SOIL WATER POTENTIAL 1
SOIL WATER POTENTIAL
(Revised 09/13/2003)
Dani Or, Department of Civil and Environmental Engineering
University of Connecticut, Storrs, Connecticut, USA
Markus Tuller, Department of Plant, Soil & Entomological Sciences
University of Idaho, Moscow, Idaho, USA
Jon M. Wraith, Department of Land Resources & Environmental Sciences
Montana State University, Bozeman, Montana, USA
Introduction
Water status in soils is characterized by both the amount of water present and its
energy state. Soil water is subjected to forces of variable origin and intensity, thereby
acquiring different quantities and forms of energy. The two primary forms of energy of
interest here are kinetic and potential. Kinetic energy is acquired by virtue of motion and
is proportional to velocity squared. However, because the movement of water in soils is
relatively slow (usually less than 0.1 m/h) its kinetic energy is negligible. Potential
energy, which is defined by the position of soil water within a soil body and by internal
conditions, is largely responsible for determining soil water status under isothermal
conditions.
Like all other matter, soil water tends to move from where the potential energy is higher
to where it is lower, in pursuit of equilibrium with its surroundings (Hillel, 1998). The
magnitude of the driving force behind such spontaneous motion is a difference in
potential energy across a distance between two points of interest. At a macroscopic
scale, we can define potential energy relative to a reference state. The standard state
for soil water is defined as pure and free water (no solutes and no external forces other
than gravity) at a reference pressure, temperature, and elevation, and is arbitrarily given
the value of zero (Bolt, 1976).
The “Total” Soil Water Potential and its Components
Soil water is subject to several force fields, the combined effects of which result in a
deviation in potential energy relative to the reference state, called the total soil water
potential (ψT) defined as: “The amount of work that an infinitesimal unit quantity of water
at equilibrium is capable of doing when it moves (isothermally and reversibly) to a pool
of water at similar standard (reference) state, i.e., similar pressure, elevation,
temperature and chemical composition”. It should be emphasized that there are
alternative definitions of soil water potential using concepts of chemical potential or
specific free energy of the chemical species water (which is different than the soil
solution termed ‘soil water’ in this chapter). Some of the arguments concerning the
definitions and their scales of application are presented by Corey and Klute (1985),
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SOIL WATER POTENTIAL
Iwata et al. (1988), and Nitao and Bear (1996). Recognizing that these fundamental
concepts are subject to ongoing debate, we have opted to present simple and widely
accepted definitions which are applicable at macroscopic scales and which yield an
appropriate framework for practical applications.
The primary forces acting on soil water held within a rigid soil matrix under isothermal
conditions can be conveniently grouped (Day et al., 1967) as: (i) matric forces resulting
from interactions of the solid phase with the liquid and gaseous phases; (ii) osmotic
forces owing to differences in chemical composition of soil solution; and (iii) body forces
induced by gravitational and other (e.g., centrifugal) inertial force fields.
The thermodynamic approach whereby potential energy rather than forces are used is
particularly useful for equilibrium and flow considerations. Equilibrium would require the
vector sum of these different forces acting on a body of water in different directions to
be zero; this is an extremely difficult criterion to deal with in soils. On the other hand,
potential energy mathematically defined as the negative integral of the force over the
path taken by an infinitesimal amount of water when it moves from a reference location
to the point under consideration is a scalar (not a vector) quantity. Subsequently, we
can express the total potential as the algebraic sum of the component potentials
corresponding to the different fields acting on soil water as:
ψ T =ψ m +ψ s +ψ p +ψ z
(1)
where the component potentials ψi are discussed below:
ψm is the matric potential resulting from the combined effects of capillarity and
adsorptive forces within the soil matrix. The primary mechanisms for these effects
include: (i) capillarity caused by liquid-gas interfaces forming and interacting within the
irregular soil pore geometry (see Capillarity chapter, this volume); (ii) adhesion of water
molecules to solid surfaces due to short-range London-van der Waals forces and
extension of these effects by cohesion through hydrogen bonds formed in the liquid;
and (iii) ion hydration and water participating in diffuse double layers (particularly near
clay surfaces). There is some disagreement regarding the practical definition of this
component of the total potential. Some consider all contributions other than gravity and
solute interactions (at a reference atmospheric pressure). Others use a device known
as a tensiometer (to be discussed later) to measure and provide a practical definition of
the matric potential in a soil volume of interest in contact with a tensiometer’s porous
cup (Hanks, 1992). The value of ψm ranges from zero when the soil is saturated to
increasingly negative values as the soil becomes drier (note that ψm=0 mm is greater
than ψm=-1000 mm; in analogy, a temperature of 00 C is greater than -100 C ).
Applied theories for flow and transport in unsaturated porous media, particularly at low
water content, commonly lump capillary and adsorptive forces without distinguishing
individual contributions to the matric potential. Based on the pioneering studies of
Edlefsen and Anderson (1943) and Philip (1977), Tuller et al. (1999) advanced a
framework that simultaneously considers the individual contributions of capillary and
adsorptive forces for calculation of liquid-vapor interfacial configurations in angular pore
spaces. They consider the liquid-vapor interface as a surface of constant partial specific
Gibbs free energy (or matric potential) made up of an adsorptive component (A) and a
SOIL WATER POTENTIAL 3
capillary component (C):
ψ m = A( h ) + C( κ )
(2)
with κ as the mean curvature of the liquid-vapor interface, and h as the distance from
the solid to the liquid-vapor interface, taken normal to the solid surface (thickness of the
adsorbed film). The capillary component C is given by the classical Young-Laplace
equation:
C (κ ) =
− 2 ⋅σ ⋅κ
ρ
(3)
where κ is positive for an interface concave outward from the liquid, σ is the surface
tension at the interface and ρ is the density of the liquid. Phenomena giving rise to
capillarity are discussed in detail in the article on “Capillarity” in this Encyclopedia.
The adsorptive component in Eq. 2 is attributed to two types of surface forces
(Derjaguin et al. 1987). The first kind includes long-range (>500 Å) electrostatic forces
(e.g., diffuse double layer, DDL), and short-range (<100 Å) van der Waals and hydration
forces, responsible for molecular interactions and structural changes in water molecules
near the solid surface. The second kind is comprised of long-range forces due to the
overlapping of two interfacial regions (e.g., mutual attraction between two clay platelets
across a slit-shaped pore space). The combined effect of interfacial interactions results
in a difference in chemical potentials between the liquid in the adsorbed film and the
bulk liquid phase. This difference in chemical potentials may be expressed as an
equivalent interfacial force per unit area of the interface, termed by Derjaguin et al.
(1987) as the disjoining pressure (Π). The disjoining pressure is a function of liquid film
thickness (h), and it can also be viewed as the difference between a normal component
of film pressure, PN (in equilibrium with the gaseous phase PN=PG), and the pressure in
the bulk liquid phase, PL,
Π ( h ) = PN ( h ) − PL = PG − PL
(4)
The disjoining pressure is related to more conventional thermodynamic quantities such
as Gibbs free energy (Adamson, 1990; Nitao and Bear, 1996). Gibbs free energy (G)
per unit area of the interface may be defined on the basis of Π(h) isotherms for constant
pressure PL, temperature T, chemical µ and electric potentials of the liquid-gaseous and
the liquid-solid interfaces as (Derjaguin et al., 1987):
h
G( h ) = − ∫ Π ( h )dh
(5)
∞
The value of G(h) is equal to the work of thinning the film in a reversible isobaric isothermal process from 4 to a finite thickness h, with Α(h)=-(∂G/∂h)T, PL, :, Ρg, Ρs. Derjaguin
4
SOIL WATER POTENTIAL
et al. (1987) point out that the use of Α(h) as the basic thermodynamic property is not a
mere change of notation, but that Α(h) has advantages in cases where Gibbs
thermodynamic theory is difficult to define, such as when interfacial zones overlap to the
extent that the film does not retain the intensive properties of the bulk phase. The use of
the disjoining pressure is advantageous from an experimental point of view because of
the relative ease in accounting for different contributions (e.g., electrostatic effects).
The disjoining pressure is a sum of several components, similar to the concept of total
soil water potential discussed above. The primary components of Π(h) in porous media
are molecular, Πm(h); electrostatic, Πe(h); structural, Πs(h); and adsorptive Πa(h):
Π (h) = Π m(h) + Π e(h) + Π s (h) + Π a(h)
(6)
Πm(h): The molecular component originates from van der Waals interaction between
macro-objects (e.g., parallel clay plates). Various expressions, with Πm(h) often
proportional to h-3 , were derived by Paunov et al. (1996) and Iwamatsu and Horii
(1996).
Πe(h): The electrostatic component of the disjoining pressure is calculated from the
solution of the Poisson-Boltzmann equation for the DDL with appropriate boundary
conditions. Approximate solutions are adequate for many applications and are available
in the literature (e.g., Paunov et al., 1996; Derjaguin et al., 1987), often with Πe(h) ∝ h-2.
Πs(h): Some controversy exists regarding the origin of the structural component; some
attribute it to changes in the structure (density) of water adjacent to solid surfaces and
deformation of hydrated shells, while others attribute this force to the presence of a
layer with a lower dielectric constant near the surface (Paunov et al., 1996). Regardless
of its exact origin, this component is responsible for the so-called hydration repulsion
which stabilizes dispersion and prevents coagulation of some colloidal particles, even at
high electrolyte concentrations (Mitlin and Sharma, 1993); Πs(h) ∝ h-1 (Novy et al.,
1989).
Πa(h): The adsorptive component of the disjoining pressure results from nonuniform
concentrations in the water film due to unequal interaction energies of solute and
solvent with interfaces in nonionic solutions. This is different than the nonuniform
distribution of charged ions. This component of the disjoining pressure is likely to
become very important for interactions between nonpolar molecules (e.g., NAPLs)
which give rise to repulsive forces in the liquid film (see discussion in Derjaguin et al.,
1987, p. 171).
The form of the disjoining pressure isotherm Π(h) is determined by the nature of surface
forces. While the molecular component Πm(h) is always present, the influence of other
components depends on surface properties, liquid polarity and its composition, and
adsorption of dissolved components. The range of the electrostatic forces in dilute
solutions of a 1:1 electrolyte (10-6-10-7 mol l-1) is in the range of 0.3 to 1.0 µm.
Consequently, thick films of water and aqueous electrolyte solutions (h > 500 Å) are
stable mainly through the Πe(h) component of disjoining pressure. The magnitude and
contribution of Πe(h) primarily depend on the charges of the film and substrate surfaces.
SOIL WATER POTENTIAL 5
Dispersion forces become appreciable in the range h < 500 Å, and their influence is
enhanced by large differences between the permittivity of the liquid and the solid. The
forces of structural repulsion may come to play in film thickness of less than 100 Å.
ψs is the solute or osmotic potential determined by the presence of solutes in soil water,
which lower its potential energy and its vapor pressure. The effects of ψs are important
when: (i) there are appreciable amounts of solutes in the soil; and (ii) in the presence of
a selectively permeable membrane or a diffusion barrier which transmits water more
readily than salts. The effects of ψs are otherwise generally negligible when only liquid
water flow is considered and no diffusion barrier exists. The two most important
diffusion barriers in the soil are: (i) soil-plant root interfaces (cell membranes are
selectively permeable); and (ii) air-water interfaces; thus when water evaporates salts
are left behind. In dilute solutions the solute potential, also called the osmotic pressure
is proportional to the concentration and temperature according to:
ψ s = - R T Cs
(7)
where ψs is in kPa, R is the universal gas constant [8.314x10-3 kPa m3/(mol K)], T is
absolute temperature (K), and Cs is solute concentration (mol/m3). A useful
approximation which may be used to estimate ψs in kPa from the electrical conductivity
of the soil solution at saturation (ECs) in dS/m is:
ψ s ≈ - 36 EC s
(8)
ψp is the pressure potential defined as the hydrostatic pressure exerted by unsupported
water that saturates the soil and overlays a point of interest. Using units of energy per
unit weight provides a simple and practical definition of ψp as the vertical distance from
the point of interest to the free water surface (unconfined water table elevation). The
convention used here is that ψp is always positive below a water table, or zero if the
point of interest is at or above the water table. In this sense non-zero magnitudes of ψp
and ψm are mutually exclusive: either ψp is positive and ψm is zero (saturated
conditions), or ψm is negative and ψp is zero (unsaturated conditions), or ψp = ψm = 0 at
the free water table elevation. Note that some prefer to combine the pressure and
matric components into a single term, which assumes positive values under saturated
conditions and negative values under unsaturated conditions. Based on operational and
explanatory considerations, we prefer to adopt the more commonly used separate
components protocol.
ψz is the gravitational potential which is determined solely by the elevation of a point
relative to some arbitrary reference point, and is equal to the work needed to raise a
body against the earth's gravitational pull from a reference level to its present position.
When expressed as energy per unit weight, the gravitational potential is simply the
vertical distance from a reference level to the point of interest. The numerical value of ψz
itself is thus not important (it is defined with respect to an arbitrary reference level) what is important is the difference (or gradient) in ψz between any two points of interest.
This value is invariant of the reference level location.
Soil water is at equilibrium when the net force on an infinitesimal body of water equals
6
SOIL WATER POTENTIAL
zero everywhere, or when the total potential is constant in the system. Though the last
statement is a logical consequence of the definitions above, it is not strictly true as
pointed out by Corey and Klute (1985). They argue that constant total potential is a
necessary but not a sufficient condition, and for thermodynamic equilibrium to prevail
three conditions must be met simultaneously: thermal equilibrium or uniform
temperature; mechanical equilibrium meaning no net convection-producing force; and
chemical equilibrium meaning no net diffusional transport of chemical reaction. In most
practical applications, however, the macroscopic definition of the total potential and
equilibrium conditions based on it are completely adequate (Kutilek and Nielsen, 1994).
The difference in chemical and mechanical potentials between soil water and pure
water at the same temperature is known as the soil water potential (ψw):
ψ w =ψ m +ψ s +ψ p
(9)
Note that the gravitational component (ψz) is absent in this definition. Soil water
potential is thus the result of inherent properties of soil water itself, and of its physical
and chemical interactions with its surroundings, whereas the total potential includes the
effects of gravity (an "external" and ubiquitous force field).
Total soil water potential and its components may be expressed in several ways
depending on the definition of a "unit quantity of water". Potential may be expressed as
(i) energy per unit of mass; (ii) energy per unit of volume; or (iii) energy per unit of
weight. A summary of the resulting dimensions, common symbols, and units are
presented in Table 1.
Table1: Units, Dimensions and Common Symbols for Potential Energy of Soil Water
Units
Symbol
Name
Dimensions*
2 2
SI Units
cgs Units
Energy/Mass
µ
Chemical Potential
L /t
J/kg
erg/g
Energy/Volume
ψ
Soil Water Potential,
Suction, or Tension
M/(Lt2)
N/m2 (Pa)
erg/cm3
Energy/Weight
h
Pressure Head
L
m
cm
* L is length, M is mass, and t is time
Only µ has actual units of potential; ψ has units of pressure, and h of head of water.
However, the above terminology (i.e., potential energy expressions rather than units of
potential, per se) is widely used in a generic sense in the soil and plant sciences. The
various expressions of soil water energy status are equivalent, with:
µ=
ψ
= gh
ρw
(10)
where ρw is density of water (1000 kg/m3 at 20 oC) and g is gravitational acceleration
(9.81 m/s2).
SOIL WATER POTENTIAL 7
Measurement of Potential Components
Water potential: A psychrometer (Fig.1) is commonly used for measurement of total
water potential (ψw) in soils. The potential of the soil solution is in thermodynamic
equilibrium with its ambient water vapor. Taking the vapor pressure above pure water at
reference state (ψw=0) as e0, the vapor pressure (e) over a salt solution or soil water
held in soil pores by matric forces is depressed relative to the reference state, i.e., e<e0.
A convenient measure obtained by the psychrometer is the relative vapor pressure of
the ambient soil atmosphere, which is related to the water potential (ψw) of soil water
through the well-known Kelvin equation (Adamson, 1990):
 Mw ψ w

e
ρ RT
RH =
= e w
eo



(11)
where e is water vapor pressure (kPa), eo is saturated vapor pressure at the same
temperature, Mw is the molecular weight of water (0.018 kg/mol), R is the ideal gas
constant (8.31 J K-1 mol-1 or 0.008314 kPa m3 mol-1 K-1), T is absolute temperature (K),
and ρw is the density of water (1000 kg/m3 at 20 oC). Rearranging and taking a logtransformation of Eq.11 yields an expression for water potential ψw:
ψw =
R T ρw  e 
ln 
Mw
 e0 
(12)
The water potential in drier soils is lower such that fewer water molecules "escape" into
the ambient atmosphere, resulting in lower relative humidity (lower relative vapor
pressure). Concentrated soil solutions having lower osmotic potentials have similar
effect on reducing vapor pressure, as more water molecules are associated with
hydrated salt molecules and are less free to “escape” the liquid state. The inability to
distinguish between matric and osmotic effects limits psychrometric measurements to
soil water potential only. In some cases where the osmotic potential is negligible,
psychrometric measurements are used to infer the matric potential.
8
SOIL WATER POTENTIAL
Figure 1: (a) A field psychrometer with porous ceramic shield (Source: Wescor Inc., Logan, UT); and (b)
SC10X sample chamber for psychrometric laboratory measurements of soil water potential
(Source: Decagon Devices Inc., Pullman, WA).
Pressure potential: Piezometers are commonly applied to measure ψp. A piezometer
(Fig.2) is a tube that is placed in the soil to depths below the water table and that
extends to the soil surface and is open to the atmosphere. The bottom of the
piezometer is perforated to allow soil water under positive hydrostatic pressure to enter
the tube. Water enters the tube and rises to a height equal to that of the unconfined
water table. The elevation of the free water table is measured relative to the soil surface
using a steel tape with bell sounder, or other electro-optic devices that indicate water
table depth. The value of pressure potential expressed as energy per weight is simply
the vertical distance from a point of interest to the surface of the free water table.
Pressure potentials above the water table surface are always zero (non-zero pressure
and matric potentials are mutually exclusive).
SOIL WATER POTENTIAL 9
Figure 2: Sketch illustrating the concept of piezometer measurements
Matric potential: Tensiometers or heat dissipation sensors are commonly applied to
measure soil matric potential. A tensiometer consists of a porous cup, usually made of
ceramic and having very fine pores, connected to a vacuum gauge through a waterfilled tube (Fig. 3). The porous cup is placed in intimate contact with the bulk soil at the
depth of measurement. When the matric potential of the soil is lower (more negative)
than inside the tensiometer, water moves from the tensiometer along a potential energy
gradient to the soil through the saturated porous cup, thereby creating suction sensed
by the gauge. Water flow into the soil continues until equilibrium is reached and the
suction inside the tensiometer equals the soil matric potential. When the soil is wetted,
flow may occur in the reverse direction, i.e., soil water enters the tensiometer until a
new equilibrium is attained. The tensiometer equation is:
ψ m = ψ gauge + ( zgauge − zcup )
(13)
with ψgauge the reading at the vacuum gauge location and z indicating depth. The vertical
distance from the gauge plane to the cup, expressed as a negative quantity, must be
added to the matric potential measured by the gauge (ψgauge) in order to obtain the
matric potential at the depth of the cup. This accounts for the positive head exerted by
the overlying tensiometer water column at the depth of the ceramic cup. Note that using
the difference in vertical elevation is appropriate only when potentials are expressed per
unit of weight. Electronic sensors called pressure transducers often replace mechanical
10
SOIL WATER POTENTIAL
vacuum gauges. The transducers convert mechanical pressure into an electric signal
which can be more easily and more precisely measured. In practice, pressure
transducers can provide more accurate readings than other gauges, and in combination
with data logging equipment are able to supply continuous measurements of matric
potential.
Figure 3: Illustration of tensiometers for matric potential measurement using vacuum gauges
and electronic pressure transducers.
The tensiometer range is limited to suctions (absolute values of the matric potential) of
less than 100 kPa, i.e., 1 bar, 10 m head of water, or ~1 atmosphere. Therefore other
means are needed for matric potential measurement under drier conditions.
Heat dissipation sensors may be applied for a matric potential range from -10 to -1000
kPa. The rate of heat dissipation in a porous medium is dependent on the medium’s
specific heat capacity, thermal conductivity, and density. The heat capacity and thermal
conductivity of a porous matrix are affected by its water content. Heat dissipation
sensors contain heating elements in line or point source configurations embedded in a
rigid porous matrix with fixed pore space. The measurement is based on application of a
heat pulse by applying a constant current through the heating element for specified time
period, and analysis of the temperature response measured by a thermocouple placed
at a fixed distance from the heating source (Phene et al., 1971; Bristow et al., 1993).
With the heat dissipation sensor buried in the soil, changes in soil water matric potential
result in a gradient between the soil and the porous ceramic matrix, inducing water flow
SOIL WATER POTENTIAL 11
between the two materials until a new equilibrium is established. The water flow
changes the water content of the ceramic matrix which, in turn, changes the thermal
conductivity and heat capacity of the sensor, and hence the measured temperature
response to the applied heat pulse.
As already mentioned above, for cases where the osmotic potential is negligible,
psychrometric measurements can be used to infer the matric potential. A typical range
for psychrometers is -800 to -10000 kPa.
Osmotic Potential: Soil water solutions contain varied quantities and compositions of
dissolved salts. The relationships between the salt concentration and ψs, and the
possibility for estimating ψs from the electrical conductivity (EC) of the soil solution were
discussed above. Conventional measurement of soil solution EC involves solution
extraction from saturated soil samples and measuring the EC using an electrical
conductivity meter (Fig. 4a).
Figure 4: (a) Handheld electrical conductivity (EC) meter. (b) TDR probes and solution EC vs.
concentration measured with TDR and EC meter (Mmolawa and Or, 2000).
Electrical conductivity meters rely on Ohm’s law:
E = I⋅R
(14)
,with E the electromotive force (volts), I the current flow (amperes), and R the resistance
(ohms). For constant voltage, the current flowing through the solution is inversely
proportional to the electrical resistance, or directly proportional to the electrical
conductance. The solution EC is thus determined from known voltage and electrode
geometry and measurement of the electric current. More recently, a variety of in-situ
methods such as time domain reflectometry (TDR) have been used to deduce soil bulk
12
SOIL WATER POTENTIAL
EC from electromagnetic signal attenuation, hence enabling simultaneous
measurements of water content and soil EC in the same undisturbed soil volume
(Dalton et al., 1984). Concurrent knowledge of θ and EC can be used to infer the soil
solution EC (Hendrickx et al., 2002), and hence to estimate ψs. Other laboratory or insitu methods to measure or infer the EC of the soil solution or the bulk soil are
discussed by Rhoades and Oster (1986) and Hendrickx et al. (2002).
Further Reading
Adamson, A.W., Physical chemistry of surfaces, Fifth edition, John Wiley and Sons, New York, 1990.
Bolt, G.H., Soil physics terminology, Int. Soc. Soil Sci. Bull. 49:16-22, 1976.
Bristow, K.L., G.S., Campbell, and K. Calissendroff, Test of a heat-pulse probe for measuring changes in
soil water content. Soil Sci. Soc. Am. J., 57:930-934, 1993.
Corey, A. T., and A. Klute, Application of the potential concept to soil water equilibrium and transport. Soil
Sci. Soc. Am. J., 49:3-11, 1985.
Dalton, F.N., W.N. Herkelrath, D.S. Rawlins, and J.D. Rhoades. 1984. Time-domain reflectometry:
Simultaneous measurement of soil water content and electrical conductivity with a single probe.
Science 224:989-990.
Day, P.R., G.H. Bolt, and D.M. Anderson, Nature of soil water. p. 193-208. In R.M. Hagan, H.R. Haise,
and T.W. Edminster (ed.) Irrigation of agricultural lands. American Society of Agronomy, Madison,
WI, 1967.
Derjaguin, B.V., N.V. Churaev, and V.M. Muller, Surface Forces, Plenum Publishing Corporation,
Consultants Bureau, New York, 1987.
Edlefsen, N.E., and A.B.C. Anderson, Thermodynamics of soil moisture, Hilgardia, 15, 31-298, 1943.
Hanks, R.J., Applied Soil Physics. 2nd Ed., Springer Verlag, New York, NY, 1992.
Hendrickx, J.M.H, J.M. Wraith, D.L. Corwin, and R.G. Kachanoski, Solute content and concentration. p.
1253-1322. In J.H. Dane and G.C. Topp (ed.). Methods of Soil Analysis. Part 4. Physical Methods.
ASA, Madison, WI, 2002.
Hillel, D., 1998. Environmental Soil Physics, Academic Press, San Diego.
Iwamatsu, M., and K. Horii, Capillary condensation and adhesion of two wetter surfaces, J. Colloid
Interface Sci., 182, 400-406, 1996.
Iwata, S., T. Tabuchi, and B.P. Warkentin, Soil water interactions. M, Dekker, New York, NY, 1988.
Kutilek, M., and D. R. Nielsen, Soil hydrology. Catena Verlag, Cremlingen-Destedt, Germany, 1994.
Mitlin, V.S., and M.M. Sharma, A local gradient theory for structural forces in thin fluid films, J. Colloid
Interface Sci., 157, 447-464, 1993.
Mmolawa, K. B., and D. Or, Root zone solute dynamics under drip irrigation: A review. Plant and Soil
222:161-189, 2000.
Nitao, J.J., and J. Bear, Potentials and their role in transport in porous media. Water Resour. Res.,
32:225-250, 1996.
Novy, R.A., P.G. Toledo, H.T. Davis, and L.E. Scriven, Capillary dispersion in porous media at low wetting
phase saturations, Chem. Eng. Sci., 44(9), 1785-1797, 1989.
Paunov, V.N., R.I. Dimova, P.A. Kralchevsky, G. Broze, and A. Mehreteab, The hydration repulsion
between charged surfaces as an interplay of volume exclusion and dielectric saturation effects, J.
Colloid Interface Sci., 182, 239-248, 1996.
SOIL WATER POTENTIAL 13
Phene, C.J., G.J. Hoffman, and S.L. Rawlins, Measuring soil matric potential in situ by sensing heat
dissipation within a porous body: 1. Theory and sensor construction. Soil Sci. Soc. Am. Proc. 35:2733, 1971.
Rhoades, J.D., and J.D. Oster, Solute content, p. 985-1006. In A. Klute (ed.). Methods of Soil Analysis.
Part 1, Physical and Mineralogical Methods, Second Edition. ASA, Madison, WI, 1986
Philip, J.R., Unitary approach to capillary condensation and adsorption, The Journal of Chemical Physics,
66(11), 5069-5075, 1977.
Tuller, M., D. Or, and L.M. Dudley, Adsorption and capillary condensation in porous media -liquid
retention and interfacial configurations in angular pores. Water Resour. Res., Vol.35, No.7, 19491964, 1999.