Potentia Revised01

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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 (ψ T ) 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),

Transcript of Potentia Revised01

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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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: ψψψψψ zpsmT + + + = (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

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capillary component (C):

)(C)h(Am κψ += (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:

( )ρ

κσκ ⋅⋅−=

2C (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,

LGLN PPP)h(P)h( −=−=Π (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

dh)h()h(G Π (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

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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):

)h()h()h()h()h( asem ΠΠΠΠΠ +++= (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.

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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:

C T R- = ssψ (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:

EC 36- ss ≈ψ (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

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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):

ψψψψ psmw + + = (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* SI Units cgs Units Energy/Mass µ Chemical Potential L2/t2 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:

gh = = wρ

ψµ (10)

where ρw is density of water (1000 kg/m3 at 20 oC) and g is gravitational acceleration (9.81 m/s2).

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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):

ρ

ψ

== RTwwwM

oe

eeRH (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 log-transformation of Eq.11 yields an expression for water potential ψw:

=

0w

ww e

elnMTR ρ

ψ (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.

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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).

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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 water-filled 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:

)zz( cupgaugegaugem −+=ψψ (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

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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

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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:

RIE ⋅= (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

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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 in-situ 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.

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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. 35:27-33, 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, 1949-1964, 1999.