0038-075X/06/171S1-S21–S28
Soil Science
Copyright * 2006 by Lippincott Williams & Wilkins, Inc.

June 2006
Vol. 171, Suppl. 1
Printed in U.S.A.

MILESTONES IN SOIL PHYSICSPeter A. C. Raats1 and Martinus Th. van Genuchten 2This special issue of Soil Science celebrates the enormous accomplishments made during the past century or more in the field of soil science,
including some of the key articles published in Soil Science during its 90
years of existence. In this article, we focus on the contributions in soil
physics, exemplified by the articles of Willard Gardner (1919) and John
Philip (1957c), both of which are reprinted in this issue. Much of theoverview is limited to the physics of water flow in unsaturated soils as
described with the Richards equation, including its mathematicalsolutions. (Soil Science 2006;171:S21–S28)

1919 Soil Science article by Willard
Gardner was an important step forward in
the studies of water flow in unsaturated soils,
leading to the discovery of the Richards
equation. About 70 years ago, Lorenzo A.
Richards formulated a general, macroscopic
theory for the flow of water in unsaturated soils(Richards, 1928, 1931). The wide applicabilityof that theory and its numerous extensions form
the core of the discipline of soil physics.
Richards’ theory combines the simplest possible
balances of mass and of forces. Assuming thedensity, D, of soil water to be constant, thebalance of mass can be expressed as a volumetricbalance equation:T

Buckingham equation (Narasimhan, 2005), isexpressed as

HE

ﬂK
ﬂKv
¼j
;
ﬂt
ﬂz

q ¼ jK

ﬂh
þ K;
ﬂz

ð2Þ

where K is the hydraulic conductivity and h is
the capillary pressure head defined as
h¼

pI jpg
pc
¼j ;
Dg
Dg

ð3Þ

where pl and pg are the pressures of the liquidand gaseous phases, respectively; pc, the capillarypressure; and g, the gravitational constant. The
pressure head and the hydraulic conductivity are
nonlinear functions of the volumetric watercontent, K. Moreover, the relationship h(K) is
hysteretic.
With the theory named after him, Richards
consolidated the efforts of his predecessors,notably Charles S. Slichter, Lyman J. Briggs,
Edgar Buckingham, Willard Gardner, and W. B.
Haines (for details, see Philip, 1974; Gardner,
1976, 1986; and Raats, 2001). Slichter (1899;
see also Wang, 1987) used the mass balance andDarcy’s law to describe the flow of water insaturated soils. He also calculated the hydraulicconductivity for a packing of spheres based on amodel of tubes with variable triangular crosssection, finding a quadratic dependency of thehydraulic conductivity on the particle size in theprocess. Briggs (1897; see also Landa andNimmo, 2003) used the Young-Laplace equation to describe the jump in pressure acrossliquid-gas interfaces in unsaturated soils, thus
discovering in essence the inverse dependence

ð1Þ

where t denotes time; z, the vertical coordinate
taken positive downward; K, the volumetricwater content (Gardner used D for this variable),and v, the velocity of water such that Kv equals
the volumetric flux. Under quite reasonableassumptions for the balance of forces, Darcy’slaw, also appropriately referred to as the Darcy-

1Wageningen University and Research Center, Wageningen, the Netherlands.

Dr. Martinus Th van Genuchten, Phd is corresponding author. E-mail: rvang@
ussl.ars.usda.gov
2George E. Brown, Jr. Salinity Laboratory, US Department of Agriculture,
Agricultural Research Service, Riverside, CA.

Received May 9, 2006; accepted May 9, 2006.
DOI: 10.1097/01.ss.0000228048.85215.bf

S21

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S22

RAATS AND VAN GENUCHTENof the capillary pressure head on the particle
size. Buckingham (1907; see also Nimmo andLanda, 2005; Narasimhan, 2005) furthermore
described hydrostatic vertical equilibrium conditions in soils, formulated in words the principle of mass balance, and extended Darcy’s law tounsaturated soils.
In the article reprinted here, Gardner (1919)
was the first to actually write the one-dimensional
form of the macroscopic mass balance equation.
For horizontal flow, he formulated, by analogy
with Stokes law for the motion of a particle fallingthrough a viscous liquid, a linear relationshipbetween the macroscopic velocity and the gradientof the curvature pressure (i.e., he postulated Darcy’slaw). Gardner used a microscopic model for awater wedge between touching spheres to relate
the curvature pressure to the water content, andused the resulting pore-scale model to derive themacroscale water retention characteristic. His
equations also implied a dependency of thehydraulic conductivity on the water content.
Willard Gardner provided much inspirationand leadership in the early and mid 1900s on thehydrodynamics of unsaturated soils, causing
Sterling Taylor (1965) and others to refer toGardner as the Bfather of modern soil physics.^
As a further example, Gardner and Widtsoe(1921) motivated Darcy’s law for water flow inunsaturated soils as follows:
BWe may therefore say that for the chosenelement [of liquid] there exists a force acting
vertically downward proportional to the mass,a pressure on each of the six sides, and africtional drag due to the relative slipping of
the element, which may be zero for any or allof the six sides, depending upon the relative
velocity at each side.^With such a motivation, the theory also fits
well within the framework of modern continuum theories of mixtures, provided that one
recognizes from the outset the existence ofseparate solid, liquid, and gaseous phases, andtreats these phases as superimposed continua (forreviews, see Raats, 1984, 2001).
Incidentally, Willard Gardner also played an
important role in early drainage theory. Afterobtaining bachelor of science, master of science,and doctor of philosophy degrees in physics atColumbia University, Don Kirkham came to
the Department of Physics at the Utah Agricultural Experiment Station to teach physics andmathematics. An analysis by Willard Gardnerand his colleagues in the late 1920s and early
1930s of tile drainage of land overlying an

SOIL SCIENCE

artesian basin prompted Don Kirkham to perform sand tank experiments and to make a more
complete mathematical analysis, resulting inthree articles in the Transactions of the American
Geophysical Union in 1939, 1940, and 1945, thus
marking the start of a highly successful 60-year
career in soil physics.
Returning to the unsaturated zone, Haines(1930; see also Keen, 1931) applied the YoungLaplace equation to ideal soils (i.e., packings ofmonodisperse spheres) to calculate water retention, including the hysteresis effect, and cohesion. The foundation of the Richards equation
based on the principles of surface tension andviscous flow was treated later very comprehensively by Miller and Miller (1956). The motivation presented by Miller and Miller (1956) andthe more recent formal derivation by Withaker(1986) based on the method of volume averaging nicely complement each other.
PHYSICAL CHARACTERIZATIONOF SOILS
Within the context of the Richards equation,
the relationships among the volumetric watercontent K, pressure head h, and hydraulicconductivity K define the hydraulic propertiesof a soil. Much effort over the years has gone
into the measurement, mathematical description, and prediction of these relationships (forexamples, see Dane and Topp, 2002). After the
publication of his theory in 1931, Richards’
main interest was the development of soundmethods for determining water retention andhydraulic conductivity characteristics and for insitu monitoring of the soil water and salinity
status. In the period from 1941 to 1964, hepublished a dozen methodological articles in Soil
Science. For anyone working with tensiometers,
Richards (1949) is still very much worth reading. In the same year, Gardner and Kirkham(1949) introduced neutron scattering as a means
to determine the soil water content. In the next
three decades, the neutron probe was the main
device for measuring the time course of watercontent profiles. Starting in the 1980s, neutron
scattering has been supplemented by a range ofelectromagnetic methods (Evett and Parkin,2005), including time domain reflectometry (forexamples, see Robinson et al., 2003), ground
penetrating radar, and capacitance methods.
Miller and Miller (1956) related the water
retention and hydraulic conductivity characteristics of geometrically similar soils, each

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VOL. 171 ~ SUPPL. 1

MILESTONES IN SOIL PHYSICS

characterized by a length scale. They establishedthat the Young-Laplace equation implies thatthe capillary pressure is inversely proportional to
the length scale for geometrically similar soils at
the same volumetric water content, whereas thelinearized Navier-Stokes equation implies thatthe hydraulic conductivity is proportional to thesquare of the length scale. In the late 1950s andthe 1960s, the Miller-Miller scaling theory wasverified in a series of clever experiments by Ed
Miller, Arnold Klute, and several coworkers.
Later, an extension of this scaling theory served asa basis for the analysis of hydraulic characteristics
of spatially variable field soils (for examples, see
Reichardt et al., 1975).
Much use has been made of mathematical
functions that match measured soil physical characteristics in a reasonable way. Some early literature was reviewed by Gardner (1974), whereas
more recent developments were presented at two
international symposia (van Genuchten et al.,
1992, 1999). Various classes of soils have been
defined by specific mathematical functions representing their physical properties. Two groupsof classes of soils can be distinguished. One ofthese groups yields flow equations that can besolved analytically, in most cases because of
linearization after one or more transformations.
Examples of classes of soils belonging to thisgroup and associated analytical solutions of flow
problems are presented below.
Another group of classes of soils, favored innumerical studies, is based on statistical pore-size
distribution models that have a relatively solid
basis in Poiseuillean flow through networks ofcapillaries (for examples, see Childs andCollis-George, 1950; Mualem, 1976). In thisgroup of relationships among K, h, and K, thehydraulic conductivity is calculated from thewater retention function using certain assumptions concerning the geometry of the poresystem. The procedure in essence links physicomathematical models at the Darcy andNavier-Stokes scales. Examples of relationships of this type are given by Brooks andCorey (1964) and van Genuchten (1980).
SOLUTIONS OF FLOW PROBLEMS
Analytical Solutions for Specific Classes of Soils
For the class of linear soils, the Richards
equation reduces to the linear convectiondiffusion equation. Neglecting the gravitationalterm, this equation further reduces to the linear
diffusion equation that was used by Childs(1936) in a study of water flow through heavy

S23

clay soils, by Kirkham and Feng (1949) in ananalysis of experiments on horizontal absorption, and by Gardner (1956) in the outflow
method for measuring the diffusivity.
In pioneering articles, Philip (1957b) andGardner (1960) also used the linear diffusion
equation to calculate the water depletion patternaround individual plant roots. Philip used his
calculations to demonstrate that the previously
held concept of an invariable wilting point of asoil is not tenable. Gardner used the linear model
as a point of departure to formulate a simple
model in which the depletion resulting fromuptake by a single root is treated as a series ofsteady flows in a cylindrical shell of soil surrounding the root, with the soil-root interface at the
inner edge and the water coming from the outeredge. This simple model has served ever since as apoint of departure for more sophisticated mesoscopic and macroscopic models for water uptake(for a review, see Raats, 2006).
In later work, the full linear convectiondiffusion equation has served as a contrast andlimit of different forms of the Richards equationfor various classes of nonlinear soils (see Philip,
1969 for an early review). The class of Green and
Ampt or delta function soils, for which the watercontent is discontinuous at wetting fronts, represents the opposite limiting behavior. In fact,
Philip (1954) showed that the early solutions ofGreen and Ampt (1911) are solutions of a limiting
form of the later Richards equation.
Other classes of soils for which analyticalsolutions (for numerous references, see Raats,
2001, 2002) can be obtained are (i) the class ofGardner (1958) soils for which steady flows aredescribed with a linear equation in terms of thematric flux potential, irrespective of the spatial
dimensionality of the flow problem; (ii) the classof Brooks and Corey (1964) power function soils
for which similarity solutions of the Richards
equation can be found; and (iii) the class ofversatile nonlinear soils for which the Richards
equation can be transformed into the Burgers
equation, which, in turn, can be transformed to alinear diffusion equation. The basis for the lastclass, in essence, goes back to the doctor ofphilosophy thesis of John Knight (1973).
Hybrid Analytical-Numerical Solutions
It is remarkable that a large number of onedimensional solutions of the Richards equation
are of the formz ¼ zðK; tÞ; z ¼ zðh; tÞ;ð4Þ

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S24

RAATS AND VAN GENUCHTEN

where z is the depth. The simplest of such
solutions are those for steady upward anddownward flow as shown in the Soil Science
articles by Youngs (1957) and Gardner (1958),
and in a review by Raats and Gardner (1974).
In the article reprinted here, Philip (1957c)
derived a series solution in the form of Eq.(4) for
vertical infiltration in powers of t2. The firsttransient solution in the form of Eq.(4) was
already published by Ludwig E. Boltzmann in1894 and first applied to horizontal absorption
by Klute (1952). Philip’s article was the first of aseries of 7 articles on the theory of infiltration in
Volumes 83 to 85 of Soil Science. The second
article of the series dealt with the long-term
traveling wave solution, which, again, is in theform of Eq.(4). The next four articles treatedsuch topics as moisture profiles and their relation
to experiments, the sorptivity and algebraic
infiltration equations, the influence of the initialmoisture content, and the effect of a water
depth on the soil. The series of articles introduced new mathematical methods and gave
clear physical interpretations, including theiruse for optimally designing or analyzing experiments. Around the same time, Philip wrote four
more Soil Science articles on the early stages ofadsorption and infiltration, on energy dissipation
during adsorption and infiltration, and on
absolute thermodynamic functions in soil-waterstudies. In a well-known Citation Classic, Philip(1969) reviewed these various articles andrelated work. These and much later contributions by John R. Philip are, in turn, reviewed byRaats et al. (2002) and Smiles (2005).
John Philip’s series of articles on the theoryof infiltration was soon widely noticed. From1960 to 1961, Gerry Bolt already discussed thetheory in his soil physics course for master ofscience students at Wageningen University. Don
Kirkham also paid much attention to it in his
lectures, eventually resulting in 60 textbook
pages that provided numerous details of thederivations (Kirkham and Powers, 1972). But
the most important early contact of John Philip
in the northern hemisphere was likely E. C.
Childs of the Agricultural Research Council
Unit of Soil Physics of the University of Cambridge (for contributions by Childs, see Youngs
et al., 1974).
In the 40-year period from 1936 to 1975,
the Cambridge group published an impressive
and varied series of 41 articles in Soil Science,
representing a quarter of the group’s output.
Among these are, in the period from 1943 to

SOIL SCIENCE

1951, a series of 6 articles by E. C. Childs on thewater table, equipotentials, and streamlines indrained lands. In 1956, as a visitor with theCambridge group, John Philip inspired a program of testing his solutions of the Richards
equation experimentally, and of seeking extensions of the theory of infiltration. The first Soil
Science article by Youngs (1957) sought to verify
experimentally the solution of the infiltration
problem in the article by Philip now reprinted.
In his next two Soil Science articles, Youngs(1958a,b) presents data on redistribution, an
important problem not dealt with in the seriesof articles by Philip. His data later inspired manyother studies of redistribution, particularly of therole of hysteresis in that process. In the period
from 1962 to 1975, A. Poulovassilis of theCambridge group published in Soil Science sevenarticles on hysteresis, exploring initially theconcept of independent domains (Poulovassilis,
1962); later, Poulovassilis also explored theconcept of dependent domains (Poulovassilis
and Childs, 1972). Going beyond the theory ofRichards, the two-fluid phase character ofunsaturated soils became apparent by three Soil
Science articles on water content profile development and air compression during absorption andinfiltration (Youngs and Peck, 1964; Peck,1965a,b). The 1974 issue of Soil Science dedicated to the life and work of E. C. Childs showsthat some 30 years ago, soil physics could
already cope with a wide range of practicalproblems in the agronomic, hydrologic, andenvironmental areas.
In the period from 1971 to 1975, Jean-Yves
Parlange published in Soil Science a series of 11
articles. The first two articles of the series(Parlange, 1971a,b) addressed the same physicalproblems as those addressed by Klute (1952) andby the series of seven articles by Philip, whereas
the other nine articles treated such topics as twoand three-dimensional absorption, two- andthree-dimensional steady infiltration, unsteady
infiltration from spherical cavities, multidimensional cavities under pressure, one-dimensional
infiltration with constant flux at the surface, thedynamics of capillary rise, and cavities withconstant flux. Typically, Parlange’s solutions
were iterative approaches, using an integral
moment balance as constraint. Parlange’s method
was noticed soon by others. For example, Knightand Philip (1973) criticized its convergence,
whereas Cisler (1974) proposed improvements
for the iterative approach. As an alternative,
Philip (1973) and Philip and Knight (1974)

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MILESTONES IN SOIL PHYSICS

developed a flux concentration method in which
the integral mass balance, not the integral
moment balance, was used as a constraint. With
numerous colleagues around the world, particularly in the United States, Australia, and France,
Jean-Yves Parlange has been pursuing solutionsof Richards equation ever since, resulting in afurther 35 contributions in Soil Science by 1995
and many more in other journals.
Numerical Solutions
In the early work of Klute (1952) and Philip(1955, 1957a,c) it was already clear that mostflow problems in the unsaturated zone require,at least in part, numerical analysis. In a comprehensive review by Breaster et al. (1971), mostearly numerical studies discussed concerned
one-dimensional flow, including complications
arising from hysteresis, ponding, and moving
water tables. Rubin (1968) was the first to numerically analyze a two-dimensional flow problem
using an alternating direction implicit method.
About a dozen studies of two-dimensional
problems followed in the next three years, allusing various finite difference methods. The firstcomprehensive three-dimensional model was
published by Freeze (1971).
The progress with especially multidimensional models was initially hampered not only by
the low speed of available computers but also by
the invoked numerical methods. In addition,
finite difference methods were awkward forhandling curved boundaries and coping withanisotropic media for which the principal axes do
not coincide with the coordinate axes. Theselimitations have been largely overcome with theintroduction of Galerkin finite element (forexamples, see Neuman, 1973), control volume
finite element (Therrien and Sudicky, 1996),
and integrated finite difference techniques(Narasimhan and Witherspoon, 1976), moreaccurate mass-conservative numerical solution
schemes for the Richards equation (Celia et al.,1990), and more powerful iterative matrix
equation solvers for multidimensional problems(for a review and numerous references, see van
Genuchten and Sudicky, 1999).
FUTURE OPPORTUNITIESThanks to Willard Gardner and John R.
Philip and their colleagues during the past
century, soil physics has slowly matured into awell-established scientific discipline. This overview of past contributions has, by necessity, been

S25

limited mostly to flow in unsaturated soils, as
described with the Richards equation, and has
been very arbitrary because many others not citedhere have equally contributed over the years.
Echoing Alvin Weinberg, John Philip(1991) pointed out 15 years ago that many
important societal problems are transscientific:
even if a problem can be stated in the languageof science, scientific analysis may not provide afull answer but, at best, provide some intellectual discipline. Smiles et al. (2000) suggested thatsuch intellectual discipline may well come fromsimulation models. We believe that this is most
likely if the availability of simulation results isaccompanied by a more thorough understandingof the underlying physical, chemical, and biological processes.
Faster computers and more advanced methods for solving the Richards equation are now
making it possible to efficiently solve transientvariably saturated flow problems in multidimensional heterogeneous subsurface systems that
may include parts of the saturated zone andoverland flow as needed (for examples, see
Panday and Huyakorn, 2004). This capability
to tackle increasingly complicated problems has
shifted research to such post-Richards problems
as swelling and shrinkage phenomena (forexamples, see reviews by Smiles, 2000 and
Raats, 2002), pore-scale flow processes, localscale nonequilibrium flow (e.g., dynamic memory effects as described by Hassanizadeh andGray, 1990, 1993), flow in structured soils or
unsaturated fractured rock (for examples, seereview by Simunek et al., 2003), unstable flow
(Van Duijn et al., 2004) and preferential flow in
general (for examples, see review by Hendrickx
and Flury, 2001), flow in the capillary fringe,root water uptake, multiphase flow, nonisothermal flow, and uncertainty. Many contributions
on nonisothermal flow actually expanded upona mechanistic theory by Philip and de Vries(1957) for the simultaneous movement of heatand moisture, although alternative formulations
within the framework of the thermodynamics ofirreversible processes are also possible.
Additional developments have been in thearea of solute transport, exemplified by the
initial work of Don Nielsen and Jim Biggar in
the early 1960s (for examples, see Nielsen andBiggar, 1962) and continuing up the present.
This article cannot do justice to the many
contributions in this area of research, which
now encompasses such topics as nonequilibriumtransport, transport in fractured rock, colloid

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S26

RAATS AND VAN GENUCHTEN

and colloid-facilitated transport, and multicomponent reactive transport. We simply refer to a
recent review by Simunek and van Genuchten(2006) that includes many references. The shear
number of topics now being tackled shows thatsoil physics has also increasingly become apartner of other disciplines in efforts to integrate
all relevant physical, chemical, and biologicalprocesses operative in the unsaturated zone,
with research ranging from the pore scale tothe field and larger scales. Soil Science has served
the scientific community well for 90 years. We
trust that this journal will remain an importantoutlet of both disciplinary and interdisciplinary
research touching upon soil physics and its
related disciplines.

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