model. 

Assumptions and Limitations: There are basically two assumptions for this model. The

first assumption is nearly saturated plug flow. An example of when this assumption may be

violated is if a coarse-textured soil with a high hydraulic conductivity is underlying a fine-textured

soil with a low hydraulic conductivity. In general, any time the infiltration rate becomes less than

the hydraulic conductivity at the wetting front, complete wetting will not occur beyond the

wetting front, and the assumption of nearly saturated flow is no longer valid. The second

assumption is that the water content is uniform within each layer behind the wetting front,

although it is nonuniform between layers. The limitation for this model is that this equation is

valid for layered soils only if the dimensionless depth (z*) is less than or equal to 1. Crusted soils

and surface seals are typical conditions when this criterion is not met.
I

Ks

'(1&

2

3

)t%

2

3

Pt%t

2%(

2&1

3

)P[ln(t%P)&lnP]

%

2

3

P[ln(t%

P

2

% Pt%t

2

)&ln(P/2)]

q

Ks

'

2

2

J

&1/2%

2

3

&

2

6

J

1/2%

1& 2

3

J

P'

(hs&hf)(2s

&20)

Ks

J'

t

t%P

12

(10)

(11)

(12)

(13)

2.3.4 Explicit Green-Ampt Model

Description: The Green-Ampt model is the first physically-based equation describing the

infiltration of water into a soil. It has been the subject of considerable development in soil physics

and hydrology owing to its simplicity and satisfactory performance for a wide variety of water

infiltration problems. This model yields cumulative infiltration and infiltration rates as implicit

functions of time (i.e., given a value of time, t, q and I cannot be obtained by direct substitution). 

The equations have to be solved in an iterative manner to obtain these quantities. Therefore, the

required functions are q(t) and I(t) instead of t(q) and t(I). The explicit Green-Ampt model

(GAEXP) for q(t) and I(t), developed by Salvucci and Entekhabi (1994), facilitated a

straightforward and accurate estimation of infiltration for any given time. This model supposedly

yields less than 2% error at all times when compared to the exact values from the implicit GreenAmpt model.

Equations: Mathematical formulations for this model are as follows:

with

and 

where q is infiltration rate (cm/h), K is saturated hydraulic conductivity (cm/h), t is time (h), h is s s
q ' r

I 'rt

q ' r

I 'rt

13

(14)

(15)

(16)

(17)

ponding depth or capillary pressure head at the surface (cm), hf is capillary pressure head at the

wetting front (cm), 2s is saturated volumetric water content (cm /cm ) and 20 is initial volumetric 3 3

water content (cm /cm ). 3 3

Assumptions and Limitations: The assumptions of the model are: (1) the water content

profile is of a piston-type with a well-defined wetting front; (2) antecedent (i.e., prior to

infiltration) water content distribution is uniform and constant; (3) water content drops abruptly

to its antecedent value at the front; (4) soil-water pressure head (negative) at wetting front is h ;

f 

(5) soil-water pressure head at the surface, h , is equal to the depth of the ponded water; and, (6) s

soil in the wetted region has constant properties. The limitations of the model are: (1)

homogeneous soil conditions and properties; (2) constant, non-zero surface ponding depth; and,

(3) in most situations of rainfall or irrigation, the surface is not at a constant water content.

2.3.5 Constant Flux Green-Ampt Model

 Description: The constant flux Green-Ampt model (GACONST) can be used to simulate the

water infiltration for non-ponding conditions, where the water flux application rate is represented

by r (cm/h). Two cases are presented, that where the application rate is less than the saturated

hydraulic conductivity r < Ks), and where the application rate is greater than the saturated

hydraulic conductivity r > Ks). When r < Ks, the infiltration rate (q) is always equal to the surface

application rate (r) and the surface is never saturated; when r > K , surface saturation first occurs s

at time t .o

Equations: Mathematical formulations for this model are given below:

For r K and t > 0, we have: s

For r > K , we have the following cases: s

(i) t < to
q'Ks

[1&(2s&20)

hf

I

]

I

0

'rt

0

Ks(t&t

0

)'I&I

0

%hf(2s&20)ln[

I&(2s

&20)hf

I

0

&(2s&20)hf

]

t

0

'

&Kshf(2s&20)

r (r&Ks)

14

(18)

(19)

(20)

(21)

(ii) t > to

with 

where q is the surface infiltration rate (cm/h), r is the constant water application rate at the surface

(cm/h), t is time (h), Ks is saturated hydraulic conductivity (cm/h), 2s is saturated volumetric

water content (cm /cm ), and 2 is initial volumetric water content (cm /cm ), and h is capillary 3 3 3 3

0 f

pressure head (< 0) at the wetting front (cm). In the case of r >Ks, before surface saturation

occurs, the infiltration rate is simply equal to r.

Assumptions and Limitations: The assumptions and limitations are the same as those

identified for the other Green-Ampt models.

2.3.6 Infiltration/Exfiltration Model

Description: The vertical movement of soil water between the surface and the water

table can be divided into two processes according to the predominant forces involved: (1)

infiltration and (2) exfiltration (evapotranspiration). An infiltration/exfiltration model (INFEXF)

developed by Eagleson (1978) was selected to estimate the water infiltration during a wetting

season and exfiltration during a drying season. Philip’s equation, which can be used to simulate

both infiltration and exfiltration, assumes the medium to be effectively semi-infinite and the

internal soil water content at the beginning of each storm and inter-storm period to be uniform at

the long term space-time average. The exfiltration equation is modified for presence of natural

vegetation through the approximate introduction of a distributed sink representing the moisture

extraction by plant roots. 
f

i

'

1

2

Sit

&1⁄2%

1

2

(K1%K0)

f

e

'

1

2

Set

&1⁄2&

1

2

(K1%K0) & MEv

15

(22)

(23)

Equations: Mathematical formulations for this model are given as:

for water infiltration, and 

for water exfiltration, where f

i

 is the infiltration rate (cm/h), Si is the infiltration, sorptivity

(cm/h ), t is the time (h), K is the hydraulic conductivity (cm/h)’s of soil at soil surface with 1/2

1

water content 2, K0 is the initial hydraulic conductivity (cm/h), f

e

 is the exfiltration rate (cm/h), Se

is the exfiltration sorptivity (cm/h ), M is the vegetated fraction of land surface, and E is the 1/2

v

transpiration rate (cm/h). During the wetting season (Infiltration), K1 and 21 might represent the

saturated hydraulic conductivity (Ks) and the saturated water content (2s) repectively for the

conditions at the soil surface. During the dry season (exfiltration), K1 might represent the

hydraulic conductivity for the dry soil surface with the water content approaches to zero.

Assumptions and Limitations: Assumptions and limitations for the model are: (1) the

water table depth is much greater than the larger of the penetration depth and the root depth; (2) 

soil water content throughout the surface boundary layer is spatially uniform at the start of each

storm period and at the start of each inter-storm period at the value s = s , where s is degree of o

saturation and s is initial degree of saturation in surface boundary layer; (3) vegetation is o

distributed uniformly and roots extend through the entire volume of soil; and (4) homogeneous

soil system.
16

CHAPTER 3. METHOD FOR SELECTION OF INFILTRATION MODEL

Mathematical modeling has become an important methodology in support of the planning

and decision-making process. Soil infiltration models provide an analytical framework for

obtaining an understanding of the mechanisms and controls of the vadose zone and ground-water

flow. Results from soil water infiltration models can be used to simulate the fate and transport of

contaminants by other models. For managers of water resources, infiltration models may provide

an essential support for planning and screening of alternative policies, regulations, and engineering

designs affecting ground-water flow and contaminant transport. A successful application of soil

water infiltration models requires a combined knowledge of scientific principles, mathematical

methods, and site characterization. In general, application of mathematical models involves the

following processes (US EPA, 1994a): (1) model application objectives; (2) project

management; (3) conceptual model development; (4) model selection; (5) model setup and input

parameter estimation; (6) simulation scenarios; (7) post simulation analysis; and (8) overall

effectiveness. In this chapter, the focus will be on methods for selecting appropriate soil

infiltration models. A detailed discussion of model selection is reported by U.S.EPA (1994a &

1994b, 1996) and van der Heijde and Elnawawy (1992).

3.1 Model Selection Criteria

In the model application process, mathematical model selection is a critical step to ensure

an optimal trade-off between project effort and result. It is the process of choosing the

appropriate software, algorithm, or other analysis techniques capable of simulating the

characteristics of the physical, chemical, and biological processes, as identified in the conceptual

model, as well as meeting the overall objectives for the model application. In general, the major

criteria in selecting a model are: (1) suitability of the model for the intended use, (2) reliability of

the model, and (3) efficiency of the model application.

A major criterion in selecting a mathematical model is its suitability for the identified task. 

A selected mathematical model must meet the needs identified in the conceptual model. These 

include initial and boundary conditions (e.g., surface ponding), hydrogeological properties (e.g.,

layered soil), biological properties (e.g., root water uptake), and availability of input parameters. 

Before a final selection is made, users should develop an understanding of the details incorporated

into the mathematical model. Also, the user should be aware that each model has its strengths

and weaknesses, often related to terms included in the governing equations and in the boundary

conditions. It should be realized that a perfect match rarely exists between desired characteristics

and the characteristics presented in the mathematical models.

 Another criterion in selecting a mathematical model is model credibility. The credibility of

the model, and that of the theoretical framework represented, is based on the model’s proven

reliability, and on its acceptance by users. Model credibility is a major concern in model use. 

Therefore, special attention should be given in the selection process to ensure the use of qualified

simulation models which have undergone adequate review and testing.

The applicability of a model is referred to as how the model can be used for given

conditions. The infiltration models chosen in this document are simple and easy to use, and yet

cover a variety of field conditions, including (1) rainfall and irrigation, (2) surface runoff, (3)
17

multiple layers, and (4) wetting and drying conditions. Table 3 provides a guidance for the

selection of a suitable infiltration model for given site conditions. It should be noted that the site

conditions listed in this table are common conditions encountered in environmental remediation

practices. It is impractical to list all conditions for each individual site; however, users are

encouraged to read the assumptions and limitations for each model discussed in the previous

section before selecting a model.

3.2 Example for Selecting the Water Infiltration Model

As an example of the selection process being implemented, the four basic steps are

suggested for the following generic scenario. For this example, the infiltration rate is needed for

use in a screening level application of a simplified fate and transport model. For this type model,

the advective portion of the model is handled by a single estimated infiltration rate. Conceptually,

the site soil profile is noticeably layered, and no surface runoff or ponding is routinely observed

during rainfall events. Due to the climatic region, and the generally protected nature of the site to

high evaporative demand, the wetting and drying cycle is not considered critical. No noticeable

cracks or other indications of preferential flow have been noted, therefore this characteristic can

be discounted.

(1) Modeling objective: The objective of the modeling project is to estimate the soil water

infiltration rate. Therefore, a soil water infiltration model is needed.

(2) Conceptual model: The soil profile is non-homogeneous and layered. Therefore, a soil

water infiltration model for layered soil is needed.

(3) Other site conditions: No surface runoff or ponding is to be considered for the

application, and preferential flow is not a concern. The wetting and

drying cycle is not significant at the site.

(4) Model selection: Based on the modeling objective and conditions listed above,

Green-Ampt model for layered systems is an appropriate choice for

the project.

Once an infiltration model has been selected, it is imperative that the user gather additional

experience with using the model. This would include the implementation of a sensitivity analysis

to establish the priority and importance of input parameters. Hopefully, the user will already be

familiar with the model description, assumptions, and limitations. Input parameters for the model

now become critical, and the user will need to establish the availability of quality data for the input

parameters. The quality of this data must be established prior to utilization in any simulations.
18

Table 3. Summary of common site conditions and the selected model’s capabilities of simulating

those conditions. Y indicates the model is suitable for the specific site condition and N

indicates that it is not.

Site

Conditions

Model Name

SCS Green- Flux Green- Philip Explicit Infiltration/

2-Term Green-Ampt Exfiltration

Layered Constant

Ampt Ampt

Surface

Ponding N N Y Y N N

Surface

Runoff Y N N N N N

Rainfall &

Irrigation Y N N N Y Y

Multiple

Layers N N Y N N N

Wetting &

Evaporation N N N N N Y

Homogeneous

Soil Profile Y Y Y Y Y Y

Vegetation

Cover N N N N N Y
19

CHAPTER 4. INPUT PARAMETER ESTIMATIONS

4.1 General Description of Parameters

An important step in a model project is input parameter estimation. Because of

improvements in computer software and hardware, the usefulness of numerical models hinges

increasingly on the availability of accurate input parameters. Approaches for obtaining input

parameters in numerical modeling include field observations and measurements, experimental

measurements, curve-fitting, and theoretical calculations. Table 4 provides a list of input

parameters required for the use of models discussed in this document. A majority of parameter

values required for these models can be found in the literature (Hillel, 1982; Carsel and Parrish,

1988; van Genuchten et al., 1989; Breckenridge et al., 1991; Leij et al., 1996). Typical

parameter values are provided in this table for common soil conditions; however, it is always a

good practice to obtain site-specific values.

Table 4. Listing of input parameters for running the infiltration models.

Parameter Symbol Data Source Typical Value or Required Model

Method Name

Statistically derived

parameter Fw 0.1 to 90 inches SCS USDA-SCS, 1972

Daily rainfall P 0.1 to 10 inches SCS Weather Stations

Duration of All models except

infiltration for SCS t Site-Specific

Sorptivity S Philip’s two term 0.1 to 1 cm/h Philip, 1969; Jury et al,

1⁄2

or Calculated 1991

Empirical Constant A Calculated Philip, 1969; Hillel, 1982 Philip’s two 

term

Saturated Hydraulic 1x10 to 60 cm/h All models except Carsel and Parrish, 1988;

Conductivity or Measured for SCS Hillel, 1982; Li et al.,

Ks

-5

Breckenridge et al., 1991;

1976

Unsaturated Breckenridge et al., 1991;

Hydraulic Site-Specific Carsel and Parrish, 1988;

Conductivity of or Measured Hillel, 1982; Li et al.,

Layer n 1976

Kn model for layered

Green Ampt

systems

Hydraulic Green Ampt

Conductivity of K model for layered

Layer i systems

i

Site-Specific Carsel and Parrish, 1988;

or Measured Hillel, 1982; Li et al.,

Breckenridge et al., 1991;

1976

Thickness of

Layer i

Zi Measured model for layered

Green Ampt

systems
Parameter Symbol Data Source Typical Value or Required Model

Method Name

20

Potential Head Breckenridge et al., 1991;

While Wetting Site-Specific Carsel and Parrish, 1988;

Front Passes or Measured Hillel, 1982; Li et al.,

Through Layer n 1976

Hn model for layered

Green-Ampt

systems 

Change in Breckenridge et al., 1991;

Volumetric Water Site-Specific Carsel and Parrish, 1988;

Content Within or Measured Hillel, 1982; Li et al.,

Layer n 1976

)2 model for layered

Green-Ampt

systems

Saturated Site-Specific

Volumetric Water 2 0.3 to 0.5 cm /cm

Content (or Table 7)

s

3 3

Green-Ampt Breckenridge et al., 1991;

explicit & Carsel and Parrish, 1988;

Constant flux Hillel, 1982; Li et al.,

Green-Ampt 1976

Initial Volumetric Site-Specific explicit & Carsel and Parrish, 1988;

Water Content or Measured Constant flux Hillel, 1982; Li et al.,

2o

Green-Ampt Breckenridge et al., 1991;

Green-Ampt 1976

Ponding Depth hs

Site-Specific Green-Ampt

or Measured explicit

Capillary Pressure

at the Wetting h Calculated

Front

f

Green-Ampt Hillel, 1982; Jury et al.,

explicit 1991

Constant Constant flux

Application Rate Green-Ampt r Measured

Vegetated Fraction M Site-Specific Eagleson, 1978 Infiltration/Exfilt

r-ation

Transpiration Rate Ev Measured Eagleson, 1978 Infiltration/Exfilt

-ration

Infiltration Infiltration/Exfilt

Sorptivity -ration Si Eagleson’s Report Eagleson, 1978

Exfiltration Infiltration/Exfilt

Sorptivity -ration Se Eagleson’s Report Eagleson, 1978

A brief description of each parameter listed in Table 4 is provided as follows. These

descriptions are not intended to be fully explanatory, and the reader is referred to cited references

for more detail.

1. Statistically derived parameter with some resemblance to the initial water content

(Fw): This parameter can be estimated according to USDA-SCS (1972). Detailed

estimation of Fw is provided in Section 4.2. This parameter is required in the SCS model.

2. Daily rainfall rate (P): The daily rainfall rate for a specific site can be obtained from

local weather station or atmospheric research institutes. This parameter is required in the

SCS model.
21

3. Duration of infiltration (t): The duration of infiltration is site specific, and depends on

rainfall rate, surface application rate, and soil infiltrability. This is also the simulation

time.

4. Sorptivity (S): The sorptivity is a function of initial and saturated water contents, and

can be obtained simply by determining the slope of I/t versus t in Equation 4. This -1/2

parameter is required in Philip’s two-term model.

5. Empirical constant (A): This parameter is required in Philip’s two-term model. When

the infiltration time is very large, the constant A is similar to hydraulic conductivity, and

can be obtained simply by determining the intercept of I/t versus t in Equation 4. -1/2

6. Saturated hydraulic conductivity (K ): This parameter measures the ability of the soil s

to conduct water when saturated. The value of the saturated hydraulic conductivity

depends on soil types and is site-specific. This parameter can be obtained from

experimental measurements and literature, and is required in Philip’s two-term model,

Green-Ampt model for layered systems, the Green-Ampt explicit model, and the Constant

flux Green-Ampt model.

7. Hydraulic conductivity of layer n (K ): This parameter measures the ability of the soil n 

to conduct water in layer n when the soil is unsaturated. The value of the unsaturated

hydraulic conductivity depends on soil types and soil water content, and is site-specific. 

This parameter can be obtained from experimental measurements or literature, and is

required in the Green-Ampt model for layered systems.

8. Hydraulic conductivity of layer i (K ): This parameter measures the ability of the soil i 

to conduct water in layer i when the soil is unsaturated. The value of unsaturated

conductivity depends on soil types and soil water content, and is site-specific. This

parameter can be obtained from experimental measurements and literature, and is required

in the Green-Ampt model for layered systems.

9. Thickness of layer i (Z ): This parameter defines the thickness of layer i when the soil is i

heterogeneous. The value of each layer’s thickness is site-specific, and is required in the

Green-Ampt model for layered systems.

10. Potential head while wetting front passes through layer n (H ): The potential head is n

the difference between the energy state of soil water and that of pure free water. The soil

water potential head is a function of soil water content, and can be determined from the

soil water retention curve. Users may need to know basic concepts of the soil water

retention curves in order to estimate this parameter. This parameter is required in the

Green-Ampt model for layered system.

11. Change in volumetric water content within layer i ()2): The volumetric water

content is expressed as a decimal fraction of the volume of the total soil sample. The

change in volumetric water content within layer i is the difference between the maximum

and minimum water content in the i layer at a given time frame. This parameter is th

required in the Green-Ampt model for layered system.

12. Saturated volumetric water content (2s): The saturated volumetric water content is the

percentage of the volume of the soil sample when all soil pore spaces are filled with water

(i.e., saturated). The value of the saturated volumetric water content is dependent on soil
22

type and is site-specific. This parameter can be obtained from experimental measurements

and the literature, and is required in the Green-Ampt explicit model. 

13. Initial volumetric water content (2 ): This parameter defines the volumetric water o

content at start of the simulation. The value of the initial volumetric water content is

dependent on soil type and is site-specific, and is required in the Green-Ampt explicit

model.

14. Ponding depth (hs): This parameter defines the thickness of water accumulated at the

soil surface during water infiltration. The extent of ponding depth depends on soil types

and is thus site-specific. This parameter is required in the Green-Ampt explicit model.

15. Capillary pressure at the wetting front (h ): The capillary pressure is the suction of f

water in the pore space due to surface tension or capillary force. This parameter is a

function of soil water content, and can be determined from experimental measurements

(Hillel, 1982) or from the following equation h = 2(/ r, where ( is the surface tension of f

water and r is the radius of capillary. The parameter is required in the Green-Ampt

explicit model, and is discussed in the section on the estimation of the Green-Ampt

parameters.

16. Constant application rate (r): This is the rate of water application to the soil surface,

and is site-specific. This parameter is required in the constant flux Green-Ampt model.

17. Vegetated fraction (M): This parameter defines the density of plant canopy coverage at

the soil surface. This parameter is site-specific, and is required in the

infiltration/exfiltration model.

18. Transpiration rate (E ): This parameter defines the plant water transpiration rate. It v

depends on type of plant and soil water availability to roots. This parameter can be

obtained from experimental measurements, and is required in the infiltration/exfiltration

model.

19. Infiltration sorptivity (Si): The infiltration sorptivity is a function of initial and saturated

water contents. It can be obtained from Eagleson’s report (1978). This parameter is

required in the INFEXF model. 

20. Exfiltration sorptivity (Se). The exfiltration sorptivity is a function of initial and

saturated water contents and surface evaporation rate. This parameter can be obtained

from Eagleson (1978), and is required in the infiltration/exfiltration model.

In summary, the saturated water content at the application surface (2 ), antecedent water s

content (2o), and respective hydraulic conductivities (Ks and Ko) are required for most infiltration

models. However, the pore size index (8) and air entry head (hb) are also required by some of the

models for the estimation of other input parameters (e.g., head at the wetting front (hf)). These

values are listed in Tables 9 and 10. In addition to the tabulations indicated, the electronic

database known as UNSODA (Leij et al., 1996) provides information on the soil hydraulic

parameters related to water retention and hydraulic conductivity. Estimates of infiltration based

on tabulated parameter values should be considered only for preliminary, and order-of-magnitude

type analyses. For a more reliable estimation of infiltration at a given site, it is recommended that

the values of 2 , 2 , K , K , and the retention parameters be measured. Johnson and Ravi (1993) s o s o

call for caution in the use of literature values for a specific application and discuss the resulting
Fw

'

1000

CNI

& 10

23

(24)

uncertainty.

4.2 Estimation of F in SCS Model w

The statistically derived parameter Fw utilized in the SCS model can be estimated based on

the following equation:

where CNI is the soil moisture condition I curve number or hydrologic soil-cover complex

number. CN is related to the soil moisture condition II curve number, CN , with the polynomial: I II

CNI = -16.91 + 1.348 CNII - 0.01379(CNII) + 0.0001177(CNII) . Fw can be considered an 2 3

estimate of the maximum potential difference between rainfall and runoff (Schwab et al., 1981). 

The CN is based on an antecedent moisture condition (AMC) II determined by the total rainfall II

in the 5-day period preceding a storm (USDA-SCS, 1972). Three levels of AMC are used: (1)

lower limit of moisture content, (2) average moisture content, and (3) upper limit of moisture

content. Table 5 lists values for CN at the average AMC. The description of hydrologic groups II

used in Table 5 is provided in Table 6.

Table 5. Curve numbers CNII for hydrologic soil-cover complex for the antecedent moisture

condition II and I = 0.2F . I is initial water abstraction (USDA-SCS, 1972). a w a

Land Use or Hydrologic

Crop Condition Treatment or Practice Hydrologic Soil Group

A B C D

Fallow Straight Row --- 77 86 91 94

Row Crops Straight Row Poor 72 81 88 91

Straight Row Good 67 78 85 89

Contoured Poor 70 79 84 88

Contoured Good 65 75 82 86

Terraced Poor 66 74 80 82

Terraced Good 62 71 78 81

Small Grain Straight Row Poor 65 76 84 88

Straight Row Good 63 75 83 87

Contoured Poor 63 74 82 85

Contoured Good 61 73 81 84

Terraced Poor 61 72 79 82

Terraced Good 59 70 78 81

Close-seeded

Legumes or

Rotation

Meadow

Straight Row Poor 66 77 85 89

Straight Row Good 58 72 81 85

Contoured Poor 64 75 83 85
24

Contoured Good 55 69 78 83

Terraced Poor 63 73 80 83

Terraced Good 51 67 76 80

Pasture or Range Poor 68 79 86 89

Fair 49 69 79 84

Good 39 61 74 80

Contoured Poor 47 67 81 88

Contoured Fair 25 59 75 83

Contoured Good 6 35 70 79

Meadow Good 30 58 71 78

Woods Poor 45 66 77 83

Fair 36 60 73 79

Good 25 55 70 77

Farmsteads --- 59 74 82 86

Roads and

Right-of-Way --- 74 84 90 92

(Hard Surface

Table 6. Description of hydrologic soil groups.

Group Description of Hydrologic Group

A

Lowest runoff potential. Includes deep sands with very little silt and clay, also

deep, rapidly permeable loess.

B loess less deep or less aggregated than group A, but the group as a whole has

Moderately low runoff potential. Mostly sandy soils less deep than Group A, and

above average infiltration after thorough wetting.

C considerable clay and colloids, though less than those of group D. The group has

Moderately high runoff potential. Comprises shallow soils and soils containing

below-average infiltration after presaturation.

D group also includes some shallow soils with nearly impermeable subhorizons near

Highest runoff potential. Includes mostly clays of high swelling percent, but the

the surface.

An example on how to determine the CNII from Table 5 is given as follows. For pasture land

with good hydrologic conditions, under moderately high runoff potential (Group C), the CN is II

74 and CN is 55, and thus F is about 8.2 inches. I w 

4.3 Estimation of Green-Ampt Model Parameters

The discussion presented here is relevant to the three models which utilize the Green-Ampt

parameters: (1) the explicit Green-Ampt (Salvucci and Entekhabi, 1994), (2) the constant flux

Green-Ampt model (Swartzendruber, 1974), and (3) the layered Green-Ampt model (Flerchinger
25

et al., 1988). The parameters 2 , 2 , K , and K are required in most models. If no measurements s o s o

are available, then 2 can be approximated based on soil texture (Table 7), and the residual or s

irreducible water content, 2r, can be used in place of 2o (Table 8), even though this would tend to

overestimate infiltration. Ks and Ko correspond to the hydraulic conductivities at the respective

water contents (i.e., K(2 ) and K(2 )). K may be estimated based on Table 11. However, s o s

Bouwer (1966) recommended that Ks be taken as one half of Ksat, and Ko may be assumed to be

zero. If the water content at the soil surface and the initial water content are quite different from

that of the saturated and residual water contents, respectively, then Ks and Ko may be estimated

from either the Brooks-Corey or the van Genuchten model for K(2) (van Genuchten et al., 1991). 

Typical values of 2 , and 2 for different textures are provided in Tables 7 and 8. Similarly, s o

typical values of h and 8, respectively, for different textures are provided in Tables 9 and 10. b

Only the magnitude of hb is provided in Table 9 (i.e., the negative sign is omitted). Table 11

provides a summary of soil hydraulic parameters commonly used in the unsaturated zone models

(Rawls et al., 1992).

Table 7. Typical values of saturated volumetric water content (2 ). s

Texture Brakensiek et al., 1981 Panian, 1987 Carsel and Parrish, 1988

Sand 0.35 0.38 0.43

Loamy Sand 0.41 0.43 0.41

Sandy Loam 0.42 0.44 0.41

Loam 0.45 0.44 0.43

Silty Loam 0.48 0.49 0.45

Sandy Clay Loam 0.41 0.48 0.39

Clay Loam 0.48 0.47 0.41

Silty Clay Loam 0.47 0.48 0.43

Silty Clay 0.48 0.49 0.36

Clay 0.48 0.49 0.38
26

Table 8. Typical values of residual volumetric water content (2 ). r

Texture Brakensiek et al., 1981 Panian, 1987 Carsel and Parrish, 1988

Sand 0.054 0.020 0.045

Loamy Sand 0.060 0.032 0.057

Sandy Loam 0.118 0.045 0.065

Loam 0.078 0.057 0.078

Silty Loam 0.038 0.026 0.067

Sandy Clay Loam 0.188 0.093 0.100

Clay Loam 0.185 0.107 0.095

Silty Clay Loam 0.155 0.089 0.089

Silty Clay 0.182 0.102 0.070

Clay 0.226 0.178 0.068

Table 9. Typical values of air-entry head (h )(cm). b

Texture Brakensiek et al., 1981 Panian, 1987 Carsel and Parrish, 1988

Sand 35.30 3.58 6.90

Loamy Sand 15.85 1.32 8.06

Sandy Loam 29.21 9.01 13.33

Loam 50.94 19.61 27.78

Silty Loam 69.55 31.25 50.00

Sandy Clay Loam 46.28 7.81 16.95

Clay Loam 42.28 31.25 52.63

Silty Clay Loam 57.78 30.30 100.00

Silty Clay 41.72 15.87 200.00

Clay 63.96 10.00 125.00
27

Table 10. Typical values of pore size index (8).

Texture Brakensiek et al., 1981 Panian, 1987 Carsel and Parrish, 1988

Sand 0.57 0.40 1.68

Loamy Sand 0.46 0.47 1.28

Sandy Loam 0.40 0.52 0.89

Loam 0.26 0.40 0.56

Silty Loam 0.22 0.42 0.41

Sandy Clay Loam 0.37 0.44 0.48

Clay Loam 0.28 0.40 0.31

Silty Clay Loam 0.18 0.36 0.23

Silty Clay 0.21 0.38 0.09

Clay 0.21 0.41 0.09
28

Table 11. Typical values for soil hydraulic property parameters (Rawls et al., 1992).

Texture Class Sample

Size

Total Residual Hydraulic

Porosity (N) Water Content Conductivity

 Effective Water Water

Porosity Bubbling Pressure (h ) Pore Size Distribution (8) Retained at Retained at

(Ne) -33 kPa -1500 kPa

b

Saturated

(K )s

cm /cm cm /cm cm /cm cm /cm cm /cm cm/hr 3 3 3 3 3 3 Arithmetic Geometric Arithmetic Geometric

cm cm

** **

3 3 3 3

Sand 762 21.00 0.437 0.020 0.417 15.98 7.26 0.694 0.592 0.091 0.033 *

(.374-.500) (.001-.039) (.354-.480) (.24-31.72) (1.36-38.74) (.298-1.090) (.334-1.051) (.018-.164) (.007-.059)

Loamy Sand 338 6.11 0.437 0.035 0.401 20.58 8.69 0.553 0.474 0.125 0.055

(.368-.506) (.003-.067) (.329-.473) (-4.04-45.20) (1.80-41.85) (.234-.872) (.271-.827) (.060-.190) (.019-.091)

Sandy Loam 666 2.59 0.453 0.041 0.412 30.20 14.66 0.378 0.322 0.207 0.095

(.351-.555) (.024-.106) (.283-.541) (-3.61-64.01) ( 3.45-62.24) (.140-.616) (.186-.558) (.126-.288) (.031-.159)

Loam 383 1.32 0.463 0.027 0.434 40.12 11.15 0.252 0.220 0.270 0.117

(.375-.551) (.020-.074) (.334-.534) (-20.07-100.3) (1.63-76.40) (.086-.418) (.137-.355) (.195-.345) (.069-.165)

Silt Loam 1206 0.68 0.501 0.015 0.486 50.87 20.76 0.234 0.211 0.330 0.133

(.420-.582) (.028-.058) (.394-.578) (-7.68-109.4) (3.58-120.4) (.105-.363) (.136-.326) (.258-.402) (.078-.188)

Sandy Clay Loam 498 4.3 0.398 0.068 0.330 59.41 28.08 0.319 0.250 0.255 0.148

(.332-.464) (.001-.137) (.235-.425) (-4.62-123.4) (5.57-141.5) (.079-.889) (.125-.502) (.186-.324) (.085-.211)

Clay Loam 366 .23 0.464 0.075 0.390 56.43 25.89 0.242 0.194 0.318 0.197

(.409-.519) (.024-.174) (.279-.501) (-11.44-124.3) (5.80-115.7) (.070-.414) (.100-.377) (.250-.386) (.115-.279)

Silty Clay Loam 689 .15 0.471 0.040 0.432 70.33 32.56 0.177 0.151 0.366 0.208

(.418-.524) (.038-.118) (.347-.517) (-3.26-143.9) (6.68-158.7) (.039-.315) (.090-.253) (.304-.428) (.138-.278)

Sandy Clay 45 .12 0.430 0.109 0.321 79.48 29.17 0.223 0.168 0.339 0.239

(.370-.490) (.013-.205) (.201-.435) (-20.15-179.1) (4.96-171.6) (.048-.398) (.078-.364) (.245-.433) (.162-.316)

Silty Clay 127 .09 0.479 0.56 0.423 76.54 34.19 0.150 0.127 0.387 0.250

(.425-.533) (.024-.136) (.334-.512) (-6.47-159.6) (7.04-166.2) (.040-.260) (.074-.219) (.332-.442) (.193-.307)

Clay 291 .06 0.475 0.090 0.385 85.60 37.30 0.165 0.131 0.396 0.272

(.427-.523) (.015-.195) (.269-.501) (-4.92-176.1) (7.43-187.2) (0.31-.293) (.068-.253) (.326-.466) (.208-.336)

*First line is the mean value, and second is one standard deviation about the mean.

**Antilog of the log mean.
hf

'

1

Ks

m

hi

0

K(h) dh

hf'

1

2Ks

m

hi

0

1 %

2(h)&20

2s&20

K(h)dh

hf'

0

0&1

he

S ' I/ t

29

(25)

(26)

(27)

(28)

The Green-Ampt model also requires two additional parameters, hs and hf, which correspond to the

soil-water pressure at the surface and at the wetting front, respectively. While hs is simply taken to be

equal to the constant surface ponding depth, the estimation of hf is difficult. It was believed for a long time

since Green and Ampt proposed the model in 1911 that this parameter had no obvious physical

significance. Bouwer (1966) was the first one to suggest that h can be related to measurable soil f

characteristics as follows:

where K(h) is the unsaturated hydraulic conductivity expressed as a function of the soil-water pressure h

(negative), and h is the initial soil-water pressure at the antecedent water content. Neuman (1976) i

provided a derivation of Equation 25 based on the Green-Ampt concept of piston displacement. He further

stated that Equation 25 may be inappropriate for small times, since it takes some time for piston type flow

to develop, and presented the following expression (Neuman, 1976) for h valid for small times: f

It is seen that if the Green-Ampt concept of an abrupt profile is invoked, then 2 = 2s, and Equation 26

immediately reduces to Equation 25. Based on a Brooks-Corey type model for K(h) and using Equation

25, Brakensiek and Onstad (1977) obtained the following form for h :

f

where 0 is the exponent of the Brooks-Corey conductivity model and he is the air exit head. 0 is calculated

as (2 + 38), where 8 is the exponent of the Brooks-Corey water retention model. The air exit head, h ,

e

may be taken as equal to one half of the air entry head, or the bubbling pressure head, h . b

4.4 Estimation of Philip's Two-Term Model Parameters

The Philip's two-term model requires two parameters, S and A. S is the sorptivity of a soil defined by

Equation 28 for the case of absorption or horizontal infiltration.

Sorptivity embodies in a single parameter the influence of capillarity on the transient infiltration process that

occurs when a soil is introduced to a step change in water content at the surface from 2o to 2s. It should be

noted that S depends strongly upon 2 and 2 , and has meaning only in relation to an initial state of the soil o s

and an imposed boundary condition (Philip, 1969). Therefore, typical S values for different soil textures

cannot be tabulated.

The primary difficulty in the application of the Philip's two-term model lies in the use of A over the

whole time range. It is seen from Equation 3 that as t approaches infinity, the infiltration rate decreases

monotonically to final infiltration rate, which is equal to K . This does not imply that A must be equal to K . s s

Philip (1969) showed that other reasonable values for A are, Ks/2, 2Ks/3, and 0.38Ks. Further, Philip

(1974) showed that A can also be equal to 0.363Ks, and that this value for A, among all the others, resulted

in the best agreement with the exact q(t) calculations for a real soil. Due to these reasons, Philip (1969)

recommended that this model be only used for times that are not very large. 
30

CHAPTER 5. EXAMPLE APPLICATION SCENARIOS

Six example application scenarios were chosen, and are illustrated in the following discussion. An

application scenario is presented for each model. The intention of these scenarios is to provide guidance

for users to apply infiltration models for various field conditions.

5.1 Semi-Empirical Model (SCS)

Statement of the Problem: This scenario was chosen to simulate water infiltration through a loamy

sand soil for conditions with rainfall and surface runoff. A hypothetical site has been contaminated with

hazardous waste materials and an estimate of the contaminant transport through the vadose zone is desired. 

Since the soil water infiltrating flux is an important factor controlling contaminant migration in the vadose

zone, it must be first estimated before performing the contaminant transport modeling.

Conceptual model: Figure 3 is a schematic diagram of the hypothetical site showing the conceptual

model for this scenario. The sandy material is bounded above by the soil surface and below by the groundwater table. Factors that affect the rate of water infiltration into the soil include initial soil moisture

content, rainfall rate and duration, and surface runoff.

In this scenario, the purpose is to calculate the daily infiltration amount into the soil profile vertically

during a process of rainfall and surface runoff. Since little is known concerning the mathematical boundary

conditions for the site and little soil physical data available, the semi-empirical model known as the SCS

model is chosen for the simulations.

Input Parameters: Input parameters for running the simulations are obtained from several sources. 

Attempts are made to obtain realistic values for the input parameters. Table 12 lists the input parameter

values used for this scenario. In this model, only two input parameters are required (refer to Equations 1

and 2). For detailed estimation of Fw under different soil conditions, readers are encouraged to read the

USDA report cited in the table. The daily rainfall amount can be obtained from a local weather station or

climatic research institutes. 

Table 12. Input parameter values for simulations with a sandy soil using the SCS model.

Parameter Symbol Value Units Reference

Statistically USDA-SCS,

Derived 1972 F 8.2 inches w

Daily Rainfall

Amount P 0.1 to 10.0 inches Assumed

 Simulation Results: Figure 4 shows the daily infiltration amount as a function of the daily rainfall

amount while Figure 5 shows the surface runoff as a function of the daily rainfall amount. Each point on

the curves of Figures 4 and 5 could represent one rainfall event. In Figure 5, the runoff does not occur in

the event where the daily rainfall amount is smaller than 1.64 inches (0.2F =1.64 inches). Before runoff w

occurs, the daily infiltration amount equals the daily rainfall amount. After that, the daily infiltration

amount increases as the daily rainfall amount increases but the rate of the increase of the daily infiltration

decreases (Fig. 4). Note that the SCS model is applicable to the situation in which the daily amounts of 

rainfall, runoff and infiltration are of interest. 
31

Figure 5. Daily runoff amount as a

function of daily rainfall amount.

Figure 4. Daily infiltration amount as a

function of daily rainfall amount.

Figure 3. Conceptual model for simulating water infiltration using the SCS model.
32

5.2 Infiltration Model for Homogeneous Conditions (PHILIP2T)

Statement of the Problem: This scenario was chosen to simulate the vertical infiltration of water into

a sandy soil with a homogeneous soil profile. A hypothetical site has been contaminated with hazardous

waste materials. In order to estimate the transport of the contaminants in the vadose zone, the infiltration

rate (water flux) must be known before performing transport modeling. Initial soil water constant at the

site is approximately uniformly distributed and constant. Soil water content at the inflow end (at the

surface) is held constant and near saturation. Other soil physical properties such as saturated water content

and saturated hydraulic conductivity values are available. 

Conceptual model: Figure 6 is a schematic diagram of the hypothetical site showing the conceptual

model for this scenario. This site is bounded above by the soil surface and below by the ground-water table.

The water depth is much greater than the water penetration depth. The free water is available in excess at

the surface and a constant water content at the surface is maintained through the infiltration period. In this

scenario, the Philip’s two-term model was selected for the simulations. 

Input Parameters: Input parameters for running the simulations are obtained from several sources. 

Attempts are made to obtain realistic values for the input parameters. Table 13 lists the input parameter

values used in the simulations. The sorptivity (S) and the constant (A) can be obtained simply by

determining the slope and intercept of I/t versus t in Equation 4, respectively. -1/2

Table 13. Input parameter values for the simulations in a sandy soil using PHILIP2T model.

Parameter Symbol Value Units Reference

Duration of t 1 to 24 h Assumed

Infiltration

Sorptivity S 1 cm/h Philip, 1969 1⁄2

Constant A 7.6 (=0.363 ks) cm/h Jury et al., 1991

Saturated Hydraulic K 21 cm/h Carsel and Parrish,

Conductivity 1988

s

Simulation Results: Figure 7 shows a typical soil infiltration pattern, with an infiltration rate

relatively high at the onset of the infiltration, then decreasing, and eventually approaching a constant rate at

t > 20 h. Figure 8 illustrates a linear relationship between the cumulative infiltration and time. By knowing

the soil water infiltration rate, the movement of contaminants in the soil could be evaluated using other

transport models.
33

Figure 7. Surface water infiltration rate

as a function of time.

Figure 8. Cumulative infiltration as a

function of time.

Figure 6. Conceptual model for simulating water infiltration using Philip’s Two-Term

Model.
34

5.3. Infiltration Model for Nonhomogeneous Conditions (GALAYER)

Statement of the Problem: Since the soil profiles are differentiated into layers, the water distribution

during infiltration is not uniform. Two scenarios were selected to calculate water infiltration into layered

soils. In Scenario 1, the soil consists of two layers with a sand layer underlain by a loam. In Scenario 2,

the soil consists of three layers with a sand layer underlain by a loam, and the loam underlain by a clay

layer. Under such conditions, the infiltration rate is initially controlled by the sand layer, and with time the

infiltration rate will become controlled by the lower fine-textured layer with the low hydraulic conductivity. 

Simulation results from these scenarios would provide information concerning water infiltrating into 

nonhomogeneous soil profiles. 

Conceptual model: Figures 9a and 9b are the schematic diagrams of the layered soils showing the 

conceptual model for the Scenario 1 and the Scenario 2, respectively. In Scenario 1, the soil profile

consists of sand and loam. The thickness of the sand layer is 10 cm. The loam layer extends from the

bottom of the first layer to the bottom of the soil profile. Scenario 1 is to estimate water infiltration into

the sand layer and continuing through the loam layer. Scenario 2 is to estimate water infiltration into three

layers (with sand, loam and clay in sequence). The thickness of the upper layers (sand and loam) is 10 cm.

The lowest layers (loam or clay) extend to the rest of the soil profile. In these scenarios, the GALAYER

(Flerchinger et al., 1988) model for layered soils was selected for the simulations. The comparison of the

two scenarios for water infiltration was also provided. 

Input Parameters: Input parameters for running the simulations are obtained from several sources. 

Attempts are made to obtain realistic values for the input parameters. Table 14 listed the input parameter

values used in the simulations. 

Table 14. Input parameter values for the simulations using the GALAYER (Flerchinger et al., 1988) 

model.

Parameter Symbol Value Units Reference

Duration of Infiltration t 1 to 24 h Assumed

Saturated Hydraulic

Conductivity (Layer 1) K1 1 cm/h Hillel, 1982

Saturated Hydraulic

Conductivity (Layer 2) K2 0.5 cm/h Hillel, 1982

Saturated Hydraulic

Conductivity (Layer 3) K3 0.1 cm/h Hillel, 1982

Change in Volumetric

Water Content in Layer 2 )2 0.2 cm /cm Assumed

for Scenario 1.

2

3 3

Change in Volumetric

Water Content in Layer 3 )2 0.1 cm /cm Assumed

for Scenario 2.

3

3 3

Suction Head of Layer 2

for Scenario 1. H2 3000 cm Hillel, 1982

Suction Head of Layer 3

for Scenario 2. H3 7000 cm Hillel, 1982
35

Figure 9. Conceptual model for simulating water infiltration into a layered soil.

Figure 10. Comparison of water

infiltration between Scenario 1

(solid line) and Scenario 2 (dotted

line).

Simulation Results: Water infiltration into the layered soil as a function of time is shown in Figure 10. 

 The infiltration rate was relatively high at the onset of the infiltration, then decreased, and eventually

approached a constant rate. Comparison of the two scenarios revealed that water infiltration is faster in the

soil with two layers (Scenario 1) than in the soil with three layers (Scenario 2).
36

5.4 Infiltration Model for Ponding Conditions (GAEXP)

Statement of the Problem: This scenario was chosen to simulate water infiltration into a sandy soil

under surface ponding conditions. A water depth of 1 cm was applied at the soil surface. The initial soil

water content of 0.05 cm /cm is uniformly distributed in the soil profile. Under such continuous ponding 3 3

conditions, the infiltration rate (equal to the infiltrability) can be expected to settle down to a steadyinduced rate which is practically equal to the saturated conductivity.

Conceptual model: Figure 11 is a schematic diagram of a sandy soil profile showing the conceptual

model for this scenario. This soil profile is bounded above by the soil surface which is under the water

ponding condition and below by a groundwater table. In this scenario, the Explicit Green-Ampt model

was selected for the simulations.

Input Parameters: Input parameters for running the simulations are obtained from several sources. 

Attempts are made to obtain realistic values for the input parameters. Table 15 lists the input parameter

values used in the simulations.

Table 15. Input parameters for the simulations in a clay soil using GAEXP Model.

Parameter Symbol Value Units Reference

Saturated Hydraulic K 21 cm/h Carsel and Parrish,

Conductivity 1988

s

Ponding Depth hs 1 cm Assumed

Saturated Volumetric 2 0.43 cm /cm Hillel, 1982

Water Content

s

3 3

Initial Volumetric 2 0.05 cm /cm Assumed

Water Content

o

3 3

Simulation Results: Figures 12 and 13 show the surface infiltration and cumulative infiltration as a

function of time, respectively. A typical soil infiltration pattern, with an infiltration rate relatively high at

the onset of the infiltration, then decreasing, and eventually approaching a constant rate was obtained for

this sandy soil (Figure 12). 
Groundwater Table

Soil surface

Vadose zone

Ponding water

(Sand)

Infiltration

1 cm

37

Figure 11. Conceptual model for simulating water infiltration using the Explicit GreenAmpt model.

Figure 13. Cumulative infiltration as a

function of time. Figure 12. Water infiltration as a function

of time.
38

5.5 Infiltration Model for Non-ponding Conditions (GACONST)

Statement of the Problem: This scenario was chosen to simulate water infiltration into a sandy

loam soil under non-ponding conditions. The initial soil water content of 0.05 cm /cm is uniformly 3 3

distributed in the soil profile. Under this non-ponding condition, water from rainfall or surface irrigation 

either enters the soil or flows over the surface as runoff. 

Conceptual model: Figure 14 is a schematic diagram of a sandy loam soil profile showing the

conceptual model for this scenario. This soil profile is bounded above by the soil surface which is under the

non-ponding conditions and below by the groundwater table. No ponding occurs after the soil is saturated

and the excess water is discharged as surface run-off. Several infiltration models for non-ponding

conditions have been developed (Philip, 1957; Swartzendruber, 1974). In this scenario, the Constant Flux

Green-Ampt Model was chosen for the simulations. 

Input Parameters: Input parameters for running the simulations are obtained from Carsel and Parrish 

(1988). Attempts are made to obtain realistic values for the input parameters. Table 16 listed the input

parameter values used in the simulations.

Table 16. Input parameters for the simulations with r >Ks in a sandy loam soil using the GACONST

model.

Parameter Symbol Value Units Reference

Saturated Hydraulic K 2.59 cm/h Carsel and Parrish,

Conductivity 1988

s

Constant Application r 3.5 cm/h Assumed

Rate

Saturated Volumetric 2 0.41 cm /cm Carsel and Parrish,

Water Content 1988

s

3 3

Initial Volumetric 2 0.05 cm /cm Assumed

Water Content

o

3 3

Air Exit Head h -13.33 cm Carsel and Parrish, e

1988

Pore Size Index 8 0.89 - Carsel and Parrish,

1988

Simulation Results: Figure 15 shows a constant-flux infiltration pattern when r>Ks

. Before the

surface saturation first occurs at time t ( 5 hours), infiltration rate is constant and equals to r, after that, 0

infiltration rate decreases with time. Figure 16 illustrates a linear relationship between the cumulative

infiltration and time.
39

Figure 14. Conceptual model for simulating water infiltration using the constant flux

Green-Ampt model.

Figure 15. Water infiltration as a function

of time. Figure 16. Cumulative infiltration as a

function of time.
40

5.6 Infiltration and Exfiltration Model (INFEXF)

Statement of the Problem: The vertical movement of soil water in subsurface environments can be

divided into two major processes: (1) infiltration and (2) exfiltration. An example was selected to calculate

water infiltration and exfiltration in a homogeneous soil profile during wetting and drying seasons. The

exfiltration process includes the capillary rise mechanism, evaporation, and water uptake by plant roots

(transpiration).

Conceptual Model: Figure 17 is a schematic diagram of a sandy loam soil profile showing the

conceptual model for this scenario. This soil profile is semi-infinite. The groundwater table is much greater

than the water penetration depth. The surface rainfall, evaporation, and transpiration are included in the

model. In this scenario, the INFEXF (Eagleson, 1978) model was chosen for the simulations. Two cases

are presented separately. The first case is for infiltration. Infiltration is simulated during the storm period.

The initial water content (2 ) before the storm is 0.07 cm /cm . This case only considers the condition that 0

3 3

rainfall intensity is greater than the infiltration capacity. In the second case, a uniform initial water content

(2 ) of 0.20 cm /cm is assumed through redistribution after the storm. Also potential evaporation is 0

3 3

assumed to be greater than the exfiltration capacity. The dry condition at the soil surface causes capillary

rise and evaporation of water out of the soil surface. This case is simulated by the exfiltration equation 23.

Input Parameters: Input parameters for running the simulations are obtained from Eagleson

(1978) and Carsel and Parrish (1988). Attempts are made to obtain realistic values for the input

parameters. Table 17 lists the input parameter values used in the simulations. For estimations of other

parameters in Equations 22 and 23, readers are referred to the report by Eagleson (1978). 

Table 17. Input parameters for the simulations using the INFEXF model.

Parameter Symbol Value Units Reference

Saturated Hydraulic Carsel and Parrish,

Conductivity 1988 Ks 2.59 cm/h

Vegetated Fraction M 0.2 Eagleson, 1978

Transpiration Rate Ev 0.1 cm/h Eagleson, 1978

Saturated

Volumetric Water 2 0.41 cm /cm

Content

s

3 3 Carsel and Parrish,

1988

Initial Volumetric

Water Content 2o 0.05 cm /cm Assumed 3 3

Pore Distribution

Index 8 .89 - Eagleson, 1978

Initial Volumetric

Water Content

During Interstorm

Period

2o 0.20 cm /cm Assumed 3 3

 Simulation Results: Figure 18 shows a typical soil infiltration pattern, with an infiltration rate

relatively high at the onset of the infiltration, then decreasing, and eventually approaching a constant rate. 

Figure 19 illustrates that exfiltration (actual evaporation) decreases with time. Negative value of 

exfiltration indicates that exfiltration has ceased (at the time of 11 hours in this case) but transpiration is

still going on.
41

Figure 18. Water infiltration as a function

of time.

Figure 17. Conceptual model for simulating water infiltration and exfiltration.

Figure 19. Water exfiltration as a

function of time.
42

REFERENCES

Ahuja, L.R. and J.D. Ross. 1983. A new Green-Ampt type model for infiltration through a surface seal

permitting transient parameters below the seal. In: Advance infiltration (Proc. Nat. Conf. On Advances

in Infiltration), ASAE. 147-162.

Bouwer, H. 1966. Rapid field measurement of air entry value and hydraulic conductivity of soil as

significant parameters in flow system analysis. Water Resour. Res., 2:729-738.

Bouwer, H. 1969. Infiltration of water into nonuniform soil. J. Irrigation and Drainage Division of ASCE

95:451-462.

Brakensiek, D. L., R. L. Engelman, and W. J. Rawls. Variation within texture classes of soil water

parameters, Transaction of the ASAE 24:335-339, 1981.

Brakensiek, D.L., and C.A. Onstad, Parameter estimation of the Green and Ampt infiltration equation.

Water Resour. Res., 13:1009-1012, 1977.

Breckenridge, R.P., J.R. Williams, and J.F. Keck. 1991. Characterizing soil for hazardous waste site

assessments. EPA Ground-Water Issue Paper. EPA/540/4-91/003.

Carsel, R.F., and R.S. Parrish. 1988. Developing joint probability distributions of soil water retention

characteristics. Water Resour. Res., 24:755-769.

Childs, E.C., and M. Bybordi. 1969. The vertical movement of water in stratified porous materials. I.

Infiltration. Water Resour. Res., 5:446-459. 

Corradini, C., F. Melone, and R.E. Smith. 1994. Modeling infiltration during complex rainfall sequences. 

Water Resour. Res., 30:2777-2784.

Eagleson, P. 1978. Climate, soil, and vegetation. 3. A simplified model of soil moisture movement in the

liquid phase. Water Resour. Res., 14:722-730.

Flerchinger, G.N., F.J. Waltts, and G.L. Bloomsburg. 1988. Explicit solution to Green-Ampt equation for

nonuniform soils. J. Irrig. and Drain. Div., ASCE, 114:561-565. 

Fok, Y. -S. 1970. One-dimensional infiltration into layered soils. J. Irrig. and Drain. Div., ASCE, 90:121-

129.

Ghildal, B.P., and R.P. Tripathi. 1987. Soil Physics. John Wiley & Sons. New York. 

Haverkamp, R, J.Y. Parlange, J.L. Starr, Schmitz, and C. Fuentes. 1990. Infiltration under ponded

conditions: 3. A predictive equation based on physical parameters. Soil Sci., 149:292-300.

Haverkamp, R., P.J. Ross, and J.Y. Parlange. 1994. Three-dimensional analysis of infiltration from the

disc infiltrometer, 2. Physically based infiltration equation. Water Resour. Res. 30:2931-2935.

Hillel, D. 1982. Introduction to Soil Physics. Academic Press. New York. 

Hillel, D., and W.R. Gardner. 1970. Measurement of unsaturated conductivity diffusivity by infiltration
43

through an impeding layer. Soil Sci., 109:149.

Johnson, J. A., and V. Ravi. 1993. Uncertainty in the use of literature soil hydraulic parameter data in

ground-water modeling of the vadose zone, Robert S. Kerr Environmental Research Laboratory, Ada,

Oklahoma (unpublished).

Jury, W.A., W.R. Gardner, and W.H. Gardner. 1991. Soil Physics. John Wiley & Sons. New York.

Leij, F.J., W.J. Alves, Martinus Th. van Genuchten, and J.R. Williams. 1996. The UNSODA unsaturated

soil hydraulic database, User’s Manual Version 1.0. National Risk Management Research Laboratory,

Office of Research and Development, U.S. EPA, Cincinnati, Ohio 45268. EPA/600/R-96/095.

Li, E. A., V.O. Shanholtz, and E. W. Carson. 1976. Estimating saturated hydraulic conductivity and 

capillary potential at the wetting front. Department of Agricultural Engineers, Virginia Polytechnic

Institute and State University, Blacksburg.

Moore, I.D. 1981. Infiltration equation modified for subsurf`ce effects. J. Irrigation and Drainage

Division, ASCE( 107:71-86.

Neuman, S.P. 1976. Wetting front pressure head in the infiltration model of Green and Ampt. Water

Resour. Res., 12:564-566.

Pajian, T. F. 1987. Unsaturated flow properties data catalog, Volume II, Publication #45061, Water

Resources Canter, Desert Research Institute, DOE/NV/10384-20.

Parlange, J. Y., R. Haverkamp, and J. Touma. 1985. Infiltration under ponded conditions. 1. Optimal

analytical solution and comparison with experimental observations. Soil Sci., 139: 305-311.

Philip, J.R. 1957. The theory of infiltration* 4. Sorptivity and algebraic infiltration equations. Soil Sci.,

84:257-264.

Philip, J.R. 1969. Theory of infiltration. Advances in Hydroscience. 5:215-296. Academic Press, New

York, NY.

Philip, J.R. 1974. Recent progress in the solution of nonlinear diffusion equations. Soil Sci,, 117: 257-264,

Ravi, V and J.R Williams. 1998. Estimation of infiltration rate in the vadose zone: Compilation of simple

mathematical models. Volume I. U.S. Environmental Protection Agency, Subsurface Protection and

Remediation Division, National Risk Management Research Laboratory, Ada, Oklahoma, 74820.

EPA/600/R-97/128a.

Rawls, W.J., L. R. Ahuja, and D. L. Brakensiek. 0992. Estimatinf soil hydraulic properties from soils data. 

In: van Genuchten, M. Th, F. J. Leij, and L. J. Lund (ed.). Indirect Methods for Estimating the

Hydraulic Properties of Unsaturated Soils. Proceedings of the international workshop on indirect

methods for estimating the hydraulic properties of unsaturated soils. Riverside, California, October 11-

13, 1989. University of California, Riverside, CA 92521, USA.

Salvucci, G.D., and D. Entekhabi. 1994. Explicit expressions for Green-Ampt (Delta function diffusivity)

infiltration rate and cumulative storage. Water Resour. Res., 30:2661-2663.

Schwab, G. O., R. F. Frevert, T. W. Edminster, and K. K. Barnes. 1981. Soil and Water Conservation
44

Engineering. 3 Edition. John Wiley and Sons, New York. rd

Swartzendruber, D. 1974. Infiltration of constant-flux rainfall into soil as analyzed by the approach of

Green and Ampt. Soil Sci., 117: 272-281. 

van der Heijde, P.K.M., and O.A. Elnawawy. 1992. Quality assurance and quality control in the

development and application of ground-water models. Robert S. Kerr Environmental Research

Laboratory, U.S. Environmental Protection Agency, Ada, OK 74820. EPA/600/R-93/011.

van Genuchten, M. Th., F.J. Leij, and L.J. Lund. 1989. Indirect methods for estimating the hydraulic

properties of unsaturated soils. Proceedings of the international workshop on indirect methods for

estimating the hydraulic properties of unsaturated soils. Riverside, California, October 11-13, 1989. 

University of California, Riverside, CA 92521, USA.

van Genuchten, M. Th., F.J. Leij, and S. R. Yates. 1991. The RETC code for quantifying the Hydraulic

Functions of unsaturated soils. Robert S. Kerr Environmental Research Laboratory, U.S.

Environmental Protection agency, Ada, OK 74820. EPA/600/2-91/065.

USDA-SCS. 1972. National Engineering Handbook, Hydrology Section 4. USDA, Washington,DC.

US EPA, 1994a. Assessment framework for ground-water model applications. OSWER Directive

#9029.00. EPA/500/B-94/003.

US EPA, 1994b. A technical guide to ground-water model selection at sites contaminated with radioactive

substances. EPA/402/R-94/012.

US EPA, 1996. Documenting ground-water modeling at sites contaminated with radioactive substances.

EPA/540/R-96/003, PB96-963302, January 1996. 
APPENDICES
APPENDIX A

User’s Instructions for MathCad Worksheets
MathCad Plus 6.0 is a registered trademark of Mathsoft, Inc., of Cambridge, 3

Massachusetts. Mention and use of this package, or any other commercial product does not

constitute endorsement or recommendation for use.

A - 2

Appendix A

User Instructions for Mathcad Worksheets

Infiltration models presented in this report have been incorporated into MathCad Plus 6.0 worksheets ®

to facilitate their ease of use. These worksheets are provided in both hardcopy and electronic formats, and 3

are also available through the U.S.EPA/NRMRL/SPRD’s Center for Subsurface Modeling Support

(CSMoS). They have been developed on systems running Windows 3.x and Windows 95. The electronic

formats from CSMoS can be obtained via diskette exchange or through the Center’s website at

http://www.epa.gov/ada/kerrlab.html. Each model is presented with a brief description, definition of

variables, equations, results, discussion, and an example sensitivity analysis.

Before using these worksheets, make a backup copy of the files for retrieval of the initially distributed

worksheets including equations and variable quantities. For greater accessibility to the worksheets, copy

them to a subdirectory of the hard disk, preferably under a Mathcad associated directory. For example, if

the main Mathcad subdirectory has the name winmcad, create a subdirectory under this directory with the

name infiltra by entering the following from a DOS prompt.

C:\>cd winmcad

C:\winmcad>md infiltra

C:\winmcad>cd infiltra

C:\winmcad\infiltra>copy a:*.mcd (Assuming that the distribution diskette is in drive A:.)

MathCad can now be executed from Windows, and the files of interest loaded from the winmcad\infiltra

directory. There are a total of six (6) files distributed including: SCS.mcd, philip2t.mcd, galayer.mcd,

gaexp.mcd, gaconst.mcd, stochmac.mcd, and infexf.mcd. Values for the various parameters can be easily

changed, and the resulting output can quickly be observed to evaluate any changes resulting in the output. 

By utilizing the tiling features within Windows, the same worksheet can be loaded several times to allow

side-by-side evaluation of results. As a word of caution, these files are not protected in any way from

changes that can be made by the user. Therefore, any changes made and saved back to the same file name

will result in the original information being lost.
B - 1

Appendix B

MathCad Worksheets for Infiltration Models

Note that all references cited in the individual worksheets are listed

in the references section of this document.
B - 1

Appendix B

MathCad Worksheets for Infiltration Models

Note that all references cited in the individual worksheets are listed

in the references section of this document.
B1. SCS Model

A. Description

The empirical approach to develop water infiltration equation consists of first finding a mathematical function 

whose shape as a function of time matches the observed features of the infiltration rate and then attempting a 

physical explanation of the process (Jury et al., 1991). In semi-empirical models, most physical processes are 

represented by commonly accepted and simplistic conceptual methods rather than by equations derived from 

fundamentally physical principles. The commonly used semi-empirical infiltration model in the fields of soil 

physics and hydrology is Soil Conservation Service (SCS) model. A scenario was chosen to simulate water 

infiltration into a soil for conditions with rainfall and surface run off by using the SCS model. Input parameters 

and simulation results are discussed below. 

B. Definition of Variables

P 0.0, 0.4.. 10.0 Daily rainfall amount (inches)

F w 8.2 Statistically derived parameter (Water retention parameter), (inches)

C. Equations

R(P) if

P 0.2.F w

2

P 0.8.F w

P>0.2. F w

0 otherwise

Daily surface runoff (inches) (1)

q(P) P R(P) (2) Daily infiltration amount (inches)

D. Results

P

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

4.4

R(P)

0

0

0

0

0

0.015

0.064

0.144

0.249

0.378

0.527

0.695

q(P)

0

0.4

0.8

1.2

1.6

1.985

2.336

2.656

2.951

3.222

3.473

3.705

B1 - 1
4.8

5.2

5.6

6

6.4

6.8

7.2

7.6

8

8.4

8.8

9.2

9.6

10

0.879

1.078

1.29

1.514

1.748

1.993

2.247

2.509

2.778

3.055

3.338

3.626

3.921

4.22

3.921

4.122

4.31

4.486

4.652

4.807

4.953

5.091

5.222

5.345

5.462

5.574

5.679

5.78

0 5 10 15 0

5

10

Daily rainfall amount (inches)

Daily infiltration amount (inches)

q( P )

P

0 5 10 0

3

6

Daily rainfall amount (inches)

Daily surface runoff (inches)

R( P)

P

Figure B1-1. Daily infiltration amount as 

a function of daily rainfall amount.

Figure B1-2. Daily runoff amount as a 

function of daily rainfall amount.

E. Discussion

Figure B1-1 shows the daily infiltration amount as a function of the daily rainfall amount while Figure B1-2 shows 

the surface runoff as a function of the daily rainfall amount. Each point on the curves of Figs. B1-1 and B1-2 

could represent one rainfall event. In figure B1-2, the runoff does not occur in the event where the daily rainfall 

amount is smaller than 1.64 inches (0.2Fw =1.64 inches). Before runoff occurs, the infiltration amount equals 

the rainfall amount. After that, the daily infiltration amount increases as the daily rainfall amount increases but 

the rate of the increase of the daily infiltration decreases (Fig. B1-1). Note that the SCS model is applicable to the 

situation in which the daily amounts of rainfall, runoff and infiltration are of interest. 

B1 - 2
B2. Philip's Two-Term Model (PHILIP2T) 

 A. Description

The Philip's two-term model (PHILIP2T) is a truncated form of the power series solution of Philip 

(1957). During the initial stages of infiltration, i.e., when t is very small, the first term of Equation 1 

below dominates. In this stage the vertical infiltration proceeds at almost the same rate as absorption or 

horizontal infiltration, because the gravity component, represented by the second terms of Equation 1, is 

negligible. As infiltration continues, the second term becomes progressively more important until it 

dominates the infiltration process. Philips (1957) suggested the use of the two-term model in applied 

hydrology when t is not too large. A scenario was chosen to simulate the water infiltration into a sandy 

soil by using the PHILIP2T model. Input parameters and simulation results were given below.

B. Definition of Variables

t 1.. 24 Duration of infiltration (h)

S 1.0 The sorptivity of a soil defined by S = I/(t)1/2 (cm/h1/2) for the horizontal 

infiltration and is a function of the boundary and initial water contents 

Θ0

 and saturated water content Θs (Philip, 1969) 

K s 21 Saturated hydraulic conductivity (cm/h)

A 0.363.K s A constant in Equation 3 (cm/h)

C. Equations

q(t) . . 1

2

S t

1

2 A Infiltration rate (1)

I(t) S.t

1

2 A.t Cumulative infiltration (2)

0 10 20 30 7.6

7.8

8

8.2

Time (h)

Surface infiltration rate (cm/h)

q(t)

t

D. Results

t

1

2

3

4

5

6

7

8

9

10

11

12

13

q(t)

8.123

7.977

7.912

7.873

7.847

7.827

7.812

7.8

7.79

7.781

7.774

7.767

7 762

I(t)

8.623

16.66

24.601

32.492

40.351

48.187

56.007

63.812

71.607

79.392

87.17

94.94

102 705

Figure B2-1. Surface infiltration as a 

function of time. 

B2 - 1
14

15

16

17

18

19

20

21

22

23

24

7.757

7.752

7.748

7.744

7.741

7.738

7.735

7.732

7.73

7.727

7.725

110.464

118.218

125.968

133.714

141.457

149.196

156.932

164.666

172.396

180.125

187.851

0 10 20 30 0

100

200

Time (h)

Cumulative infiltration (cm)

I(t)

t

Figure B2-2. Cumulative Infiltration as a 

function of time.

E. Discussion

Figure B2-1 shows a typical soil infiltration pattern, with an infiltration rate relatively high at the 

onset of the infiltration, then decreasing, and eventually approaching a constant rate. The 

infiltration rate decreased by about 4% within the first 5 hours, then the infiltration rate decreased 

by 1.4% from from 5 to 20 hours, and finally decreased to 0.1% from 20 to 24 hours. Figure B2-2 

illustrates that the cumulative infiltration increased with time. This occurred because more water 

accumulated in the soil as the time increase.

F. Sensitivity and Relative Sensitivity of Infiltration Rate to the Sorptivity of Soil

 A sensitivity analysis is an estimation for how important an input parameter is to affecting the simulation results. 

 Mathematically the sensitivity coefficient, Ss, is defined as:

 Ss =df/dx (3) 

 

 where f represents the output of interest and x represents the input parameter (McCuen, 1973). The value of Ss 

 calculated from this equation has units associated with it. This make it difficult to compare sensitivities for 

 different input parameters. This problem can be overcome by using the relative sensitivity, Sr, given by

 Sr = S (x/f) (4)

 The relative sensitivity Sr gives the percentage change in response for each one percent change in the input 

 parameter. If the absolute value of Sr is greater than 1, the absolute value of the relative change in model output 

 will be greater than the absolute value of the relative change in input parameter. If the absolute value of Sr is less 

 than 1, the absolute value of the relative change in model output will be less than the absolute value of the relative 

 change in input (Nofziger et al., 1994).

This section shows the sensitivity, Ss, and the relative sensitivity, Sr, of the surface infiltration rate,

q, to the sorptivity of a soil, S. The expressions were obtained by applying Equations 3 and 4 to 

Equation 1 .

B2 - 2
F.1. Input Data

t 5 S 0, 0.1.. 2

F.2. Sensitivity

S s(S) . 1

2

t

1

2 (5)

F.3. Relative Sensitivity

(6) S r

(S) S.t

1

2

S. t

1

2 2.A

0 1 2 3 0.2232

0.2234

0.2236

0.2238

0.224

Sorptivity 

Sensitivity of infiltration rate

S s( S)

S

F.4. Results

S

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

S s(S)

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

0.224

S r(S)

0

2.925.10 3

5.832.10 3

8.723.10 3

0.012

0.014

0.017

0.02

0.023

0.026

0.028

0.031

0.034

0.037

0.039

0.042

0.045

0.047

0.05

0.053

0.055

Figure B2-3. Sensitivity of infiltration rate for 

different values of sorptivity of the soil at t = 5 h.

0 1 2 3 0

0.05

0.1

Soprtivity 

Relative sensitivity of infiltr. rate

S r( S)

S

Figure B2-4. Relative sensitivity of infiltration 

rate for different values of sorptivity of the soil at 

t = 5 h.

B2 - 3
F.5. Discussion

Figure B2-3 shows a sensitivity of the infiltration rate for different values of sorptivity of a soil, which 

has a constant value of 0.224 for all sorptivities of the soil at t = 5 h. This implies that a 10-fold increase 

in sorptivity will have a 2.24-fold increase in infiltration rate. Figure B2-4 shows the relative sensitivity 

of infiltration rate for different values of sorptivity of the soil. The relative sensitivity increased as the 

sorptivity of the soil increased. 

B2 - 4
B3. Green-Ampt Model for Layered Systems (GALAYER)

A. Description

The Green-Ampt Model has been modified to calculate water infiltration into nonuniform soils by 

several researchers (Bouwer, 1969; Fok, 1970; Moore, 1981; Ahuja and Ross, 1983). In this section, 

the Green-Ampt model for layering systems (GALAYER) developed by Flerchinger et al. (1989) was 

selected to calculate water infiltration over time in vertically heterogeneous soils. Two simulation 

scenarios were selected in the application. The first scenario was to estimate water infiltration into a 

soil with two layers (sand and loam), while the second scenario was to estimate the water infiltration 

into a soil with three layers (with sand, loam, and clay in sequence). Comparison of the two scenarios 

for water infiltration was also provided.

B. Definition of Variables

K1 1 Hydraulic conductivity (cm/h) for layer 1 (Top layer)

K2 0.5 Hydraulic conductivity (cm/h) for layer 2 (Middle layer) 

K3 0.1 Hydraulic conductivity (cm/h) for layer 3 (Bottom layer) 

t 1.. 24 Time for infiltration (h)

∆θ n1 0.2 Change in volumetric water content as wetting front passes in last 

layer for Scenario 1 (cm3/cm3)

K n1 K2 Hydraulic conductivity for last layer in Scenario 1 (cm/h)

H n1 3000 Potential head (cm) as the wetting front passes last layer for Scenario 1. 

n 2 Number of soil layers

Z1 10 Depth of layer 1 (cm)

Z2 10 Depth of layer 2 (cm)

∆θ n2 0.1 Change in volumetric water content as wetting front passes in last 

layer for Scenario 2 (cm3/cm3)

K n2 K3 Hydraulic conductivity for last layer in Scenario 2 (cm/h)

H n2 7000 Potential head (cm) as the wetting front passes last layer for Scenario 2. 

Z3 10 Depth of layer 3 (cm) for scenario 2 with n=3

C. Equations and Results

a. Scenario 1

n 2

n1 n

B3 - 1
td(t)

K . n1 t

∆θ . n1 H n1

= 1

n 1

i

Zi

Dimensionless time since wetting 

front penetration. (1)

Dimensionless depth accounting 

for the thickness and conductivity 

of layers behind wetting front.

zd . K n1

H n1

= 1

n 1

i

Zi

= 1

n 1

i

Zi

Ki

(2)

Dimensionless accumulated 

infiltration. (3) Fd(t) . 1

2

td(t) 2.zd (td(t) ) 2.zd 2 8.td(t)

fd(t) Fd(t) 1

Fd(t) zd Dimensionless infiltration rate. (4)

f1(t) . fd(t) K n1 Infiltration rate. (5)

t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

f1(t)

12.618

9.036

7.448

6.501

5.855

5.378

5.007

4.708

4.461

4.251

4.071

3.914

3.775

3.652

3.541

3.441

3.349

3.266

3.189

3.118

3.052

2.99

2.933

0 10 20 30

0

5

10

15

Time (h)

Infiltration Rate (cm/h)

f1( t)

t

Figure B3-1. Water infiltration as a function 

of time through the layered soil profile in 

Scenario 1.

B3 - 2
24 2.879

b. Scenario 2

n 3

n2 n

td(t)

K . n2 t

∆θ . n2 H n2

= 1

n 1

i

Zi

Dimensionless time since wetting 

front penetration. 

(6)

Dimensionless depth accounting 

for the thickness and conductivity 

of layers behind wetting front.

zd . K n2

H n2

= 1

n 1

i

Zi

= 1

n 1

i

Zi

Ki

(7)

Dimensionless accumulated 

Fd(t) . infiltration. 1

2

td(t) 2.zd (td(t) ) 2.zd 2 8.td(t) (8)

fd(t) Fd(t) 1

Fd(t) zd

Dimensionless infiltration rate.

(9)

f2(t) . fd(t) K n2 Infiltration rate. (10)

B3 - 3
t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

f2(t)

5.996

4.262

3.494

3.036

2.724

2.493

2.314

2.169

2.049

1.948

1.861

1.785

1.718

1.658

1.604

1.556

1.512

1.471

1.434

1.4

1.368

1.338

1.31

1.284

0 10 20 30

0

2

4

6

Time (h)

Infiltration Rate (cm/h)

f2( t)

t

Figure B3-2. Water infiltration as a function 

of time through the layered soil profile in 

Scenario 2.

0 10 20 30

0

5

10

15

Time (h)

Infiltration Rate (cm/h)

f1( t)

f2( t)

t

Figure B3-3. Comparison of water 

infiltration between Scenarios 1[solid line] 

and 2 [dash line].

E. Discussion 

Water Infiltration into a two-layer soil (Scenario 1) and a three-layer soil (Scenario 2) as a 

function of time is given in Figures B3-1 and B3-2, respectively. These figures show that the

infiltration rate is relatively high at the onset of the infiltration, then decreasing, and eventually 

approaching a constant rate at t > 20 h. Comparison of the two scenarios revealed that water 

infiltration is faster in the two-layer soil than in the three-layer soil. 

B3 - 4
F. Sensitivity of Infiltration Rate to ∆Θ

This section shows the sensitivity coefficient (Ss) and the relative sensitivity (Sr) of the surface 

infiltration rate to the change in volumetric water content. The expressions were obtained by 

applying Equations 3 and 4 in Section B2 (PHILIP2T model) to Equation 5 in this section. The input 

parameters were the same as in Scenario 2 except for ∆Θ and t as shown below.

F.1. Input Data

n 3

n2 n

∆θ n 0.1, 0.11.. 0.2 t 5

F.2. Sensitivity Calculation Equations 

td ∆θ n

K . n2 t

∆θ .

n H n2

= 1

n 1

i

Zi

Dimensionless time since 

wetting front penetration.

(11)

Dimensionless depth accounting 

for the thickness and conductivity 

of layers behind wetting front. 

(12) zd ∆θ n . K n2

H n2

= 1

n 1

i

Zi

= 1

n 1

i

Zi

Ki

(13) Fd ∆θ n . 1

2

td ∆θ n 2.zd ∆θ n td ∆θ n 2.zd ∆θ n

2 8.td ∆θ n

fd ∆θ n

Fd ∆θ n 1

Fd ∆θ n zd ∆θ n

(14)

f ∆θ n . fd ∆θ n K n2 (15)

B3 - 5
S s ∆θ n

d

d∆θ n

f ∆θ n Sensitivity (16)

F.3. Results

0.1 0.15 0.2 0.25

8

10

12

14

Change in water content (cm3/cm3)

Sensitivity of infiltration

S s ∆θ n

∆θ n

∆θ n

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

S s ∆θ n

13.24

12.624

12.086

11.612

11.189

10.81

10.466

10.154

9.867

9.604

9.361

Figure B3-4. Sensitivity of infiltration rate for 

different values of the change in volumetric 

water content at the wetting front at t = 5 h.

F.4. Discussion

Figure B3-4 shows a sensitivity of the infiltration rate for different values of ∆ θ n. The sensitivity 

decreased as ∆ θ n increased. A ten-fold increase in ∆ θ n resulted in 30% decrease in sensitivity 

for conditions used in this application. 

B3 - 6
B4. Green-Ampt Explicit Model (GAEXP)

A. Description

The Green-Ampt model is the first physically-based equation describing the infiltration of water into 

a soil. It has been the subject of considerable developments in soil physics and hydrology owing to its 

simplicity and satisfactory performance for a great variety of water infiltration problems. This model 

yields cumulative infiltration and infiltration rate as implicit functions of time, i.e., given a value of 

time, t, q and I cannot be obtained by direct substitution. The equations have to be solved in an 

iterative manner to obtain these quantities. Therefore, the required functions are q(t) and I(t) instead 

of t(q) and t(I). The Green Ampt explicit model (GAEXP) for q(t) and I(t), developed by Salvucci and 

Entekhabi (1994), facilitated a straight forward and accurate estimation of infiltration for any given 

time. This model supposedly yield less than 2% error at all times when compared to the exact values 

from the implicit Green-Ampt model. A scenario was chosen to simulate the water infiltration into a 

sandy soil under ponding conditions by using the GAEXP model. A ponding depth of 1 cm was applied 

at the soil surface. Input parameters and simulation results were given below.

B. Definition of Variables 

h e 6.9 Air exit head and is equal to one half of the bubbling pressure head (cm)

λ 1.68 The exponent of the Brooks-Corey water retention model

θ s 0.43 Saturated volumetric water content (cm3/cm3)

θ 0 0.05 Initial volumetric water content (cm3/cm3)

K s 21 Saturated hydraulic conductivity (cm/h)

h s 1 Ponding depth or capillary pressure head at the surface (cm)

t 1.. 24 Duration of infiltration (h)

Values given above were obtained from Carsel and Parrish (1988) for a sandy soil.

C. Equations

(1) η ( 2 ) 3.λ

h f . η

(η 1)

h e Capillary pressure head at the wetting front (2)

χ

h . s h f θ s θ 0

K s

(3)

τ(t) t

(t χ)

(4)

Infiltration rate

q(t) . . 2

2

τ(t)

1

2 2

3

. 2

6

τ(t)

1

2 . 1 2

3

τ(t) K s (5)

B4 - 1
I(t) 1 . . 2

3

t . 2

3 χ.t t

2 . . 2 1

3 χ (ln(t χ) ln(χ)) . . 23 χ ln t χ2 χ.t t

2 ln χ

2

K s (6)

Cumulative infiltration

D. Results

0 10 20 30 21

22

23

Time (h)

Infiltration rate (cm/h)

q( t)

t

t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

q(t)

22.94

22.01

21.68

21.51

21.41

21.34

21.3

21.26

21.23

21.21

21.19

21.17

21.16

21.15

21.14

21.13

21.12

21.12

21.11

21.1

21.1

21.09

21.09

21.09

I(t)

28.01

50.38

72.2

93.79

1.15.102

1.37.102

1.58.102

1.79.102

2.102

2.22.102

2.43.102

2.64.102

2.85.102

3.06.102

3.28.102

3.49.102

3.7.102

3.91.102

4.12.102

4.33.102

4.54.102

4.75.102

4.96.102

5.18.102

Figure B4-1. Soil infiltration as a function of 

time. 

0 10 20 30 0

200

400

600

Time (h)

Cumulative infiltration (cm)

I(t)

t

Figure B4-2. Cumulative Infiltration as a 

function of time.

E. Discussion

Figures B4-1 and B4-2 show the soil water infiltration rate and cumulative infiltration as a function of 

time, respectively. A rapid decrease in water infiltration rate was observed within the first 5 hours. From 

5 to 24 hours, the water infiltration rate decreased gradually and finally approached a constant rate at t > 

20 hours.

B4 - 2
F. Sensitivity Analysis of Infiltration Rate to Saturated hydraulic Conductivity

This section shows the sensitivity coefficient (Ss) and the relative sensitivity (Sr) of the surface 

infiltration rate, q, to the saturated hydraulic conductivity (Ks) at the time of 5 hours. The 

expressions were obtained by applying Equations 3 and 4 in Section B2 (PHILIP2T model) to 

Equation 5 in this section.

F.1. Input Data

K s 20.5, 20.6.. 21.5 t 5

F.2. Sensitivity Calculation Equations

χ K s

h . s h f θ s θ 0

K s

(7)

τ K s

t

t χ K s

(8)

(9)

q K s . . 2

2

τ K s

1

2 2

3

. 2

6

τ K s

1

2 . 1 2

3

τ K s K s

(10) S s K s

d

dK s

q K s Sensitivity

(11) Relative Sensitivity S r K s . d

dK s

q K s

K s

q K s

F.3.Results

20.5 21 21.5 22 1.0003

1.00032

1.00034

Saturated hydrualic conductivity (cm/h)

Sensitivity of infiltration rate S s K s

K s

K s

20.5

20.6

20.7

20.8

20.9

21

21.1

21.2

21.3

21.4

21.5

q K s

20.91

21.01

21.11

21.21

21.31

21.41

21.51

21.61

21.71

21.81

21.91

S s K s

1.0003354

1.0003322

1.0003291

1.000326

1.000323

1.00032

1.000317

1.0003141

1.0003113

1.0003084

1.0003057

S r K s

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98

0.98 Figure B4-3. Sensitivity of infiltration rate for 

different values of saturated hydraulic 

conductivity at t = 5 hours.

B4 - 3
20.5 21 21.5 22 0.9805

0.981

0.9815

Saturated hydraulic conductivity (cm/h)

Rel. sensitivity of infiltration rate S r K s

K s

Figure B4-4. Relative sensitivity of 

infiltration rate for different values of 

saturated hydraulic conductivity at t = 5 hours.

F.4. Discussion

Figure B4-3 shows a sensitivity of the infiltration rate for different values of saturated hydraulic 

conductivity. The sensitivity decreased as the saturated hydraulic conductivity increased. Figure B4-4 

shows the relative sensitivity of the infiltration rate for different values of saturated hydraulic 

conductivity. The relative sensitivity increased as the saturated hydraulic conductivity increased.

B4 - 4
B5. Constant Flux Green-Ampt Model (GACONST)

 A. Description

Let the water flux application rate at the surface be r (cm/h). Here two cases need to be considered, 

first where r >Ks, and second where r < Ks. When r > Ks, surface water saturation first occurs at 

time t0; when r < Ks, the infiltration rate (q) is always equal to the surface application rate (r) and the 

surface is never saturated. A scenario was chosen to simulate the water infiltration into a sandy 

soil under non-ponding conditions (a negligible thin layer of water ponds on the soil surface) by 

using the Green-Ampt model with constant application rate (GACONST). Input parameters and 

simulation results were given below.

B. Definition of Variables

t 1, 2.. 20 Duration of infiltration (h)

r 3.5 Constant water application rate at the surface (cm/h)

K s 2.59 Saturated hydraulic conductivity (cm/h)

θ s 0.41 Saturated volumetric water content (cm3/cm3)

θ 0 0.05 Initial volumetric water content (cm3/cm3)

λ 0.89 The exponent of the Brooks-Corey water retention model

h e 13.33 Air exit head and is equal to one half of the bubbling pressure head (cm)

I

0 0.01 Initial guess for cumulative infiltration (cm)

C. Equations

(1) η ( 2 ) 3.λ

h f . η

(η 1)

h e (2)

t 0 if

K . . s h f θ s θ 0

.r r K s

r>K s

∞ otherwise

t = 0 4.966 (3)

Q t r,K , , s t t 0 r if . r K s (t>0)

r if > . r K s t t 0

(4)

I 1 r,K , , s t t 0 if .r t . r K s (t>0)

if .r t > . r K s t t 0

(5)

B5 - 1
I 0 if .r t 0 r>K s

∞ otherwise

(6)

f 1 K , , , , s t t 0 I I 0 I I 0 h . . f θ s θ 0 ln

I θ . s θ 0 h f

I 0 θ . s θ 0 h f

K .

s t t 0 (7)

f(t,I) root f , 1 K , , , , s t t 0 I I 0 I

(8)

f(t,I) root f , if 1 K , , , , s t t 0 I I 0 I > . r K s t t 0

I otherwise 1 r,K , , s t t 0 (9)

I

t f t,I if t > . r K s t t 0

I otherwise 1 r,K , , s t t 0 (10)

qt if K .

s 1 θ . s θ 0

h f

f t,I

t

> . r K s t t 0

Q otherwise t r,K , , s t t 0

(11)

0 10 20 2.5

3

3.5

Time (h)

Surface infiltration rate (cm/h)

qt

t

D. Results 

t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

qt

3.5

3.5

3.5

3.5

3.494

3.346

3.243

3.167

3.108

3.06

3.021

2.989

2.961

2.937

2.916

2.898

2.881

2.867

2.853

2.841

I

t

3.5

7

10.5

14

17.5

20.915

24.207

27.41

30.546

33.629

36.67

39.674

42.649

45.597

48.524

51.43

54.319

57.193

60.053

62.9

Figure B5-1. Surface infiltration rate as 

a function of time. 

0 10 20 0

50

100

I

t

t

Figure B5-2. Cumulative Infiltration as 

function of time.

B5 - 2
r 2.5

t 0 if

K . . s h f θ s θ 0

.r r K s

r>K s

∞ otherwise

t = 0 1 10307 (12)

Q t r,K , , s t t 0 r if . r K s (t>0)

r if > . r K s t t 0 (13)

I 1 r,K , , s t t 0 if .r t . r K s (t>0)

if .r t > . r K s t t 0 (14)

I 0 if .r t 0 r>K s

∞ otherwise (15)

f 1 K , , , , s t t 0 I I 0 I I 0 h . . f θ s θ 0 ln

I θ . s θ 0 h f

I 0 θ . s θ 0 h f

K .

s t t 0 (16)

f(t,I) root f , if 1 K , , , , s t t 0 I I 0 I > . r K s t t 0

I otherwise 1 r,K , , s t t 0

(17)

I

t f t,I if t > . r K s t t 0

I otherwise 1 r,K , , s t t 0

(18)

qt if K .

s 1 θ . s θ 0

h f

f t,I

t

> . r K s t t 0

Q otherwise t r,K , , s t t 0 (19)

B5 - 3
0 10 20 2.496

2.498

2.5

2.502

2.504

Time (h)

Surface infiltration rate (cm/h) qt

t

t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

qt

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

2.5

I

t

2.5

5

7.5

10

12.5

15

17.5

20

22.5

25

27.5

30

32.5

35

37.5

40

42.5

45

47.5

50

Figure B5-3. Surface infiltration rate as 

a function of time. 

0 10 20 0

20

40

60

I

t

t

Figure B5-4. Cumulative Infiltration as 

function of time.

E. Discussion

Figure B5-1 shows a constant-flux infiltration pattern when r>Ks. Before the surface saturation first 

occurs at time t0 ( 5 hours), infiltration rate is constant (=r) , after that, infiltration rate decreases 

with time. Figure B5-3 illustrates a soil infiltration with a constant rate equal to r. Figures B5-2 and 

B5-4 deplict a nearly linear relationship between the cumulative infiltration and time during the 

infiltration period.

B5 - 4
B6. Infiltration/Exfiltration Model (INFEXF)

A. Description

The vertical movement of soil water between the surface and the water table can be divided into two 

processes according to the predominant forces involved: (1) infiltration and (2) exfiltration 

(evaportranspiration). An infiltration/exfiltration model (INFEXF) developed by Eagleson (1978) was 

selected to estimate the water infiltration during wetting season and exfiltration during drying 

season. Infiltration and exfiltration are described by Philip's equation, which assumes the medium to 

be effectively semi-infinite and the internal soil water content at the beginning of each storm and 

interstorm period is assumed to be uniform at its long term space-time average. The exfiltration 

equation is modified for presence of natural vegetation through the approximate introduction of a 

distributed sink representing the moisture extraction by plant roots. Two cases were presented in 

this application. The first case is the water infiltration into a sandy loam during storms. A boundary 

condition with saturated water content on the soil surface is assumed. In this case, rainfall 

intensity is assumed to be greater than the infiltration capacity. The second case is the exfiltration 

during drying surface (interstorm) periods with a dry soil surface and under the assumption that 

potential evaporation is greater than the exfiltration capacity.

B. Definition of Variables

θ s 0.41 Saturated volumetric water content at the soil surface (cm3/cm3)

θ 0 0.07 Initial volumetric water content (cm3/cm3)

K s 2.59 Saturated hydrualic conductivity (cm/h)

λ 0.89 Pore size distribution index

c

2 3.λ

λ

Soil index based on the pore distribution index

C. Equations and Results

a. Infiltration during Rainy Seasons

t 1.. 24 Time (h)

θ 1 0.41 Soil water content at surface (cm3/cm3)

S( θ)

θ

θ s

Degree of saturation 

K(θ) K .

s (S(θ))

c Hydraulic conductivity (cm/h)

Ψ1 13.33 Near saturated soil water suction (cm) 

B6 - 1
Soil water suction (cm) (1) Ψ(θ) Ψ1.S(θ)

1

λ

D(θ) K . (θ) d

dθ

(Ψ( θ)) Soil water diffusivity (cm2/h) (2)

Di . . 5

3

θ 1 θ 0

5

3 d

θ 0

θ 1

θ θ . θ 0

2

3 D(θ) Diffusivity over range θ1-θ0, (cm/h1/2) (3)

Di = 38.218

S i 2. θ . 1 θ 0

Di

π

1

2

Sorptivity (cm/h1/2) (4)

fit . . 1

2

S i t

1

2 . 1

2

K θ 1 K θ 0 Infiltration rate (cm/h) (5)

K θ = 1 2.59 K θ = 0 2.427 10 4

t

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

fit

2.481

2.134

1.98

1.888

1.825

1.779

1.743

1.714

1.69

1.67

1.653

1.637

1.624

1.612

1.601

1.592

1.583

1.575

1.567

1.56

1.554

1.548

1.542

1.537

0 10 20 30 1.5

2

2.5

Time (h)

Infiltration Rate (cm/h)

fit

t

Figure B6-1. Water infiltration as a 

function of time during rainy season.

B6 - 2
b. Exfiltration During Drying Season

t 1.. 24 Time (h)

θ 1 0.0001 Soil water content at dry surface (cm3/cm3)

θ 0 0.15 Initial soil water content (cm3/cm3)

Ψ1 13.33 Near saturated soil water suction (cm)

Ev 0.05 Transpiration rate (cm/h)

M 0.2 Vegetated fraction of land surface

Ψ(θ) Ψ1.S(θ)

1

λ Soil water suction (cm) (6)

D(θ) K . (θ) d

dθ

(Ψ( θ)) Soil water diffusivity (cm2/h) (7)

De . . ( 1.85) θ 0 θ 1

( 1.85) d

θ 1

θ 0

θ . θ 0 θ 0.85 D(θ) Desorption diffusivity (cm2/h) (8)

De = 0.439

Se 2. θ . 0 θ 1

De

π

1

2

Exfiltration sorptivity (cm/h1/2) (9)

Se = 0.112

fet . . 1

2

Se t

1

2 . 1

2

K θ 1 K θ 0 M.Ev Exfiltration rate (cm/h) (10)

t

1

2

3

4

5

6

7

8

9

10

11

12

fet

0.039

0.023

0.016

0.011

8.427.10 3

6.245.10 3

4.549.10 3

3.182.10 3

2.049.10 3

1.091.10 3

B6 - 3
12

13

14

15

16

17

18

19

20

21

22

23

24

2.669.10 4

4.52.10 4

1.086.10 3

1.651.10 3

2.159.10 3

2.618.10 3

3.036.10 3

3.419.10 3

3.771.10 3

4.096.10 3

4.398.10 3

4.679.10 3

4.942.10 3

5.188.10 3

0 10 20 30 0.02

0

0.02

0.04

Time (h)

Exfiltration rate (cm/h)

fet

t

Figure B6-2. Water exfiltration as a 

function of time during drying season.

D. Discussion

Water Infiltration into and exfiltration out of soil surface as a function of time is shown in 

Figures B6-1 and B6-2, respectively. Figure B6-1 shows a typical soil infiltration pattern, with 

an infiltration rate relatively high at the onset of the infiltration, then decreasing, and eventually 

approaching a constant rate. Figure B6-2 (Case 2) illustrates that exfiltration (actual 

evaporation) decreases with time. Nagative value of exfiltration indicates that exfiltration has 

ceased (at the time of 11 hours in this case) but transpiration is still going on.

B6 - 4
C - 1

APPENDIX C

Sensitivity Analysis Discussion and Demonstration
Si,j 
 0yi

0xj

C - 2

(C1)

Sensitivity Analysis Discussion and Demonstration

1. Introduction

Sensitivity analysis should be considered an essential step to all mathematical-based

model applications. To the users of these models, the sensitivity analysis can provide a basis for

greater understanding of the processes identified in a site conceptualization. This appendix is

provided to model users as (1) an explanation of the purpose of the sensitivity analysis, (2) a

working definition of sensitivity coefficients, (3) an explanation of approaches to the sensitivity

analysis, (4) an explanation of the role of sensitivity in an parameter uncertainty analysis, and (5)

a set of tools, or procedures for conducting a sensitivity analysis. As an example, a sensitivity

analysis is given for the Philip’s two-term model using a published dataset from an infiltration

experiment. Uncertainty estimation of model outputs due to uncertain input parameters is also

analyzed using another set of reported field experimental data. Implications of the case studies of 

sensitivity analysis using these two sets of experimental data for Philip’s two-term model are also

discussed.

2. Purpose of Sensitivity Analysis

The purpose of conducting a sensitivity analysis is two-fold. That is, first to evaluate

model’s response to changes in input parameters, and secondly, to quantify the likely uncertainty

of the calibrated model resulting from uncertainties associated with the input parameters,

stresses, and boundary conditions (Anderson and Woessner, 1992; Zheng and Bennett, 1995). 

The sensitivity analysis can be used to estimate model parameters, design experiments, and

analyze ground-water management problems (Skaggs and Barry, 1996). In field sampling,

sensitivity analysis can assist with the identification of parameters requiring more accurate

measurements. This will be driven by the need to reduce overall uncertainty in the model results,

as well as to identify parameters requiring less precision and thereby saving time, money and

effort in the data collection process. In the model development process, sensitivity analysis can

be utilized to determine the need for specific parameters in the final model. This would be

dependent on the significance of the parameter to the model output (Hamby, 1994).

3. Sensitivity Coefficients

Sensitivity (S) is a measure of the impact of change in one input parameter on an output

result. The sensitivity (S ) of the j model dependent variable (y ) to the i model input i,j j

th th

parameter (x ) is the partial derivative of the dependent variable with respect to the input i

parameter, and is expressed as

where i is the index for i model dependent variable (i.e., model output), j is the index for j th th
S r

i,j 
 0yi

0xj

xj

yi

Si,j 
 0yi

0xj

 yi

xj

C - 3

(C2)

(C3)

model input parameter. For a model with n output variables and m input parameters, the number

of sensitivity coefficients that could be generated is given by nm. The units for S are the units i,j

of yi i over the units of x . For convenience of comparing the sensitivity coefficients of varying

parameters and dependent variables, as well as comparing different models, a normalized form

of the sensitivity coefficient defined in Equation C1 is utilized. The normalized form is given by

Equation C2.

The normalized sensitivity, Si,j , is referred to as the relative sensitivity of dependent variable yi

r

with respect to independent input parameter x . The normalized sensitivity coefficient S is j i,j

r

dimensionless. 

4. Approaches to Computing Sensitivity Coefficients

Three approaches for computing sensitivity coefficients are commonly recognized and

used. They are the direct, the perturbation, and the adjoint methods (Skaggs and Barry, 1996). 

The direct method computes the sensitivity coefficients analytically through the solving of the

coefficients’ governing equations. These equations are obtained by differentiating the governing

equation and associated boundary and initial conditions with respect to model input parameters

of interest. Sensitivity coefficients are calculated from the sensitivity equations using a base set

of model parameters. The perturbation method, also known as the divided difference method,

computes a numerical approximation to the sensitivity coefficients for the model input

parameters. The perturbation method is the simplest of the sensitivity methods and is widely

used. The adjoint method also makes use of sensitivity equations, but rather than solving them

directly, the sensitivities are obtained by solving the adjoint problem. In the adjoint method, the

functionals (e.g., concentration in solute transport problem), that dependent on state variables

(e.g., velocity and head) and model parameters, are formulated, and the sensitivity of functionals

are computed without evaluating the sensitivities of the state variables. As a result, it can lead to

computational savings. The details of this method can be seen in Skaggs and Barry (1996).

Complex models, for which analytical sensitivity equations are not easily derived,

generally result in the need for using the perturbation method. The sensitivity coefficients are

calculated using the approximation forms (i.e., the difference equations) of Equations C1 and C2.

The difference forms for these equations are given by
S r

i,j 
 0yi

0xj

xj

yi

 yi

xj

xj

yi

f
f(x)	f(xo)
 (x	xo)

1!

fx(xo) (x	xo)sup2

2!

fxx(xo) ...  (x	xo)

n

n!

f(n)(xo)Rn(xo)

y 
 f(x)

C - 4

(C4)

(C5)

(C6)

and

where y is the change of y due to an infinitesimal change (x ) in x . i i jj

In addition to these three methods of calculating sensitivity coefficients, other approaches

are also used based on differing needs and objectives. Furthermore, the reader must be aware

that slightly different definitions of “sensitivity coefficients” could be defined in literature. In

this regard, Hamby (1994, 1995) has provided a detailed review and evaluation of more than ten

sensitivity methods.

5. Role of Sensitivity in Uncertainty Analysis

To examine the deviation of model output (f) due to the deviation (x-x ) of input o

parameters, the Taylor’s formula can be used

where f(x )o is the function value in the neighborhood of a point x=xo x o xx o (n) o ; f (x ), f (x ), f (x ), are the

first, second, and n derivatives of f(x) with respect to x at x=x , and R (x ) is the remainder. th

o no

Here f (x ) x o is the sensitivity coefficient as defined in Equation C1 and evaluated at x=xo. When

high-order terms are ignored, f is approximated by the first term, (x-x )·f o x, in Equation C5. In

other words, the deviation of the model output (f) can be approximated by the product of

deviation of input parameter (x-x ) and sensitivity (f =S ) o x i,j

Equation C5 is useful for the analysis of a single set of input parameters (i.e., only one

realization). The common way to quantify uncertainty is to treat the input parameters as random

variables. For examining model uncertainty (in terms of variance or covariance) due to uncertain

input parameters, the First-Order Second-Moment (FOSM) technique can be used. It provides a

method of calculating the mean, variance, and covariance of model outputs from means,

variances, covariance and sensitivity coefficients for the model inputs. The following equations

are used to determine these quantities (Dettinger and Wilson, 1981).
m

y 
 f(mx)  1

2

02f

0x 2

var(x)

cov(y) 
 Si,j cov(x)[Si,j]

T

C - 5

(C7)

(C8)

where the variables are defined as follows:

n is the number of model outputs,

k is the number of model input parameters,

y is a (n x 1) vector of model outputs,

f(x) is the (n x 1) vector of model outputs with inputs of x, x is a (k x 1) vector of model

parameters,

m is a (n x 1) vector of mean model outputs, y

f(m ) is a (n x 1) vector of model outputs where the model is evaluated at m , x x

m is a (k x 1) vector of mean parameter values, x

0 f/0x is a (n x k) matrix of 2 partial derivatives of f(m ), 2 2 nd

x

cov(x) is a (k x k) covariance matrix of the input parameters,

cov(y) is a (n x n) covariance matrix of model outputs,

var(x) is a (k x 1) vector of variances of parameters, and

Si,j is a (n x k) matrix of sensitivity coefficients of model outputs to input parameters. The

First-Order Second-Moment analysis is most appropriate when the model is not too nonlinear

with respect to its input parameters, and the coefficients of variation for the parameters are small. 

When the model is highly nonlinear, or the model is so complex that sensitivity coefficients are

not easily obtained, a Monte Carlo method could be used. The methodology of conducting

uncertainty analysis using the Monte Carlo method for transport models can be seen in Nofziger

et al., (1993). Using Equations C6 through C8, uncertainty of model outputs, cov(y), can be

obtained when uncertainty of input parameters, cov(x), and sensitivity coefficients of model

outputs to input parameters, S , are known. i,j

6. Procedures for Sensitivity Analysis

An optimal set of input parameters for the problem of interest should be identified before

the sensitivity analysis is conducted. This set of input parameters is referred to as the base set. 

The basic procedures for the sensitivity analysis are outlined as follows:

1. Gathering of input parameters: When site-specific input parameters are not available,

the base set of parameters could be obtained either from indirect estimation using

other data at the site (e.g., to estimate conductivity using soil texture data) or from the
C - 6

“best justified” mean values of published data which are “appropriate” for the site

conditions. When experimental data are available, calibration of the model to obtain

an optimal set of input parameters (i.e., base set) should be performed first before

sensitivity analysis.

2. After an optimal set of parameters has been identified, sensitivity analysis is

performed to quantify the sensitivity of the model results with respect to model

parameters. It is typically performed by changing one parameter at a time while

keeping the rest constant. By changing the parameter over a range (e.g., above and

below a certain percent or standard deviation of the base set value), the responses of

model output to input parameters can be examined. The results are usually presented

by a) presenting the model outputs for several values of input parameters; b) plotting

(absolute or relative) sensitivity coefficients versus input parameters; and c) providing

the ranking descriptors, such as high, medium, and low.

If the sensitivity results show that the model results are highly sensitive to a particular

parameter, the uncertainty associated with that parameter will significantly affect the ability of

the model to make meaningful interpretations and predictions. This is specifically true when that

parameter exhibits great spatial variability in the field, and/or great uncertainty due to

experimental measurement/estimation of parameters. On the other hand, if the model results are

not sensitive to a given parameter, the uncertainty in that parameter will have little impact on the

model’s interpretative or predictive capabilities and that parameter might be eliminated. 

7. Case Studies

In this section, two sets of infiltration experimental data are used for performing

sensitivity analysis with the Philip’s two-term infiltration model. The model outputs are

cumulative infiltration (I) and infiltration rate (q). The model input parameters are sorptivity (s)

and a constant, A, that depends on the soil properties, saturated water content and initial water

content. 

Data Set 1: The experimental data for infiltration into the Grenoble sand of Barry et al., (1995) is

used for illustration of model calibration and performing sensitivity analysis. The infiltration

experiment was conducted on a vertical, ponding column (93.5 cm in length and 6 cm in

diameter) filled with air-dried, graded Grenoble sand. The reported cumulative infiltration (I) is

shown in Figure C1a. From cumulative infiltration data, infiltration rates are calculated from the

infiltration data (Figure C1a) and presented in Figure C1b. The experimental conditions and

hydraulic properties of the Grennoble sand is given in Table C1.
C - 7

Table C1. Experimental conditions and hydraulic properties for the Grenoble sand infiltration

experiment (Barry et al., 1995).

Quantity Value

Porosity 0.37

Initial soil pressure head within the soil profile 89 cm

Head at the soil surface 23 cm

Initial moisture content 0.082 cm /cm 3 3

Saturated moisture content 0.31 cm /cm 3 3

Saturated hydraulic conductivity 15.4 cm/h

Estimated sorptivity 9.54 cm/h1/2

To calibrate Philip’s two-term model, either the cumulative infiltration data, infiltration

rate or both can be used. Model parameter estimation can be accomplished through either

regression techniques or graphical methods. Table C2 presents six methods (Cases 1, 2, 3, 4, 5,

and 6) which illustrate the combinations of the estimation methods and the datasets used in the

calibration. In Case 1, the estimated value (9.54 cm/h ) of sorptivity by Barry et al., (1995) is 1/2

used and calibration is only conducted for estimating A. For Cases 2, 3, and 4, calibration is

carried out to estimate s and A using a regression method. In Case 5 and Case 6, a graphical

method is used to estimate s and A. The estimates of s and A are given in Table C2. The

standard error is also given in Table C2. Calculated infiltration and infiltration rate using these

six sets of estimated parameters are depicted in Figures C2a and C2b respectively. These

figures indicate that experimental infiltration data are reasonably described using the values of

estimates obtained from Case 4 and Case 5. Figures C2a and C2b also depict that variation of

the estimates of input parameters can result in significant differences in model prediction. 
C - 8

Table C2. Calibration of Philip’s two-term model parameters using the experimental infiltration

data of Barry et al., (1995).

Case s, (cm/h ) A, (cm/h ) 1/2 2

Case 1: Reported sorptivity of 9.54 cm/h1/2

(Barry et al., 1995) is used. Calibration is

performed on A only using cumulative

infiltration data. 

[9.54] 10.62±0.08†

Case 2: Calibration is performed on both s and

A using cumulative infiltration data. 8.92±0.19 11.78±0.36 † †

Case 3: Calibration is performed on both s and

A using infiltration rate data. 12.65±1.19 7.35±2.13 † †

Case 4: Calibration is performed on both s and

A using both cumulative infiltration and 11.17±0.99 8.13±1.74 

infiltration rate data. 

† †

Case 5: Graphical estimation of s and A from

the plot of I(t)/t versus t . -1⁄2 11.0±0.46 9.00±0.15 † †

Case 6: Graphical estimation of s from the plot

of q(t) versus t . -1⁄2 8.0±0.41 -- †

†.. Estimated value ±Standard error 

Figures C2a and C2b also indicate that using the estimates of s and A of Case 4 and Case

5 provides better fits of both the observed cumulative infiltration and infiltration rate data than

using those estimates from Case 1, 2, 3, and 6. From this evaluation, the estimates of either Case

4 or Case 5 can be chosen as the optimal set. However, the estimates of s and A obtained from

Case 4 is arbitrarily selected as an optimal set (i.e., the base set).

To compute sensitivity and relative sensitivity, the direct method is selected for this

example due to its simplicity. The sensitivity and relative sensitivities equations are given in

Table C3. Table C3 gives the analytically derived sensitivities for infiltration rate, q(t), and

cumulative infiltration, I(t), with respect to sorptivity (s) and the constant term (A). Note that the

sensitivities of I(t) with respect to s (or A) and q(t) with respect to s (or A) are independent with s

(or A). In other words, the sensitivities are constant at given particular time over the whole range

of s (or A). However, sensitivities vary with time for Philip’s two-term model.
q(t) 
 1

2

st

	 1

2 A

I(t) 
 st

1

2 At

1

2

t

	 1

2

s

s2A t

2A t

st2A t

t

1

2

s t

s tAt

At

s tAt

C - 9

Table C3. Sensitivity and relative sensitivity of infiltration rate, q(t), and cumulation infiltration,

I(t) with respect to sorptivity, s and soil-dependent constant, A, of Philip’s two-term model.

Model Output Sensitivity Output with Input Sensitivity

Parameter Si,j

Relative Derivatives of

S respect to input i,j

r

Second

Parameter

s 0

A1 0

s 0

At 0

Using the identified base set of estimates, sensitivity analysis is conducted and presented

in the following four areas.

1. Examining the time changes of infiltration by changing one parameter while keeping 

the rest of parameters fixed (Figures C3a and C3b); 

2. Examining sensitivity changes of infiltration with respect to s or A over a range of

input parameter at different time periods, t. The range of input parameter is presented

in terms of relative ratios of that parameter to the base value (Figures C4a and C4b); 

3. Examining time changes of sensitivities of infiltration with respect to s or A over time

using 60%, 100%, and 140% of base values of input parameters (Figures C4c and

C4d); and

4. Examining the changes of relative sensitivities over a range of one particular

parameter at different time periods (Figures C5a and C5b).

The results indicate:

1. sensitivities of infiltration with respect to s (or A) are constant over a range of 60% to

140 % of s (or A) at any given time (Figures C4a and C4b);

2. sensitivities of infiltration with respect to s (or A) vary with time but they are

independent of s (or A) (Figures C4c and C4d);

3. Relative sensitivities of infiltration with respect to sorptivity, s, are greater than those

with respect to constant A (Figures C5a and C5b). 
I
I(s,A)	I(so,Ao)
 (s	so)

1!

I

s

(so,Ao) (s	so)

2

2!

I

ss(so)(s	so)(A	Ao)IsA(so,Ao)

 (A	Ao)

1!

I

A(so,Ao) (A	Ao)

2

2!

I

AA(so,Ao)R3(so,Ao)

I
I(s,A)	I(so,Ao)
I

s

(so,Ao)(s	so)I

A(so,Ao)(A	Ao)

C - 10

(C9)

(C10)

4. Similar results (not shown) are obtained for sensitivities of infiltration rate with

respect to s and A.

The results of sensitivities shown in Figures C4a, C4b, C5a, and C5b, indicate that s is

the most sensitive parameter in Philip’s two-term model for the range of input parameters

examined. Similar results are also observed for infiltration rates (not shown).

To examine the impact of deviation of the estimates of input parameters, s and A from the

values of the base set on the deviation of infiltration, Taylor’s formula (Equation C5) is applied

to Philip’s two-term infiltration model. Taking up to second-order terms, deviation (difference)

of infiltration (I) is expressed as

where

• I (s ,A ), I (s ,A ) are the first, and second derivatives of I(s,A) with respect to s at s=s s o o ss o o o

and A=A respectively; o

• I (s ,A ), I (s ,A ) are the first, and second derivatives of I(s,A) with respect to A at A o o AA o o

s=s and A=A respectively; o o

• I (s ,A ) sA o o is the partial derivative of I(s,A) with respect to s and A; and

• R (s ,A ) is the remainder for the terms in the order of 3 and up. 3o o

Here, I (s ,A ) so o Ao o and I (s ,A ) are the sensitivity coefficients of I(t) with respect to s, and A

respectively. It is worth noting that analytical values of I , I and I and higher-order ss AA sA

derivatives are zeros (Table C3). Accordingly, the deviation (I) of I(s,A) from I(s ,A ), (I(s,A)- o o

I(s ,A )) o o is given by

Equation C10 states that deviation of I(s,A) from I(s ,A ) o o can be estimated from the sum of the

products of sensitivity and deviation of a parameter value from its base value.

Assuming s and A are the optimal values (Case 4 in Table C2) for s and A in Philip’s o o

two-term model, there is interest in how the sensitivities of I with respect to s and A will impact

the prediction of infiltration. Taking Case 5 (e.g., using the graphical method to estimate s and A

from infiltration data) as an example, deviation (difference) of predicted infiltration, due to

estimated s and A, from using the optimal data set is shown in Figure C6. The differences

between predicted infiltration are approximately equal to the sum of the products of sensitivity
C - 11

and the difference between estimated and optimal values. Figure C6 indicates that: (a)

contribution of deviation of infiltration due to deviation of the constant A term (i.e., the second

term on the right-hand-side of Equation C10, IA o #(A-A )) is greater than contribution of deviation

of infiltration due to deviation of sorptivity (i.e., the first term on the right-hand-side of equation,

I #(s-s )) and (b) the difference of predicted infiltration (maximum value is around 0.1 cm) is s o

negligible compared to the magnitudes of infiltration during the first 0.4 hours (Figures C6 and

Figure C2a). It is worth noting that the difference of predicted infiltration is dependent on

sensitivities as well as the deviation of input parameters from base set values. 

8. Uncertainty Analysis using Sensitivity Results

After obtaining sensitivity coefficients of input parameters, uncertainty of model outputs

due to uncertainty of input parameters can be quantified using Equation C8. Here uncertainty is

expressed in terms of variance (or covariance) which is the expected value of the square of the

difference between a random variable and its mean (e.g., Variance of sorptivity is E((s- *s) ). Here 2

*s is the mean of sorptivity.). To demonstrate quantifying uncertainty of predicted infiltration due

to uncertainty of input parameters (s and A), another set of reported experimental data is used.

Data Set #2: A set of field measurement data for sorptivity (s) and conductivity at the surface

(Ko) in a field study using disk permeameters had been reported by Lien (1989). Field

measurements of infiltration were conducted at four locations in a field of about 35 hectares. The

dominant soils in the field are Casa Grande - Trix fine loamy soils. The Cassel ring and the disc

permeameters at a 2 cm positive head, and the 10 cm and the 5 cm tension disc permeameters

were used. For this case study, values of K were used to estimate constant A (A=0.363 K ) o o

(Philip, 1974). The reported values and their statistics of sorptivity (s) and the calculated values

of A are given in Table C4. Note that this data set includes uncertainty due to spatial variability

as well as differences among the measurement methods. 
C - 12

Table C4. Statistics of parameters used in FOSM uncertainty analysis for Casa Grande/Trix fineloamy soils .

Sorptivity, s Constant, A

Mean 4.03 2.38

Coefficient of Variance 0.93 1.16

log Mean 1.08 0.44 †

log Coefficient of Variance 0.73 2.11 †

Correlation Matrix

log (s) log (A)

log (s) 1.00

log (A) 0.69 1.00

 Natural logarithm; The sample size is 16. †

The infiltration phenomena is examined during the first hour infiltration period. The

means of sorptivity and the constant A (Table C4), second derivatives (all zeros for the two-term

model, Table C3), and variances are used to calculate the means of infiltration (*I) and infiltration

rates (*q) (Equation C7). The sensitivity coefficients and covariances for s and A (Table C4) are

used to calculate covariances (cov(I) and cov(q)) of infiltration (I) and infiltration rates (q) using

Equation C8. The means of infiltration, 97.5% confidence values are given in Figures C7a and

C7b. Here, 97.5% confidence values for infiltration are obtained as (*I±#) ) and (*q ±#) ) I q

respectively. Here ’s are confidence coefficient (±1.96) at 97.5% confidence percentiles. )i and

) are standard deviations (square root of variance) for infiltration (I) and infiltration rate (q) q

respectively. Since sorptivity, the constant A, the predicted infiltration and infiltration rates

follow log-normal distributions, calculation is performed using the log-transformed values for s,

A, I and q. The First-Order Second-Moment (FOSM) technique used is simple and straight

forward. For comparison, the Monte Carlo technique is also used to quantify uncertainty of

infiltration. The results are shown in Figures C7a and C7b. It indicates that great variability of

infiltration exists due to the uncertainty of parameter measurement and their spatial variabilities

in the field. Figures C7a and C7b also show that both FOSM and Monte Carlo approaches give

very close results for uncertainty estimation.
var(I) 
 I

2

s var(s)I

2

A var(A)I

sA cov(s,A)

C - 13

(C11)

Table C5. Statistics of parameters used in FOSM uncertainty analysis for Grenoble sand.

Sorptivity, s Constant, A

Mean 10.2 9.38

Coefficient of Variance 0.169 0.167

Correlation Matrix

s A

s 1.00

A 1.00 -0.69

Remark: The sample size is 6.

To examine the impact of uncertainty of estimates using different methods on predicted

infiltration, the same procedures are repeated for Grenoble sand except normal distributions are

used for the input parameters, s and A, and output parameters, I and q. The estimates of s and A

obtained from different parameterization methods (Cases 1, 2, 3, 4, 5 and 6 in Table C2) are

found to be better described by the normal distribution. The statistics of these estimates are

given in Table C5. As a result, predicted values of I and q form normal distributions. 

Uncertainty results for Grenoble sand are shown in Figures C8a and C8b. It indicates that 

a) The predicted infiltration and infiltration rates satisfactorily described the observed

data using estimates of s and A. Although differences exist due to different parameter

methods, prediction of infiltration using any set of estimates in Table C2 are

reasonably good and the predicted values are fallen within the 95% confidence

intervals (Figures C2a, C2b, C8a, and C8b).

b) Uncertainty of predicted infiltration using data set #2 is much greater than using data

set #1 since data set #2 possess greater uncertainty in measured values of s and A.

Furthermore, sensitivity results are also used to examine predicted uncertainty of

infiltration due to uncertainty of estimates of input parameters. Contribution of uncertainty of

predicted infiltration is quantified by using Equation C8. Expanding Equation C8 for infiltration

(I), we can describe uncertainty of I as follows

where var(s) and var(A) are variances of s and A respectively, and cov(s,A) are covariances of s

and A. The term, var(I), in the left-hand side of Equation C11 represents uncertainty of predicted

infiltration due to uncertainty of s and A. The term I s #var(s) represents the contribution resulting 2
C - 14

from s. The term I #var(A) represents the amount of contribution resulted from A. The term I A sA

2

#cov(s,A) represents the amount of contribution resulted from both s and A. The case study is

given for the Casa Grande - Trix fine-loamy soils using data set #2. The results are shown in

Figure C9a. It indicates that contribution of uncertainty of predicted infiltration due to sorptivity

is much greater than that due to the constant A. The similar results are also found for infiltration

rate particularly in the earlier infiltration time period (t < 0.4h, Figure C9b). In this case, 

infiltration is more sensitive to s than to A and variability of s is greater than variability of A. As

the result, the model parameter s is more important than the constant A.

9. Implications

In Philip’s two term model, the first term (Equation C3) represents the contribution of the

influence of the matric suction, and the second term. represents the contribution of the influence

of gravity (Philip, 1969). The analysis has illustrated that

a) Infiltration rate dramatically decreases with time and gradually reaches a constant

value. In other words, matric suction (sorptivity) will mostly contribute to infiltration

at the early stage. On the other hand, gravity (the constant A) increasingly contributes

more to infiltration at later stage. This phenomena is clearly observed from the results

of the sensitivity/uncertainty analysis.

b) For Grenoble sand, uncertainty of infiltration due to uncertainty of sorptivity is about

two thirds or greater of overall total uncertainty of predicted infiltration during the

first hour of infiltration (Figure C9). Therefore, more effort to accurately measure

sorptivity is needed to reduce uncertainty of the predicted infiltration.

c) The estimates of input parameters using different estimation methods (Table C2) can

be used to provide reasonable prediction on infiltration. Using the mean values of the

estimates would be appropriate for uncertainty analysis since these estimates obtained

from different parameterization methods follow normal distributions (Figures C8a,

and C8b).

10. Summary

The procedures of conducting sensitivity analysis are illustrated using two sets of reported

experimental data for Philip’s two-term model. Sensitivity results are used to quantify

uncertainty of model prediction due to uncertainty of input parameters. The results indicate that

sorptivity is the most sensitive and important input parameter to predicted infiltration and

infiltration rate; and uncertainty of sorptivity contributes significantly to uncertainty of predicted

infiltration at the early stage of infiltration. 
C - 15

References

Anderson, M.P., and W.W. Woessner. 1992. Applied groundwater modeling -- Simulation of

flow and advective transport. Academic Press Inc., New York.

Barry, D.A., J.-Y. Parlange, R. Haverkamp, and P.J. Ross. 1995. Infiltration under ponded

conditions: 4. An explicit predictive infiltration formula. Soil Sci. 160:8-17.

Dettinger, M.D., and J.L. Wilson. 1981. First order analysis of uncertainty in numerical models

of groundwater. Part 1. Mathematical development. Water Resour. Res. 17:149-161.

Hamby, D.M. 1994. A review of techniques for parameter sensitivity analysis of environmental

models. Environmental Monitoring and Assessment. 31:135-154.

Hamby, D.M. 1995. A numerical comparison of sensitivity analysis techniques. Health Phys.

68:195-204.

Lien, B.K. 1989. Field measurement of soil sorptivity and hydraulic conductivity, M.S. thesis,

Univ. of Ariz., Tucson.

Nofziger, D.L., J.-S. Chen, and C.T. Haan. 1993. Evaluation of unsaturated/vadose zone models

for superfund sites. EPA/600/R-93/184. Robert S. Kerr Environmental Research

Laboratory, Ada, OK. 

Philip, J.R. 1974. Recent progress in the solution of nonlinear diffusion equations. Soil. Sci.

116:257-264.

Philip, J.R. 1969. Theory of infiltration. Adv. Hydrosci. 5:215-290.

Skaggs, T.H., and D.A. Barry. 1996. Sensitivity methods for time-continuous, spatially discrete

groundwater contaminant transport models. Water Resourc. Res. 32:2409-2420.

Zheng, C., and G.D. Bennett. 1995. Applied contaminant transport modeling: theory and

practice. Van Nostrand Reinhold, New York.
C - 16 

Figure C1a. Experimental infiltration data for Grenoble sand (Barry et al., 1995)
C - 17 

Figure C1b. Calculated infiltration rates from the experimental infiltration data for Grenoble

sand (Barry et al., 1995).
C - 18 

Figure C2a. Prediction of infiltration in Grenoble sand using the calibrated input parameters in

Philip’s two-term model.
C - 19 

Figure C2b. Prediction of infiltration rates in Grenoble sand using the calibrated input

parameters in Philip’s two-term model.
C - 20 

Figure C3a. Sensitivity of infiltration with respect to sorptivity (s) in Grenoble sand for Philip’s

two-term model.
C - 21 

Figure C3b. Sensitivity of infiltration with respect to the constant A in Grenoble sand for

Philip’s two-term model.
C - 22 

Figure C4a. Sensitivity of infiltration with respect to sorptivity (s) in Grenoble sand at different

time periods for Philip’s two-term model.
C - 23 

Figure C4b. Sensitivity of infiltration with respect to constant A in Grenoble sand at different

time periods for Philip’s two-term model.
C - 24 

Figure C4c. Sensitivities of infiltration with respect to sorptivity (s) vary with time for 60%,

100%, and 140% of base values of s in Grenoble sand. Note that time curves (solid and dash

lines) are coincident each other.
C - 25 

Figure C4d. Sensitivity of infiltration with respect to constant A vary with time for 60%, 100%,

and 140% of base values of A in Grenoble sand. Note that time curves (solid and dash lines)

are coincident each other.
C - 26 

Figure C5a. Relative sensitivity of infiltration with respect to sorptivity (s) in Grenoble sand at

different time periods for Philip’s two-term model.
C - 27 

Figure C5b. Relative sensitivity of infiltration with respect to constant A in Grenoble sand at

different time periods for Philip’s two-term model.
C - 28 

Figure C6. Deviation (difference) of predicted infiltration as the results of contribution from

deviation of the values of input parameters from base values.
C - 29 

Figure C7a. Uncertainty of infiltration due to uncertainty of input parameters in Casa GrandeTrix fine-loamy soils. The mean predicted infiltration (solid line) as well as 95% confidence

limits are shown.
C - 30 

Figure C7b. Uncertainty of infiltration rate due to uncertainty of input parameters in Casa

Grande-Trix fine-loamy soils. The mean predicted infiltration rate (solid line) as well as 95%

confidence limits are shown.
C - 31 

Figure C8a. Uncertainty of infiltration due to uncertainty of input parameters in Grenoble sand.

The mean predicted infiltration (solid line) as well as 95% confidence limits are shown.
C - 32 

Figure C8b. Uncertainty of infiltration rate due to uncertainty of input parameters in Grenoble

sand. The mean predicted infiltration rate (solid line) as well as 95% of confidence limits are

shown.
C - 33 

Figure C9a. Uncertainty of predicted infiltration as the results of contribution from input

parameters in Casa Grande-Trix fine-loamy soils.
C - 34 

Figure C9b. Uncertainty of predicted infiltration rate as the results of contribution from input

parameters in Casa Grande-Trix fine-loamy soils.