Water potential plays a key role in water flow theory similar to the role played by tem- perature in heat flow problems, or voltage in electrical circuit theory. Water flows in response to gradients in water potential. Darcy’s law (Darcy, 1856) states that
fw= –Kdψ
dx (5.15)
wherefwis the water flux density [kg m–2s–1],Kis the hydraulic conductivity [kg s m–3], ψ is the water potential [J kg–1], x is the space dimension [m] and dψ/dxis the water potential gradient that drives the flow.
98 Soil Liquid Phase and Soil–Water Interactions
The soil water potential is the potential energy of water in soil. Since we cannot define an absolute scale for potential energy, the soil water potential is quantified relative to a standard state where water has no solutes, is free from external forces except gravity, and is at a reference pressure, a reference temperature and a reference elevation. The soil water potential is then defined as the energy state of water in soil with respect to the energy of water at the standard state. The driving force for water flow is the uneven distribution of water potential.
When pure free water at a standard state is brought into contact with soil water, across a semipermeable membrane, a pressure difference develops across the membrane.
The pressure difference corresponds to the water potential. The standard-state values, usually specified, are temperatureT0, pressureP0and vertical positionz0. The value of the water potential in these conditions is equal to zero.
Soil water potential represents energy per unit quantity of water. The quantity can be energy per unit mass [J kg–1] or per unit volume of water [J m–3]. The former is preferable because there is then no need to include in the computation the changes of water volume with temperature. Soil water potential is also expressed as energy per unit weight, which is equivalent to a head of water. The energy is equivalent to the pressure exerted by a water column of a given height. For instance, a column of water of about 10 m correspond to a pressure of 100 kPa. This unit is common because it appears simpler and allows visualization of the gravitational and pressure potentials, which are often expressed in metres. The base 10 logarithm of the head expressed in centimetres is called pF, which is another common unit for water potential.
The unit J m–3 is the same as N m–2, which is the SI unit of pressure, the pascal [Pa].
Energy per unit volume is therefore a pressure. Pressure units have long been used in soil physics to measure the soil water potential in tensiometers and pressure plate apparatus (Richards, 1948). Table 5.2 shows water potential in various units for a range of water potentials.
The soil water potential is usually expressed as a negative number, because it repre- sents the energy required to transfer the soil water to the reference state of pure, free water described above. The terms ‘suction’ and ‘tension’ are definitions developed to avoid using the negative sign and to represent the soil water potential as a positive num- ber. They are common terms used in geotechnical engineering and soil mechanics. Soil water potential can range over several orders of magnitude, from a few joules per kilo- gram when the soil is close to saturation to minus thousands of joules per kilogram when the soil is very dry.
The total soil water potentialψtis determined by a variety of forces acting on the soil water, including gravitational (ψg), matric (capillary and adsorptive,ψm), osmotic (ψo), hydrostatic (ψh) and overburden pressure (ψ) components:
ψt =ψg+ψm+ψo+ψh+ψ (5.16) Usually, only one or two of the component potentials needs to be considered in any given flow problem, but gradients in any of these potentials can result in water flow when conditions are right.
Soil Water Potential 99 Table 5.2Water potential in various units for a range of water potentials
Water potential Head
[cmH2O]
pF Pore
diameter [µm]
hra
FPD.b [◦C]
[J kg–1] [MPa] [bar]
–1 –0.001 –0.01 10 1.0 290.80000 0.99999 –0.001
–10 –0.01 –0.1 102 2.0 29.08000 0.99993 –0.008
–30 –0.03 –0.3 306 2.5 9.69333 0.99978 –0.025
–100 –0.1 –1 1 020 3.0 2.90800 0.99926 –0.082
–1000 –1 –10 10 204 4.0 0.29080 0.99262 –0.820
–1500 –1.5 –15 15 306 4.2 0.19387 0.98895 –1.230
–10 000 –10 –100 102 041 5.0 0.02908 0.92860 –8.197
–100 000 –100 –1 000 10 20 408 6.0 0.00291 0.47676 (na) –1 000 000 –1000 –10 000 10 204 082 7.0 0.00029 0.00061 (na)
ahr= relative humidity.
bFPD = freezing-point depression.
5.2.1 Gravitational Potential
The gravitational component of the water potential is fundamentally different to any of the other components, since it is the result of ‘body forces’ applied to the water as a consequence of the water being in a gravitational field. The gravitational potential is calculated from
ψg =g(z–z0) (5.17)
wheregis the gravitational acceleration (9.8 m s–2) andzis height. The reference level isz0, at whichψgis taken as zero. The reference level is usually taken as the soil surface or the surface of a water table. In flow problems, we are interested in the gradient of the gravitational potential, which isdg/dz=g, a constant.
5.2.2 Matric Potential
The matric potential is one of the most important components of the water potential in soil and plant systems. It is defined as the amount of work, per unit mass of water, required to transport an infinitesimal quantity of soil water from the soil matrix to a reference pool of the same soil water at the same elevation, pressure and temperature.
The reduction in potential energy of water in porous materials is primarily the result of physical forces that bind the water to the porous matrix. The water potential under a
100 Soil Liquid Phase and Soil–Water Interactions
curved air–water interface, such as might exist in a capillary tube or a soil pore, is given by the capillary rise equation
ψm= 2γ cosβ
ρlrc (5.18)
where β is the contact angle between the water and the wetted surface, γ is the sur- face tension of water [N m–1],ρl is the density of water [kg m–3] andrc is the radius of curvature [m]. Both the contact angle and the surface tension of the liquid phase were described earlier.
Equation (5.18) can be used to find the equivalent diameter of pores in a soil that cor- responds to a given matric potential. These are shown in Table 5.2 for 20◦C and zero contact angle. For example, a soil atψm = –100 J kg–1, according to the capillary equa- tion, will have pores larger than 2.9µm filled with air and pores smaller than this value filled with water. At some water potential, in the range –3000 J kg–1 < ψm <–100 000 J kg–1, the capillary analogy breaks down because most of the water is absorbed in lay- ers on particle surfaces rather than being held in pores between particles. The potential where the capillary equation is no longer valid is not a fixed value but depends on various soil properties such as structure and texture.
Although the total amount of adsorbed water is typically small when compared with the volumetric contribution of capillary water, its contribution is important for processes such as microbial activity, plant water uptake and evaporation in dry environments. The adsorption of water on soil particles is mainly due to van der Waals forces that promote the formation of liquid films around soil particles. Clearly, adsorbed water is strictly linked to the soil specific surface area and is important in determining processes related to contaminant adsorption, ion exchange reactions, microbial attachment to solid par- ticles and heat transfer. Based on the contribution of van der Waals forces controlling adsorbed water films in soils, it is possible to postulate a relationship between the amount of soil water in the ‘dry end’ and the soil specific surface area (Tuller and Or, 2005):
θd =h∗Amρl (5.19)
whereθdis volumetric water content in the ‘dry end’,Am[m2kg–1] is the specific surface area andh∗[m] is the thickness of the water film (Grismer, 1987). The thickness of the water film can be obtained from knowledge of the measured water potential (Iwamatsu and Horii, 1996):
h∗= 3
Asvl 6πgρlψm
(5.20) where Asvl [J] is the Hamaker constant for solid–vapour interactions and ψm is the ab- solute value of the matric potential [m]. This equation is valid for planar surfaces and neglects contributions of capillary condensation. Tuller and Or (2005) utilized the rela- tionship to estimate the soil specific surface area from measurements of soil water content
Soil Water Potential 101 and soil water potential at low matric potential. Substituting eqn (5.20) into eqn (5.19) leads to
θd = 3
Asvl 6πgρlψm
Amρl (5.21)
The methodology is based on measurements of soil water contentθdand matric poten- tialψmat low water contents, and derivation of the soil specific surface areaAmby fitting eqn (5.21) to experimental data, using onlyAm as a fitting parameter. For the Hama- ker constant, the authors recommend using a value of –6×10–20J. Limitations of these techniques could be due to additional effects such as formation of water molecule clus- ters around cationic charge sites determining other adsorption mechanisms in addition to the van der Waals effects or different values of the Hamaker constant for different materials (Tuller and Or, 2005). Moreover measurement of the matric potential in dry soil is performed with dew-point techniques where the measurement is affected by both the matric and the osmotic potential. In soils with high solute concentration, error may be introduced by the osmotic contribution.
5.2.3 Osmotic Potential
The osmotic potential is equivalent to the work required to transport water reversibly and isothermally from a solution to a reference pool of pure water at the same elevation.
In practical terms, it is the energy one must add to a solution to equilibrate the solution with pure water across a perfect semipermeable membrane. If the concentration of solute in a solution is known, the osmotic potential can be calculated from
ψo= –cναRTK (5.22)
wherecis the solute concentration [mol kg–1],νis the number of particles in solution per molecule of solute (ν= 1 for non-ionizing solutes;ν= number of ions per molecule for ionizing solutes),αis the osmotic coefficient,Ris the gas constant (8.31 J mol–1K–1) and TK is temperature in kelvins. The osmotic coefficient is a function of solution concentra- tion and solute species. Osmotic coefficients for common solutes are given by Robinson and Stokes (1965).
If mixtures of solutes are present, the total osmotic potential is the sum of the con- tributions from the components. In other words, the interaction between species is apparently small. When detailed data on the chemical composition of the soil solution are not available, the osmotic potential can still be estimated if the electrical conduct- ivity of the saturation extract is known. A rule of thumb for a soil solution of typical composition is
ψos= –36σ (5.23)
102 Soil Liquid Phase and Soil–Water Interactions
where ψos is the osmotic potential of the saturation extract [J kg–1] and σ is the elec- trical conductivity [dS m–1] of the saturation extract. If the soil dries without a change in the amount of solutes present, then (ignoring changes inc, anion exclusion effects and precipitation of sparingly soluble salts) we can write
ψo =ψos
θs
θ (5.24)
where θs is the saturation water content. These are rough approximations, but are adequate for many purposes if used with care.
The osmotic potential is an important component of water potential in plant cells and affects plant water uptake. Osmotic potential gradients in soil are usually unimportant as driving forces for flow, because the salts move with the water. Osmotic potential is always negative or zero.
5.2.4 Hydrostatic Potential
The hydrostatic potential describes the effects on water of changing the hydrostatic or pneumatic pressure applied to the water. This pressure changes the energy of soil water relative to the reference level. The relationship between potential and pressure is
ψh= P ρl
(5.25) whereP is pressure [Pa]. Pressure can be either higher or lower than the reference pres- sure, so ψh can be either positive or negative. The pressure potential is an important component of the water potential below a water table, in plant cells and in tensiometers, which are used for measuring matric potential. The pressure component is used to de- scribe the status of water under various laboratory conditions when pressure or suction is applied to equilibrate an external phase with soil or plant water. The suction plate, pres- sure plate (Richards, 1948) and pressure bomb (Scholanderet al., 1965) are examples of equipment that utilize this principle.
5.2.5 Overburden Potential
The overburden potential is similar to the pressure potential, in that it is the increase in potential of water in a porous system resulting from the application of pressure to the water in the system. The difference is that with the pressure potential, the pressure is applied directly to the water. This can only occur if the system is saturated. With the overburden potential, the pressure is applied to the water by the matrix. When mechan- ical pressure is applied to the matrix, some of the pressure is borne by the solid structure of the matrix itself, but part of it may be transferred to the water in the matrix. The overburden potential is a function of the pressure applied to the matrix and the fraction
Water Potential–Water Content Relations 103 of the load that is transmitted to the water in the matrix. If the matrix has no resistance to deformation, then
ψ= P ρl
(5.26) whereP is the applied pressure. The overburden potential can be present in an unsat- urated porous medium. It is mainly important in soil at depth, where the pressure of the overburden can be substantial. It is relatively unimportant in sand, but can be an appreciable fraction of the overburden pressure in wet clay.