United National Risk Management EPA/600/R-97/128a
Environmental Protection Research Laboratory February 1998
Agency Ada, OK 74820
V>EPA Estimation of Infiltration
Rate in the Vadose Zone:
Compilation of Simple
Mathematical Models
Volume I
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EPA/600/R-97/128a
February 1998
ESTIMATION OF INFILTRATION RATE IN THE VADOSE ZONE:
COMPILATION OF SIMPLE MATHEMATICAL MODELS
Volume I
by
Varadhan Ravi
Dynamac Corporation
and
Joseph R. Williams
U.S. EPA
Contract No. 68-C4-0031
Project Officer
David S. Burden
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
Ada, OK 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
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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). 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.
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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.
Infiltration estimation methods are an integral part of the assessment of contaminant transport
and fate. This document provides an extensive compilation of methods, in addition to a
categorization of those methods. Appendices are included for methods based on the Green-Ampt
model and the Richards Equation, and for the estimation of soil hydraulic properties.
Clinton W. Hall, Director
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
in
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ABSTRACT
The unsaturated or vadose zone provides a complex system for the simulation of water movement
and contaminant transport and fate. Numerous models are available for performing simulations
related to the movement of water. There exists extensive documentation of these models. However,
the practical application of these infiltration models has not been adequately addressed in the
literature. In recent years, the use of vadose zone models has increased for the purpose of estimating
contaminant levels in soils related to different types of remedial decision-making. The rate of
infiltration of water is generally the most important parameter required in such models. Often these
models use an over-simplified estimate of the infiltration rate, which has little basis in reality and
actual field conditions. This document presents a compilation of simple infiltration models for
quantifying the rate of water movement. A great majority of the compiled models are based on
widely-accepted concepts of soil physics. These models, represented by simple mathematical
expressions, can be readily implemented in a spreadsheet environment. Proper use of these models
should provide a rational and scientific basis for remedial decision-making related to soil
contaminant levels.
IV
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CONTENTS
age
NOTICE ii
FOREWORD iii
ABSTRACT iv
1.0 Introduction 1
2.0 Estimation Methods for Infiltration Rate 2
3.0. Empirical Models 4
3.1 Kostiakov's Equation 4
3.2 Norton's Equation 5
3.3 Mezencev's Equation 5
3.4 SCS Equation 5
3.5 Holtan's Equation 6
3.6 Boughton's Equation 6
4.0. Green-Ampt Models 6
4.1 Basic Concepts 6
4.2 Compilation of Green-Ampt Models 7
4.3 Parameter Estimation for the Green-Ampt Models 9
5.0. Richards Equation Models 10
5.7 Basic Concepts 10
5.2 Simplistic Models of Infiltration 13
5.3 More Realistic Models of Infiltration 14
5.4 Parameter Estimation for the Richards Equation Models 18
6.0 Summary and Recommendations 19
REFERENCES 20
APPENDICES (Annotated Bibliography)
A. Green-Ampt Models A-l
B. Richards Equation Models B-l
C. Soil Hydraulic Property Estimation C-l
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LIST OF FIGURES
Page
Figure 1. Conceptualization of water content profiles during
infiltration, redistribution, and drainage 3
Figure 2. Illustration of the impact of hysteresis on wetting
and drying curves for water content 4
Figure 3. Illustration of Green-Ampt parameters and the conceptualized
water content profile, which demonstrates the sharp wetting front 7
LIST OF TABLES
Table 1. Models of Soil Water Movement Based on the Green-Ampt Concept 8
Table 2. Analytical Solutions of the Richards Equation 15
VI
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1.0 Introduction
The vadose zone is an integral component of the hydrological cycle, directly influencing
infiltration, storm runoff, evapotranspiration, interflow, and aquifer recharge. Understanding the
nature of water movement in the vadose zone and its quantification is essential to solving a
variety of problems. Examples of such problems are: prediction of runoff from given
precipitation events for the purposes of erosion control; sediment transport and flood control;
estimation of water availability for plant growth; estimation of water recharge to the underlying
aquifer; and assessment of the potential for aquifer contamination due to migration of water-
soluble chemicals present in the vadose zone. Consequently, the study of soil-water movement
has interested scientists from diverse disciplines such as soil science, hydrology, agriculture, civil
engineering, and the environmental sciences for several years.
The use of vadose zone models for the determination of contaminant cleanup levels, preliminary
remediation goals (PRG), soil screening levels (SSL), and other related terms has proliferated in
recent years. In providing technical support to various consultants, EPA Regional Offices, and
the state agencies, it has become apparent that there exists a lack of guidance in the estimation of
model parameters. In an attempt to address this problem, an EPA Issue Paper was prepared by
Breckenridge et al. (1991), which was later published as a journal article (Breckenridge et al.,
1994). This paper discussed the techniques for characterizing soils for their chemical, physical,
and hydraulic properties. It also provided a list of available field and laboratory measurement
techniques and look-up methods for these parameters. In addition to the identification of the
parameters, the document provided a table of common unsaturated zone models and their
parameter requirements. An extensive survey of unsaturated zone models was provided by van
derHiejde(1994).
A related effort is the release of the Soil Screening Guidance (U.S.EPA, 1996). This Guidance
provides the public with a tool for determining risk-based, site-specific, soil screening levels
(SSLs) for the evaluation of the need for further investigation at NPL (National Priority List)
sites. The release of this document with an accompanying technical background document has
resulted in a growth in emphasis by several states on the development of state soil screening
levels, and remediation goals and thus, a renewed interest in the prediction of contaminant
movement in the unsaturated zone, in most cases with simple analytical models for the
estimation of water movement. Often these models utilize an over-simplified estimate of the
infiltration rate, which may have very little basis in physics and actual site characteristics.
This document presents a compilation of simple mathematical models for quantifying the rate of
soil-water movement due to infiltration. The rate of infiltration is usually a required input
parameter for simpler screening level models which describe contaminant migration in the
vadose zone. There are numerous methods available for the measurement/estimation of
infiltration rate. In an activity and report supported by the American Petroleum Institute, Daniel
B. Stephens & Associates provided a good review of various field and laboratory methods, as
well as numerical models (API, 1996). The distinguishing feature and primary motivation for the
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present study is the compilation of relatively simple mathematical expressions (models),
implementable in a spreadsheet or symbolic math software environment. A great majority of the
compiled models are based on widely-accepted concepts of soil physics. Proper use of these
models should provide a rational and scientific basis for the determination of site-specific PRGs
and SSLs. The primary intended audience for this document would be applied hydrologists, soil
scientists, and modelers working on field problems. Soil physicists and hydrologists in the
research community with a need to locate infiltration models and mathematical solutions may
also find this document useful.
The mathematical models compiled here have several advantages over other estimation
techniques:
(a) They are relatively easy to use.
(b) They are based on widely-accepted concepts of soil physics (except for the empirical
models).
(c) Required hydraulic parameters can be easily obtained from the published literature as
well as electronic databases.
(d) Site-specific measurement of all the parameters, although recommended, is not necessary
for obtaining preliminary estimates of water flux to be used in screening level
contaminant transport models.
(e) The models presented here are ideal for simulating the infiltration component in the
watershed models since they incorporate all the essential physics of the soil-water flow
phenomena without being complicated.
(f) Spatial variability of the soil parameters can be more easily incorporated into the
analytical models either deterministically or statistically (e.g., a Monte-Carlo type
approach).
(g) The mathematical solutions compiled here may also be useful as benchmarks for
establishing the validity and accuracy of the finite-difference or finite-element
discretization schemes.
2.0 Estimation Methods for Infiltration Rate
There are numerous techniques available for the estimation of water infiltration rate through the
vadose zone. API (1996) provides a good review of such techniques identified as: 1) soil-water
balance, 2) lysimeter measurements, 3) Darcy flux method, 4) plane of zero flux method, 5) soil
temperature methods, 6) electromagnetic methods, 7) ground-water basin outflow method, 8)
water-level fluctuations, 9) stream gauging, 10) chemical tracers and isotopes, 11) chloride mass
balance, 12) water balance models, and 13) numerical models based on the Richards equation.
However, none of these methods offers a quick and easy way to obtain an estimate of infiltration
rate for the purposes of preliminary analysis and decision-making. Therefore, the objective of
this study is to compile estimation methods which are easy-to-use and yield scientifically-based
estimates using soil-hydraulic and climatic parameters representative of the prevailing site
conditions. The compiled methods can be divided into three broad categories of: a) empirical
models, b) Green-Ampt models, and c) Richards equation models. Numerous formulations of
these models have then been developed for utilization with specific site conceptualizations, or
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boundary conditions. Before discussing these models, a brief introduction to the processes
involved in soil-water movement will be presented. For more detailed discussion of these
processes the reader is referred to several excellent soil physics and hydrology textbooks such as
Hillel (1980), Hanks and Ashcroft (1980), Guymon (1994), Stephens (1996), Dragun (1988), and
Fetter (1988).
Water movement in the vadose zone is generally conceptualized as occurring in the three stages
of infiltration, redistribution, and drainage or deep percolation as illustrated in Figure 1. For this
conceptualization, infiltration is defined as the initial process of water entering the soil resulting
from application at the soil surface. Capillary forces, or matric potentials, are dominant during
this phase. Redistribution occurs as the next stage where the infiltrated water is redistributed
within the soil profile after the cessation of water application to the soil surface. During
redistribution, both capillary and gravitational effects are important. Simultaneous drainage and
wetting takes place during this stage, and the impact of hysteresis may be important. Hysteresis
is the phenomenon illustrated in Figure 2, which implies that the wetting and drying curves of a
specific soil will not be identical. Evapotranspiration takes place concurrently during the
redistribution stage, and will impact the amount of water available for deeper penetration within
the soil profile. The final stage of water movement is termed deep percolation or recharge, which
occurs when the wetting front reaches the water table. For the purposes of this report, the term
"infiltration" will lump all the three stages of water movement through vadose zone. Also the
terms, "water flux," "infiltration rate," and "rate of water movement" will be used
interchangeably.
Surface
Water Content
Water Content
Water Content
I
Q
Infiltration
Redistribution
Drainage
Figure 1. Conceptualization of water content profiles during infiltration, redistribution, and
drainage (deep percolation).
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Water Content
03
O
1
Wetting
Saturation
\ ....*i-;
Figure 2. Illustration of the impact of hysteresis on
wetting and drying curves for water
content (Hillel, 1980).
3.0. Empirical Models
Empirical methods are usually in the form of simple equations, the parameters of which are
derived by means of curve-fitting the equation to actual measurements of cumulative water
infiltration. These equations only provide estimates of cumulative infiltration and infiltration
rates, and do not provide information regarding water content distribution. Most are derived on
the basis of a constant water content being available at the surface. A few of the commonly used
infiltration equations, which have no apparent physical basis, are presented below. Detailed
discussions of these empirical models are provided by Philip (1957), Swartzendruber and Hillel
(1973), Dunin (1976), and Parlange and Haverkamp (1989).
3.1 Kostiakov 's Equation
Kostiakov (1932) proposed the following equation for estimating infiltration,
/(O = 0) and (3 (0
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The constants arand /?can be determined by curve-fitting Eq. 2 to experimental data for
cumulative infiltration, I(t). Since infiltration rate, /', becomes zero as t -> °°, rather than approach
a constant non-zero value, Kostiakov proposed that the Eqs. 1 and 2 be used only for t < tmax,
where tmax is equal to (cc/K^11®, andKs is the saturated hydraulic conductivity of the soil.
Kostiakov's equation describes the infiltration quite well at smaller times, but becomes less
accurate at larger times (Philip, 1957; Parlange and Haverkamp, 1989).
3.2 Morton's Equation
Horton (1940) proposed to estimate infiltration in the following manner,
i(f) = if+(iQ-if)e-v (3)
and
7(0 =ift + -(i0-if)(l-e-^ (4)
where i0 and /yare the presumed initial and final infiltration rates, and /is an empirical constant.
It is readily seen that i(t) is non-zero as t approaches infinity, unlike Kostiakov's equation. It does
not, however, adequately represent the rapid decrease of/' from very high values at small t
(Philip, 1957). It also requires an additional parameter over the Kostiakov equation. Parlange
and Haverkamp (1989), in their comparison study of various empirical infiltration equations,
found the performance of Horton' s equation to be inferior to that of Kostiakov's equation.
3. 3 Mezencev 's Equation
In order to overcome the limitations of Kostiakov's equation for large times, Mezencev (Philip,
1957) proposed the following as modifications to Eqs. 1 and 2. Mezencev proposed infiltration
estimated by
7(0 = if+at-V (5)
and
where /yis the final infiltration rate at steady state.
3.4 SCS Equation
The USDA Soil Conservation Service (1957) developed an equation for rainfall-runoff
relationship based on daily rainfall data as input:
(P-0.2F )2
R =
P+Q.8F
w
where P is the daily rainfall, R is the runoff, and Fw is a statistically derived parameter bearing
some resemblance to the initial soil moisture deficit. Infiltration is calculated as the excess of
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rainfall over runoff:
/ = P-R (8)
3.5 Holt an's Equation
The empirical infiltration equation devised by Holtan (1961) is explicitly dependent on soil water
conditions in the form of available pore space for moisture storage:
/(O = if + ab(u-I)lA (9)
where a is a constant related to the surface conditions varying between 0.25 and 0.8, bis a
scaling factor, a> is the initial moisture deficit or the pore space per unit area of cross section
initially available for water storage (cm), and / is the cumulative infiltration (cm) at t. This
equation has been found to be suitable for inclusion in catchment models because of soil water
dependence, and satisfactory progress has been reported for runoff predictions (Dunin, 1976).
3.6 Bough ton's Equation
Boughton (1966) modified the rainfall-runoff relationship of USDA-SCS (1957) given by Eq.(7)
as follows:
where Fr is an empirical parameter; however, some success has been reported when interpreted
as the initial soil moisture deficit (Dunin, 1976). Infiltration is calculated using Eq. (8).
4.0. Green-Ampt Models
4.1 Basic Concepts
Green and Ampt (1911) derived the first physically based equation describing the infiltration of
water into a soil. The Green-Ampt model has been the subject of considerable developments in
applied soil physics and hydrology owing to its simplicity and satisfactory performance for a
great variety of hydrological problems. For many hydrological problems the use of more
sophisticated approaches (e.g., the models based on the nonlinear Richards equation,) is both
impractical and inefficient due to more information on soil hydraulic parameters (e.g. water
retention and hydraulic conductivity functions) being required. In these methods, the entire soil
moisture-pressure profile is generally evaluated, even though the main quantity of interest is the
flux at one or both of the boundaries. Therefore, not surprisingly, the Green-Ampt equation has
been the choice model of infiltration estimation in many physically-based hydrologic models
(Freyberg et al., 1980). Also, the USDA's Agricultural Research Service (ARS) has done
extensive work to develop empirical relations for obtaining the Green-Ampt model parameters
(Brakensiek and Onstad, 1977; McCuen et al., 1981; Rawls and Brakensiek, 1982; Springer and
Cundy, 1987), thus providing additional impetus for inclusion in many watershed models
(Goldman, 1989).
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Green and Ampt assumed a piston-type water content profile (Figure 3) with a well-defined
wetting front. The piston-type profile assumes the soil is saturated at a volumetric water content
of 6S (except for entrapped air) down to the wetting front. At the wetting front, the water content
drops abruptly to an antecedent value of 60, which is the initial water content. The soil-water
pressure head at the wetting front is assumed to be /^(negative). Soil-water pressure at the
surface, /z,, is assumed to be equal to the depth of the ponded water.
•W/WAWVW
e
o
e<
Soil
Surface
Depth Z -
Wetting Front
Pressure Head
Actual.
*
' Green-Ampt
Model
a) Green-Ampt Parameters
b) Water Content Profile
Figure 3. Illustration of Green-Ampt parameters and the conceptualized water content
profile, which demonstrates the sharp wetting front.
At any time, t, the penetration of the infiltrating wetting front will be Z. Darcy's law can be
stated as follows:
q . 4L .-K\OdW*
^ dt Z
(11)
where Ks is the hydraulic conductivity corresponding to the surface water content, andlft) is the
cumulative infiltration at time t, and is equal to 2(6-6^. Using this relation forl(t) to eliminate
Z from Eq. 11, and performing the integration yields,
I --
(V/0(0-GO)
(12)
Equation 12 is precisely the statement of the Green-Ampt model. Philip (1957) demonstrated
that the Green-Ampt equation can also be obtained as an exact solution of the Richards equation
when the diffusivity function is assumed to be a Dirac Delta-type function with a non-zero value
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only at the saturated water content. Philip called the Green-Ampt model the "delta-function"
model.
4.2 Compilation of Green-Ampt Models
Even though originally developed for idealized conditions (i.e., homogeneous soil and constant
surface ponding depth), the Green-Ampt model has been extended to take into account more-
realistic features. Refer to Table 1 for a list of methods to estimate water flux based on the
Green-Ampt model. Also, Appendix A is provided as an annotated bibliography pertaining to
the Green-Ampt approach. The primary utility of the Green-Ampt approach lies in the
estimation of the water flux, and it must be emphasized that the actual water content distribution
with depth, 6(z), cannot be simulated, since the model formulation assumes a sharp wetting front.
Table 1. Models of Soil Water Movement Based on the Green-Ampt Concept1
Model Author(s)
Green and Ampt (1911)
Bouwer(1969)
Childs and Bybordi (1969)
Mein and Larson (1973)
Swartzendruber (1974)
Morel-Seytoux and Khanji
(1974)
James and Larson (1976)
Important Features / Limitations
Equation 9 for cumulative infiltration, implicit in time, is the Green-Ampt
model; sharp wetting front; constant ponding depth; homogeneous soil;
uniform antecedent water content.
Layered soils; non-uniform antecedent water content; constant ponding
depth.
Implicit equation (Eq. 5) for cumulative infiltration in layered soils; constant
ponding depth; uniform antecedent water content.
Pre-ponding (Eq. 6) and ponded infiltration (Eq. 8); Eq. 8 has to be
integrated after replacing^, with dF/dt and the lower limits of integration
(tyFJ; constant rainfall rate greater than Ksat; homogeneous soil; uniform
antecedent water content.
Constant surface water flux greater than saturated hydraulic conductivity;
pre-ponding and ponding infiltration; Eqs. 4 and 5 yield time to ponding and
cumulative infiltration prior to ponding, respectively; Eq. 8 or 12 gives
cumulative infiltration, implicitly in time, after ponding; two approximate
explicit equations (Eqs. 23 and 30) for cumulative infiltration as a function
of time; homogeneous soil; uniform antecedent water content.
Infiltration under the two-phase flow of air and water (Eq. 17 for total
velocity); a rigorous derivation of expressions (Eqs. 18 and 26) for the
wetting front suction, hf, and a viscous correction factor, /?, in terms of
initial water content; homogeneous soil; uniform antecedent water content.
Intermittent (piecewise-constant) rainfall; infiltration (Eqs. 2 and 3) and
redistribution (Eq. 1); homogeneous soil; uniform antecedent water content.
1 Equation numbers referenced in this table correspond to those in the cited article. They do not
correspond to the equation numbers in this report.
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Model Author(s)
Lietal. (1976)
Smith and Parlange (1978)
Chu (1978)
Flerchinger et al. (1988)
Philip (1992)
Philip (1993)
Salvucci and Entekhabi
(1994)
Important Features / Limitations
Explicit approximations for calculating the Green- Ampt model cumulative
infiltration and infiltration rate as functions of time (Eqs. 11, 25, and 6);
homogeneous soil; constant head ponding at the surface; uniform antecedent
water content.
Two 2-parameter models for ponding time (Eqs. 6 and 9) and infiltration
rate (Eqs. 20 and 26); arbitrary transient rainfall; homogeneous soil; uniform
antecedent water content.
Transient rainfall; pre-ponding (Eqs. 18, 19, and 23-26) and ponded
infiltration (Eqs. 18, 19 and 27); homogeneous soil; uniform antecedent
water content.
An explicit equation for cumulative infiltration for layered soils; extension
of Li et al. (1976) approach for layered soils; constant head at the surface.
A solution for falling head ponded infiltration (Eq. 4). Solution form is the
same as that for constant head infiltration; only the values of the constants
change.
Variable-head ponded infiltration (Eq. 1 1) due to constant or arbitrarily
transient rainfall; Eq. 26 for constant rainfall, and Eq. 35 for piecewise
constant rainfall; homogeneous soil; uniform antecedent water content.
Four-term expression explicit approximations for Green-Ampt infiltration
rate and cumulative infiltration (Eqs. 9 and 10) as a function oft;
homogeneous soil; uniform antecedent water content.
4.3 Parameter Estimation for the Green-Ampt Models
The popularity of the Green-Ampt models is primarily due to simplicity, adaptability to varying
scenarios, and the availability of characteristic parameter values for various soil textures and
conditions. Extensive studies conducted by the USDA's Agricultural Research Service (ARS)
have resulted in the development of empirical relations for the model parameters in terms of
easily-measurable variables. This has provided an additional impetus for the inclusion of Green-
Ampt models in many watershed models (Goldman, 1989).
Bouwer (1966) demonstrated that^ in Eq. 12 is not equal to the saturated hydraulic conductivity
of the soil (Ksat), but can be approximated as 0.5*Ksat. He also suggested that hfcan be treated as
air-exit suction head. Neuman (1976) derived expressions for hfi valid for small, intermediate,
and large times. Useful empirical expressions and various statistical correlations are available
for K, and /^(Brakensiek and Onstad, 1977; McCuen et al., 1981; Rawls and Brakensiek, 1982).
Appendix C is provided to include additional references pertaining to the estimation of soil
hydraulic parameters.
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5.0. Richards Equation Models
5.7 Basic Concepts
The Darcy -Buckingham law (Eq. 13), which is the vadose zone analog of Darcy's law, for soil
water flux is given as follows:
q = -*(0)Vi|r(0) (13)
where q is the water flux (cm/s), $is the volumetric water content as a function of location and
time t, K is the unsaturated hydraulic conductivity of soil (cm/s) as a function of volumetric water
content, and ^is the total soil-water head (cm) as a function of the volumetric water content, and
z is the vertical coordinate. For this formulation, z is positive in the direction of gravity with z=0
being the top surface. The primary difference between the Darcy -Buckingham law and its
counterpart in the saturated flow is the dependence of hydraulic conductivity and total head on
the volumetric water content. The total head ^is the sum of capillary head, h(which is
dependent on moisture content 0), and the elevation head, z. The Darcy-Buckingham law can be
combined with the continuity (i.e., the differential water balance) equation to obtain a general
form of the Richards equation:
f=v (^(0)^(0))-^ 04)
Sometimes, specifically when solving multi-dimensional problems, it is helpful to make a change
of dependent variable known as the Kirchoff transformation:
h
U = fK(a)da n$\
^
where the lower limit can be chosen as arbitrarily as is convenient. C/is known as the matric flux
potential. With this transformation Eq. 14 reduces to:
where D = Kdh/d0is the diffusivity of the soil. This is the f/-based formulation of the Richards
equation.
A vast majority of the analyses of Richards equation has considered only the vertical soil water
movement, treating the problem as essentially one-dimensional. In which case, Eq. 14 can be
rewritten as
= _
dt dz( dz ) dz ^ '
Eq. 17 involves the two dependent variables h and 0, and can be rewritten either in terms of dor
with h as the dependent variable. The $-based formulation of the one-dimensional Richards
model is:
10
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dt dz( dz) dQ dz
Similarly, an //-based formulation can be written as follows:
dt dz\ dz) dh dz
where C(h) = dd(h)/dh.
Equations 18 and 19 are not completely equivalent (Philip, 1969). As an example, Eq. 19 may
still apply when h exceeds the air-entry value (i.e., the value at which air enters an initially
saturated soil), hb, or when positive, as may be the case when a depth of free water is ponded
over the soil. Eq. 18 cannot be applied to these scenarios. This results from the water
diffusivity, D(6), being infinite due to dh/ddbecoming infinite for all h > hb. Under such
saturated conditions Eq. 19 reduces to the Laplace equation, since C(h) = 0 andK(h) = constant.
Also, Eq. 18 cannot apply to layered soils due to the existence of discontinuities in water content
at the layer interfaces.
In the formulation of Eqs. 18 and 19 the assumption is tacitly made that 6(h) andK(h) are single-
valued functions such that the inverse functions of h(6) and K(6) exist, and are single-valued.
For soils exhibiting hysteresis these functions are two-valued, and special care should be
exercised in the solution of the Richards equation. Soil hysteresis is usually understood in terms
of the curve depicting the equilibrium relationship between water content and capillary pressure.
This equilibrium relationship can be obtained in two ways: (1) in drying, by taking an initially
saturated soil sample and applying increasing capillary pressure to dry the soil while recording
equilibrium water content at each applied suction; and (2) in wetting, by gradually wetting up an
initially dry soil sample while reducing capillary pressure. Each of these methods yields a
continuous curve, but the two curves generally will not be identical (refer to Figure 2). The
equilibrium soil water content at an applied suction is greater in drying than in wetting. This
dependence of the equilibrium water content upon the direction of the process is referred to as
hysteresis (Hillel, 1980).
Several limitations exist to the general applicability of the Richards model as represented by Eqs.
18 and 19. Important limitations are (refer to Philip (1969) for a detailed discussion of these) as
follows:
• The specification of a representative elementary volume or a Darcy scale may not
be possible (e.g., preferential pathways or macropores may be present).
• Colloidal swelling and shrinking of soils may demand that the water movement be
considered relative to the movement of the soil particles; this phenomenon may
also cause significant changes in soil permeability.
• Two-phase flow involving air movement may be important when air pressures
differ significantly from atmospheric pressure.
• Thermal effects may be important, especially for evaporation during the
redistribution of infiltrated water, in which case the simultaneous transfer of both
11
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heat and moisture needs to be considered.
• Soil hysteresis may be very significant after infiltration ceases and redistribution
begins; wetting and drying occur simultaneously; different points in the soil
follow different scanning (h versus 6) curves; and there is no definite relationship
between gradients of h and gradients of 6.
• Sources and sinks (e.g., root extraction) are neglected, but may be readily
included.
• Flow is one-dimensional; this is reasonable for rainfall or irrigation infiltration
over a large area.
The following system inputs are needed to obtain soil water flux by solving the Richards
equation, Eqs. 18 or 19:
• A boundary condition (BC) at the water-supply surface, z = 0; either a
concentration-type, 6= 0s(t) or h = H(t), or a flux-type, K-D d6/dz = R(t) or K(l -
dh/dz) =R(t).
• An initial condition for all z, 6(z,t=0) = 60(z) or h(z,t=0) = h0(z), and R(t) is the
rate of water entry at the surface as a function of time.
• Soil hydraulic parameters, K(6) and h(6).
The boundary and initial conditions are formulated according to the nature of the problem to be
solved. Different conditions are required for infiltration, redistribution, evaporation, and
drainage problems (Refer to page 83 of Hanks and Ashcroft, 1980). Except for drainage
problems where the soil water profile drains to the ground water, a second boundary condition is
not explicitly formulated because the soil domain is generally assumed to be semi-infinite.
Where the water depth over the supply surface (or the hydrostatic head under which water is
supplied) is not negligibly small, and also where the air-entry value of the soil is (arithmetically)
large, the //-based formulation of the Richards equation, Eq. 19, along with appropriate boundary
conditions in terms of h, is the correct model. In all other cases, the $-based formulation is more
convenient to solve.
The mathematical analysis of the Richards equation has almost exclusively dealt with the
absorption (moisture movement without gravity) and infiltration problems. Relatively little, if
any, attention has been paid to the redistribution and drainage problems. Philip (1991) gives
three reasons for this relative paucity of mathematical-physical studies of redistribution. First,
the initial conditions for the redistribution process tend to be complicated. Second, various
mathematical techniques useful in the infiltration context do not, generally, carry over to
redistribution problems. Third, redistribution processes involve the very considerable
complication of capillary hysteresis. Therefore, the analyses of redistribution and drainage
processes using the Richards model have almost totally relied upon numerical techniques.
Depending on the simplicity (or complexity) of these input parameters, the Richards equation can
be solved exactly or numerically. Therefore, the models presented here will be separated
according to such a classification as simple infiltration equations, analytical or quasi-analytical
models and numerical models. The infiltration equations only relate the cumulative infiltration
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to the time; they do not provide information on the moisture profile or water flux distribution.
There are numerous analytical/quasi-analytical and numerical solutions to Richards equation.
These provide estimation of moisture and flux distribution as well as infiltration rates. Only
some of the more significant approaches will be presented here; however, all are subject to the
limitations previously discussed, unless stated otherwise.
5.2 Simplistic Models of Infiltration
There are numerous simple infiltration equations which are solutions to the Richards equation
under highly ideal conditions. They are quite restrictive since they only describe infiltration from
a water-ponded surface into a semi-infinite, homogeneous soil with a uniform antecedent water
content. Swartzendruber and Clague (1989) list several such infiltration equations, some of
which are presented here. Each of these equations are given in terms of the dimensionless time
(T), and dimensionless cumulative infiltration (Y), which are defined as follows:
T = K2t/S2 (20)
and
Y = KsI/S2 (21)
where Ks is the saturated hydraulic conductivity and S is the sorptivity defined by Philip (1957).
Philip (1957)
Y = T1'2 + XT (22)
where A is a constant between 0 and 1.
Philip (1969)
Y = I[2r1/2exp(-777i) + (2r + 7i)er/[(777i)1/2] + 27J
Knight (1973)
(23)
Y = - In [l + erf(4T/nf2] + T (24)
Parlange (1975)
27-[l -exp(-2r1/2)] = IT (25)
Brutsaert (1977)
Tl/2
7 = T + '
(26)
where a is either 2/3 or 7.
Collis-George (1977)
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7 = r + l[tanh(7V27)]1/2 (27)
where TV is a dimensionless constant that varies between 1 and 4.
Swartzendruber and Clague (1989)
Y = r + l[l-exp(-ar1/2)] (28)
1/C
where
-------
Table 2. Analytical Solutions of the Richards Equation2
Model Author(s)
Important Features / Limitations
Philip (1969)
Boltzmann similarity transformation used to obtain a power-series solution in t1'2
(Eqs. 83,87,88,91 and 92); the coefficients, which are functions of 6, are
obtained (numerically) as the solutions of a set of linear, ordinary integro-
differential equations; homogeneous soil; constant water content at the supply
surface ; uniform initial water content; there is a practical time limit beyond
which the series may not converge; an asymptotic solution valid for large times
is also provided; a computer code, INFIL, based on this solution is available (El
Kadi, 1983); it can accept various K(6) and h(6) functions as input.
Philip (1969)
A steady-state (large t) solution (Eq. 117) to Kirchoff-transformed infiltration
equation (a "quasilinear" equat-ion); solution (Eq. 122) is also provided for
infiltration from a point source into a three-dimensionally infinite region;
homogeneous soil; K(h) is equal to Ks exp(ah).
Philip (1969)
Solutions (Eqs. 146,132 and 145) to a linearized form of the Richards equation;
homogeneous soil; constant water content at the supply surface ; uniform initial
water content; D(6)=Df and dK/dd=k are constants; D* is obtained as a function
of sorptivity, S, by matching the linear and nonlinear one-dimensional absorption
solutions; k is the velocity of the "profile at infinity."
Philip (1969)
Solutions to linearized Richards equation describing multi-dimensional
infiltration from the surfaces of cylindrical (Eqs. 156 and 155), and spherical
(Eqs. 160 and 155) cavities; source radius is small; homogeneous soil; constant
water content at the supply surface; uniform initial water content; linearized
solutions are accurate at large t when applied to multi-dimensional problems.
Philip (1972)
Solutions for quasilinear, steady infiltration in heterogeneous soils; Eq. 2
describes K(h,z); solutions for buried, surface, and perched point sources, Eqs.
29, 34, and 38, respectively; solutions for buried, surface, and perched line
sources, Eqs. 41, 43, and 46, respectively; source strengths are known.
Raats (1972)
A solution (Eq. 8 or 9) for matric flux potential, U(h), for steady infiltration from
a single point source (of known strength) at arbitrary depth; exponentialK(h);
homogeneous soil; also, Eq. 6 (Philip, 1971) relates the solution for surface
source (point, line, or areal) to that for a source buried in an infinite soil mass
which is easily obtainable; Philip's (1971) superposition theorem for surface
sources (Eq. 6) is generalized to an arbitrary distribution of sources at arbitrary
depths (Eq. 12); Eq. 14 provides the inverse transformation for obtaining the
pressure head h(U).
2 Equation numbers referenced in this table correspond to those in the cited article. They do not
correspond to the equation numbers of this report.
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Model Author(s)
Important Features / Limitations
Philip (1974)
Exact solutions to the Burger's equation which is a minimally nonlinear form of
Richards equation; these were developed by Knight (1973); a constant ZY$ and
quadratic K(6) are assumed; Eqs. 24 and 25 provide the moisture profile and
infiltration rate for constant moisture content BC; Eq. 29 provides the moisture
profile for constant flux BC for pre-ponding when the flux is greater thanKsat.
Warrick(1974a)
An analytical solution (Eq. 11) for steady, quasilinear, one-dimensional
infiltration for an arbitrary plant-water uptake function; K(h) is exponential;
semi-infinite and finite soil domains; constant surface flux; homogeneous soil.
Lomen and Warrick
(1974)
Analytical solutions (Eqs. 12 and 17) for linearized, unsteady, two-dimensional
infiltration from buried and surface line sources; D(6) is constant and K(h) is
exponential; uniform initial water content; zero-flux at the surface except at the
source; homogeneous soil.
Warrick (1974b)
Analytical solutions (Eqs. 13 and 16) for linearized, unsteady, three-dimensional
infiltration from buried and surface point sources; D(6) is constant and K(h) is
exponential; uniform initial water content; zero-flux at the surface except at the
source; homogeneous soil.
Warrick (1975)
An analytical solution (Eqs. 13-15) for one-dimensional, linearized Richards
equation; time-varying surface flux or water content and arbitrary initial
conditions; D(0) is constant and K(h) is exponential.
Warrick and Lomen
(1976)
Analytical solutions for linearized, unsteady, infiltration from strip and disc
sources; Eqs. 10 and 14 for buried and surface strip sources, respectively; Eqs.
20 and 25 for buried and surface disc sources, respectively; D(6) is constant and
K(h) is exponential; uniform initial water content; zero-flux at the surface except
at the source; homogeneous soil.
Eagleson(1978)
Equations for soil-water flux during dry and wet periods; Eq. 59 for flux due to
capillary rise from the water table during dry periods; Eq. 60 for infiltration
during a storm; Eq. 61 for exfiltration or upward movement due to
evapotranspiration and capillary rise from the water table; Eq. 62 for deep
percolation to water table; Eqs. 60 and 61 are based on Philip's (1969) series
solution; moisture profile is assumed to be uniform before storm and during
interstorm periods at an appropriate average value; homogeneous soil.
Lomen and Warrick
(1978)
An analytical solution (Eqs. 21, 17 and 19) for the linearized one-dimensional
Richards equation with plant-water uptake; D(6) = constant and K(h) is
exponential; the sink (water uptake) function is a sequence (for successive time
intervals) of depth-dependent functions; surface flux can be time-dependent;
homogeneous soil.
/. (1980)
Equations for redistribution and drainage based on kinematic wave theory; only
gravity effects (unit total potential gradient) are considered; a first-order,
hyperbolic PDE results; solution is based on the method of characteristics;
Table 1 provides the procedure and actual examples for obtaining moisture
profiles, 6(z,t), for realistic K(0> functions; uniform initial moisture profile
before redistribution; homogeneous soil.
16
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Model Author(s)
Important Features / Limitations
Parlange et al. (1982)
A three-parameter equation (Eq. 13) for estimating cumulative infiltration;
constant surface water content (non-ponding); uniform initial water content;
homogeneous soil.
Dagan and Bresler
(1983)
A sharp front or a rectangular profile model for one-dimensional infiltration and
redistribution; Eq. 36 for constant water content BC; Eqs. (47) or (49), and (53)
for time-varying or constant flux infiltration; for redistri-bution Eqs. (68) and
(53) are valid; ponding time is given by Eq. (55); uniform initial water content.
Smith (1983)
Similar to Sisson's (1980) approach, but a little more general; does not neglect
capillary gradients; assumes q = K(6) - D(6) dd/dz to be only a function of 6, z/t
= dq/dd; either moisture content or flux specified at the surface; homogeneous
soil.
Philip (1984a)
A solution (Eq. 28) for quasilinear (Kirchoff-transformed), steady infiltration
from circular cylindrical cavities; non-zero source radius; water content or
suction specified at the source boundary; K(h) is exponential; homogeneous soil.
Philip (1984b)
A solution (Eq. 20) for quasilinear, steady infiltration from spherical cavities;
finite source radius; water content or suction specified at the source boundary;
K(h) is exponential; homogeneous soil.
Morel-Seytoux (1984)
A rectangular profile model for redistribution based on Brooks-Corey functions
for h(6) and K(6); soil is at residual saturation prior to a wetting event; water
content within the actual profile is equal to that within the rectangular profile;
cumulative infiltration prior to redistribution is known; Eqs. 16, 17 and 18
provide the moisture profile, front velocity (in the Darcy sense) and front
location, respectively, during redistribution; evaporation is neglected;
homogeneous soil.
Charbeneau(1984)
Eq. 7 yields the moisture profile based on kinematic wave theory; surface
moisture content is specified asgft); capillary gradients are neglected; an
inconsistency in Smith's (1983) formulation relating to the non-neglect of
capillary gradients is shown; homogeneous soil.
Parlange etal. (1985)
An equation, Eq. 23, along with Eqs. 1 and 19b, for estimating cumulative
infiltration, based on Parlange et al. (1982); ponded conditions at the surface;
uniform initial water content; homogeneous soil.
Sander etal. (1988)
An exact, parametric solution (Eqs. lOa-lOd), with parameter £ for unsteady,
one-dimensional, constant flux infiltration; both D(6) and K(6) are nonlinear,
but of special forms (Eqs. 4 and 6); Eq. 8 provides the h(6) relation; uniform
initial water content; homogeneous soil.
Broadbridge and
White (1988)
An exact, parametric solution (Eqs.41-44) with parameter (, for unsteady, one-
dimensional, constant flux infiltration; almost identical to that of Sander et al.
(1988); D(6) andK(0) are of the form given by Eqs. 4 and 5, respectively,
enabling the Richards equation to be reduced to the minimally nonlinear Fokker-
Planck equation (Philip, 1974); the actual expressions ofD(0) andK(0) are,
respectively, Eqs. 21 and 11; Eq. 22 provides the h(6) relation; uniform initial
water content; homogeneous soil.
17
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Model Author(s)
Important Features / Limitations
Broadbridge et al.
(1988)
The results of Broadbridge and White (1988) for semi-infinite domain are
extended to a finite soil column with zero-flux BC at the bottom; Eqs. (A27-
A30) provide the parametric solutions; there appears to be a slight discrepancy
between Eqs. 16 & 17 and Eqs. A28 & A30.
Swartzendruber and
Clague (1989)
A listing of various, simple infiltration equations (Eqs. 5-24) in terms of two
dimensionless variables which are defined by Eqs. 3 and 4; constant water
content at the surface; uniform initial water content; homogeneous soil.
Warrick(1990)
An analytical solution (Eqs. 26-28) for one-dimensional soil-water movement
based on Broadbridge and White (1988); arbitrary initial water content; constant
surface flux, R, but it can be positive, zero, or negative; when,R is negative, Eqs.
36 and 39 may have to be used instead of Eq. 27; homogeneous, semi-infinite
soil.
Barry and Sander
(1991)
A parametric, analytical solution (Eqs. 13a, 13b, 1 la, and 1 Ib), with parameter
for one-dimensional soil-water movement; arbitrarily time-varying flux at the
surface; arbitrary initial water content; D(0) and K(6) given by Eqs. 2b and 2c;
homogeneous, semi-infinite soil.
Warrick etal. (1991)
An analytical solution (Eqs. 26-30) for one-dimensional, time-varying
infiltration; based on Broadbridge and White (1988); K(6) and D(6) functions
are the same as in Broadbridge and White (1988); arbitrary initial water content;
surface flux is treated as a series of constant rates stepped over arbitrary time
intervals; homogeneous, semi-infinite soil.
Sander etal. (1988)
An exact solution to nonlinear, nonhysteretic redistribution of water in a vertical
soil column of finite depth; both D(6) and K(0) are nonlinear, but of special
forms (Eqs. 3 and 4); no flow at the surface; arbitrary initial water content
distribution; homogeneous soil.
Protopapas and Bras
(1991)
Solutions to the linearized, one- (Eq. 8) and two- (Eq. 15) dimensional, unsteady
unsaturated water flow in terms of matric flux potential; K(h) is exponential and
D(6) is constant; Ksat is assumed to vary exponentially in space; uniform initial
water content; constant matric flux potential BC at the surface; soil is infinite
laterally and semi-infinite vertically.
Hills and Warrick
(1993)
An exact solution to Burger's equation for soil water flow in a finite length
domain (Eqs. 37, 32, and 21); D is constant and K(0> is a quadratic function (Eq.
7); constant flux at the top surface; constant water content at the bottom;
homogeneous soil.
5.4 Parameter Estimation for the Richards Equation Models
The solution of Richards equation requires, among other things, the specification of soil
characteristic functions, K(6) and h(6). Based on experiments, these functions can be estimated
in a tabulated form; however, algebraic forms are preferred in order to facilitate analytical or
numerical solutions of the Richards equation. There are several empirical functional forms of the
water retention function, h(0), and the hydraulic conductivity function, K(6) (El-Kadi, 1985).
18
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The coefficients of water retention function can be determined from experimental data of h
versus 6, using nonlinear regression techniques (van Genuchten et al., 1991). Even though K(6)
can be estimated in a similar manner, the measurement of unsaturated hydraulic conductivity for
a fixed water content is very difficult. Therefore, h(6) is customarily used in conjunction with
theoretical models such as those of Burdine (1953) and Mualem (1976) to obtain K(6). For each
of these models an additional parameter, the saturated hydraulic conductivity (Ksal) is required.
The most widely used h(6) functions are those developed by Brooks and Corey (1964) and van
Genuchten (1980). Numerous sources in the soil science literature provide estimates of the
model parameters to calculate h(6) and K(6). The three most comprehensive sources for these
model parameters are Brakensiek etal. (1981), Panian (1987), and Carsel and Parrish (1988).
These papers provide estimates of saturated hydraulic conductivity and water retention
parameters for soils of various textures. Johnson and Ravi (1993) evaluated the basis of these
databases and discussed their potential limitations and possible uncertainties with respect to use
in simulating water movement in the vadose zone, van Genuchten et al. (1992) presented
various indirect methods for estimating the hydraulic properties of unsaturated soils. An
electronic database called IMSODA (Leij et al., 1996) is another good source for soil hydraulic
parameters. Also refer to Appendix C for an annotated bibliography on soil hydraulic parameter
estimation.
6.0 Summary and Recommendations
A compilation of simple mathematical models has been presented for quantifying the rate of soil-
water movement due to infiltration. The rate of infiltration is usually required as an input
parameter by the simpler, screening level models which describe contaminant migration in the
vadose zone. Even though numerous techniques are available for estimating infiltration rate
(API, 1996), none offers a quick and easy way to obtain an estimate for the purposes of
preliminary analysis and decision-making. The estimation methods presented here are easy-to-
use, and yield scientifically-based estimates using soil-hydraulic and climatic parameters
representative of the prevailing site conditions. The compiled methods were divided into three
types: a) empirical models, b) Green-Ampt models, and c) Richards equation models. These
methods (except the empirical models) are based on widely-accepted concepts of soil physics.
Proper use of them should provide a rational and scientific basis for the determination of site-
specific PRGs and SSLs.
Evaluation of selected models is presented in an accompanying document (Ouyang et al., 1998).
These models should be adequate for quantifying infiltration rates under conditions which are
commonly-encountered at many hazardous waste sites. The document includes examples
illustrating the proper application of these models under different scenarios. Also, sources of
information for the model parameters are identified.
19
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Resource Center, Desert Research Institute. DOE/NV/10384-20.
Parlange, J.-Y. 1975. A Note on the Green and Ampt Equation. Soil Science 119(6): 466-467.
Parlange, J.-Y., I. Lisle, R.D. Braddock, and R.E. Smith. 1982. The Three-Parameter Infiltration
23
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Equation. Soil Science 1(6): 337-341.
Parlange, J.-Y., R. Haverkamp, and J. Touma. 1985. Infiltration Under Ponded Conditions: 1.
Optimal Analytical Solution and Comparison with Experimental Observations. Soil
Science 139(4): 305-311.
Parlange, J.-Y. and R. Haverkamp. 1989. Infiltration and Ponding Time. In Unsaturated Flow in
Hydrologic Modeling, Theory and Practice, ed. HJ. Morel-Seytoux. 95-126. Kluwer
Academic Publishers. Boston, MA.
Philip, J.R. 1957. The Theory of Infiltration: 4. Sorptivity and Algebraic Infiltration Equations.
Soil Science 84: 257-264.
Philip, J.R. 1969. Theory of Infiltration. Advances in Hydroscience 5: 215-296. Academic
Press, New York, NY.
Philip, J.R. 1971. General Theorem on Steady Infiltration from Surface Sources, with
Application to Point and Line Sources. Soil Science Society of America Proceedings 34:
709-714.
Philip, J.R. 1972. Steady Infiltration from Buried, Surface, and Perched Point and Line Sources
in Heterogeneous Soils: I. Analysis. Soil Science Society of America Proceedings 36:
268-273.
Philip, J.R. 1974. Recent Progress in the Solution of Nonlinear Diffusion Equations. Soil
Science 117(5): 257-264.
Philip, J.R. 1984a. Steady Infiltration from Circular Cylindrical Cavities. Soil Science Society
of America Journal 48:270-278.
Philip, J.R. 1984b. Steady Infiltration from Spherical Cavities. Soil Science Society of America
Journal 48: 724-729.
Philip, J.R. 1991. Horizontal Redistribution with Capillary Hysteresis. Water Resources
Research 27(7): 1459-1469.
Philip, J.R. 1992. Falling Head Ponded Infiltration. Water Resources Research. 28(8): 2147-
2148.
Philip, J.R. 1993. Variable-Head Ponded Infiltration Under Constant or Variable Rainfall. Water
Resources Research 29(7): 2155-2165.
Protopapas, A.L. and R.L. Bras. 1991. The One-Dimensional Approximation for Infiltration in
Heterogeneous Soils. Water Resources Research 27 (6): 1019-1027.
24
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Raats, P.A.C. 1972. Steady Infiltration from Sources at Arbitrary Depth. Soil Science Society of
America Proceedings 36: 399-401.
Rawls, WJ. and D.L. Brakensiek. 1982. Estimating Soil Water Retention from Soil Properties.
Journal of Irrigation and Drainage, Division Proceedings of ASCE 108: 166-171.
Salvucci, G.D., and D. Entekhabi. 1994. Explicit Expressions for Green-Ampt (Delta Function
Diffusivity) Infiltration Rate and Cumulative Storage. Water Resources Research 30(9):
2661-2663.
Sander, G.C, J.-Y. Parlange, V. Kuhnel, W.L. Hogarth, D. Lockington, and J.P.J. O'Kane. 1988.
Exact Nonlinear Solution for Constant Flux Infiltration. Journal of Hydrology 97: 341-
346.
Sisson, J.B., A.H. Ferguson, and M. Th. van Genuchten. 1980. Simple Method for Predicting
Drainage from Field Plots. Soil Science Society of America Journal 44: 1147-1152.
Smith, R.E. 1983. Approximate Soil Water Movement by Kinematic Characteristics. Soil
Science Society of America Journal 47: 3-8.
Smith, R.E. and J.-Y Parlange. 1978. A Parameter-Efficient Hydrologic Infiltration Model.
Water Resources Research 14(3): 5-538.
Springer, E.P. and T.W. Cundy. 1987. Field-Scale Evaluation of Infiltration Parameters from
Soil Texture for Hydrologic Analysis. Water Resources Research 23(2): 325-334.
Stephens, D.B. 1996. Vadose Zone Hydrology. CRC Press, Inc. Boca Raton, FL.
Swartzendruber, D. 1974. Infiltration of Constant-Flux Rainfall into Soil as Analyzed by the
Approach of Green and Ampt. Soil Science. 117:272-281.
Swartzendruber, D. and D. Hillel. 1973. The Physics of Infiltration, in Physical Aspects of Soil
Water and Salts in Ecosystems, (ed.) A. Hadas. Springer-Verlag.
Swartzendruber, D. and E.G. Youngs. 1974. A Comparison of Physically-Based Infiltration
Equations (Note) Soil Science. 117 (3): 165-167.
Swartzendruber, D. and F.R. Clague. 1989. An Inclusive Infiltration Equation for Downward
Water Entry into Soil. Water Resources Research 25 (4): 619-626.
USDA Soil Conservation Service. 1957. Hydrology. National Engineering Handbook. Sect. 4.
Supplement A.
25
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U.S.EPA. 1996. Soil Screening Guidance: User's Guide. U.S.Environmental Protection Agency,
Office of Solid Waste and Emergency Response. OSWER Directive 9355.4-23.
EPA/540/R-96/018, April 1996 (PB96-963505).
van der Heijde, P.K.M. 1994. Identification and Compilation of Unsaturated/Vadose Zone
Models. EPA/600/R-94/028, March 1994 (PB94-157773).
van Genuchten, M. Th. 1980. A Closed-Form Equation for Predicting the Hydraulic
Conductivity of Unsaturated Soils. Soil Science Society of America Journal 44: 892-898.
van Genuchten, M. Th., FJ. Leij, and S.R. Yates. 1991. The RETC Code for Quantifying the
Hydraulic Functions of Unsaturated Soils. U.S. EPA, Robert S. Kerr Environmental
Research Laboratory. EPA/600/2-91/065.
van Genuchten, M. Th., FJ. Leij, and LJ. Lund. 1992. Indirect Methods for Estimating the
Hydraulic Properties of Unsaturated Soils. University of California, Riverside, CA.
Warrick, A.W. 1974a. Solution to the One-Dimensional Linear Moisture Flow Equation with
Water Extraction. Soil Science Society America Proceedings 38: 573-576.
Warrick, A.W. 1974b. Time-Dependent Linearized Infiltration. I. Point Sources. Soil Science
Society of America Proceedings 38: 383-386.
Warrick, A.W. 1975. Analytical Solutions to the One-Dimensional Linearized Moisture Flow
Equation for Arbitrary Input. Soil Science 120(2): 79-84.
Warrick, A.W., and D.O. Lomen. 1976. Time-dependent Linearized Infiltration: III. Strip and
Disc Sources. Soil Science Society of America Journal 40: 639-643.
Warrick, A.W. 1990. An Analytical Solution to Richards Equation for a Draining Soil Profile.
Water Resources Research 26(2): 253-258.
Warrick, A.W., A. Islas, and D.O. Lomen. 1991. An Analytical Solution to Richards Equation
for Time-Varying Infiltration. Water Resources Research 27(5): 763-766.
26
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APPENDIX A
Green-Ampt Models
Annotated Bibliography
A- 1
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Green-Ampt Based Models
GREEN-AMPT BASED METHODS
1. Aggelides, S., E.G. Youngs. The Dependence of the Parameters in the Green and Ampt Infiltration
Equation on the Initial Water Content in Draining and Wetting States. Water Resources Research, 14:857-
862 (1978).
The parameters in the Green and Ampt infiltration equation were determined from infiltration
experiments in a sand column at various uniform initial water contents in both draining and wetting
states and were compared with various estimates obtained from the soil water properties measured on
the same experimental column. The estimates of the soil water pressure head at the wetting front
were generally more negative than the values obtained from the directly measured parameters. It was
found that the calculated cumulative infiltration as a function of time was fitted better by using
Bouwer's crude 'water entry' estimate than values deduced by approximating Richards' flow equation.
2. Ahuja, L.R. Applicability of the Green-Ampt Approach to Water Infiltration through Surface Crust. Soil
Science, 118:283-288 (1974).
Applicability of the Green-Ampt approach to describe water infiltration through surface crust after
the rapid initial stage is evaluated by testing the equations of Hillel and Gardner (1970) and two
modifications of these against numerical-solution data for Yolo soil. The modifications were
designed to approximately account for the transient nature of soil-water content at the crust-soil
boundary. One of the modifications involved piece-wise application of Hillel and Gardner's (1970)
equations, and the other treated the boundary water content as a continuous variable. The results of
tests showed that the unmodified equations of Hillel and Gardner (1970) were fairly good for a
smaller crust resistance but involved considerable error for a larger resistance. Piecewise application
of these equations with as small as four subranges gave results in close agreement with the numerical
solution for both resistances. The other modifications involving continuously variable boundary
water content showed very good agreement for both resistances in early stages. In the intermediate
stages, it showed errors of large relative magnitude, which decreased with increase in resistance.
3. Barry, D.A., J.-Y. Parlange, and R. Haverkamp. Short Communication. Falling-Head Ponded Infiltration
with Evaporation—Comment. Journal of Hydrology, 162:211-213 (1994).
4. Bouwer, H. Rapid Field Measurement of Air Entry Value and Hydraulic Conductivity of Soil as Significant
Parameters in Flow System Analysis. Water Resources Research, 2:729-738 (1966).
Field measurements of air entry value and hydraulic conductivity of soil are obtained with a covered
cylinder infiltrometer equipped with standpipe and vacuum gage. Tests are normally completed in
approximately 30 minutes. The resulting data are used to construct step functions relating hydraulic
conductivity to (negative) soil water pressure for sorption and desorption. These functions may be
used as simplified hydraulic conductivity characteristics to include negative pressure flow in the
analysis of subsurface water movement. Several field and laboratory studies demonstrate the validity
of the concepts and the technique.
5. Brakensiek, D.L., C.A. Onstad. Parameter Estimation of the Green and Ampt Infiltration Equation. Water
Resources Research , 13:1009-1012 (1977).
Infiltrometer data are utilized to test a fitting procedure for the Green and Ampt infiltration equation
parameters. The spatial variation of the estimated parameters is averaged to give lumped parameter
values for watershed modeling. An appropriate scale for parameter averaging is discussed, which
transforms parameter values to normal deviates. A sensitivity analysis for the equation parameters is
performed for both infiltration estimates and runoff hydrograph volume and peak rates. The capillary
pressure parameter shows least sensitivity; however, variation in the finable porosity and effective
conductivity parameters has a major influence on infiltration and runoff amounts and rates.
A-2
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Green-Ampt Based Models
6. Brakensiek, D.L. Estimating the Effective Capillary Pressure in the Green and Ampt Infiltration Equation.
Water Resources Research, 13:680-682 (1977).
A considerable amount of research indicates that the Green and Ampt infiltration equation can be
used in hydrologic modeling with confidence. The parameters, however still require considerable
field and or laboratory effort to determine estimated values. Reported here are alternatives for
determining the average wetting front capillary pressure parameter. Only a moisture characteristic
would be required.
7. Charbeneau, RJ. andR.G. Asgian. Simulation of the Transient Soil Water Content Profile for a
Homogeneous Bare Soil. Water Resources Research, 27(6): 1271-1279 (1991).
Characterization of the fate and transport of solutes through the vadose zone requires estimation of
average water contents and travel times through the unsaturated profiles. This paper reviews a model
for long-term simulation of the soil water content profile for a homogeneous bare soil using physically
based parameters. For an arbitrary rainfall record, the model calculates the cumulative infiltration,
runoff, evaporation, and recharge components, as well as the time average reduced saturation and its
variance as a function of depth. The model is computationally efficient and easily allows long-term
simulation for parameter sensitivity investigations.
8. Childs, E.G., M. Bybordi. The Vertical Movement of Water in Stratified Porous Material 1. Infiltration.
Water Resources Research, 5:446-459 (1969).
The infiltration law of Green and Ampt has long been established as an adequate approximation in
uniform soil profiles. It is here extended to include heterogeneous profiles. Formulas are developed
to express three relationships. Firstly, the infiltration law is derived for a profile consisting of a
succession of n different layers with conductivity decreasing from the surface. Secondly, an
expression is derived for the conductivity profile that will give a specified infiltration law, and in
particular a linear infiltration law with specified parameters. Thirdly, this expression serves to
determine the conductivity when the infiltration law is observed. Experiments are described to
measure the infiltration rates into layered columns or porous material, the results of which confirm
the theory.
9. Chu, S.T. Infiltration During an Unsteady Rain. Water Resources Research, 14(3):461-466 (1978).
Infiltration during a rainfall event can be divided into two distinct stages: a stage with surface
ponding and a stage without surface ponding. Few of the infiltration models in current use are
suitable to describe infiltration for both stages. In this paper the Green and Ampt equation was
applied to determine the time that separates these two stages so that infiltration for the different
stages can be treated separately. To obtain an integrated form of the Green and Ampt equation, it is
traditionally assumed that the cumulative infiltration is zero at the time when surface ponding starts.
But in a rainfall event the cumulative infiltration equals the water infiltrated into the soil profile prior
to the ponding time, which is usually not zero. Therefore, a modification in the traditional Green and
Ampt equation is needed to describe infiltration during a rainfall event. It is shown in this paper that
this modification is equivalent to a shift of the time scale by an amount which is referred to as the
pseudotime by the author. The modified version of the Green and Ampt equation was applied to
determine rainfall excess and to predict total runoff for three major storms recorded by the
Agricultural Research Service from 1957 to 1959 on a watershed near Oxford, Mississippi. A
comparison of the prediction and the measured total runoff appeared to be promising.
10. Freyberg, D.L., J.W. Reeder, J.B. Franzini, I. Remson. Application of the Green-Ampt Model to Infiltration
Under Time-Dependent Surface Water Depths. Water Resources Research, 16:517-528 (1980).
The performance of the Green-Ampt model for infiltration problems where the depth of water above
the ground surface is varying with time is investigated. In order to yield infiltration rates that agree
with those predicted by the Richards equation for flow in a homogeneous, nondeforming, and
nonhysteretic soil the effective suction head parameter in the Green-Ampt model must be considered
A-3
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Green-Ampt Based Models
a function of time, surface water depth, initial moisture content, and soil type. However, when a
constant value of the effective suction head is assumed, the response of the Green-Ampt model to
variations in surface water depth is qualitatively equivalent to the response of the Richards model.
The effectiveness of a number of definitions of the suction head parameter that have been proposed in
the literature is investigated. While the differences In predicted infiltration rates and cumulative
infiltration obtained by using the different definitions are, in general, of marginal significance, the
best choice of the value of the effective suction head depends upon the particular problem to which
the model is being applied.
11. Green, W.H. and C.A. Ampt. Studies on Soil Physics I. The Flow of Air and Water through Soils. J.Agr.
ScL, IV (Part I): 1-24 (1911).
12. Hillel, D., W.R. Gardner. Transient Infiltration into Crust-Topped Profiles. Soil Science, 109:69-76 (1970).
13. Mein, R.G. and C.L. Larson. Modeling Infiltration during a Steady Rain. Water Resources Research,
9(2):384-394 (1973).
Few of the infiltration models in current use are suitable for the situation in which the rainfall
intensity is initially less than the infiltration capacity of the soil. In this paper a simple two-stage
model is developed for infiltration under a constant intensity rainfall into a homogeneous soil with
uniform initial moisture content. The first stage predicts the volume of infiltration to the moment at
which surface ponding begins. The second stage, which is the Green-Ampt model modified for the
infiltration prior to surface saturation, describes the subsequent infiltration behavior. A method for
estimating the mean suction of the wetting front is given. Comparison of the model predictions with
experimental data and numerical solutions of the Richards equation for several soil types shows
excellent agreement.
14. Morel-Seytoux, HJ. and J. Khanji. Derivation of an Equation of Infiltration. Water Resources Research,
10(4):795-800 (1974).
Since the publication of the article of Green and Ampt (1911) the two constants appearing in their
'equation of infiltration' have been considered to be empirical constants. In fact, they can be deduced
simply from the soil characteristics, and they have a very precise physical meaning. However, the
assumption of an abrupt front separating a saturated zone from the zone at the initial water content
can lead to prediction errors of the order of 10-70%. Derived formulas display explicitly the
functional dependence of the effective (or flowing) capillary head and of the viscous correction factor
as a function of the initial water content. These formulas are both simple and accurate and of
practical value to the hydrologist.
15. Neuman, S.P. Wetting Front Pressure Head in the Infiltration Model of Green and Ampt. Water Resources
Research, 12:564-566 (1976).
A theoretical expression relating the wetting front pressure head in the infiltration model of Green
and Ampt (1911) to soil characteristics is derived. This expression is identical to that previously
suggested by Bouwer (1964) on the basis of an analogy with horizontal flow. It differs from a more
complete expression recently proposed by Morel-Seytoux and Khanji (1974) in that the effect of air
mobility is neglected, the need for determining the functional relationship between the relatively
permeability of air and water saturation thus being avoided.
16. Philip, J.R. Variable-Head Ponded Infiltration Under Constant or Variable Rainfall. Water Resources-
Research, 29(7):2155-2165 (1993).
This study of the effect of excess rainfall, ponded without runoff, on the dynamics of infiltration into a
deep homogeneous soil uses the Green-Ampt, or delta function, approximation. Exact solutions are
given for constant and for piecewise constant rainfall. An inverse method yields solutions for
arbitrary hyetographs. Ponding can increase markedly both the total duration of the infiltration event
and the total amount infiltrated. The analysis suggests that capillary hysteresis emerges during the
A-4
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Green-Ampt Based Models
ponding recession phase toward the end of a storm. In long-duration heavy rainstorms infiltration
rate may increase with time, contrary to the classical picture. The asymptotic large time infiltration
rate under continuous constant rainfall at rate R, with R > K (the conductivity parameter in the
Green-Ampt formulation) is an increasing function of both K and R. (With ponding neglected, it is
simply K). An appendix gives some exact results on the influence of hyetograph shape on time to
ponding.
17. Philip, J.R. Short Communication. Falling-Head Ponded Infiltration with Evaporation-Reply. Journal of
Hydrology, 162:215-221 (1994).
18. Salvucci, G.D. andD. Entekhabi. Explicit Expressions for Green-Ampt (Delta Function Diffusivitv)
Infiltration Rate and Cumulative Storage. Water Resources Research, 30(9):2661-2663 (1994).
The sharp wetting front model of infiltration (Green and Ampt, 1911; Philip, 1954) yields through
simple integration an exact solution relating the infiltration rate (i), cumulative infiltration (I), and
time (/). The relation, however, is implicit for /' or/; i.e., it is of the form t = A[l - B In (1 + I/B)].
Numerical iteration is required to find the infiltration rate, and furthermore, analytic manipulations
are limited using this traditional formulation of the Green-Ampt infiltration. In this note we present
an accurate expression for the infiltration rate in the form of a rapidly converging series in the
variable T= t/(t + %). Truncating the series at four terms yields a useful expression for i(t). The
proposed four-term expression gives less than 2% error at all times and is readily integrated to yield
the cumulative infiltration I ft). In conjunction with the exact expression for time ft) given /' or /, the
proposed expression is useful in infiltration/runoff calculations that necessitate the time compression
approximation (TCA).
19. Smith, R.E. and J.-Y. Parlange. A Parameter-Efficient Hydrologic Infiltration Model. Water Resources
Research, 14(3):533-538 (1978).
By adopting two extreme assumptions concerning the behavior of unsaturated soil hydraulic
conductivity K near saturation, we derived a two-branched model for ponding time and infiltration
rate decay for arbitrary rainfall rates. One assumption was that Ovaries slowly near saturation and
leads to an expression for ponding time and infiltration decay. For initially ponded conditions,
ponding time is zero, and with rainfall rate r 6 4, the familiar Green and Ampt (1911) expression
results. The other, rather opposite assumption was that Ovaries rapidly, e.g., exponentially, near
saturation. This model also holds for both rainfall and ponded surface conditions, and for ponded
conditions the expression is identical to that of Parlange (1971). Each model uses only two
parameters, saturated soil conductivity Ks and a parameter that is roughly related to sorptivity and
responds nearly linearly to variations in initial saturation. Both parameters are physically related to
measurable soil properties. Methods are presented to estimate parameters of either model from
infiltrometer tests. The two models are compared with a precise numerical solution of the
unsaturated soil water diffusion equations for three soils that represent a range of soil behaviors near
saturation. Our results show that either assumption would be an excellent model for most hydrologic
purposes, and the relative goodness of fit of each model is generally consistent with the appropriate
behavior of K(Q 6 Qs).
20. Swartzendruber, D. Infiltration of Constant-Flux Rainfall into Soil as Analyzed by the Approach of Green
and Ampt. Soil Science, 117:272-281 (1974).
Infiltration of water into soil from an application of constant water flux r is analyzed theoretically by
treating the resulting soil-water profiles as step functions, as assumed in the approach of Green and
Ampt (1911). Important features of the infiltration process then follow in fairly straightforward
mathematical form. As r decreases toward the near-saturated soil hydraulic conductivity Kb, the time
to of onset of surface-water excess is sufficiently increased so that the curves of infiltration flux /'
versus cumulative time t are shifted toward larger times, progressively away from the curve of /' versus
A-5
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Green-Ampt Based Models
t for continuous ponding. But / is not a simple function of /, primarily because the cumulative water
infiltration / is an implicit function of t. Two explicit-form equations are then proposed for / versus /,
both equations being integrable into explicit forms of / versus /. The one explicit equation is matched
with the implicit function within ±11 percent, over the whole range from r = Kb to r .
A-6
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APPENDIX B
Richards Equation Models
Annotated Bibliography
B- 1
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Richards Equation Models
RICHARDS' EQUATION (ANALYTICAL / QUASI-ANALYTICAL)
1. Babu, D.K. Infiltration Analysis and Perturbation Methods 1. Absorption with Exponential Diffusivitv.
Water Resources Research, 12(l):89-93 (1976).
Simple perturbation methods are employed to analyze the horizontal absorption of moisture in
unsaturated soils. The special case treated assumes the diffusivity to be an exponential function of the
concentration and the concentration at the boundaries to be constant. The solution emerges as an
explicitly determined power series in the Boltzmann variable.
The resulting profiles are compared with some others found in the existing literature. A discussion
about the relevance and advantages of this type of analysis forms the concluding part of the paper.
2. Babu, D.K. Infiltration Analysis and Perturbation Methods 2. Horizontal Absorption. Water Resources
Research, 12(5): 1013-1018 (1976).
Perturbation methods are employed to analyze the horizontal absorption of moisture in unsaturated
soils. The present study applies to absorption under the general diffusivity function, subject to
constant boundary concentration. The concept of the wetting front moving with a finite velocity plays
a fundamental role in this set of papers. The solution emerges as a series of terms that can be
explicitly calculated in terms of the integrals involving diffusivity functions. The examples of
exponential and constant diffusivity functions, as well as the case of Yolo light clay, are computed to
one or two terms to exhibit the advantages over, and the good agreement with, other published data of
Parlange, Philip, etc. The method possesses great potential, and in a subsequent paper the study of
vertical infiltration will be presented by utilizing the perturbation mechanism.
3. Babu, D.K. Infiltration Analysis and Perturbation Methods 3. Vertical Infiltration. Water Resources
Research, 12(5): 1019-1024 (1976).
Simple perturbation methods continue to yield good results in the solution of the problem of vertical
infiltration of water into unsaturated soils. The initial approximation is valid for all times and gives
useful information about important flow characteristics. The require computation are always direct
and simple. Some numerical work pertaining to Yolo light clay is presented to exhibit the
advantages, relevance, and simplicity of the present analysis compared to other exiting works on this
topic.
4. Barry, D.A., et al. A Class of Exact Solutions for Richards' Equation. Journal of Hydrology, 142:29-46
(1993).
A new solution satisfying Richards' equation is derived. The solution, which may be applied for
infiltration or capillary rise, is valid for the condition of an arbitrary moisture tension imposed at the
soil surface. When written in terms of the moisture tension, the new result is very simple, being
derived m terms of a similarity variable. The solution applies when the form of the soil moisture
characteristic curve is a particular weighted integral of the gradient of the unsaturated hydraulic
conductivity. Thus, if the soil moisture characteristic curve is selected a priori, then this condition
determines the hydraulic conductivity. The converse of this statement also applies. The cumulative
infiltration derived from the solution is of the form of the Green-Ampt infiltration equation; however,
there is no need to assume a steep wetting front as Green and Ampt did. Finally, using the
correspondence between Richards' equation and the convection-dispersion equation with a non-linear
solute adsorption isotherm, a new exact solution for adsorptive solute transport is derived.
B-2
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Richards Equation Models
5. Barry, D.A., and G.C. Sander. Exact Solutions for Water Infiltration with an Arbitrary Surface Flux or
Nonlinear Solute Adsorption. Water Resources Research, 27:2667-2680 (1991).
Several authors have now used the Backlund transformation to linearize Richards' equation and so
obtain analytical solutions for infiltration and redistribution processes for the linearizable class of
diffusivities and hydraulic conductivities. After transformation a convection-diffusion equation is
obtained. To date, all the analytical results making use of the Backlund transform have been based on
a specified, constant flux that is assumed to exist at the surface boundary. We generalize the previous
work by removing this restriction and obtain a general solution for an arbitrary temporal variation in
the surface flux. The solution contains a function that is specified by a Volterra integral equation. In
practice, the Volterra equation must be evaluated numerically. This task is accomplished using a
relatively straightforward algorithm. The analytical results for infiltration can be applied to the
process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are
equivalent under another mapping. Our new analytical results, therefore, also apply to this class of
solute transport processes. The general form of the adsorption isotherm for which exact results can be
obtained is derived. This isotherm appears capable of representing a relatively large family of
physically relevant cases, although this topic remains open for future investigation.
6. Batu, V. Time-Dependent. Linearized Two-Dimensional Infiltration and Evaporation from Nonuniform
and Nonperiodic Strip Sources. Water Resources Research, 18:1725-1733 (1982).
Using a linearized equation describing unsaturated homogeneous and isotropic porous media flow, a
general, two-dimensional, time-dependent mathematical model is presented for infiltration and/or
infiltration-evaporation from nonuniform and nonperiodic strip sources located at the soil surface.
The analysis is based on an exponential relationship between the unsaturated hydraulic conductivity
and the soil water pressure head and also assumes a constant value for the derivative of unsaturated
hydraulic conductivity with respect to water content. In the mathematical analysis, Laplace transform
and Fourier analysis techniques are used simultaneously, and a general equation in an integral form
for the distribution of matric flux potential has been obtained. The result of Warrick and Lomen
(1976) for a single strip source is shown to be a special case of this general model. The solution for
infiltration from two strip sources is presented as another special case. Also, a third special case is
presented for infiltration from two strip sources and evaporation from another strip located in the
middle part. The horizontal and vertical flux components for these cases are presented. All results
are expressed in integral forms and calculated with a computer using a numerical integration method.
The solutions predict the matric flux potential and flux components as functions of space and time.
Examples that illustrate the applicability of some of the solutions are presented. These solutions are
of interest in the design of both sprinkler and furrow irrigation systems. The results may also be used
for different purposes in civil and agricultural engineering applications.
7. Ben-Asher, I, D.O. Lomen, and A.W. Warrick. Linear and Nonlinear Models of Infiltration from a Point
Source. Soil Sci. Soc. Am. J., 42:3-6 (1978).
Numerical and analytical solutions for water flow from a point source are compared. The numerical
solution was for the nonlinear moisture flow equation and the analytical solution for the
corresponding linearized form. For cyclic conditions, results were approximately the same with
regard to range in values between the wettest and driest values. However, the numerical results show
a faster response both for wetting and drying. Computational times required of the analytical solution
were of the order of 1/20 to l/200th of that required for the finite difference solutions.
8. Bolt, G.H. Discussion on "Absorption and Infiltration in Two- and Three Dimensional Systems." In:
Water in the Unsaturated Zone, IASH/AIHS - Unesco (1969).
B-3
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Richards Equation Models
9. Boulier, J.F., J. Touma, and M. Vauclin. Flux-Concentration Relation-Based Solutions of Constant-Flux
Infiltration Equation: I. Infiltration into Nonuniform Initial Moisture Profiles. Soil Sci. Soc. Am. J.,
48:245-251 (1984).
The quasi-analytical solution of the infiltration equation based on the flux-concentration relation for
constant flux condition and initially uniform water content profile is extended here to nonuniform
moisture profiles for fluxes either smaller or greater than the saturated hydraulic conductivity. In the
latter case, the solution is developed for the postponding stage. It is shown that the measured flux-
concentration relation is well-approximated by F(6*) = 6* and that no significant time dependence
can be observed. The quasi-analytical solution is then successfully compared with both laboratory
experiments performed on a sandy soil column and numerical solution of the Richards equation for
the nonponding, preponding, and postponding stages of infiltration.
10. Broadbridge, P., J.H. Knight, and C. Rogers. Constant Rate Rainfall Infiltration in a Bounded Profile:
Solutions of a Nonlinear Model. Soil Sci. Soc. Am. J., 52:1526-1533 (1988).
We present new exact solutions to a versatile analytic nonlinear model of single phase vertical
unsaturated flow during constant rate rainfall infiltration in a bounded soil profile with an
impermeable base. The nonlinear flow equation in a fixed finite region is transformed to a linear
diffusion problem with boundary conditions on a shrinking domain. The linear problem is treated by
King's method of Laplace-transform boosts. The analytic solutions illustrate the theoretical
differences in the basement moisture build-up when the water content dependence of the soil
hydraulic properties varies from strong to weak. When the rainfall rate exceeds a critical value which
is estimated here, surface ponding precedes basement saturation. In this case, time to ponding is
approximated well by the expression for the infinite column. When the rainfall rate is less than the
critical value (which is greater than the conductivity at saturation), basement saturation precedes
surface ponding. In this case, the time to basement saturation is close to the time taken for the
rainfall to fill the available pore space.
11. Broadbridge, P. and I. White. Constant Rate Rainfall Infiltration: A Versatile Nonlinear Model 1.
Analytic Solution. Water Resources Research, 24(1): 145-154 (1988).
Analytic solutions are presented for a nonlinear diffusion-convection model describing constant rate
rainfall infiltration in uniform soils and other porous materials. The model is based on the Darcy-
Buckingham approach to unsaturated water flow and assumes simple functional forms for the soil
water diffusivity D(8) and hydraulic conductivity K(8) which depend on a single free parameter C
and readily measured soil hydraulic properties. These D(8) and K(8) yield physically reasonable
analytic moisture characteristics. The relation between this model and other models which give
analytic solutions is explored. As C -> , the model reduces to the weakly nonlinear Burgers'
equation, which has been applied in certain field situations. At the other end of the range as C -> 1,
the model approaches a Green-Ampt-like model. A wide range of realistic soil hydraulic properties is
encompassed by varying the C parameter. The general features of the analytic solutions are
illustrated for selected C values. Gradual and steep wetting profiles develop during rainfall, aspects
seen in the laboratory and field. In addition, the time-dependent surface water content and surface
water pressure potential are presented explicitly. A simple traveling wave approximation is given
which agrees closely with the exact solution at comparatively early infiltration times.
12. Broadbridge, P. and I. White. Time to Ponding: Comparison of Analytic. Quasi-Analytic, and
Approximate Predictions. Water Resources Research, 23(12):2302-2310 (1987).
An analytic expression for time to ponding is introduced using the nonlinear model of Broadbridge
and White (1987). The hydraulic properties of this model can encompass properties ranging from
those of a highly nonlinear Green-Ampt-like soil to those satisfying the weakly nonlinear Burgers'
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equation. Because of its versatility, this analytic solution is used as a benchmark against which extant
analytic, quasi-analytic, and approximate expressions are compared. Time to ponding is
parameterized here in terms of the readily measured field properties, sorptivity and hydraulic
conductivity. In the limit of Green-Ampt-like properties the analytic solution reduces exactly to the
Parlange and Smith (1976) approximation. A similar functional dependence of time to ponding on
rainfall rate is found from quasi-analytic approximations. Based on this, a modified approximation is
suggested which should give time to ponding for most soils to within ±10%. Some existing
approximations are found to have unacceptable deviations from the analytic solution, and their
continued use appears unwarranted. Finally, we address the field problem of predicting time to
ponding at any antecedent water content, given sorptivity measured at only one initial water content.
13. Brutsaert, W. More on an Approximate Solution for Nonlinear Diffusion. Water Resources Research,
10:1251-1252(1974).
A method of solution developed earlier (1970) for a diffusivity, that is, a power function of
concentration, is shown to be more generally applicable. One of the conditions of its applicability is
the existence of a sharp diffusing front. As an illustration the solution is worked out herein for a
diffusivity that has been useful in the analysis of soil water problems.
14. Brutsaert, W., and R.N. Weisman. Comparison of Solutions of a Nonlinear Diffusion Equation. Water
Resources Research, 6:642-644 (1970).
Four solutions of a general nonlinear diffusion equation are given. A numerical analysis serves as
reference for one series solution and for two more simple approximate solutions. The accuracy of the
methods is discussed and presented graphically; the results of the numerical analysis are given in
tabular form. It was found that the usefulness of the two approximate solutions lies in their
mathematical simplicity and the accuracy with which they describe infiltration.
15. Charbeneau, R.J. Kinematic Models for Soil Moisture and Solute Transport. Water Resources Research,
20(6):699-706 (1984).
The kinematic theory of soil moisture and solute transport in the vertical direction for unsaturated
groundwater recharge is considered. The general theory of kinematic models is reviewed and applied
for an isolated wetting event wherein the soil starts at and eventually drains to field capacity.
Analytical expressions are developed for the water content and moisture flux as a function of depth
and time. The approach is extended for an arbitrary sequence of surface flux or water content
boundary conditions. The transport of a solute with a general nonlinear sorption isotherm is also
considered. It is shown that for the general isotherm the vertical displacement of a solute isochore
during an arbitrary wetting sequence depends only on the total depth of water infiltrated and not on
how the infiltration rate varies with time.
16. Cisler, J. Note on the Parlange Method for the Numerical Solution of Horizontal Infiltration of Water into
Soil. Soil Science, 117:70-73 (1974).
Parlange's approximate analytical solution for the horizontal infiltration of water into soil is
numerically tested for the case of two-step diffusivity function and also for the Yolo Light Clay
diffusivity of Philip (1957). It is shown that Parlange's second approximation for the Boltzman
variable (|> = x«t"1/2 as a function of moisture content gives in both cases too high values, as compared
to the analytical solution for two-step diffusivity, or Philip's solution for the soil-water case. The error
decreases as the diffusivity function approaches more closely to the delta-function type of diffusivity.
The improved second approximation of Parlange is, however, very precise, even for the diffusivities,
which are not of delta-function type. The increase of labor for this improved approximation is very
small. A recurrence formula for the successive (|>'s is also examined, but it is found, however, that
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this formula does not give convergent values and the computational effort is larger than the improved
second approximation.
17. Clothier, B.E., J.H. Knight, and I. White. Burgers' Equation: Application to Field Constant-Flux
Infiltration. Soil Science, 132(4) (1981).
We present the analytical constant-flux solution to Burgers' equation. Burgers' equation is a
minimally nonlinear Fokker-Planck diffusion equation, applicable to infiltration into soils with a
constant diffusivity and quadratic conductivity-water content relationship. Field Bungendore fine
sand has a near-constant diffusivity-water content relationship, and analytical solutions of Burgers'
equation are in good agreement with field profiles of water content obtained in situ with a rainfall
simulator. The solutions and experimental data all relate to nonponding infiltration with fluxes less
than the saturated hydraulic conductivity; however, a wide range of elapsed times and flux rates are
covered. Predicted wet-front penetration is in good agreement with the field experimental data. The
complete analytical solution can be evaluated using a programmable hand calculator. The simpler
"profile-at-infinity" solution is shown to be surprisingly accurate over a wide range of times, giving
useful results easily. An expression for the time to ponding is also presented.
18. Corradini, C., F. Melone, and R.E. Smith. Modeling Infiltration During Complex Rainfall Sequences.
Water Resources Research, 30(10):2777-2784 (1994).
An extension of the conceptual model earlier developed by Smith et al. (1993) is presented. Their
basic model considered the problem of point infiltration during a storm consisting of two parts
separated by a rainfall hiatus, with surface saturation and runoff occurring in each part. The model is
here extended toward further generality, including the representation of a sequence of infiltration-
redistribution cycles with situations not leading to soil surface saturation, and rainfall periods of
intensity less than the soil infiltration capacity. The model employs at most a two-part profile for
simulating the actual one. When the surface flux is not at capacity, it uses a slightly modified version
of the Parlange et al. (1985) model for description of increases in the surface water content and the
Smith et al. (1993) redistribution equation for decreases. Criteria for the development of compound
profiles and for their reduction to single profiles are also incorporated. The extended model is tested
by comparison with numerical solutions of Richards' equation, carried out for a variety of experiments
upon two contrasting soils. The model applications yield very accurate results and support its use as
part of a watershed hydrologic model.
19. Drake, R.L., et al. Similarity Approximation for the Radial Subsurface Flow Problem. Water Resources
Research, 5(3):673-684 (1969).
A useful mathematical technique in the study of radial flow of underground water is the application of
similarity transformations, such as Boltzmann's transformation. The transformations lead to
approximations that are shown to be valid if certain physical parameters satisfy given limiting
conditions.
A nonlinear partial differential equation is formulated describing the radial flow, of soil moisture. A
similarity approximation is used to transform the partial differential equation into a nonlinear
ordinary differential equation. Criteria by which to judge the applicability of the similarity
approximation are derived.
Several monotonicity properties of the transformed moisture as a function of the similarity variable
are derived. The properties are used to develop a numeric procedure for the solution of the
transformed equation. The procedure is used to study the radial flow of soil moisture to a cylindric
sink for diffusivities which vary linearly, quadratically, and exponentially with soil moisture. The
solutions show that the moisture front steepens with increasing nonlinearity.
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20. Drake, R.L. and C.P. Peterson. Application of a Local Similarity Concept in Solving the Vertical
Subsurface Flow. Water Resources Research, 7(5): 1241-1255 (1971).
A technique is described for solving various problems in continuum mechanics through the
application of similarity transformations when all the requirements for the application of such
transformations are not satisfied. The transformed equations usually involve a stray time variable
that is interpreted as a system parameter. When this parameter is properly interpreted, the solutions
of the transformed systems are very good numerical approximations for the solutions of the original
systems. The advantages of considering the transformed systems are usually two-fold: numerical
solutions can be obtained with less computer time, and qualitative analysis of the transformed systems
is usually easier than that of the original systems. The problems discussed in this paper include
certain previously published results and some new results obtained for the nonlinear Fokker-Planck
equation in which the vertical movement of soil moisture under the influence of gravity is described.
21. Gardner, W.R. Some Steady-State Solutions of the Unsaturated Moisture Flow Equation with Application
to Evaporation from a Water Table. Soil Science, 4(85):228-232 (1957).
22. Ghosh. R.K. Modeling Infiltration. Soil Science, 130(6):297-302 (1980).
The equation I = at + Kst is proposed as a model of infiltration. This is a combination of the Lewis-
Kostiakov equation and Philip's two-term infiltration equation, but it has the advantage of eliminating
the lapses of both those equations. The formulas by which a and b can be predicted in advance are
also presented.
23. Ghosh. R.K. A Note on the Infiltration Equation. Soil Science, 136(6):333-338 (1983).
Philip's two-term infiltration equation / = St"2 + At, sometimes fails to describe field results precisely.
Though physically less appropriate, /= mf, the Lewis-Kostiakov equation, is advantageous in that it
can accommodate varieties of field results. For the case where n is greater than 1/2, the term f can be
expanded suitably, and if only two terms of this expanded time series are considered, the equation / =
m0t'/2 + m,t3/2 is formed. This equation describes the field results more precisely than Philip's two-
term equation. Two sets of published data (Lewis 1937) and six sets of experimentally observed data
for various time ranges are examined here. The agreement between the observed and generated data
(using the present model) is found to be very good.
24. Gilding, B.H. Qualitative Mathematical Analysis of the Richards Equation. Transport in Porous Media,
(5):651-666(1991).
The Richards equation is widely used as a model for the flow of water in unsaturated soils. For
modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a
specific nonlinear partial differential equation with a time derivative and one spatial derivative. This
paper is a survey of recent progress in the pure mathematical analysis of this last equation. The
emphasis is on the interpretation of the results of the analysis. These are explained in terms of the
qualitative behavior of the flow of water in an unsaturated soil which is described by the Richards
equation.
25. Haverkamp, R., et al. Infiltration Under Ponded Conditions: 3. A Predictive Equation Based on Physical
Parameters. Soil Science, 149(5):292-300 (1990).
We derived a new infiltration equation that takes into account the possibility of an infinite diffusivity
near saturation. Using the example of two soils (clay and coarse sand), we showed that this new
infiltration equation has a sound physical basis. In particular, all parameters used are true soil
properties that are constant with time and independent of the water depth imposed as a surface
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boundary condition. Compared with analytical, numerical, and experimental results, the equation
shows a great precision (a2 < 5.10"3 cm2) at all times.
The present law introduces a significant improvement over the law obtained in part 1 of this series
dealing with ponded infiltration by introducing the physical effect of an infinite diffusivity at
saturation.
26. Haverkamp, R., et al. Three-Dimensional Analysis of Infiltration from the Disc Infiltrometer 2.
Physically Based Infiltration Equation. Water Resources Research, 30(11):2931-2935 (1994).
In situ measurement of soil hydraulic properties may be achieved by analyzing the unconfined efflux
from disc tension infiltrometers, once consistent infiltration equations can be derived. In this paper
an analytical, three-dimensional infiltration equation is developed, based on the use of parameters
with sound physical meaning and adjustable for varying initial and boundary conditions. The
equation is valid over the entire time range. For practical purposes, a simplified solution is also
derived. The full and simplified equations give excellent agreement with published experimental
results and are particularly useful for determining soil hydraulic properties through application of
inverse procedures.
27. Irmay, S. A Linearization Technique for the Study of Infiltration. In: Water in the Unsaturated Zone,
IASH/AIHS - Unesco (1969).
The partial differential equation (PDE) of unsaturated flow of liquids in porous media, especially of
water in stable soils, is discussed: steady and unsteady flows, with or without gravity, in one-, two-,
and three-dimensional cases, in cartesian and cylindrical coordinates. The initial and boundary
conditions are described, and the hodograph sphere (circle) method of saturated flow is extended to
unsaturated media. The inverse PDE are also developed.
Various auxiliary functions are used: effective concentration C, hydraulic conductivity K, capillary
potential i|r, and diffusivity D, but the equations become much simplified when expressed in terms of
the diffusivity potential F, when they often become linear in steady flow. It is shown that K1/3 is linear
in C, and i|r is linear in log C in a wide range.
A number of steady flow solutions is studied analytically and graphically, either transforming the
PDE into an ordinary differential equation (ODE), finding a first integral or the general solution.
The cases treated include: horizontal parallel, radial and two-dimensional flows; vertical flows and
flows in a vertical plane and axi-symmetric. In downflow two cases are possible: from higher to
lower C, and conversely.
In unsteady flows the PDE are transformed into an ODE by the Boltzmann transformation, by a linear
wave-like transformation, by similarity transformations, by separation of variables and by other
methods. Among the new solutions the effect of a rising or falling aquifer is described, and unsteady
linear inclined flow, evaporation and infiltration effects.
The method outlined above can be used in solving similar diffusion equations in other fields.
28. Knight, J.H. and J.R. Philip. On Solving the Unsaturated Flow Equation: 2. Critique of Parlange's
Method. Soil Science, 116(6):407-415 (1974).
The physical content of Parlange's method of solving the flow equation is explored. His first
approximation, which was developed by Macey (1969), satisfies continuity in the integral sense, but
the second and higher approximations do not. Approximations beyond the first lack any constraining
link between the separate steps of 'satisfying continuity' and 'satisfying Darcy's law', which make up
each iteration. There is, consequently, nothing in the procedure to ensure convergence.
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A detailed investigation establishes the nonconvergence of the method when applied to one-
dimensional sorption. It is found that the first approximation is best, and that the higher
approximations make oscillations of increasing magnitude about the exact solution. Two illustrative
examples are given. There is no reason to expect the procedure to be any more useful in other cases.
The utility of Parlange's method is thus simply the utility of the first approximation: the dependence
of this on the shape of the diffusivity function and on the flux-concentration relation is discussed.
Contrary to Parlange's claim, the method cannot be applied to two- and three-dimensional systems
other than radially symmetrical ones.
29. Kobayashi, H. The Mathematics of Unsaturated Flow. A Theoretical Analysis and Numerical Solutions of
Unsaturated Flow in Soil. In: Water in the Unsaturated Zone, Chapter 7, IASH/AIHS - Unesco (1969).
Vertical flow is investigated by Forward type difference equation, using S-function, because the error
of numerical calculation can be reduced in this way.
In the case where the fundamental equation can be transformed to an ordinary differential equation,
the author shows the necessary conditions that the auxiliary function has to satisfy and indicates that
there are two kinds of auxiliary functions, i.e., the product from and the linear form. Furthermore a
generalized Boltzmann's transformation of the product form has been shown in three dimensional
form
= {(ax + by + cz)2'
'
By using the generalized Boltzmann's transformation, the author solved the problem for one
dimensional horizontal flow by the iterative method and compared the solution with the solution of
the fundamental solution by the difference method by Hanks et al. Next, the author derived the
Forward difference equation of vertical flow, investigated the variation of computed results by the
mesh size Ax of the difference method and compared the solution with the solution by the above
iterative method.
These results were computed at relatively small values of /, but they seem to be reasonable with the
influence of the gravity term.
30. Koussis. A.D. Nonlinear Sorption of Water in Soil. Soil Science, 132(4):262-266 (1981).
We derive analytical solutions, based on integral approximations of the governing equation, for
sorption of water in power law soil. The parameters of the solution are functions of the soil-
characterizing exponent and are determined by considering the extreme medium behavior of "linear"
and "delta" type. A particular parameter choice reproduces an earlier solution due to Brutsaert. We
compare several solution configurations, including that of Brutsaert, to other approximate methods
and to exact numerical integration.
31. Lomen, D.O. and A.W. Warrick. Time-Dependent Linearized Infiltration: II. Line Sources. Soil Sci. Soc.
Amer. Proc., 38:568-572 (1974).
Water flow from line sources is analyzed using a linearized form of the moisture flow equation. Both
single and parallel line sources are considered. Results are particularly relevant for high-frequency
irrigation, such as by trickle sources, for which the soil moisture at any particular point varies over a
relatively small range. Numerical calculations include lines of constant matric flux potential (or
equal moisture content ) as a function of time and the time-dependent response to a cyclic input.
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Although the results are developed for surface sources, the analysis may easily be extended to buried
sources.
32. Lomen, D.O. and A.W. Warrick. Time-Dependent Solutions to the One-Dimensional Linearized Moisture
Flow Equation with Water Extraction. Journal of Hydrology, 39:59-67 (1978).
An exact, analytical solution is developed for one-dimensional soil water flow. Assumptions are that
the unsaturated conductivity is exponential with pressure head and that the soil-moisture diffusivity is
constant. The surface boundary condition is taken as a time-dependent specified-intake velocity. A
sink function for plant-water uptake is described by a sequence of depth-dependent functions which
change at specified times. Numerical examples include drainage, infiltration, steady uptake and
cyclic surface flux and uptake patterns. The linearizing assumptions are the most realistic for short
cycles, but in any case the exact nature of the answers make them ideal for checking numerical
approximations, such as finite differencing.
33. Molz, F.J., et al. Soil Moisture Availability for Transpiration. Water Resources Research, 4(6): 1161-1169
(1968).
Potential transpiration is a measure of the rate at which water can be transmitted from evaporation
sites in plant leaves to the atmosphere. Potential soil-moisture availability is defined as a measure of
the capacity of a soil to transmit water to a root site. A differential equation is presented describing
radial flow of soil moisture to a single vertical sink (root) in an infinite soil mass which is initially at
a uniform moisture content. The relationship between moisture content and diffusivity for the soils
studied may be represented by an exponential function. A numerical solution of the differential
equation is used to determine the soil-moisture flux. The results show that for scientific soils the
decrease in soil moisture with time occurs mainly in the immediate vicinity of the sink. Moisture flux
increases with initial moisture content but is essentially time independent. In natural systems the flux
would probably decrease with time because of multiple root interference. At large soil moisture
contents, actual transpiration is limited by and equivalent to potential transpiration. At small soil-
moisture contents, actual transpiration is limited by and equivalent to potential soil-moisture
availability.
34. Morel-Seytoux, HJ. Analytical Results for Prediction of Variable Rainfall Infiltration. Journal of
Hydrology, 59:209-230 (1982).
The basic equations that govern water movement in unsaturated soils are presented for a boundary
condition of variable rainfall rate at the soil surface. In particular, a differential equation for the
water content at the soil surface is derived. Solutions are then obtained for the time evolution of
water content at the soil surface assuming a power law form for the relative permeability to water as a
function of normalized water content. Ponding time formulae are obtained and compared with other
previously published relations. Water content profiles are also obtained and their shapes are
displayed graphically for the case of an exponent of 2 in the power law form of the relative
permeability to water for a constant rainfall rate. Formulae are also derived for the situation of a
variable rainfall pattern. A methodology for the coupling of the analytical procedures with an
implicit numerical solution of the water content at the soil surface is suggested as a cost-effective
alternative to the strict numerical solution of the partial differential equation that governs the water
content profile evolution. Postponding infiltration formulae and water content profile equations are
also provided.
35. Morel-Seytoux, HJ. Drainage Rates from a Vertical Column. Hydrological Sciences Bulletin, 20(2):249-
255 (1975).
When both water and air movements are included in the formulation of infiltration or drainage
problems, the problems of infiltration and drainage exhibit a natural symmetry. Methods of solution
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for one problem are applicable to the other type of problem, an equation of drainage was derived and
it had precisely the same functional form as the equation derived for infiltration in a previous paper.
36. Morel-Seytoux, HJ. From Effective Infiltration to Aquifer Recharge: A Derivation Based on the Fluid
Mechanics of Unsaturated Porous Media Flow. Colorado State University, Interim Report for FY 83-84,
CER83-84HJM6 (1983).
An approximate unit hydrograph of aquifer recharge due to effective percolation is derived from the
fluid mechanics of unsaturated flow. The parameters appearing in the expression of the unit
hydrograph have clear physical meaning. Comparison of theory with field observations indicates that
the theory works well. A new tool for estimation of the temporal variation of aquifer recharge from
that of effective infiltration is now available. The tool is practical (simple), physical and apparently
works well as evidenced by comparison with field observations.
37. Morel-Seytoux, HJ. From Excess Infiltration to Aquifer Recharge: A Derivation Based on the Theory of
Flow of Water in Unsaturated Soils. Water Resources Research, 20(9): 1230-1240 (1984).
An approximate unit of hydrograph of aquifer recharge due to effective percolation is derived from a
theory describing flow of water in unsaturated soils. The parameters appearing in the expression of
the unit hydrograph have physical meaning. Comparison of the theory with field observations
indicates that the theory works well. A new tool for estimation of the temporal variation of aquifer
recharge from that of excess infiltration is now available. The tool is practical (simple) and physical
and apparently works well, as evidenced by comparisons with field observations.
38. Morel-Seytoux, HJ. Two-Phase Flows in Porous Media. In: Advances in Hydroscience, V.T. Chow, Ed.,
Academic Press, New York, NY. pp. 119-202 (1973).
With the proliferation of scientific and technological journals it is becoming more and more difficult
to read the complete literature in one's own field. Treatises that synthesize the current state of the art
are welcome additions since they extract the essential information from hundreds of articles, permit
the reader to bypass the task of actually reading most of these articles, and direct him to the really
fundamental papers in his field of interest. Two more such treatises in the field of flow through
porous media (Bear, 1972) and soil and water (Hillel, 1971) have recently appeared.
When the books cover a broad field they run two risks: one of being at times rather shallow and the
other of being partly obsolete by the time of publication. It is hoped that the present article will avoid
these two hurdles because it is both narrow in scope and limited in length.
This article was conceived with three purposes in mind. The first purpose was to present a review of
what had already been done in the field of two-phase flow of interest to the soil scientist and the
hydrologist. The second purpose was to review the "petroleum industry know-how" in the area of
two-phase flow in porous media in a form which would appeal to the hydrologist or soil scientist.
Finally, the third purpose was to show how these particular techniques and their extensions can be
used fruitfully for a better understanding and description of the movement of water and air in soils.
In summary, the basic philosophy of this article gravitates around the following central themes: (1)
water movement in the soil is (at least) a two-phase phenomenon and may not be at times adequately
described or understood by the prevailing current unsaturated (one-phase) approach, and (2) much of
the petroleum literature know-how can be used effectively to solve two-phase flow problems.
39. Morel-Seytoux, H J. and C. Miracapillo. Groundwater Flow in Unsaturated/Saturated Zones. In:
Appropriate Methodologies for Development and Management of Groundwater Resources in Developing
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Countries. Volume III. Proceedings of an International Workshop held February 23-March 4; 1989,
Rotterdam (1989).
The unsaturated zone plays a crucial role in the hydrologic cycle as the link between surface and
groundwater. Though there are many aspects to this interaction, examination of a single aspect, that
of aquifer recharge, will illustrate nevertheless most essential elements in the linkage. Typically
intermittent streams (wadis) and (artificial) infiltration basins are not in permanent hydraulic
connection with the underlying water table aquifer. As water becomes available in the wadi or in the
basin, infiltration proceeds. Due to the frequent presence of a clogged layer the water flow below the
wadi (or basin) bed is unsaturated. Recharge occurs only after the unsaturated wetting front reaches
the water table. On the other hand, long after infiltration has ceased at the soil surface, recharge
continues from the unsaturated storage in excess of field capacity, which has accumulated above the
water table during the infiltration phase. The three-dimensional character of the flow is rendered by a
technique that consists of matching a flow solution in a vertical cross-section with a flow pattern in a
horizontal plane. Practical examples illustrate the influence of the parameters on the water table level
below the infiltrating area and on the lateral recharge rate into the part of the aquifer which is not
overlain by the river or basin bed. Comparison with field observations shows the technique to be
accurate, inexpensive, and easy to apply.
40. Morel-Seytoux, HJ. and C. Miracapillo. Prediction of Water Table Mound Development and Aquifer
Recharge from an Infiltrating Area. In: Unsaturated Flow in Hydrologic Modeling, Theory and Practice,
HJ. Morel-Seytoux, Ed., Kluwer Academic Publishers, Boston, MA. pp. 241-272 (1989).
Typically intermittent streams (wadis) and zones of infiltration during and shortly after a rain or a
flood are not in permanent hydraulic connection with the underlying water-table aquifer. Due to the
frequent presence of a clogged layer the water flow below the infiltration zone is often unsaturated.
Recharge occurs only after the unsaturated wetting front reaches the water-table. On the other hand,
long after infiltration has ceased at the soil surface, recharge continues from the unsaturated storage
which has accumulated above the water table during the infiltration phase. This distinct phase of
recharge after infiltration has stopped is called the drainage phase. The problem is two or three-
dimensional in nature. To simplify the predictive tool and make it very practical, the multi-
dimensional character of the flow problem is rendered, approximately but effectively, by a special
technique that consists of matching two unidimensional flows, a vertical and a horizontal one.
Practical examples illustrate the influence of the parameters on the water table level below the
infiltrating area and on the lateral recharge rate into the part of the aquifer which is not overlain by
the river or infiltration basin bed. Comparison with field observations shows the technique to be
accurate, inexpensive and easy to apply.
41. Novak, M.D. Quasi-Analytical Solutions of the Soil Water Flow Equation for Problems of Evaporation.
SoilSci. Soc. Am. J., 52:916-924 (1988).
Quasi-analytical solutions of the one-dimensional soil water flow equations applied to problems of
evaporation are derived and compared with exact numerical solutions available in the literature.
Constant-concentration and constant-flux surface boundary conditions and semi-infinite and finite
soil columns are considered. For the semi-infinite soils the quasi-analytical technique of Philip and
Knight (1974) is very successful in predicting the water content profiles for the constant-
concentration condition, and only moderately so for the constant-flux condition. Their iterative
procedure for the flux concentration function converges rapidly in the constant-concentration case but
an analogous procedure diverges in the constant-flux case.
For finite soils the simple assumption that the rate of drying of the column is independent of depth in
the flow equation leads to accurate predictions of the water content profiles for the constant-
concentration cases and for the constant-flux case if the potential evaporation is low, the soil is
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shallow, and/or the initial hydraulic diffusivity is high. For conditions other than these roughly
accounting for the higher rate of drying that occurs near the surface in this case greatly improves the
agreement.
42. Parlange, J.-Y. Convergence and Validity of Time Expansion Solutions: A Comparison to Exact and
Approximate Solutions. Soil Sci. Soc. Am. Proc., 39:3-6 (1975).
The convergence of series solutions for the diffusion equation by time expansion is discussed
quantitatively, on the basis of the linear and delta function solutions for a spherical cavity. It is
shown that convergence alone is a poor criterion to justify the validity of the series solutions. A
counter example, diffusion in the presence of an impervious wall, shows that the series may converge
for all times but be entirely erroneous. By comparison an approximate integral technique yields a
solution which agrees very well with the exact result.
43. Parlange, J.-Y. Theory of Water-Movement in Soils: 1. One-Dimensional Absorption. Soil Science,
111(2):134-137(1971).
44. Parlange, J.-Y. Theory of Water-Movement in Soils: 2. One-Dimensional Infiltration. Soil Science,
45. Parlange, J.-Y. Theory of Water-Movement in Soils: 3. Two and Three-Dimensional Absorption. Soil
Science, 112(5):313-317 (1971).
46. Parlange, J.-Y. Theory of Water-Movement in Soils: 4. Two and Three-Dimensional Steady Infiltration.
Soil Science, 113(2):96-101 (1972).
47 . Parlange, J.-Y. Theory of Water-Movement in Soils: 5. Unsteady Infiltration from Spherical Cavities .
Soil Science, 113(3):156-161 (1972).
48. Parlange, J.-Y. Theory of Water-Movement in Soils: 6. Effect of Water Depth Over Soil. Soil Science,
113(5):308-312(1972).
49. Parlange, J.-Y. Theory of Water-Movement in Soils: 7. Multidimensional Cavities Under Pressure . Soil
Science, 113(6):379-382 (1972).
50. Parlange, J.-Y. Theory of Water-Movement in Soils: 8. Multidimensional Cavities Under Pressure . Soil
Science, 114(l):l-4 (1972).
51. Parlange, J.-Y. Theory of Water-Movement in Soils: 9. The Dynamics of Capillary Rise. Soil Science,
114(2):79-81(1972).
52. Parlange, J.-Y. Theory of Water Movement in Soils: 10. Cavities with Constant Flux. Soil Science, 116:1-
7 (1973).
Analytical expressions are derived for the absorption of water from cylindrical and spherical cavities,
when the flux of water at the cavity is imposed. The solution reduces to the one-dimensional case in
the short time limit. After a certain time the soil around the cavity may become saturated and the
growth of the saturated region with time is investigated. Correction terms due to gravity are also
derived in the case of a spherical cavity.
53. Parlange, J.-Y., R.D. Braddock, and B. T. Chu. First Integrals of the Diffusion Equation: An Extension of
the Fuiita Solutions. Soil Sci. Soc. Am. J., 44:908-91 1 (1980).
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Richards Equation Models
In general, solutions of the nonlinear diffusion equation have to be obtained by numerical or
analytical iterative integration. It is shown here that if the diffusivity has a dependence on the water
content which obeys a power law, then iterations can be avoided, and the solution obtained at once.
This unique case results from the existence of a first integral of the diffusion equation. Previously
known analytical results, i.e., the constant diffusivity solution, the delta-function solution, and the
Fujita solutions belong to that general class.
For this general class of solutions it is also possible to show that the sorptivity has a dependence on
the surface water content which obeys a power law. This represents the only known case when such
an analytical relationship exists. This relationship is used to discuss the representation of the square
of the sorptivity as an integral of the diffusivity, when the latter has an arbitrary dependence on the
water content.
54. Parlange, J.-Y., R. Haverkamp, and J. Touma. Infiltration Under Ponded Conditions: 1. Optimal
Analytical Solution and Comparison with Experimental Observations. Soil Science, 139:305-311 (1985).
Under ponded infiltration the cumulative infiltration is a function of soil properties, initial conditions,
and water layer thickness above the soil surface. For arbitrary soil properties and arbitrary
dependence of water layer thickness on time but uniform initial water content, an optimization
technique predicts the cumulative infiltration from the integration of an ordinary differential
equation. If the thickness of the water layer is constant with time, a fully analytical result is obtained.
Careful experimental observations carried out in the laboratory illustrate the validity and accuracy of
the result.
55. Parlange, J.-Y., I. Lisle, R.D. Braddock, and R.E. Smith. The Three-Parameter Infiltration Equation. Soil
Science, 133:337-341 (1982).
A new infiltration equation is obtained by introducing a new parameter in addition to the usual
sorptivity and conductivity at saturation. This third parameter can be related to the conductivity of the
soil. In particular, if this parameter is equal to zero, then the infiltration equation reduces to the well-
known Green and Ampt equation. Comparison with accurate simulations carried out earlier for a
sand and clay indicates that sorptivity, conductivity at saturation, and this additional parameter are
sufficient to describe cumulative infiltration accurately. Finally, the two examples discussed in detail
suggest that this parameter is not very sensitive to soil structure, i.e., it has about the same value for
the clay and the sand. Hence, in practice, use of the new law may require a sorptivity and
conductivity determination only, i.e., two rather than three parameters. Finally, this new law has
about the same precision as existing laws that require knowledge of up to six parameters.
56. Parlange, M.B., et al. Optimal Solutions of the Bruce and Klute Equation. Soil Science, 155(l):l-7
(1993).
A simple analytical expression is derived to describe soil water profiles satisfying the Bruce and Klute
equation. Two parameters have to be determined from integral conditions. The simplicity and
accuracy of the technique is checked when the soil-water diffusivity obeys an exponential or a power
law function. For those two examples no numerical work is necessary as the results are expressed in
terms of tabulated functions only. The method extends to the profile determination the optimal
technique used earlier for the determination of the sorptivity, with the same precision.
57. Perroux, K.M., D.E. Smiles, and I. White. Water Movement in Uniform Soils During Constant-Flux
Infiltration. Soil Sci. Soc. Am. J., 45:237-240 (1981).
An analysis is presented for constant-flux infiltration of water in soil based on the flux-concentration
relation. The analysis is compared with laboratory experiments on constant-flux infiltration into
columns of fine sand and silty clay loam. It is shown that the effect of gravity is small for the early
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stage and that during this stage sufficiently accurate predictions of moisture profile development can
be made by using the simpler absorption analysis of I. White, D. E. Smiles, and K. M. Perroux.
58. Philip, J.R. Absorption and Infiltration in Two- and Three-Dimensional Systems. In: Water in the
UnsaturatedZone, IASH/AIHS - Unesco (1969).
The theory of the dynamics of one-dimensional infiltration is well developed, but many practical
problems involve two- and three-dimensional systems. This study is directed towards the two-
dimensional problem of infiltration from a semi-circular furrow and the three-dimensional problem of
infiltration from a hemispherical cavity. The latter problem conserves the principal elements of the
problem of infiltration from a shallow ring infiltrometer.
The paper begins by examining the problems of absorption in two- and three dimensional radial
systems from furrows or cavities of finite radius. 'Exact' series solutions of these non-linear problems
are found which are appropriate for small times. For the three-dimensional case a steady large-time
solution exists, and is evaluated. The corresponding linear and 'delta function' solutions are also
found; they agree well with the exact solutions. These various results support the author's argument
that previous attempts to apply the 'Boltzmann transformation' to this group of problems have been
erroneous.
The corresponding infiltration problems are formulated and the linear solutions developed. The
results are of considerable practical interest. It is shown that: 1) the wetting region for both two- and
three-dimensions is finite no matter how great the time (unlike for one dimension); and 2) the final
steady state infiltration rate depends not only on the hydraulic conductivity, but also on the capillary
properties of the soil. The influence of gravity in distorting the pattern of wetting decreases as the
radius of the cavity or furrow decreases; and it is very much less in three dimensions than in two.
59. Philip, J.R. Comments on the Paper by R. Singh and J.B. Franzini. 'Unsteady Flow in Unsaturated Soil-
Water Movement from a Cylindrical Source.' Journal of Geophysical Research, 73:3968-3970 (1968).
60. Philip, J.R. The Dynamics of Capillary Rise. In: Water in the UnsaturatedZone, IASH/AIHS - Unesco
(1969).
The problem of the dynamics of capillary rise associated with the sudden immersion the bottom of a
soil column in free water (and of the corresponding sub-irrigation problem) examined. An 'exact'
method developed for analysing the dynamics of one-dimensional infiltration is appropriate at small
times, but problems of convergence set limits to its applicability. The equilibrium situation
approached at large times is, of course, readily found; but the 'exact bridging of the gap between the
'small-time' solution and the final equilibrium appears to require the use of high-speed computers.
61. Philip. J.R. The Function Inverfc 8. Australian Journal of Physics, 3:13-20 (1960).
The function inverfc 6 arises in certain diffusion problems when concentration is taken as an
independent variable. It enters into a general method of exact solution of the concentration-
dependent diffusion equation. An account is given of the properties of this function, and of its
derivatives and integrals. The function
B(9)=(2/7t1/2) exp [—(inverfc 8)2]
is intimately connected with the first integral of inverfc 8 and with its derivatives. Tables of inverfc 8
and B(8) are given.
62. Philip, J.R. General Method of Exact Solution of the Concentration-Dependent Diffusion Equation.
Australian Journal of Physics, 13:1-12 (1960).
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Richards Equation Models
Only three forms of D(6) have previously been known to yield exact solutions of the equation
ae/at = a/ax (D(6) as/at),
subject to the conditions 6 = 0, x > 0, t = 0; 6 = 1, x = 0, t > 0. The present paper reports a general
method of establishing a very large class of D(6) functions which yield exact solutions. A similar
method enables exact solutions of the same equation subject to the conditions 6 = 0, x > 0, and 6=1,
x<0,t = 0; I1,, xd6 = 0, t > 0. In this case also a very large class of D(6) functions yield exact
solutions. Examples are given for both cases.
Many of the exact solutions which are most readily found tend to lead to zero or infinite values of D
at one or two points of the 6-range. Means of avoiding this difficulty are devised. Practical use of the
method is discussed.
63. Philip, J.R. Horizontal Redistribution with Capillary Hysteresis. Water Resources Research, 27(7): 1459-
1469 (1991).
Exact and relatively simple analysis is possible for one hysteretic redistribution process that in long
horizontal columns with the two parts, x < 0 and x > 0, at different uniform moisture contents B! and
62 (0! > 62) at time t = 0. We formulate and solve this problem. The key to its simplicity is its
similarity character, with moisture profile x(6) proportional to t"2. At the interface x = 0 there is, for t
> 0, a permanent and constant moisture content jump 63 to 64 (63 > 64). The operative moisture
potential is the primary drying potential for x < 0, 6; > 6>63 and the primary wetting potential for x
> 0, 64 > 6 > 62. The operative moisture diffusivity is the primary drying diffusivity for x < 0 and the
primary wetting diffusivity for x > 0. The drying diffusivity is greater than the wetting diffusivity near
6 = 0! and less near 8 = 82 Solutions are found for illustrative examples corresponding to four values
of H, the relative magnitude of the hysteresis loop. The resorptivity R, the total wet-to-dry exchange
of water across x = 0 in reduced form, is analogous to the sorptivity S in absorption and infiltration.
For the calculated examples, R is about one third of S for the corresponding absorption process and
varies only mildly with H. Nonhysteretic calculations may thus give reasonable estimates of R for
horizontal hysteretic redistribution, though they cannot give details such as the moisture jump at x =
0. Desorption moisture profiles for x < 0 are very gradual, with a large depth of penetration.
Absorption profiles for x > 0 are relatively steep, and the penetration depth small. It follows that in
experiments the initially wet section of the column should be made some 5-10 times longer than the
initially dry sector.
64. Philip, J.R. A Linearization Technique for the Study of Infiltration. In: Water in the Unsaturated Zone,
IASH/AIHS - Unesco (1969).
The paper introduces a linearization technique for the solution of non-linear problems in infiltration
and other problems of water movement in unsaturated soils. The method consists, essentially, in
matching exactly linear and non-linear solutions at small times, and in matching the solutions in
some integral sense at large times. The method is worked out for one-dimensional infiltration. It is
shown to yield results agreeing closely with the "exact" non-linear solution and to provide algebraic
infiltration equations equally valid for small and large times.
The parallel use of similarity (or better, "delta function") solutions provides a good method of
approximate analysis, since the linear and the "delta function" solutions represent extremes of soil
characteristics and, by implication, of hydrological behaviour. We may, therefore, obtain insight into
the general character of the phenomenon and the extent to which it may vary, and we may obtain, at
the same time, an indication of the accuracy of the results. For one-dimensional infiltration, the
similarity (or "delta function") solution agrees well with the linear solution.
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65. Philip, J.R. On Solving the Unsaturated Flow Equation: 1. The Flux-Concentration Relation. Soil
Science, 116(5):328-335 (1973).
The flux-concentration relation [F((-))J expresses in reduced form the dependence of flux density on
moisture content during unsteady flow phenomena in unsaturated soils. In this first paper of a series,
the general properties of F((-)) are established for various phenomena: one-dimensional sorption and
infiltration with constant concentration conditions and one-dimensional sorption and infiltration with
constant concentration conditions and one-dimensional sorption (for the 'linear' soil only) with
constant and variable flux and variable concentration conditions. These studies provide a guide to the
properties of F((-)) in more complicated phenomena in one-, two-, and three-dimensional systems.
F((-)) is independent of time for some phenomena and it varies only mildly with time for some others;
and for some phenomena, the dependence of F((-)) on the 'shape' of the moisture diffusivity function
lies within fairly narrow bounds, which are fixed by reference to the extreme cases of the 'delta
function' and the 'linear' soil. This relative stability of F((-)) makes it a useful aid to the study of
methods of solving the unsaturated flow equation. It is employed in later papers in a critique of
Parlange's method and in the development of a new quasi-analytical technique.
An appendix demonstrates that the approximation of step function moisture profiles implies a delta
function moisture diffusivity function.
66. Philip, J.R. Recent Progress in the Solution of Nonlinear Diffusion Equations. Soil Science, 117:257 '-264
(1974).
Important contributions of the late Dr. E. C. Childs included early recognition of the nonlinear
diffusion form of the nonhysteretic flow equation for unsaturated nonswelling soils, and of the
importance of a predictive system based on the flow equation. The full equation is a nonlinear
Fokker-Planck equation, and reduces to a diffusion equation when gravity may be neglected.
Equations of the same form describe water movement and volume change in swelling soils. This
paper reviews recent work on quasianalytical and analytical methods of solving these equations.
Quasilinear solutions of the steady equation in two and three dimensions have received much
attention. The extension to heterogeneous soils can be taken further to include sloping stratification.
Theorems enable solutions for surface sources, and sources at arbitrary finite depth, to be deduced
from mathematically simpler solutions in a region extending infinitely in all directions. A
generalization to the case of sloping soil surface is given.
In many real-world problems where similarity methods do not apply, integral methods offer the same
possibility of effectively reducing the number of independent variables. Green and Ampt introduced a
simple integral method; and more sophisticated methods have entered soil-water studies through the
work of Parlange. Recent studies reveal the importance of preserving integral continuity in integral
methods involving iteration.
For diffusivity D = a(b — 8)"2, all problems of one-dimensional nonlinear diffusion subject to arbitrary
initial conditions plus a flux boundary condition can be linearized. Exact solutions are therefore
available for various problems, some of them in explicit form.
Burgers' equation
ae/at = D a2q/az2 - (Aq + B> aq/az
is of interest as an especially simple nonlinear Fokker-Planck equation. The nonlinearity of the term
in 3q/3z suffices to give a 'profile at infinity' solution for one-dimensional infiltration. The linear
Fokker-Planck equation, on the other hand, has no such feature. Knight has found the exact Burgers
solutions corresponding to constant-concentration infiltration and to constant-flux infiltration. Both
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solutions yield the same 'profile at infinity' behavior at large times, as one expects on physical
grounds. The Burgers model is more informative and more accurate than simple linearization.
67. Philip, J.R. Steady Infiltration from Buried. Surface, and Perched Point and Line Sources in Heterogeneous
Soils: I. Analysis. Soil Sci. Soc.Amer. Proc., 36:268-273 (1972).
The quasilinearized steady infiltration equation is generalized to apply to heterogeneous soils with
conductivity depending exponentially on both moisture potential and depth. Mathematical
developments, including a theorem connecting surface and buried source solutions, follow closely
those established previously for homogeneous soils. Solutions are found for buried, surface, and
perched point and line sources. Physically relevant solutions are limited to the following ranges of p,
the dimensionless coefficient of dependence of conductivity on depth: for buried and surface sources,
P > 0; for perched point sources, p < -1; for perched line sources, p < -1. It is of interest that the
homogeneous medium (p = 0) is an extreme case for existence of buried and surface source solutions;
and that perched source solutions (relevant to subirrigation) exist only for soils with conductivity
increasing rapidly enough with height above the impermeable base. Detailed mapping of flows and
general discussion is left to Part 2.
68. Philip, J.R. Steady Infiltration from Circular Cylindrical Cavities. Soil Sci. Soc. Am. J., 48:270-278 (1984).
The problem of quasilinearized steady infiltration from circular cylindrical- cavities, with the
moisture potential fixed at the cavity surface, is solved exactly. Solutions are presented numerically
and graphically for values of the dimensionless cavity radius, R0, ranging from 0.01 to 10. The
dependence onR0 of the average infiltration rate around the cavity surface, and of the distributions of
moisture content and potential, are examined. As R0 increases, the effects of gravity on the
phenomenon increasingly dominate those of capillarity. Gravity very strongly distorts the moisture
distribution from the symmetry produced by capillarity alone: for R0 as small as 0.01, the depth of the
effectively wetted region exceeds 250 times its horizontal width on the cavity center line; and the ratio
increases rapidly with R0, exceeding 50 000 for R0 = 5. An earlier small R0 approximation proves
useful only for R0 < 0.2. The cavity flow, evaluated by the present analysis, may be combined with the
line source solution to give approximate results useful for the region deeper than 50 radii below the
cavity.
69. Philip, J.R. Steady Infiltration from Spherical Cavities. Soil Sci. Soc. Am J., 48:724-729 (1984).
The problem of quasilinearized steady infiltration from spherical cavities, with the moisture potential
fixed at the cavity surface, is solved exactly. Solutions are presented numerically and graphically for
values of the dimensionless cavity radius R0 in the range 0 to 10. The dependence on R0 of total
cavity flow, of the variation of infiltration rate around the cavity surface, and of the distributions of
moisture content and potential, are examined. As R0 increases, gravity increasingly distorts the
moisture distribution from the symmetry produced by capillarity alone. This distortion, although
marked, is less than one-hundredth that for infiltration from circular cylindrical cavities: a vivid
illustration of the stronger dominance of gravity over capillarity in two-dimensional systems than in
three-dimensional ones. An earlier approximate solution of this problem proves accurate only for R0 <
0.27. The cavity flow, evaluated by the present analysis, may be combined with the point source
solution to give approximate results useful for the region deeper than 20 radii below the cavity.
70. Philip, J.R. Theory of Infiltration. Academic Press, New York, NY, Vol. 9, pp. 215-295 (1969)
Introduction~A very large fraction of the water falling as rain on the land surfaces of the earth moves
through unsaturated soil during the subsequent processes of infiltration, drainage, evaporation, and
the absorption of soil-water by plant roots. Hydrologists, and their textbooks and handbooks, have
tended, nevertheless, to pay relatively little attention to the phenomenon of water movement in
unsaturated soils. Most research on this topic has been done by soil physicists, concerned ultimately
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Richards Equation Models
with agronomic or ecological aspects of hydrology; but their colleagues in engineering hydrology
have exhibited an increasing interest in this field in recent years.
The present article deals with the theory of infiltration which is one important outcome of the
mathematical-physical approach to the study of water movement in unsaturated soils which has been
developed over the past 15 years or so, principally in the U.S., England, and Australia. We shall
consider briefly the physical basis for the general formalism of this approach and the limits of its
applicability; but we shall be concerned principally with developing the general flow equation (a
nonlinear Fokker-Planck equation), with describing methods for its solution, and with presenting the
solutions and discussing their physical significance
We define infiltration as the process of the entry into the soil of water made available (under
appropriately defined conditions at its surface. This "surface" may be the natural, more or less
horizontal upper surface of the soil; or it may be the bed of a natural or artificial furrow or stream, or
the walls of a natural or artificial tunnel or cavity. As we shall see, a fundamental point of departure
for the study of infiltration is the study of absorption, the particular case of infiltration when the effect
of gravity may be neglected, as in horizontal systems, in the early stages of infiltration, and in fine-
textured soils in which the influence of moisture gradients dominates that of gravity.
71. Philip, J.R. The Theory of Infiltration: 1. The Infiltration Equation and Its Solution. Soil Science,
83:345-357(1957).
72. Philip. J.R. The Theory of Infiltration: 2. The Profile of Infinity. Soil Science, 83:435-448 (1957).
73. Philip, J.R. The Theory of Infiltration: 3. Moisture Profiles and Relation to Experiment. Soil Science,
84:163-178(1957).
74. Philip, J.R. The Theory of Infiltration: 4. Sorptivitv and Algebraic Infiltration Equations. Soil Science,
84:257-264 (1957).
Many situations in applied hydrology require that the dynamics of infiltration be characterized by a
small number of parameters. These parameters are most appropriately the coefficients of an algebraic
equation representing the variation of i or v0 with t.
Parts 1 and 2 of this series provided a detailed analysis of infiltration and part 3 a general discussion
of the physical significance of the analytical results. In the present paper we use the preceding work
as the basis for a study of the available (generally empirical) algebraic infiltration equations.
This study is facilitated if we first give some attention to a new physical property of porous media
which enters the subsequent developments here and which we shall have occasion to use also in later
papers of this series.
75. Philip, J.R. The Theory of Infiltration: 5. The Influence of the Initial Moisture Content. Soil Science,
84:329-339 (1957).
76. Philip. J.R. The Theory of Infiltration: 6. Effect of Water Depth Over Soil. Soil Science, 85:278-286
(1957).
77. Philip, J.R. The Theory of Infiltration: 7. Soil Science, 85:333-337 (1958).
78. Philip, J.R. and J.H. Knight. On Solving the Unsaturated Flow Equation: 3. New Quasi-Analytical
Technique. Soil Science, 117(1): 1-13 (1974).
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Richards Equation Models
We present a new quasi-analytical technique for solving the flow equation. It has affinities with
Parlange's method, but offers the following advantages: freedom to choose an initial assumed flux-
concentration relation, Fb greatly improves the possible accuracy of the first approximation, and the
higher approximations preserve integral continuity and therefore behave more stably. The first of
these advantages is of practical importance, but the second is more basic. This paper treats only
solutions subject to concentration conditions; the related technique for solutions subject to flux
conditions will be developed in a later paper.
The technique is studied analytically and numerically for one-dimensional sorption subject to constant
concentration conditions. It is found to be convergent for a wide range of shapes of the diffusivity
function. For the unfavorable ease of the 'linear' soil, the mean error is 3 percent after two iterations
and 1 percent after three. For absorption in Yolo light clay the corresponding figures are 0.57 percent
and 0.07 percent.
The general iterative scheme for one-dimensional infiltration subject to constant concentration
conditions is presented. Three choices of F{ should yield useful first approximations: (A) F^ - limt_>0
F (t is time) = Fabs, the F for the analogous absorption process; (B) F1B = limt-> F = 6; and (C) F1C, an
interpolation function which is exact in the limits as t->0 and t-> . F1A should lead to a good lower
bound for the infiltration rate function q(t), F1B an upper bound, and F1C close upper bound for all
except very large t, and the quality of the estimates of moisture profiles should be comparable.
Detailed calculations for Yolo light clay bear out these expectations; the three estimates are wholly
consistent with the power series solution of Philip (1957b). The error of the approximation based on
F1C increases from 0 percent at t = 0 to about 1 percent at t =106 sec. This first approximation is
accurate enough to render iteration unnecessary for most purposes. Parallel calculations confirm the
nonconvergence of Parlange's method when applied to infiltration.
General iterative schemes are given also for two- and three-dimensional sorption subject to constant
concentration conditions.
79. Poulovassilis, A., et al. An Investigation of the Relationship Between Ponded and Constant Flux Rainfall
Infiltration. Water Resources Research, 27(7): 1403-1409 (1991).
The physics of the rainfall infiltration under a constant rainfall flux greater than the hydraulic
conductivity at saturation are investigated and compared with those pertaining to the infiltration
process under flooding. From the arguments developed theoretically the following conclusions are
reached: The moisture profile prevailing at the time the soil surface becomes saturated during the
rainfall process (t = T) is identical with the profile which has been developed during flooding at an
earlier time at which the infiltration rate is equal to the rainfall flux (t-tc
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Richards Equation Models
front advance, and soil water profile. Infiltration rates beyond ponding include the effect of time
dependent ponding height and saturation depth.
81. Protopapas, A.L. and R.L. Bras. Analytical Solutions for Unsteady Multidimensional Infiltration in
Heterogeneous Soils. Water Resources Research, 27(6): 1029-1034 (1991).
The problem of unsteady multidimensional infiltration in heterogeneous soils is approached
analytically. By assuming that the hydraulic conductivity is an exponential function of the soil matric
potential and a linear function of the soil moisture, the governing flow equation is linearized for a
special variation of the saturated hydraulic conductivity in space. By using integral transform
methods, analytical solutions to the one- and two-dimensional flow problem are derived and shown to
be generalized versions of known results for homogeneous soils. The development aims at obtaining
quantitative criteria for the validity of the one-dimensional approximation, which is commonly used
in hydrologic applications.
82. Protopapas, A.L. and R.L. Bras. The One-Dimensional Approximation for Infiltration in Heterogeneous
Soils. Water Resources Research, 27(6): 1019-1027 (1991).
By using analytical solutions to the problem of one- and two-dimensional linearized unsteady
infiltration from a strip source in a soil with saturated hydraulic conductivity varying exponentially in
space, explicit criteria for the validity of the one-dimensional approximation are derived, in terms of
parameters describing the soil type, the uniformity of the infiltrating source at the surface, and the
soil heterogeneity. Numerical solutions for the linearized infiltration problem are developed for any
variation of hydraulic conductivity in space. Extensive simulations suggest that for a sizable strip
source and moderate variation of hydraulic conductivity, a set of independent neighboring soil
columns represents well the multidimensional infiltration problem. The possibility of defining
sufficient parameters controlling the accuracy of the one-dimensional approximation is discussed.
83. Pullan, AJ. The Quasilinear Approximation for Unsaturated Porous Media Flow. Water Resources
Research, 26:1219-1234 (1990).
This paper gives a review of the quasilinear approximation, an exponential relationship between
hydraulic conductivity and moisture potential, and the uses this has been put to since the time of its
creation. One of the major attractions of this approximation is that under steady regimes the highly
nonlinear Richards equation governing unsaturated flow can be linearized. This simplification has
enabled analytic and semianalytic solutions to be found for many unsaturated flow problems. These
are reviewed, and the current research areas, in which the quasilinear approximation is being used,
are discussed.
84. Raats, P.A.C. Steady Infiltration from Sources at Arbitrary Depth. Soil Sci. Soc. Amer. Proc., 36:399-401
(1972).
This discussion of steady infiltration is based on the assumption that the hydraulic conductivity is an
exponential function of the pressure head. The solution for infiltration from a single point source at
arbitrary depth is presented. On the basis of this solution, J. R. Philip's superposition theorem for
surface sources is generalized to an arbitrary distribution of sources at arbitrary depths. General
expressions for the pressure head, the total head, and the components of the flux are also given.
85. Remson, I., et al. Vertical Drainage of an Unsaturated Soil. Journal of the Hydraulics Division,
Proceedings of the American Society of Civil Engineers, 91(HYl):55-74 (1965).
The general equation describing vertical unsaturated flow of soil water is solved by numerical
approximation for several cases. The computer programs are described. The results show: (1) The
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intermediate belt of a well-drained unsaturated soil approximates field capacity during times of small
evapotranspiration and intermittent penetrating rainfall; (2) infiltrated rainfall moves as a pulse with
a steep wetting front; (3) the belt of soil water drains from saturation to approximately field capacity
within two days for zero evapotranspiration.
86. Salvucci, G.D. An Approximate Solution for Steady Vertical Flux of Moisture Through an Unsaturated
Homogeneous Soil. Water Resources Research, 29(11):3749-3753 (1993).
An approximate solution is found to the differential equation governing the steady state vertical flux
of moisture through an unsaturated homogeneous soil for which the dependence of hydraulic
conductivity on tension head is given in the form K(t|/) = A(B + t|/n)'. The solution expresses the
depth from the water table as a function of the capillary tension for a given rate of percolation or
capillary rise. It is easily inverted to obtain explicit expressions both for the capillary tension as a
function of depth and flow rate, and for the steady flow rate in terms of the capillary tension at a
given depth. The latter is shown to compare favorably both to existing limiting case solutions of
percolation to a water table at infinite depth, and to capillary rise from a shallow water table to an
infinitely dry surface. More importantly, this solution applies to the intermediate condition where the
magnitude and direction of flow are sensitive to both the near-surface capillary tension and the water
table depth.
87. Sander, G.C., et al. Exact Nonlinear Solution for Constant Flux Infiltration. Journal of Hydrology,
97:341-346 (1988).
Recently, an analytical nonlinear solution to the problem of two phase oil and water infiltration under
a constant flux boundary condition was derived. We show that this solution also applies to the
problem of constant infiltration of water by introducing a very simple change in the independent
variables of space and time.
88. Sander, G.C., J.-Y. Parlange, V. Kuhnel, W.L. Hogarth, and J.P.J. O'Kane. Comment on "Constant Rate
Rainfall Infiltration: A Versatile Nonlinear Model 1. Analytic Solution" by P. Broadbridge and I. White.
Water Resources Research, 24:2107-2108 (1988).
89. Sisson, IB., A.H. Ferguson, and M.T. van Genuchten. Simple Method for Predicting Drainage from Field
Plots. SoilSci. Soc. Am. J., 44:1147-1152 (1980).
When the one-dimensional moisture flow equation is simplified by applying the unit gradient
approximation, a first-order partial differential equation results. The first-order equation is
hyperbolic and easily solved by the method of P. D. Lax. Three published K(8) relationships were
used to generate three analytical solutions for the drainage phase following infiltration. All three
solutions produced straight lines or nearly straight lines when log of total water above a depth was
plotted versus log of time. Several suggestions for obtaining the required parameters are presented
and two example problems are included to demonstrate the accuracy and applicability of the method.
90. Smettem, K.R.J., et al. Three-Dimensional Analysis of Infiltration from the Disc Infiltrometer 1. A
Capillary-Based Theory. Water Resources Research, 30(11):2925-2929 (1994).
The hydraulic properties of an unsaturated homogenous and isotropic soil can be obtained from the
unconfined flux out of a disc infiltrometer into the soil over the depth of wetting. The disc
infiltrometer is becoming increasingly popular, but methods of analysis have generally relied on the
restrictive assumptions of one-dimensional flow at early times or quasi-steady state flow at large
times. We provide an approximate analytical expression for three-dimensional unsteady, unconfined
flow out of a disc infiltrometer, and this includes the geometric effect of the circular source but
ignores gravity. This physically based solution is tested against data obtained from laboratory
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Richards Equation Models
experiments on repacked material. The results illustrate that the difference between three-
dimensional and one-dimensional flow is linear with time.
91. Smith, R.E. Approximate Soil Water Movement by Kinematic Characteristics. Soil Sci. Soc. Am. J., 47:3-
8 (1983).
Using the Richard's equation in the Fokker-Planck nonlinear diffusion form, unsaturated soil water
flow may be treated as a diffusion-convection wave process. If a6/az is assumed a function of 6
alone, the unsaturated flow equation may be solved by the method of characteristics, and when aQ/az
becomes sufficiently small, the Peclet number is assumed large enough to treat unsaturated flow
kinematically. Changes in 6 with depth in the soil profile are treated as waves, moving downward.
Advancing and receding "waves" are treated differently in the approximate analytical technique
described here, with advancing wetting fronts described by kinematic "shocks." The method is
compared to the complete solution to Richard's equation for a complex rain pattern and found to
predict well the location of deeper moving fronts and also general 6 patterns. The kinematic method
is also shown to apply to root water extraction zones and to layered soil situations.
92. Smith, R.E., C. Corradini, and F. Melone. Modeling Infiltration for Multistorm Runoff Events. Water
Resources Research, 29(1): 133-144 (1993).
We present a relatively simple analytical/conceptual model for rainfall infiltration during complex
storms. It is an approximate but physically based model which can treat intervals of either no rain,
low rain, or evaporation. The infiltration model is based on the very general three-parameter analytic
model of Parlange et al. (1982), extended to treat soils with very high initial water content. The
redistribution model is based on profile extension with shape similarity. A wide range of soil types
can be simulated. The model is tested by comparison with numerical solutions of Richards's equation
carried out for a variety of events upon four selected soils. The model simulates the solution to
Richards's equation quite accurately, provided basic soil retention relations are parametrically
represented. It simulates redistribution particularly well for redistribution intervals up to 20 hours.
The model usefulness in comparison with the common and simple approach which disregards soil
water redistribution is also shown.
93. Sposito, G. and J.V. Giraldez. On the Theory of Infiltration in Swelling Soils, pp. 107-118.
The generalization of the one-dimensional Richards Equation to the case of swelling soil is presented.
Although the generalized infiltration equation has the same mathematical form as the classical
Richards Equation, it differs physically from the latter in containing material coordinates and in
having an additional component in the total soil water potential that is due to applied pressure (e.g.,
overburden pressure). A new material coordinate transformation is introduced to cast the solution of
the generalized infiltration equation into a coordinate frame that is fixed with respect of the
experimenter. This transformation can be determined empirically through dual-energy gamma-ray
measurements of the bulk density of the swelling soil. Lastly, a model expression for the shrinkage
curve is presented that may be employed, for a wide variety of swelling soils, to describe
mathematically the contribution of overburden pressure to the gradient of the total soil water
potential.
94. Swartzendruber, D. A Quasi-Solution of Richards' Equation for the Downward Infiltration of Water into
Soil. Water Resources Research, 23:809-817 (1987).
A new equation is presented that expresses depth Z of soil water content 8 at time t as an explicit
function of 8 and t and holds for all t > 0. The new Z(8, t) includes the classical J. R. Philip (1957b,
c) solution for asymptotically large times. At moderate times, the new Z(8, t) is matched to the
classical Philip solution expressed as (^t"2 + x' + t|rt3'2 + S>t2 + . . ., where (|>, %, i|r, B>,. . . are different
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Richards Equation Models
functions of 6 alone. The matching process enables the characterizing functions (or parameters)
within Z(6, t) to be evaluated in straightforward fashion from the moderate-time Philip functions (((>,
X, t|/, S>,. . .). When evaluated for either Philip's data or the Green and Ampt step-function solution,
the new Z(6, t) in each case describes the water content profiles with excellent accuracy for all times.
Also, for all t > 0, the new Z(6, t) can be directly integrated to provide a new equation for cumulative
quantity of water infiltrated or flux, both as explicit functions of time.
95. Swartzendruber, D., andF.R. Clague. An Inclusive Infiltration Equation for Downward Water Entry into
Soil. Water Resources Research, 25:619-626 (1989).
A new infiltration equation, derived exactly from a recent quasi-solution of the governing differential
flow equation, is capable of expressing eleven previously published infiltration equations. On the
basis of least squares fitting, the new equation in three-parameter dimensionless form expressed each
of the eleven previous equations within ±0.31% over all times. If only one parameter was retained
and fitted in the dimensionless form, nine of the eleven equations were still expressed within ±2.5%.
This one-parameter dimensionless form implies the dimensional infiltration equation I = (S/A0)[1 -
exp (-A0t1/2)] + Kt, which is therefore proposed as a realistic blend of rigor and utility for the complete
range of time t (0 < t < x), where I is the cumulative quantity of water infiltrated, S is the sorptivity,
A0 is a constant arising from the quasi-solution, and K is the sated (satiated, near-saturated) hydraulic
conductivity. Because of its broad matching capability, this equation is deemed to hold considerable
promise for describing and fitting experimental infiltration data, whether from the laboratory or the
field.
96. Swartzendruber, D. and W.L. Hogarth. Water Infiltration into Soil in Response to Ponded-Water Head.
SoilSci. Soc. of Am. J., 55(6):1511-1515 (1991).
The pressure head of water ponded on the soil surface can increase the infiltration of water into soil,
but the effect has often been complicated to describe mathematically. This study was conducted to
devise a simpler mathematical description without undue sacrifice of accuracy. A new three-
parameter infiltration equation was examined for its capability in describing the effect of soil-surface-
ponded water head, h, on the cumulative quantity of water infiltrated into the soil with time. The
infiltration equation, as reduced to two-parameter dimensionless form, was fitted by nonlinear least
squares to dimensionless data generated from mathematical descriptions of infiltration that included
the effect of h in somewhat complicated parametric form. The fitted two-parameter equation gave an
excellent description of all the generated data, in terms of both goodness of fit and in recovery of the
dimensionless ponded head p used as an input into the generated data. From an overall perspective,
recovery of h was achieved within a relative error of ±1.4% across the complete range of the
generated data, thus validating the new and relatively simple equation in its description of the general
effect of ponded head on the cumulative infiltration process.
97. Swartzendruber, D., and E.G. Youngs. A Comparison of Physically-Based Infiltration Equations (Note).
Soil Science, 117:165-167 (1974).
Three infiltration equations, each derived in some manner from a consideration of the physics of flow
through porous media, are compared quantitatively over the whole time range from zero to infinity.
Compared with the Green and Ampt equation, the two-term Philip equation is never more than 15.1
percent higher, while the equation arising out of Philip's linearization analysis is never less than 9.2
percent lower. For practical purposes, these discrepancies may frequently be ignored. Because of its
mathematical simplicity and adaptability, the two-term Philip equation, however, would be the
preferred choice.
98. Wallach, R. Soilwater Distribution in a Nonuniformly Irrigated Field with Root Extraction. Journal of
Hydrology, 119:137-150 (1990).
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Richards Equation Models
A steady-state solution is developed for two-dimensional distributions of irrigation water flux at the
soil surface and water extraction by plant roots. The solution of the linear equation is based on the
small perturbations method, in which a small parameter is introduced into the governing differential
equation. For the sprinkler irrigation method, this parameter is determined by scaling the soil
characteristic length, a, by the distance between the sprinklers. Linearization is attained by assuming
that the unsaturated hydraulic conductivity is exponentially related to the pressure head. Vertical
distribution of the two-dimensional plant-root uptake is based on the one-dimensional model, which
meets the approximation that the extraction pattern of plant roots is 40, 30, 20, and 10% of the total
transpiration requirements in each successively deeper quarter of the root zone. Water extraction by
plant roots increases lateral unsaturated flow beyond that which occurs as a result of irrigation
nonuniformity. However, for a homogeneous soil profile, this lateral flow component increases water
availability to those parts of the field where water shortages occur because of irrigation
nonuniformity. Indeed, actual irrigation uniformity increases as measured by the decrease in the
amplitude of the vertical flux around the mean. Younger plants suffer more from spatial fluctuations
of irrigation water than do older plants, which have deeper roots. Actual irrigation uniformity, as
measured by water availability to roots, is also affected by the distance between sprinklers and by soil
type.
99. Warrick, A.W. Additional Solutions for Steady-State Evaporation from a Shallow Water Table. Soil
Science, 146:63-66 (1988).
In this paper, I consider evaporation from a shallow water table and present analytical solutions. An
unsaturated conductivity function K of the form K = a/(Sn + b) is used, with S the matric suction and
a, b, and n constants. The results of Gardner and Fireman (1958) are generalized to all n > 1,
including fractional values and b = O, for which K reduces to the form commonly referred to as that
of Brooks and Corey. A table is presented from which evaporation rates follow for a wide range of
depths and hydraulic conductivity functions of the above form.
100. Warrick, A.W. An Analytical Solution to Richards' Equation for a Draining Soil Profile. Water Resources
Research, 26(2):253-258 (1990).
Analytical solutions are developed for the Richards' equation following the analysis of Broadbridge
and White. Included here is the solution for drainage and redistribution of a partially or deeply
wetted profile. Additionally, infiltration for various initial conditions is examined as well as
evaporation at the upper boundary. In all cases the surface flux is constant, whether it be zero for
drainage, positive for infiltration, or negative for evaporation. The solutions assume specific forms
for the soil water diffusivity and hydraulic conductivity functions: a(b -B)2 and /? + y(b - 6) + A/[2(b
- 6)], respectively. Here 8 is the water content and a, b, ft y, and A are constants.
101. Warrick, A.W. Analytical Solutions to the One-Dimensional Linearized Moisture Flow Equation for
Arbitrary Input. Soil Science, 120:79-84(1975).
Analytical solutions to the one-dimensional linearized moisture flow equation are presented. The
surface boundary condition is taken either as a time-varying flux, time-varying water content, or with
the flux as a linear function of the water content. A semi-infinite vertical flow medium is considered
with arbitrary initial conditions. Calculations for a cyclic flux input are compared to the associated
nonlinear finite difference solution for five simulations using data from two soils. The results are
encouraging in that the same general profiles are obtained. The solution is believed to have particular
merit to describe moisture regimes resulting from a high frequency irrigation schedule.
102. Warrick, A.W. Inverse Estimations of Soil Hydraulic Properties with Scaling: One-Dimensional
Infiltration. Soil Sci. Soc. Am. J., 57(3):631-636 (1993).
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Richards Equation Models
Inverse estimations of hydraulic properties are of immense importance to effectively provide input for
water flow modeling and descriptions of soil systems. Here inverse estimations were made using
scaled forms of Richards' equation and infiltration measurements in two steps. In Step 1, transient
experimental data of cumulative infiltration rate, or wetting front position were "best-fit" using
Philip's quasi-analytical, algebraic forms. This was demonstrated with linear regression although a
maximum likelihood method could have been used. The first step does not depend on the form of the
hydraulic functions and is valid for all uniform soils with constant initial and boundary conditions.
Step 2 related the algebraic coefficients to the soil properties and was dependent on the model
assumed for the hydraulic functions. Computations were generally algebraic and extremely easy
relative to alternative methods that require a numerical simulation for each combination of properties
considered. Clearly, the number of estimated parameters cannot exceed the significant number of
terms fit to the algebraic forms. For small times or "noisy" data, perhaps only one parameter can be
estimated that, for the examples presented, leads to an estimate of the ratio of saturated conductivity
to an inverse characteristic length. For this situation, further measurements are needed to provide
additional information if more parameters are to be found. Three commonly used forms of hydraulic
functions were used, but the methodology is not limited to these specific forms. Scaling could also he
used for more complex geometries and moisture regimes; however, the convenience of the algebraic
forms will not generally be available in those cases.
103. Warrick, A.W. Solution to the One-Dimensional Linear Moisture Flow Equation with Water Extraction.
SoilSci. Soc.Amer. Proc., 38:573-576 (1974).
The one-dimensional, steady-state moisture flow equation is solved for arbitrary plant water
withdrawal functions using the matric flux potential of earlier investigators to obtain a linearized
form. A semi-infinite flow medium and a finite-depth medium overlying a shallow water table are
considered with the surface boundary condition taken as a flux. Tables are presented giving the
matric flux potential (from which the pressure head is easily determined) for several withdrawal
functions. Numerical examples show the effects of different surface fluxes and rooting depths on the
matric flux potential and pressure-head profiles. The results are particularly relevant for high-
frequency irrigation.
104. Warrick, A.W. Time-Dependent Linearized Infiltration. I. Point Sources. Soil Sci. Soc.Amer. Proc.,
38:383-386 (1974).
Water flow from a point source is analyzed using a linearized form of the moisture flow equation.
Time-dependence is assumed with the results simplifying to those of previous investigators for steady-
state conditions. Discrete time-distributed inputs such as might occur for trickle or high frequency
irrigation is amenable to the solution. Numerical simulations include (i) the advance of a wetting
front during infiltration, (ii) moisture variation resulting from a cyclic input as during irrigation, and
(iii) the matric flux potential field for a two-source problem.
105. Warrick, A.W. Traveling Front Approximations for Infiltration into Stratified Soils. Journal of
Hydrology, 128:213-222 (1991).
Two approximations are developed for infiltration into a stratified soil profile. Both satisfy the initial
conditions, give the correct water content at a given depth for large time, obey mass conservation and
give the correct rate of advance for large time. The first is referred to as the "square front" solution
and is analogous to Green-Ampt theory in that the profile is either wet or dry without a transition
zone. The second solution is based on a homogeneous analog for which Philip's solution for a profile
at large time is invoked. Examples are given for surface boundary conditions both as a specified head
and as a flux density.
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Richards Equation Models
106. Warrick, A.W. and A.A. Hussen. Scaling of Richards' Equation for Infiltration and Drainage. Soil Sci.
Soc.Am.J., 57:15-18 (1993).
Scaling techniques allow a single solution to Richards' equation to suffice for numerous specific cases
of water flow in unsaturated soils. Here, such solutions are presented for the well-known hydraulic
functions of Brooks and Corey. Infiltration for a constant water content or a constant flux at the soil
surface is considered as well as drainage for a semi-infinite profile. Although limited to a single form
of hydraulic properties, the results are otherwise more general than those previously available. Of
particular interest is that the hydraulic conductivity and air-entry values are embedded within the
reduced variables. This is an advantage for computational efficiency, such as when using parameter
optimization techniques. Three examples are presented. In the first, a single solution is
presented,valid for four combinations of saturated hydraulic conductivity and surface water content.
A second example relates to drainage, also with one solution to describe four combinations of
physical parameters. The final example is for constant flux infiltration.
107. Warrick, A.W., A. Mas, and D.O. Lomen. An Analytical Solution to Richards' Equation for Time-Varying
Infiltration. Water Resources Research, 27:763-766 (1991).
An analytical solution is developed for Richards equation for time-varying infiltration. The
infiltration rates must be specified as piecewise constants with time. The answer is expressed as a
sum of two terms, the first is a function of the instantaneous infiltration and the second an integral
which accounts for the water distribution within the profile for the previous infiltration history. The
solutions assume specific forms for the soil water diffusivity and hydraulic functions used earlier by
Broadbridge and White (1988).
108. Warrick, A.W., D.O. Lomen, and A. Ammozegar-Fard. Linearized Moisture Flow with Root Extraction for
Three-Dimensional Steady Conditions. Soil Sci. Soc. Am. J., 44:911-914 (1980).
A mathematical model is presented to simulate plant water uptake assuming steady state conditions
and an unsaturated hydraulic conductivity as an exponential function of pressure head. By choosing
the conductivity to be of this form, the moisture flow equation is linear, allowing analytical solutions
to be developed in a closed form. Geometries considered possess radial symmetry and are appropriate
to describe trickle irrigation. These include matrix potentials for points, discs, and cylinders which
can be taken either as sources or sinks. These provide the basic building blocks to simulate varied
steady-state withdrawal patterns. The solution for the cylinder is newly derived; the other solutions
were already available. Nonsymmetric cases can result from superpositions. Examples include
pressure head distributions for four sink-source combinations.
109. Warrick, A.W., and D.O. Lomen. Time-Dependent Linearized Infiltration: III. Strip and Disc Sources.
5-0/7 Sci. Soc. Amer. J., 40:639-643 (1976).
Water flow from strip and disc surface sources is analyzed using an approach similar to previous
investigations with line and point sources. A line of closely spaced trickle irrigation emitters often
wets a surface strip of finite width; similarly, for a single emitter the surface wetted pattern is a disc.
Lines of equal matric flux potential are wider and more shallow for these sources than wetting
patterns for the line and point. The moisture regime is independent of the source shape and depends
only on the total flow rate for regions beyond approximately two times the strip width or disc radius.
110. Warrick, A.W., D.O. Lomen, and S.R. Yates. A Generalized Solution to Infiltration. Soil Science Society
of America Journal, 49:34-38 (1985).
A generalized solution the moisture flow (Richards') equation is developed for infiltration. The
equation is expressed in terms of dimensionless time, depth, and water content. The reduced form
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Richards Equation Models
applies for the hydraulic functions of van Genuchten or of Brooks and Corey as well as to other scaled
forms and the solution utilizes the procedure of Philip. Answers are presented in concise tables
which, in principle, allow for finding moisture profiles, wetting front, intake rate and cumulative
intake for a variety of soils and for varying initial water contents. For application, specifically
measured hydraulic conductivities can be best-fitted to the forms used here or generalized empirical
values can be used from the literature. Thus, either specific or generic parameters may be used,
depending on availability for a given problem.
111. White, I. and P. Broadbridge. Constant Rate Rainfall Infiltration: A Versatile Nonlinear Model 2.
Applications of Solutions. Water Resources Research, 24(1): 155-162 (1988).
In paper 1 (Broadbridge and White, this issue) an analytical nonlinear model for constant rate rainfall
infiltration was proposed which promised considerable versatility. In it a wide range of soil hydraulic
properties are generated through the variation of a single free parameter C. Here three techniques are
advanced for determining this parameter. The first, a one-dimensional technique, involves
simultaneous determination of sorptivity and wetting front position. The second uses measured
values of surface water content at long infiltration times for rainfall rates less than the saturated
conductivity. In the third, three- and one-dimensional flow rates are measured on the same soil
sample. All are suitable for field applications. The practical range of the C parameter, for a variety
of repacked and in situ soils, is found to be restricted to between 1 and 2. The hydraulic
conductivities and diffusivities of the model are in general agreement with independent
measurements. Its moisture characteristics t|/(B), which are not matched in any way to measured
characteristics, follow closely those observed. Also, the model permits predictions of the dependence
of sorptivity on antecedent water content given a single measurement of sorptivity. The analytic
solutions for constant flux infiltration given in paper 1 (Broadbridge and White, this issue) describe
satisfactorily the evolution of water content profiles and surface water pressure potential in the
laboratory and field without a posteriori adjustments. The mathematically simple, traveling wave
approximation agrees well with observations at comparatively short infiltration times. Finally, field
and laboratory measured times to ponding are predicted satisfactorily by the model's analytic
expression.
112. White, I., D.E. Smiles, and K.M. Perroux. Absorption of Water by Soil: The Constant Flux Boundary
Condition. Soil Sci. Soc. Am. J., 43:659-664 (1979).
Absorption of water into soil as the result of a constant flux condition at the soil surface is examined.
Experiments for a fine sand show that the surface water content, movement of the wetting front, and
the water content profiles may be predicted from the soil water diffusivity function using the notion of
the flux-concentration relation of Philip (1973).
Reduced space and time variables X = V0x and T = V02t are introduced. Use of these variables greatly
simplifies treatment of the system and reveals that the surface water content and the reduced position
of the wetting front are uniquely defined by T while the water content profiles at any value of T are
unique in terms of X.
113. Whisler, F.D. and A. Klute. Numerical Analysis of Ponded Rainfall Infiltration. In: Water in the
UnsaturatedZone, IASH/AIHS - Unesco (1969).
A numerical solution of the flow equation for water in soil was obtained for a flow system consisting
of a vertical column of stratified soil materials which had been drained from saturation to equilibrium
with a water table. Water was assumed to be applied at the top of the column as either rainfall or
ponded water. Infiltration into such a system involves hysteresis; i e., the soil at each point wets
along a different wetting scanning curve. This part of the hysteresis phenomena was taken into
account in the solution of the problem. The solution of the equation depicts the time and depth
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Richards Equation Models
distributions of water content and pressure head during the resulting infiltration as well as infiltration
rate and accumulated water content.
The properties of the soil that influence the flow are the water capacity and hydraulic conductivity.
Tables of x, a dimensionless conductivity, and y, a dimensionless water capacity, were calculated
from tables of 0, a dimensionless water content, and (p, a dimensionless pressure head, for a set of
moisture characteristic curves. These tables were fed into a digital computer which solved the
problem.
Several hypothetical cases were examined. These were coarse-textured soils overlying fine-textured
soils, fine-textured lenses in coarse-textured soils, and coarse-textured lenses in fine-textured soils.
Comparisons were made between these cases and also nonstratified cases on the basis of pressure
head-time profiles, moisture content-time profiles, infiltration rate changes, and accumulated water
content changes The numerical solution predicts a hold up effect on the wetting front as it goes from
a fine-textured layer into a coarse-textured one. As one would expect, the rate of advance of the fonts
is faster for ponded infiltration than for rainfall infiltration.
114. Youngs, E.G. and A. Poulovassilis. The Different Forms of Moisture Profile Development During the
Redistribution of Soil Water After Infiltration. Water Resources Research, 12(5):1007-1012 (1976).
Two different forms of moisture profile development may occur during the redistribution of soil water
after infiltration. In one of the profile shape continues to be similar to that during the infiltration
process, with a fairly uniform water content region near the soil surface above a steep wetting front,
and the rate of redistribution is inversely related to the infiltration depth. In the other, water drains
from near the surface to appear as a step in the moisture profile below the wetting front at the
cessation of infiltration, and the rate of redistribution is directly related to infiltration depth.
Classical theory of soil water movement is shown to account for both forms of moisture profile
development, which depend on the moisture profile shape at the cessation of infiltration and on the
soil water properties of the porous material. Experimental results of moisture profile development in
a sand column illustrate how different profile forms develop as the depth of infiltration is increased
and, by inclining the experimental column, how they depend on the relative importance of the soil
water pressure head gradients compared with that of gravity.
115. Zimmerman, R.W., G.S. Bodvarsson, and E.M. Kwicklis. Absorption of Water into Porous Blocks of
Various Shapes and Sizes. Water Resources Research, 26(11):2797-2806 (1990).
Approximate solutions are presented for absorption of water into porous spherical, cylindrical, and
slablike blocks whose characteristic curves are of the van Genuchten-Mualem type. The solutions are
compared to numerical simulations of absorption into blocks of the Topopah Spring member
(Paintbrush tuff) from the site of the proposed nuclear waste repository at Yucca Mountain, Nevada.
Guided by these results, a scaling law, based on the ratio of surface area to volume, is then proposed
for predicting the rate of absorption into irregularly shaped blocks. This scaling law is tested against
a numerical simulation of absorption into an irregularly shaped, 2-dimensional polygonal block and is
shown to be a good approximation.
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APPENDIX C
Soil Hydraulic Property Estimation
Annotated Bibliography
C-l
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5*o/7 Hydraulic Parameter Estimation
ESTIMATION OF SOIL HYDRAULIC PARAMETERS
1. Block, L.O., M.J. Hendry, and G.W. Chan. MRCON: Computer Program for Estimating Soil Moisture
Retention and Unsaturated Hydraulic Conductivity. Ground Water, 26(2):218-221 (1988).
MRCON is an interactive computer program designed to approximate soil moisture retention (6-
i|0 and/or unsaturated hydraulic conductivity (KB) for soils. MRCON, which is written in
BASIC, uses empirical methods obtained from the literature to calculate K-6 and a modified
method to calculate 6-t|/. Input data for the program consists of a saturated moisture content and
a minimum of three values of 6-t|/. A measured value of saturated hydraulic conductivity can be
used as input to better approximate the K-6 curves to field conditions.
2. Bohne, K., et al. Rapid Method for Estimating the Unsaturated Hydraulic Conductivity from Infiltration
Measurements. Soil Science, 155(4):237-244 (1993).
A method is proposed for estimating the unsaturated hydraulic conductivity from observed
infiltration data. Infiltration measurements are generally easier to obtain than experimental data
required for in situ determination of the hydraulic conductivity. The problem was formulated in
terms of a nonlinear least-squares parameter optimization method which combines Philip's two-
term infiltration equation with an analytical description of the unsaturated soil hydraulic
properties according to Mualem and van Genuchten. Reliable estimates for the hydraulic
parameters could be obtained with an inverse procedure when independently measured water
retention data were included. The results indicate that soil water content measurements at very
low values of the soil water pressure head are especially important to ensure parameter
uniqueness. The method provides rapid and cost-effective estimates of the hydraulic properties of
field soils.
3. Brutsaert, W. Some Methods of Calculating Unsaturated Permeability. In: Winter Meeting of the
American Society of Agricultural Engineers, Chicago, IL (1965).
4. Clapp, R.B. and G.M. Hornberger. Empirical Equations for Some Soil Hydraulic Properties. Water
Resources Research, 14(4):601-604 (1978).
The soil moisture characteristic may be modeled as a power curve combined with a short
parabolic section near saturation to represent gradual air entry. This two-part function—together
with a power function relating soil moisture and hydraulic conductivity—is used to derive a
formula for the wetting front suction required by the Green-Ampt equation. Representative
parameters for the moisture characteristic, the wetting front suction, and the sorptivity, a
parameter in the infiltration equation derived by Philip (1957), are computed by using the
desorption data of Holtan et al. (1968). Average values of the parameters, and associated
standard deviations, are calculated for 11 soil textural classes. The results of this study indicate
that the exponent of the moisture characteristic power curve can be predicted reasonably well
from soil texture and that gradual air entry may have a considerable effect on a soil's wetting
front suction.
5. Clothier, B.E. and I. White. Measurement of Sorptivity and Soil Water Diffusivity in the Field. Soil Sci.
Soc. Am. J., 45:241-245 (1981).
An analysis for scaling the exponential soil water diffusivity function, D(8), from concurrent
measurements of sorptivity and wet front advance is presented. This analysis provides a field
C-2
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5*o/7 Hydraulic Parameter Estimation
method for measuring p, the slope of the exponential D(6). For Bungendore fine sand in the
field it was found that p = 3. This is contrary to absorption experiments on a variety of repacked
soils that have consistently found p = 8 and led to the suggestion that p = 8 may be the
"universal" value applicable to all soils. Although the exponential function with p = 3 provided a
usable approximation over the entire range, D(6) data from undisturbed cores showed that at
high water contents D was effectively constant, and for lower water contents the undisturbed data
were well described by an exponential function with p = 8. Measurements of sorptivity, wet front
advance, and hydraulic conductivity were made in the field using a device that supplied water to
the soil surface at the small tension of 4 cm. This gives the physical properties of the soil matrix,
since larger channels and voids (>0.75 mm in diam) do not affect the measurements made at a
small tension. Field heterogeneity due to such macropores is consequently eliminated. This
sorptivity of Bungendore fine sand is about 0.4 that from ponded infiltration affected by the
macropores.
6. Cook, FJ. and A. Broeren. Six Methods for Determining Sorptivitv and Hydraulic Conductivity with
Disc Permeameters. Soil Science, 157(1):2-11 (1994).
The disc permeameter has become a popular apparatus for measuring in situ the sorptivity, S,
and hydraulic conductivity, K, of the soil at some prescribed potential. A number of different
methods have been proposed for calculating S and K using the flow rate, Q(t), from the disc.
Measurements of Q(t) on a Kokotau silt loam soil were made using discs with radii of 60 and 102
mm. Measurements were made at potentials of -20, -40, and -100 mm with the 102-mm disc
and at potentials of -20 and -40 mm with the 60 mm disc.
S and K were calculated using six different methods. Three of the methods use Wooding's
equation of flow from a disc:
Q4 /7tr2 = AK[l+4A.c/7tr]
where Q4 is the steady-state flow rate; r is the radius of the disc source; AK = K(t|/0) - K(t|/n)
(where +t|/0 and t|/n are, respectively, the potential of the source and the initial potential of the
soil); and A.c is the macroscopic capillary length scale. These three methods and a method based
on a linear diffusion model gave similar values of S and/or K. The method of Youngs, which
uses the early- to medium-time infiltration data to calculate K, gave very low and inconsistent
values of K.
Calculations of K were also made from values of S at two different potentials. Here S as S0 was
first calculated by using the early-time values of Q(t1/2). These values of K were similar to those
calculated using the methods based on Wooding's equation. This suggests that measuring S at
two or more different potentials, using the disc permeameter, would provide a very rapid method
for characterizing the hydraulic properties of soils, provided the soil is fine textured and S can be
measured either from the early-time behavior or using the method of Warrick.
7. El-Kadi, A.I. On Estimating the Hydraulics Properties of Soil. Part 1. Comparison Between Forms to
Estimate the Soil-Water Characteristic Function. Advances in Water Resources, May 1985 (1985).
The hydraulic properties of soil include the soil-water characteristic function [h(8) in which 8 is
water content and h is pressure head (suction)] and the hydraulic conductivity function [K(8) or
K(h)]. These functions are essential to the solution of unsaturated groundwater flow problems.
A number of empirical and semi-empirical forms have been proposed in the literature to estimate
these functions. The present paper employs a nonlinear least-square analysis for comparison
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between some of the available forms, using a large number of experimental measurements of h(6)
for different classes of soil. Suitability of the forms for predicting the hydraulic conductivity
function is examined. In the absence of accurate measurements, the paper facilitates modeling
providing estimates for the parameters of the soil-water characteristic function.
8. Ghosh, R.K. Estimation of Soil-Moisture Characteristics from Mechanical Properties of Soils. Soil
Science, 130(2):60-63 (1980).
Soil-moisture characteristics can be established directly from the physical properties of soils and
a single measurement of water potential, i|r, in bars, at some moisture content, 6, on a volume
basis. The method is applicable for those soils for which the i|r - 6 relations can be expressed as
i|r = t|/e (6/60)-p, where t|/e is the air-entry water potential in bars, 60 is the saturated water content
in volume per volume, and p is an empirically determined constant. The procedure is useful
when a reliable soil moisture characteristic curve is not available. We noted considerable
agreement between estimated values using this procedure and those experimentally determined
and published for eight soils.
9. Gupta, S.C. and W.E. Larson. Estimating Soil Water Retention Characteristics from Particle Size
Distribution. Organic Matter Percent and Bulk Density. Water Resources Research, 15(6): 1633-1635
(1979).
Regression models are presented for estimating soil-water-retention curves from particle-size
distribution, percentage of organic matter, and bulk density. Models were developed from the
measured soil-water-retention curves of artificially packed cores (7.6 X 7.6 cm) of 43 soil
materials. These soil materials included 13 agricultural soils. Curves predicted with these
models approximated reasonably well the measured water retention of 61 Missouri soils.
Because conventional methods of obtaining retention curves are expensive and time consuming,
these equations will be valuable for modeling salt and water flow in soils and for estimating
available water capacities.
10. Hills, R.G., et al. Modeling One-Dimensional Infiltration into Very Dry Soils 2. Estimation of the Soil
Water Parameters and Model Predictions. Water Resources Research, 25(6): 1271-1282 (1989).
Using a water content based one-dimensional finite difference algorithm, we model infiltration
into a 6-m-lysimeter column containing alternating layers of air dried clay loam and sand. We
use van Genuchten's equation to model the h-8 relationship and both Campbell's and Mualem's
equations to model the K-8 relationship. Several sets of model parameters are estimated using
data generated from various combinations of laboratory drainage experiments, laboratory
measurements of saturated hydraulic conductivity, lysimeter observations of initial conditions,
and lysimeter observations of water redistribution. We find that predictions of infiltration based
on field redistribution data give the best agreement with the observed infiltration into the
lysimeter. We also find that the use of Campbell's K-8 relationship results in closer agreement
between the infiltration model predictions and the lysimeter observations than does the use of
Mualem's K-8 equation. The results of this study show that infiltration model predictions for a
carefully controlled field scale lysimeter are very sensitive to the field and laboratory techniques
used to estimate the soil water parameters.
11. Hussen, A. A. and A.W. Warrick. Alternative Analyses of Hydraulic Data From Disc Tension
Infiltrometers. Water Resources Research, 29(12):4103-4108 (1993).
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Hydraulic conductivity values were compared based on alternative analyses of data from disc
tension infiltrometers. The first method was based on a single disc and tension and depends on
the estimate of sorptivity and steady state flow. A second method used steady state flow
measurements for two different disc radii, 52 and 118 mm. A third method used a single disc
with multiple tensions from which steady state flow was obtained at three or more tensions which
could be at the same location. A fourth method used a single disc with two tensions from which
steady state flow was obtained at two tensions. Finally, values based on soil cores were
compared. The results show reasonable agreement between methods for the hydraulic
conductivity with the largest differences for data collected for zero tension. For the most part,
there were no significant differences in hydraulic conductivity due to the disc infiltrometer
radius. A single-disc method with multiple tensions (more than 3 points) and large disc radius
gave results which were the most stable, accurate, and repeatable.
12. Klute, A. The Determination of the Hydraulic Conductivity and Diffusivitv of Unsaturated Soils. Soil
Science, 113(4):264-276 (1972).
Quantitative application of the theory of unsaturated flow to field or laboratory flow systems
requires knowledge of the hydraulic conductivity and water retention characteristics of the soils
involved. The purpose of this paper is to survey the various methods available for the
measurement of hydraulic conductivity and diffusivity and to identify the principles, advantages,
and disadvantages of each. Space will not permit a complete description of each method. The
reader is advised to consult the original references for more information on any given method of
interest.
13. Kool, J.B., J.C. Parker, and M.T. van Genuchten. Parameter Estimation for Unsaturated Flow and
Transport Models—A Review. Journal of Hydrology, 91:255-293 (1987).
This paper reviews the current status of parameter estimation techniques and their utility for
determining key parameters affecting water flow and solute transport in the unsaturated (vadose)
zone. Historically, hydraulic and transport properties of the unsaturated zone have been
determined by imposing rather restrictive initial and boundary conditions so that the governing
flow and transport equations can be inverted by analytical or semi-analytical methods. Contrary
to these direct methods, parameter estimation techniques do not impose any constraints on the
model, on the stipulation of initial and boundary conditions, on the constitutive relationships, or
on the treatment of inhomogeneities via deterministic or stochastic representations. While
parameter estimation analyses of subsurface saturated flow are increasingly common, their
application to unsaturated flow and transport processes is a relatively new endeavor.
Nevertheless, a number of laboratory and field applications currently exist that show the great
potential of parameter estimation techniques for improved designs and analyses of vadose zone
flow and transport experiments. Several practical examples for determining unsaturated soil
hydraulic functions and various transport parameters are presented, and advantages and
limitations of the estimation process are discussed. Specific research areas in need of future
investigation are outlined.
14. Malik, R.S., C. Laroussi, and L.W. De Backer. Physical Components of the Diffusivitv Coefficient. Soil
Sci. Soc. Am. J., 43(4):633-637 (1979).
Isothermal horizontal infiltration experiments were conducted in 53-63 um, 74-88 um, or 105-
125 um glass bead fractions with water, 6% ethyl alcohol water solution, 20% ethyl alcohol water
solution, ethyl alcohol, and methyl alcohol. From these experiments, the diffusivity coefficient
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was determined using the Bruce and Klute (1956) procedure and D(6) was established for the
fluid content, 6, from 0% to saturation.
The D(6) variation in the fluid content range between 3 and 30% was investigated. The
functional relationships between D(6) and the properties of the fluid, the geometry and the solid
matrix were investigated. The physical components of D(6) were determined using:
D(6) = ( r*2 /R*2 • a cos a l\a - 4.10'4 AHV + 5) • 10'2 6
where r* and R* are the mean radius of pores and beads, respectively, a, \i and AHV. are the
surface tension, viscosity and molar heat of vaporization of the fluid, respectively, and a is the
fluid-solid contact angle.
The physical significance of the diffusivity coefficient given by Laroussi and De Backer (1975)
defining this coefficient as the ease with which the fluid particles spread from a wetting front, is
clarified and confirmed.
15. Marion, J.M., et al. Evaluation of Methods for Determining Soil-Water Retentivitv and Unsaturated
Hydraulic Conductivity. Soil Science, 158(1): 1-13 (1994).
The transport of dissolved contaminants through the vadose zone is a major source of soil and
groundwater contamination. Soil hydraulic properties must be determined to accurately describe
water and contaminant transport and potential environmental impacts. Comparisons were made
of three field and three laboratory methods for estimating soil-water retention, 8(t|/) and
unsaturated hydraulic conductivity functions K(8). Instrumentation was installed in 36 field
plots and two redistribution cycles were conducted. Field data obtained from each cycle were
utilized in three outflow-based field methods: (i) instantaneous profile method (ii) Libardi's
method and (iii) a nonlinear least squares approach. Undisturbed soil cores were extracted from
24 field plots at six depths and used in laboratory tests. Techniques consisted of (i) a multi-step
outflow approach (a) coupled with (a) "inverse methodology" for transient conditions and (b) a
least-squares approach for equilibrium conditions and (ii) a particle size distribution model.
Parametric models were coupled with the modeling efforts. The results obtained by the in situ
instantaneous profile method for both soil hydraulic functions were considered to hold the
greatest validity. However the multi-step outflow methods produced feasible 8(t|/) curves and the
inverse methodology was time efficient. Libardi's method for determining K(8) relationships was
accurate at deep profile depths but failed at shallow ones.
16. McCuen, R.H., WJ. Rawls, and D.L. Brakensiek. Statistical Analysis of the Brooks-Corey and the
Green-Ampt Parameters Across Soil Textures. Water Resources Research, 17(4): 1005-1013 (1981).
Infiltration is a major component of the hydrologic cycle for most watersheds. Therefore, it is
important to have a method that can provide infiltration estimates for soils within ungaged
watersheds. Both the analysis of variance and the multivariate analysis were used to examine
whether estimated Brooks-Corey and Green-Ampt parameters differ singularly or collectively
across soil texture classes. The analysis indicated that the parameters for the two models
examined varied collectively across soil texture classes. Mean parameter values and standard
errors for soil textures are presented for possible use on ungaged watersheds.
17. Mishra, S. and J.C. Parker. Methods of Estimating Hydraulic and Transport Parameters for the
Unsaturated Zone. In: Conference on New Field Techniques for Quantifying the Physical and Chemical
Properties of Heterogeneous Aquifers, Dallas, TX, National Water Well Association (1989).
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Methods of estimating hydraulic and transport parameters for the unsaturated zone are presented.
Two approaches to parameter estimation are discussed. The first approach is based on inversion
of the governing initial-boundary value problem for unsaturated flow and/or transport using data
from transient experiments. The second approach utilizes particle size distribution data to
obtain estimates of soil hydraulic properties. Example applications of the two different
approaches are presented and their advantages and limitations are discussed. Procedures for
identifying and quantifying sources of uncertainty in parameter estimates are examined.
Ongoing research on the scale-up of parameters for applications in large-scale numerical
simulations is reviewed.
18. Mishra, S. and J.C. Parker. Effects of Parameter Uncertainty on Predictions of Unsaturated Flow.
Journal of Hydrology, 108:19-33 (1989).
This study uses first-order first and second moment analysis to evaluate the error in predictions of
unsaturated flow models caused by parameter uncertainty, expressed in terms of a mean value for
each parameter and an error covariance matrix. These are related to the mean and variance in
the model predictions by a first-order Taylor expansion. Two applications of the methodology
are presented. The first involves the estimation of parameters and their error covariance matrix
from particle size distribution data for a layered field soil. These are used to predict water
contents and associated errors for a static retention test and a dynamic drainage experiment. The
second case involves parameter estimation using a numerical inversion method from a transient
flow experiment for a hypothetical homogeneous soil. Predictions of water contents and their
errors for a rainfall-redistribution event using the estimated parameters compare well with
predictions corresponding to actual parameters. Uncertainty predictions with the first-order error
analysis procedure also agree well with results of a Monte Carlo simulation study.
19. Mishra, S., J.C. Parker, andN. Singhal. Estimation of Soil Hydraulic Properties and Their Uncertainty
from Particle Size Distribution Data. Journal of Hydrology, 108:1-18(1989).
A unified approach to the estimation of soil hydraulic properties and their uncertainty from
particle size distribution data is presented. Soil hydraulic properties are represented by the
parametric models of Van Genuchten and/or Brooks and Corey Particle size distribution data are
used to generate theoretical soil-water retention data using a modified form of the model
proposed by Arya and Paris, which was calibrated in this study using a data set of 250 soil
samples. Parameters in the van Genuchten model are fitted to the predicted water
content—capillary pressure data by nonlinear regression methods and may be optionally
converted to equivalent Brooks-Worey retention parameters using an empirical procedure.
Saturated conductivity is estimated from particle size data using a modified Kozeny-Carman
equation which was developed from the data set of 250 soil samples. Uncertainty in parameter
estimates is evaluated using first-order error analysis methods. Application of the proposed
methodology to three soils which were not in the calibration set indicated water content-capillary
pressure relations can be predicted with reasonable accuracy and precision. Uncertainty in
predicted saturated hydraulic conductivity will be rather large making direct measurement of this
variable highly desirable.
20. Mishra, S. and J.C. Parker. On the Relation Between Saturated Conductivity and Capillary Retention
Characteristics. Ground Water, 28(5):775-777 (1990).
A simple closed-form expression relating saturated hydraulic conductivity to the van Genuchten
capillary retention model parameters is derived. Application of this equation to an experimental
data set shows reasonable agreement between measured and predicted saturated conductivity
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values. The proposed equation provides a consistent theoretical basis for estimating both
saturated and unsaturated hydraulic conductivity from statistical pore structure models.
21. Moldrup, P., et al. Estimation of the Soil-Water Sorptivitv from Infiltration in Vertical Soil Columns.
Soil Science, 157(1): 12-18 (1994).
This paper presents a simple method for estimating the soil-water sorptivity (S) from vertical
infiltration experiments at ponding conditions. The proposed estimation method is based on the
Haverkamp et al. (R. Haverkamp, J.-Y. Parlange, J. L. Starr, G. Schmitz, and C. Fuentes. 1990.
Infiltration under ponded conditions. 3. A predictive equation based on physical parameters.
Soil Sci. 149:292-300) predictive infiltration equation. This equation is simplified, so that a
simple analytical solution is obtained for the sorptivity. The new S estimation method requires
measurement of cumulative infiltration versus time and independent measurement of the
saturated hydraulic conductivity on the same soil cores as used for the infiltration experiment.
Criteria for avoiding numerical instability when using the new estimation method are given. Use
of the proposed method is illustrated on infiltration data measured on undisturbed sand cores.
The method gave good and robust estimates of the soil-water sorptivity (coefficient of variation
<25% for all S estimates). Horizontal and vertical infiltration tests for homogeneous packed soil
columns showed good agreement between the new S estimation method and the classical Philip
method for horizontal infiltration at nonponding conditions.
22. Mous, S.L.J. Identification of the Movement of Water in Unsaturated Soils: The Problem of
Identifiabilitv of the Model. Journal of Hydrology, 143:153-167(1993).
The estimation of model parameters using non-linear regression techniques is one of the aspects
of inverse modelling and is known as the identification problem. The solution of which may be
non-unique. The main causes of this non-uniqueness are the structure of the model and the
design of the input signal. It will be shown that the parameters can be estimated only if a model
with different parameter values yields different output signals. A model that has this feature is
called identifiable. As an example, the identification of a model for the movement of water in
unsaturated soils is used. This model appears to be non-identifiable, which results in non-
unique solutions.
23. Paige, G.B. and D. Hillel. Comparison of Three Methods for Assessing Soil Hydraulic Properties. Soil
Science, 155(3):175-189 (1993).
Three methods for assessing soil hydraulic properties were conducted and their results compared
for two soils in Western Massachusetts. The methods compared are: the Instantaneous Profile
Method, the Guelph Permeameter, and laboratory determination using intact soil cores. The
saturated hydraulic conductivity and unsaturated conductivity function, as well as the moisture
retention relationship when possible, were determined and the results compared with respect to
their ranges of applicability and the respective limitations of each method. We found close
agreement for the moisture retention relationships determined by the instantaneous profile
method and the soil cores for the ranges of pressures and moisture contents they have in
common. In addition, there was also close agreement between the K(t|/) relationship measured
using the instantaneous profile method and that predicted using the van Genuchten and Mualem
models. The field saturated conductivity results determined using the Guelph Permeameter were
one to three orders of magnitude less than the saturated conductivity results determined from soil
cores and those determined by the instantaneous profile method. The unsaturated K(t|/)
relationship using Gardner's definition of matric potential and the results from the Guelph
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5*o/7 Hydraulic Parameter Estimation
permeameter predicted hydraulic conductivity values three to four orders of magnitude less then
the other two methods at 200 cm of pressure.
24. Ragab, R., J. Feyen, and D. Hillel. Comparative Study of Numerical and Laboratory Methods for
Determining the Hydraulic Conductivity Function of a Sand. Soil Science, 131(6):375-388 (1981).
We compared experimentally measured hydraulic conductivity-water content relationships with
those predicted by three models. Sand was the medium used for this comparative study. The
laboratory methods used included: instantaneous profile-internal drainage; infiltration through
crust; hot air drying; pressure plate outflow; and unit gradient drainage. For the numerical
approach we used the Irmay, the Jackson and the Mualem models. The moisture desorption
curve and the saturated hydraulic conductivity were used as inputs. Additional parameters were
estimated from the literature. We found close agreement between the K(6 ) relationship
measured according to the instantaneous profile method and estimates based on Irmay's equation;
Jackson's prediction method with a pore interaction exponent of 0.5; and Mualem' s model, with
an effective saturation term exponent of 0.5. In addition, the predicted K(6 ) relationship with
the Mualem model using 0.75 as exponent of the effective saturation term, and the predicted
values with the Millington Jackson's equation (being Jackson's model with p = 4/3)
corresponded with the experimental relationship obtained by the gypsum-sand crust technique.
The predicted K(6 ) relationships obtained by the above models also matched the K(8 ) values
obtained by the hot air method in the wet range.
25. Ragab, R. and J.D. Cooper. Variability of Unsaturated Zone Water Transport Parameters: Implications
for Hydrological Modelling. 2. Predicted vs. In Situ Methods and Evaluation of Methods. Journal of
Hydrology, 148:133-147 (1993).
Two methods to estimate the saturated and unsaturated hydraulic conductivity of soil were
compared in a catchment in south-west England. The catchment consists of three land use
domains-arable, permanent grass and woodland. The methods used were in situ measurements
using a Guelph permeameter and the predictive model of Rawls and Brakensiek, which uses
percentage of sand, percentage of clay and porosity as input. In grassland and woodland no
significant differences were found between the mean saturated and unsaturated hydraulic
conductivity estimated by the two methods. It is suggested that a significant difference found for
the arable land is due to the effect of farming operations on soil permeability. On a catchment
scale there were no significant differences between the methods the results obtained using the
predictive model are encouraging and worth further investigation. The advantages and
disadvantages of the two methods are evaluated.
26. Ragab, R. and J.D. Cooper. Variability of Unsaturated Zone Water Transport Parameters: Implications
for Hydro logical Modelling. 1. In Situ Measurements. Journal of Hydrology, 148:109-131 (1993).
Physically based hydrological models require values of saturated and unsaturated hydraulic
conductivity to solve the equations used to describe subsurface flow. Variability of these
parameters within and between various land uses and soil types is an important consideration. If
hydrological behaviour is to be described realistically.
A catchment in south-west England has been chosen to validate the European Hydrological
System Systeme Hydrologique Europeen SHE). The catchment includes three land use domains--
arable including temporary grass), permanent grassland and woodland. For the unsaturated
zone component of the SHE both saturated hydraulic conductivity, Ks, and unsaturated hydraulic
conductivity as a function of matric suction, K(i|r), are required. The aim of this investigation
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was to measure the parameters for input to the model to assess the variability of these parameters
and to determine whether or not parameters should be used for each domain independently. A
Guelph permeameter was used to obtain Ks and a, which describes the slope of the logarithmic
unsaturated hydraulic conductivity function (K(t|/) = Ks exp (at|/)). Measurements at four depths
were carried out. Domains were chosen according to land use arable land (including temporary
grass), permanent grassland and woodland. Both Ks and a were found to be lognormally
distributed. The Ks values were highest in the grassland and lowest in arable land. The
difference between means was significant for Ks but not for a in the three domains. High values
of Ks were attributed to preferential flow along slate faces and root-induced channels in the
grassland and woodland, respectively, as indicated by dye tests. Finally it was concluded that, as
Ks and K(t|/) differ significantly between domains, different hydrological behaviour would be
expected. It is therefore recommended that different parameter sets be used for modelling the
individual domains.
27. Rawls, W.J., D.L. Brakensiek, and K.L. Saxton. Soil Water Characteristics. In: 1981 Winter Meeting of
American Society of Agricultural Engineers, Paper No. 81-2510, Chicago, IL, ASAE (1981).
Relationships of soil vater tension and conductivity with soil water content are needed to quantify
plant available water and to model the movement of water and solutes in and through soils. Field
and laboratory measurement of these hydraulic soil properties ia very difficult, laborious, and
costly. To provide the best estimates possible from previous analyses, a comprehensive search of
the literature and data sources for hydraulic conductivity and related soil-water data was made in
1978. From this search, data for 1,323 soils with about 5,350 horizons from 32 states were
summarized. Reported here are summaries of the soil profile descriptions, soil textures, particle
size distributions, organic carbon contents, bulk densities, selected chemical data, hydraulic
conductivities, soil water retention data, sample location, and the specific reference or data
source. The Brooks and Corey equation parameters, soil water retention volumes at 0.33 bar and
15 bar, and saturated conductivities for the major soil textures classes are reported. Relationships
for predicting water retention volumes for particular tensions and saturated hydraulic
conductivities based on soil properties are also presented.
28. Rawls, WJ. and D.L. Brakensiek. Estimating Soil Water Retention from Soil Properties. Journal of
Irrigation and Drainage, Proceedings of the American Society of Civil Engineers, 108(IR2):166-171
(1982).
29. Rawls, WJ. and D.L. Brakensiek. Prediction of Soil Water Properties for Hydrologic Modeling. In:
Symposium on Watershed Management, ASCE (1985).
Water flow in soils can be characterized for many boundary and initial conditions by solving
governing differential equations. Thus, a physically consistent means of quantifying water flow
in soils in terms of the soil properties governing the movement of water and air exists. There are
several reasons why this state-of-the-art technology is not yet fully utilized. One reason may be
complexity and expense of computer-based numerical solutions. However, a more important
reason is the difficulty of obtaining the required inputs which are the relationships between
matric potential and hydraulic conductivity as a function of soil water content. Approximate
water flow models based on physical principles or empirical results may simplify the
computational requirements, however, inputs are still required. Therefore, it is the objective of
this paper to present procedures to (1) estimate input parameters for the Brooks-Corey, Campbell
and Van Genuchten soil water retention and hydraulic conductivity functions based on readily
available soil properties, and (2) estimate infiltration parameters.
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30. Reynolds, W.D. and D.E. Elrick. A Method for Simultaneous In Situ Measurement in the Vadose Zone
of Field-Saturated Hydraulic Conductivity. Sorptivitv and the Conductivity-Pressure Head Relationship.
Ground Water Monitoring Review GWMRDU, 6(l):84-95 (1986).
The Guelph Permeameter (GP) method for simultaneous, in situ measurement in the vadose zone
of field-saturated hydraulic conductivity (K sub fs), sorptivity (S) and the conductivity-pressure
head relationship (K(theta)) is described and discussed. The method involves measuring the
steady-state liquid recharge, Q, necessary to maintain a constant depth of water, H, in an
uncased, cylindrical well of radius, a, finished above the water table. An 'in-hole' Mariotte bottle
device is used to maintain H and to measure Q. Step-by-step procedures with example
calculations are given for obtaining K sub fs, S and K(theta). Techniques for assessing the
results are also given. The GP method will (theoretically) yield simultaneous, in situ
measurements of K sub fs and K(theta) for infiltration of any wet liquid, and yields a 'point'
measurement requiring replication. The GP apparatus is inexpensive, simple and easily operated
by one person. The method can be used to obtain vertical profiles of K sub fs, S and alpha,
requiring less time per measurement than most other methods.
31. Ross, P.J. and K. Smettem RJ. Describing Soil Hydraulic Properties with Sums of Simple Functions.
Soil Sci. Soc. Am. J., 57:26-29 (1993).
Simple functions do not adequately describe the hydraulic properties of many field soils,
particularly those with substantial macroporosity. By considering the soil pore-size distribution
f(ft) = ^,=1 & fi( ft) corresponding to the effective saturation S (ip) =2fr,=i fiSJ ft) , where ft
is matric pressure head, the (f>t are fractions of effective porosity, the St( ip) are simple water
retention functions in common use, and // ft) =St( ft) , we show that the relative hydraulic
conductivity according to the Mualem model is Kr(ip) = SP[2?VJ=1 $g/$) /2Fl=l (f>t g,(O)]2, where
gi(ft) = f-x ft' fi( ft) dft and/) is a pore interaction index. If the pores of the distributions do
not interact, the appropriate relation is K(ip) =2Fl=l KsiKri (ip) where Ksi is the saturated
conductivity of distribution /' and Kri = Sf fgi(ft)/gi(0)J2. We note that the van Genuchten function
S(ft) = [1 + (-aft)"]'m with the restriction m = 1 - lln leads to an infinite slope K'(ft) at ft = O
unless n > 2, which is unrealistic for field soils if a wide range of matric pressure heads is
considered. Hydraulic conductivity near saturation is often expressed asK'(ft) = Ks exp(aft). We
introduce the function S(ft) = (1 - aft) exp(aft), which gives, according to Mualem's model, a
conductivity K(ip) = Ks(l - aip)p exp[(/? + 2)aft\ that approximates Ks exp(aft) near saturation if
a = la and is exactly equal if p = O. As an example, a function using this model for one pore-
size distribution and the van Genuchten model for the other was compared with a function using
two van Genuchten distributions. The latter gave a slightly improved fit to water content and
conductivity data for an aggregated soil.
32. Ross, P.J. and J.-Y. Parlange. Comparing Exact and Numerical Solutions of Richards' Equation for One-
Dimensional Infiltration and Drainage. Soil Science, 157(6):341-344 (1994).
Analytical solutions to Richards' equation are valuable in testing numerical models of water
movement in soils, but few such solutions are available, especially for general soil hydraulic
properties. We present such a solution, valid for restricted boundary conditions during
infiltration and drainage, and illustrate its use by comparing numerical and analytic solutions. In
the case of infiltration the solution is based on an earlier result which is extended here to describe
drainage from soil with a uniform initial water content. Accurate numerical solutions could be
obtained with a coarse spatial grid of eleven points.
33. Ross, P.J. and J.-Y. Parlange. Investigation of a Method for Deriving Unsaturated Soil Hydraulic
Properties from Water Content Profiles. Soil Science, 157(6):335-340 (1994).
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Hydraulic properties of unsaturated soil profiles are needed for quantitative analysis of water
movement. A recently proposed method of estimating some of these properties from water
content profiles after drainage was examined for a uniform soil satisfying certain constraints that
allowed a new exact analytical solution to be obtained. A new analytical and numerical
examination of the method, assuming correct property functional forms and boundary conditions,
established that it can give good estimates of unsaturated hydraulic conductivities. If a matric
potential value is available at one depth in the profile, matric potentials at other depths can also
be estimated accurately. The method seemed to work reasonably well even when assumed
property functions and boundary conditions had an incorrect form, although hydraulic
conductivities were then in error up to 25 percent. This seemingly large error reflects the
difficulty in measuring a highly variable dynamic property, like conductivity, in soils.
34. Satterwhite, M.B. Evaluating Soil Moisture and Textural Relationships Using Regression Analysis. U.S.
Army Corps of Engineers, ETL-0226 (1980).
Soil moisture and textural conditions are described for 179 soil samples from an arid to semiarid
climate. Stepwise multiple regression analysis of these data produced four regression equations
that related (1) the percent sand and clay and (2) the percent fines, with the percent soil water
held at 0.33 bar (FC) and the 15 bar (WP) potentials. Evaluation of these equations showed no
differences between the estimates at the 0.33 bar potential using either the percent sand and clay
or the percent fines. Better estimates for the WP were obtained when the percent sand and clay
were used instead of the percent fines. The differences between the estimated soil moisture at FC
or WP varied less than 30 percent from the measured soil moisture values for 161 (90 percent) of
the 179 soil samples. The differences between the estimated and the measured soil moisture
values were not significant at the 95 percent level of confidence.
The regression equations provide a method by which the potential percent soil water held at the
FC or WP can be estimated from soil textural data. The accuracy and precision of the results of
applying these equations to soils of other areas has not been determined. It would seem,
however, that they would be applicable in those instances where only general working estimates
are needed.
35. Shamsai, A. andN. Sitar. Method for Determination of Hydraulic Conductivity in Unsaturated Porous
Media. Journal of Irrigation and Drainage Engineering, 117(l):64-78 (1991).
A new procedure for in situ determination of saturated hydraulic conductivity of an initially
unsaturated soil is proposed. The procedure is based on the measurement of the rise of a ground-
water mound in an initially unsaturated zone above an impervious layer receiving uniform
recharge from an infiltrometer. In this paper analytical solutions of equations describing the
growth of a groundwater mound above an impervious layer are compared with laboratory and
field experiments to demonstrate the applicability of the method to the determination of saturated
hydraulic conductivity of soils in shallow subsurface.
36. Springer, E.P. and T.W. Cundy. Field-Scale Evaluation of Infiltration Parameters from Soil Texture for
Hydrologic Analysis. Water Resources Research, 23(2):325-334 (1987).
Recent interest in predicting soil hydraulic properties from simple physical properties such as
texture has major implications in the parameterization of physically based models of surface
runoff. This study was undertaken to (1) compare, on a field scale, soil hydraulic parameters
predicted from texture to those derived from field measurements and (2) compare simulated
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overland flow response using these two parameter sets. The parameters for the Green-Ampt
infiltration equation were obtained from field measurements and using texture-based predictors
for two agricultural fields, which were mapped as single soil units. Results of the analyses were
that (1) the mean and variance of the field-based parameters were not preserved by the texture-
based estimates, (2) spatial and cross correlations between parameters were induced by the
texture-based estimation procedures, (3) the overland flow simulations using texture-based
parameters were significantly different than those from field-based parameters, and (4)
simulations using field-measured hydraulic conductivities and texture-based storage parameters
were very close to simulations using only field-based parameters.
37. Stolte, J., et al. Comparison of Six Methods to Determine Unsaturated Soil Hydraulic Conductivity. Soil
Sci. Soc. Am. J., 58:1596-1603 (1995).
Knowledge of soil hydraulic properties is required for soil-water flow models. Although many
studies of individual methods exist, comparisons of methods are uncommon. Therefore, we
compared application ranges and results for six laboratory methods for determining hydraulic
conductivity or diffusivity on eolian sand, eolian silt loam, marine sandy loam, and fluviatile silt
loam. The methods, hot air, sorptivity, crust, drip infiltrometer, Wind's evaporation, and one-
step outflow, fall into three groups: (i) those that only yield a conductivity curve; (ii) those that
yield a simultaneous estimate of conductivity, diffusivity, water content, and pressure head; and
(iii) those that yield a diffusivity curve. Diffusivities were converted to conductivities with a water
retention curve. One main difference between the methods was the pressure head-water content
range. Despite the large differences between the methods, the results for the first two groups
tended to be similar. The results of the third group did not match well with those of the first two.
It proved difficult to compare these methods correctly due to hysteresis.
38. Tseng, P.-H. and W.A. Jury. Simulation of Field Measurement of Hydraulic Conductivity in Unsaturated
Heterogeneous Soil. Water Resources Research, 29(7):2087-2099 (1993).
In this study the procedures used in common methods for field measurement of unsaturated
hydraulic conductivity in heterogeneous soil were simulated through numerical experiments. It
was assumed that the heterogeneity of the field soil can be described by a scaling factor 8 which
is treated as a lognormally distributed second-order stationary random variable in space. After
the hypothetical random field was generated using property variability and correlation length
scales typical of those observed in field measurements it was used to test various field
measurement methods currently in practice. One detailed and three simplified unsteady drainage
flux methods were used to estimate the hydraulic conductivity under plausible field measurement
conditions. In addition the soil water retention data were used to predict hydraulic conductivity
from a pore-size distribution model. It was found that even under the assumption that there were
no reading or instrument errors other sources of errors can still be created due to the limitations
of the observation techniques. These error sources include limited number of readings in space
and time separation distances between the readings of water content and pressure head and
volume-averaged readings rather than point estimates with common instrumentation. Under the
conditions studied in the simulations the detailed method of estimation produced reasonable
estimates of hydraulic conductivity, but only when very dense measurements of water content and
matric potential were used in the estimation. None of the approximate methods yielded accurate
measurements of the true hydraulic conductivity of the field region measured.
39. Valiantzas, J.D. Analysis of Outflow Experiments Subject to Significant Plate Impedance. Water
Resources Research, 26(12):2921-2929 (1990).
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A new method is developed for the determination of the diffusivity-water content relationship
from pressure-outflow experiments when the impedance of the supporting porous plate has to be
taken into account. The classic one-step outflow procedure for the determination of soil water
diffusivity does not use the early linear part of the outflow versus the square root of time, t, curve,
where the effect of plate impedance is not negligible. The present method can be applied also to
the linear part of the outflow curve which is considerably extended when the effect of the plate
impedance is significant. The suggested procedure requires additional information, namely, the
retention curve and the plate impedance value. The method was tested by simulating outflow
experiments with significant plate impedance and using data obtained to calculate diffusivities, as
well as by analyzing experimental outflow data for a real soil.
40. van Genuchten, M.T., F.J. Leij, and S.R. Yates. The RETC Code for Quantifying the Hydraulic Functions
of Unsaturated Soils. U.S. EPA, Robert S. Kerr Environmental Research Laboratory, EPA/600/S2-91/065
(1991).
This summary describes the RETC computer code for analyzing the soil water retention and
hydraulic conductivity functions of unsaturated soils. These hydraulic properties are key
parameters in any quantitative description of water flow into and through the unsaturated zone of
soils. The program uses the parametric models of Brooks-Corey and van Genuchten to represent
the soil water retention curve, and the theoretical pore-size distribution models of Mualem and
Burdine to predict the unsaturated hydraulic conductivity function from observed soil water
retention data. Some of the analytical expressions used for quantifying the soil water retention
and hydraulic conductivity functions are presented. A brief review is also given of the nonlinear
least-squares parameter optimization method used for estimating the unknown coefficients in the
hydraulic models. The program may be used to predict the hydraulic conductivity from observed
soil water retention data assuming that an estimate of the saturated hydraulic conductivity is
available. The program also permits one to fit analytical functions simultaneously to observed
water retention and hydraulic conductivity data, or to predict the hydraulic curves from specified
model parameters if no retention and conductivity data are available. The actual report,
summarized here, serves as both a user manual and reference document.
This Project Summary was developed by EPA's Robert S. Kerr Environmental Research
Laboratory, Ada, OK, to announce key findings of the research project that is fully documented
in a separate report of the same title.
41. Vereecken, H., J. Maes, and J. Feyen. Estimating Unsaturated Hydraulic Conductivity from Easily
Measured Soil Properties. Soil Science, 149(1): 1-12 (1990).
We measured hydraulic conductivity (saturated and unsaturated) on 127 soil cores, which were
taken in different horizons of a wide variety of Belgian soil series. The hot air method (Arya et
al. 1975) and the crust method (Bouma et al. 1983) were combined to obtain the complete range
of hydraulic conductivity from saturation to air-dry.
The textural composition in nine fractions, the organic carbon content, and the dry bulk density
were determined for each of the sample horizons as well. Four different empirical models were
evaluated on their performance in describing the measured hydraulic conductivity curves. The
model parameters were estimated by linear and nonlinear regression techniques. It is concluded
that the Gardner equation (1958) with three parameters best described the hydraulic conductivity
for the given soils.
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Regression equations for estimating the Gardner parameters were established from simple soil
properties, such as soil texture, carbon content, bulk density, and saturated hydraulic
conductivity. We found that the three parameters can reasonably well be estimated from the
textural composition and the saturated hydraulic conductivity. A one-dimensional sensitivity
analysis indicates that the n parameter, representing the slope of the hydraulic conductivity
pressure head relation in log-log scale, is most sensitive.
42. Warrick, A.W. Soil Water Diffusivitv Estimates from One-Dimensional Absorption Experiments. Soil
Sci. Soc. Am. J., 58:72-77 (1992).
A method to estimate soil water diffusivity from experimental absorption data was explored. The
purpose was to directly match measured sorption curves (water content vs. distance or the
Boltzmann transform variable) to scaled forms of the solutions and thus provide consistent
hydraulic properties. Evaluation of such parameters is critical to prediction of water and solute
transport within the vadose zone. The method is relevant to several, if not most, of the
commonly used hydraulic functional relationships. Generally, the results gave two independent
relationships of the parameters. An example was considered and calculations performed for
three alternative models. The best choice of each model was easily found and the evaluated
parameters were shown to be consistent. The soil water diffusivities were compared for the
estimated parametric relationships. Individual models provide alternative shapes for the
diffusivity function, but tend to converge for the wetter range of water contents.
43. Weaver, J.W., et al. Evaluating Parameter Estimation Techniques Applied in Vadose Zone Modeling. In:
First International Ground Water Modeling Conference, Golden, CO (1993).
The paper discusses vadose zone parameter estimation techniques that may be used in lieu of
measured parameter values. In particular, two different data bases that provide generic
parameter estimates are reviewed. The effects of parameter variability on vadose zone NAPL
flow are illustrated through usage of the Kinematic Oily Pollutant Transport (KOPT) model. The
model simulated a laboratory experiment where gasoline was released into a one-meter long
chromatography column. The model input parameters were measured, and the position of the
leading edge of the gasoline was recorded as a function of time from the experiment. Simulations
that use measured or tabulated parameter sets were compared with the experimental data, which
represents the correct response of the system to the gasoline input. To assess the impact of using
tabulated parameters Monte Carlo simulations were performed using correlated parameter sets
generated from the tabulations. These results show that wide distributions of the parameters are
required to capture likely variability in input parameter values.
44. Zhang, R. and M.T. van Genuchten. New Models for Unsaturated Soil Hydraulic Properties. Soil
Science, 158(2):77-85 (1994).
Two relatively simple models are proposed for describing the soil water retention curve. The
expressions define sigmoidal or bimodal type retention functions with four or five parameters,
respectively. The sigmoidal retention model may be combined with predictive pore-size
distribution theories to yield closed-form equations for the unsaturated hydraulic conductivity.
Parameters in the proposed hydraulic functions were estimated from observed retention data
using a nonlinear least-squares optimization process. The models were tested on hydraulic data
for more than 20 soils. Good agreement between predicted values and measured retention and
conductivity data was found for most of the soils. The soil hydraulic models can be effectively
utilized as inputs for numerical models of water flow and solute transport.
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