John G. Fe.rris

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Review of Some Elements of 

Soil-Moisture Theory

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-D
Review of Some Elements of 

Soil-Moisture Theory

By IRWIN REMSON and J. R. RANDOLPH

FLUID MOVEMENT IN EARTH MATERIALS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-D

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C.
CONTENTS

List of symbols.____________________________________

Abstract...________________________________________

Introduction.______________________________________

Zone of aeration._-_--_--__________-__________-_____

Background physics __--__---.-____________________

Surface tension.________________________________

Molecular causes of surface tension___________

Definition of surface tension._________________

Pressure differential across a liquid-gas interface _ 

Angle of contact.___________________________

Rise of water in a capillary tube..____________

Water vapor in unsaturated soils_________________

Molecular causes of vapor pressure.___________

Vapor pressure and temperature._____________

Vapor pressure and amount of dissolved solute _

Vapor pressure and hydrostatic pressure in the

liquid ---------_---------__-_____________

Barometric equation ________________________

Kelvin equation. ___________________________

Derivation.____________________________

Significance._ __________________________

Diffusion. _____________________________________

Osmosis. ______________________________________

Force, work, energy, potential, and gradient______

Potentials in soil water______________________________

Gravitational potential._________________________

Hydrostatic-pressure potential.___________________

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Potentials in soil water Continued

Osmotic-pressure potential.______________________ D20

Adsorption potential-______________------------- 21

Thermal potential-________-_____--------------- 21

Chemical potential..---------------------------- 22

Total potential and its measurement______________ 22

Equilibrium distribution of potential and water in soil__ 24

Distribution in the zone of saturation _____________ 24

Distribution in the capillary fringe_______________ 24

Distribution in pinching and swelling tubes

without horizontal connections____-____-_-_ 25

Distribution in tightly packed spheres-________ 27

Distribution above the capillary fringe.----------- 28

Distribution in tightly packed spheres.- - ______ 28

Distribution in randomly distributed pores____- 29

Movement of soil moisture.-.------------------------ 29

Physical picture. __________________-----------_- 29

Pore underlain by a cavity_______-__--------- 30

Pore underlain by pores of similar size ______ 31

Pore underlain by pores of dissimilar size_ _ __ __ 31

Randomly distributed pores________-_-------_ 32

Differential equation for unsaturated flow _ _ _ _ _ _ _ 33

Historical background.-_________-----------_ 33

Equation of continuity. _____-__-_-___--_---- 34

Darcy's law.____________-_____-_----------- 36

Flow equation_________________-_------_---- 36

References.-__--___-_-___---_-------------__--__-_- 37

ILLUSTRATIONS

Page

FIGURE 1. Intermolecular force as a function of molecule separation._______________________-_-----_--_-------_---_-_ D2

2. Bubble of air blown at the end of a capillary tube______________________________

3. Contact angle of a liquid-vapor surface near a solid. __________________---_-_-__--

4. Rise of water in a capillary tube._________________________________-----_-----_.

5. Column of vapor in a gravitational field-_______-__-____-____-____---__--_-_----

6. A curved liquid-vapor surface in contact with a free flat liquid surface._________--_.

7. Position of the meniscus in a capillary tube and adjacent vapor pressures.----------

8. Position of the meniscus in a wedge-shaped pore and adjacent vapor pressures_-__-_-

9. Ground-water gradient------_______________________________-__--_--__-----_--

10. Approximate flow net in the vicinity of an irrigation pond_______________________

11. Moisture characteristic of a fine sand__------_______-_-__-_----_-------_----__--

12. Plot of the capillary-rise equation for water. _________-_________----__----_-_----

13. Hypothetical variation of pore radius with height above the water table.__-_--_--_-

14. Plot Of the capillary-rise equation on the same scale used in figure 13_--------------

15. Unit rhombohedron of a mass of spheres in the most compact packing........______

16. Unit element of pore space in a mass of spheres packed in the most compact manner.

17. Dewatering of a main pore and the reentrants_____________-__._________-___-----

18. Pore underlain by a cavity.___________________________________________________

19. Pore underlain by a pore of similar size.________________________________________

20. Pore underlain by a smaller pore__----_-_-__-_-_---_-----_--___-_--------------

21. Pore underlain by a larger pore________________________________________________

22. Unit rectangular parallelepiped-________________-__________-__._--_---_---__---

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in
LIST OF SYMBOLS

a acceleration

A area of a liquid surface

b mole fraction of dissolved material

C a constant

d a digit or group of digits

dS area of an infinitesimal element of capillary surface

D density of a liquid or gas

D(tt} diffusivity as a function of concentration of ions

or molecules

e the exponential base, 2.71828 . . . 

/ a digit

F force or resultant force 

Fa force of adhesion 

Fc force of cohesion 

Fg force of gravity 

g acceleration of gravity 

G gravitational potential

Gm gravitational potential in ergs per grain of water 

G, gravitational potential in ergs per cubic centimeter

of water

h water-table elevation 

AA ^difference in water-table elevation 

i, j, k a system of unit direction vectors 

K coefficient of capillary conductivity 

I length of a capillary tube 

m mass

Al molecular weight 

o a digit 

p (a) pressure; (b) pressure on the concave side of a

curved liquid surface; (c) pressure of the vapor in a

capillary opening; (d) pressure of the vapor a distance

above a free flat water surface; or (e) vapor pressure

of a solution 

Po (a) pressure of the vapor immediately above a flat

water surface; (b) pressure on the convex side of a

curved liquid surface; or (c) vapor pressure of a free

pure solvent 

p' pressure of the liquid on the convex side of a

meniscus

pa pressure just over an element of capillary surface 

p'a pressure just under an element of capillary surface 

P hydrostatic pressure 

Q discharge or flow rate

r radius of a capillary tube

r0 critical distance of separation between the molecules

of a liquid

R radius of curvature of a hemispherical liquid surface 

RI and R2 principal radii of curvature of a liquid

surface

R the gas constant

dS area of a small element of capillary surface 

T absolute temperature 

t time

u a scalar point function such as potential 

v volume of a mass of water 

Vi specific volume of a liquid 

vv specific volume of a vapor 

V mass flow of moisture (a vector) 

Vx, Vv, Vz components of the mass flow of moisture in

the x, y, z directions 

Vv volume flow of moisture (a vector) 

W work or energy

x, y, z distances in a rectangular coordinate system 

2 distance above or below a datum 

a angle between a force and direction of movement 

)8 moisture content on a dry-weight basis

 ~ specific moisture capacity

O2j

7 density of a gas or vapor

A operator indicating change in value of a quantity

e osmotic-pressure potential

77 coefficient of viscosity

6 (1) angle of contact; or (2) angle between the incli- 

	nation of the water table and the horizontal 

K diffusivity

X heat of vaporization at constant temperature

£ adsorption potential

TT 3.14159 . . .

p density of liquid water

pa bulk density of a soil

o- surface tension

I, capillary potential

0 total potential

^ hydrostatic-pressure potential

ft concentration of ions or molecules

V the vector operator del, or nabla
FLUID MOVEMENT IN EARTH MATERIALS

REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY

By IRWIN REMSON and J. R. RANDOLPH

ABSTRACT

This review was assembled from the existing literature to make 

available a convenient introduction to this subject, which is of 

interest to workers in many diverse fields of hydrology.

Surface tension at the liquid-vapor interfaces largely controls 

the occurrence and movement of moisture in unsaturated soils. 

Surface tension is defined as the amount of work or energy required to produce a unit increase in the area of a liquid surface. 

The pressure on the concave side of a liquid-vapor surface in a 

round capillary tube exceeds the pressure on the convex side by 

an amount equal to twice the surface tension divided by the tube 

radius. The rise of liquid in a round capillary tube, if the angle 

of contact is assumed to be zero, is predicted by the capillary2<r rise equation pgz = ; where p is the density of the liquid, g is

the acceleration of gravity, z is the height of capillary rise, a is 

surface tension, and r is the radius of the tube.

Important soil-moisture phenomena result from the transfer 

of water molecules across liquid-vapor surfaces and from the 

diffusion of water vapor within the system. Vapor pressure 

decreases with increasing height above a free water surface 

according to the barometric equation. The vapor pressure over 

a water meniscus is less than the vapor pressure over a flat 

water surface by an amount that increases with surface curvature 

according to the Kelvin equation. When the barometric and 

Kelvin equations are compared, the capillary-rise equation shown 

above is obtained. This equation holds at equilibrium throughout the zone of aeration.

The total potential of soil water is the minimum energy per 

gram needed to transport a test body of water from a water-table 

datum to any point within the liquid phase of a soil-water 

system at equilibrium. It is the scalar sum of six main component potentials: (a) gravitational potential, which increases 

with height above the water table; (b) hydrostatic-pressure 

potential, which is associated with surface tension at liquidvapor interfaces; it becomes increasingly negative with increasing 

height above the water table at equilibrium, and the force 

associated with its negative gradient exerts a suction on any 

external body of water with which it is in contact; (c) osmoticpressure potential, which is present in soil that shrinks upon 

drying; it is produced by osmotic pressures resulting from 

differences in ion concentrations at surfaces of soil particles 

having electric (Gouy) double layers; the force associated with 

its negative gradient also exerts a suction on externally applied 

bodies of water; (d) adsorption or adhesion forces, which strongly 

retain water near the surfaces of soil particles; adsorption 

potential is important only in soils that are drier than those 

normally encountered in nature; (e) gradients of thermal potential, which occur when temperature differences occur in the 

system; they are relatively unimportant in causing moisture 

movement except when the temperature gradients are large; 

(f) finally, chemical potential, which is due to the osmotic energy 

of free ions in the aqueous solution; it is usually unimportant in 

nonsaline soils because solutes generally move more readily than 

the water with respect to the soil.

In a hypothetical distribution of vertically swelling and 

pinching horizontally disconnected capillary tubes, there are 

positions of both minimum and maximum capillary rise. These 

are the most common positions of the top of the capillary fringe, 

which is generally at the maximum or minimum position according to previous fluctuations of the water table, whether rising or 

falling. A hypothetical model of rhombohedrally packed 

spheres differs from the vertically pinching and swelling tubes 

mainly in that isolated wedges of water remain above the capillary fringe as the water table falls. Water is either added to or 

taken away from the water wedges by vapor diffusion so that 

the capillary-rise equation is satisfied at equilibrium.

Water may move in the zone of aeration as a liquid, a vapor, 

or both a liquid and a vapor. Vapor transfer occurs where a 

vapor continuum fills the pores. Where only the pore necks are 

filled with water, movement occurs by vapor transfer in the 

pores and by liquid transfer in the necks. Where the pore necks 

and parts of the pores are filled with liquid water, movement 

may occur by discontinuous jumps of liquid water from neck to 

neck. Finally, where a moisture continuum fills the pores, 

water may move completely as a liquid.

The hydraulic conductivity of a wet soil decreases very rapidly 

as the moisture content decreases from its saturation value. 

Field-capacity and wetting-front phenomena are largely a 

result of this decrease of hydraulic conductivity with moisture 

content.

The differential equation for unsaturated flow is derived from 

the equation of continuity and Darcy's law.

INTRODUCTION

This report is not an exhaustive treatise on soil 

moisture. It is not meant to display a full understanding of the physics of the basic phenomena. Indeed, 

the writers do not possess such a thorough knowledge 

of this adjacent but alien field. Rather, it is intended 

to provide for the hydrologist a concise and easily 

mastered picture of some of the important phenomena 

and practical manifestations of soil-moisture theory.

Dl
D2 FLUID MOVEMENT IN EARTH MATERIALS

The Seabrook investigation of the U.S. Geological 

Survey, of which this report is a product, was made 

under the supervision of Alien Sinnott, district geologist, U.S. Geological Survey, Trenton, N.J. The cooperation of the Seabrook Farms Co. in all phases 

of the investigation is acknowledged with gratitude.

ZONE OF AERATION

The zone of aeration is between the water table and 

the land surface. It is "the zone in which the interstices of the functional permeable rocks are not filled 

(except temporarily) with water. The water is under 

pressure less than atmospheric" (Am. Geol. Inst., 1957, 

p. 325; after Meinzer, 1923b, p. 31). Thus, three 

physical phases exist in an unsaturated soil in the zone 

of aeration: the solid soil matrix, the liquid water, and 

the soil air which includes the vapor of the liquid. 

Because of the presence of these phases and the interfaces between them, the occurrence and movement of 

water in the zone of aeration are more complex than 

in saturated flow.

The zone of aeration (vadose zone), according to 

Meinzer (1923a, p. 81; 1923b, p. 26), has three divisions: the capillary fringe, the intermediate belt, and 

the belt of soil water. The lowermost or capillary 

fringe is "a zone, in which the pressure is less than 

atmospheric, overlying the zone of saturation and containing capillary interstices some or all of which are 

filled with water that is continuous with the water in 

the zone of saturation but is held above that zone by 

capillarity acting against gravity" (Am. Geol. Inst., 

1957, p. 44; after Meinzer, 1923b, p. 26). The intermediate belt is "that part of a zone of aeration that 

lies between the belt of soil water and the capillary 

fringe. It contains intermediate vadose water" (Am. 

Geol. Inst., 1957, p. 151; after Meinzer, 1923b, p. 26). 

The uppermost belt, or belt of soil water, is "that part 

of the lithosphere, immediately below the surface, from 

which water is discharged into the atmosphere in perceptible quantities by the action of plants or by soil 

evaporation" (Am. Geol. Inst., 1957, p. 29; after 

Meinzer, 1923b, p. 23).

This report reviews some of the elements of soilmoisture theory, which deals largely with the unsaturated occurrence and movement of water in the zone 

of aeration. It reviews also some of the background 

physics needed for an understanding of this theory.

BACKGROUND PHYSICS 

SURFACE TENSION

Surface-tension phenomena associated with the 

liquid-vapor interfaces within the soil matrix have 

important effects upon the occurrence and movement

of soil water. Part of the following description of 

these surface-tension phenomena is based upon Sears 

(1950, p. 319-330).

MOLECULAR CAUSES OF SURFACE TENSION

Whereas the average distance between the molecules 

of a gas at atmospheric pressure is about 10 times the 

size of each molecule, the molecules of a liquid are almost 

touching. The forces between the individual liquid 

molecules are largely electrical. As shown in figure 1,

FIGURE 1 Intermolecular force as a function of molecule separation. (After Sears, 

F. W., 1850, Mechanics, heat, and sound; 2d ed., Addison-Wesley Pub. Co., Inc., 

Reading, Mass., p. 321.)

an attractive force exists between two molecules that 

are separated by a distance greater than some value, 

r0. (See "List of symbols.") As the separation distance increases from r0, the attractive force first 

increases to a maximum and then gradually decreases.

When the separation between two molecules of a 

liquid is less than r0 , there is a large repulsive force. 

This repulsive force is responsible for the high pressures 

that are needed to compress a liquid. When the 

separation distance is r0, the repulsive and attractive 

forces equate to zero, and the molecules are in a state 

of equilibrium.

A molecule in the interior of a liquid is surrounded 

by similar molecules. The molecule has thermal energy 

that tends to make it move (to become displaced from 

the equilibrium separation, r0). However, when displaced from r0, it is attracted by the adjacent molecules on one side and repelled by the adjacent molecules 

on the other side in accordance with the relation shown 

on figure 1. The combination of attractive and repulsive forces tends to restore the molecule to the equilibrium separation. As a consequence of the displacing 

effect, due to the thermal energy, and the restoring 

effect, due to the intermolecular forces, the molecules
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D3

in the interior of a liquid tend to oscillate about their 

equilibrium separation, r0.

A molecule at the surface of a liquid generally has 

a component of vibration normal to that surface. When 

the molecule is moving outward from the surface, it 

is attracted back to the surface by the molecules below. 

However, there are no molecules above the liquid surface to give the repulsive force shown in figure 1. 

Therefore, the molecule may move out a greater distance than molecules within the body of the liquid. 

If the surface molecule has sufficient thermal energy 

it may escape (evaporate) from the liquid. The molecule performs a series of excursions out to a distance 

slightly greater than that of normal separation, and 

spends most of its time in a region where an inward 

force of attraction is exerted on it in accordance with 

the relation shown in figure 1. "The fact that the 

environment of those molecules in or very near the 

surface differs from that of the molecules in the interior gives rise to the surface effects we are now 

considering" (Sears, 1950, p. 3223.

DEFINITION OF SURFACK TKNSION

A definite amount of work per unit area is required 

to increase the area of a liquid surface. The work can 

be recovered when the area decreases, so that the surface layer appears capable of storing potential energy. 

The enlarged liquid surface is not stretched like a balloon. Instead, additional molecules move into the 

surface layer and the final molecular spacing is unchanged. Unlike a balloon or rubber membrane, the 

work done, or energy of the surface, is directly proportional to the increase in the surface area of the liquid. 

The proportionality constant, or the work per unit 

area, is known as the surface tension, a. It is expressed 

in units of energy per unit area such as ergs per square 

centimeter, which is equivalent to dyne-centimeters 

per square centimeter or dynes per centimeter. Expressed mathematically,

dW=adA (1)

where dW is the work, or energy, necessary to increase 

the area of the liquid surface by an amount dA.

The free energy of a liquid-vapor surface is a fundamental property of the surface (Adam, 1930, p. 4). 

"The free energy of a substance is a property that 

expresses the resultant of the energy (heat content) of 

the substance and its inherent probability (entropy)" 

(Pauling, 1958, p. 650). It exists because work must 

be done to extend the surface, as the extension is accomplished by bringing molecules from the interior to the 

surface of the liquid against the inward attractive 

forces. As a mathematical device, a hypothetical

tension parallel to the surface and equal to the free 

energy is sometimes substituted for the free energy. 

This is known also as the surface tension, but it is 

merely a convenient mathematical fiction that does 

not really exist. Surface tension is correctly defined 

as the proportionality constant in equation 1 (Sears, 

1950, p. 322). Actually, W in equation 1 should be 

the free energy for a rigorous definition of surface 

tension.

A system is in stable equilibrium when its potential 

energy is a minimum (Sears, 1950, p. 322). Liquidvapor interfaces tend toward a condition of minimum 

potential energy and minimum free energy. Because 

the energy is proportional to the area, as shown in 

equation 1, the condition of minimum energy corresponds to a surface of minimum area for a given volume. 

Thus, the inward attraction causes the number of 

molecules in the surface to dimmish to a minimum by 

contraction of the surface. "This tendency is shown 

in the spherical form of small drops of liquid, in the 

tension exerted by soap films as they tend to become 

less extended, and in many other properties of liquid 

surfaces" (Adam, 1930, p. 1).

In discussing the work of Plateau, Adam (1930, p. 1) 

points out that liquid surfaces always assume a curvature such that

---=Constant (2)

when the disturbing effect of gravity is absent. RI and 

E2 are the principal radii of curvature at any point. 

It is a geometrical fact that surfaces for which this 

relation holds are surfaces of minimum area.

It is customary to show RI and R2 as positive when 

the liquid surface is convex to the vapor as for a water 

bubble in air. In this report it will be more convenient 

to take RI and R2 as positive when the liquid surface 

is concave to the vapor, as for a capillary meniscus for 

water.

PRESSURE DIFFERENTIAL, ACROSS A LIQUID-GAS 

INTERFACE

The pressures above and below a flat liquid-gas 

interface are equal at equilibrium. For convenience 

the discussion will be limited to the case where the 

liquid is water. In a vacuum the pressure of the water 

vapor above the interface equals the very small hydrostatic pressure immediately below the interface. Where 

the system is open to the atmosphere the pressure 

above the surface is the sum of the partial pressures 

of the water vapor and the atmospheric gases. The 

hydrostatic pressure immediately below the surface is 

the result of the combined weight of the overlying 

water vapor and atmosphere.
D4 FLUID MOVEMENT IN EARTH MATERIALS

"If a liquid surface be curved the pressure is greater 

on the concave side than on the convex, by an amount 

which depends on the surface tension and on the curvature. This is because the displacement of a curved 

surface, parallel to itself, results in an increase in area 

as the surface moves toward the convex side, and work 

has to be done to increase the area. This work is supplied by the pressure difference moving the surface" 

(Adam, 1930, p. 12). "The magnitude of this pressure 

differential across a curved surface can be derived at 

once from the fact that work must be done to create 

fresh surface" (Sears, 1950, p. 323).

In figure 2 the pressure, p, of the air plus its water

FIGURE 2. A bubble of air blown at the end of a capillary tube, po is the normal 

pressure of the air and its water vapor, and p is increased pressure in the tube of 

the air and its water vapor. (After Sears, F. W., 1950, Mechanics, heat and sound: 

2d ed., Addison-Wesley Pub. Co., Inc., Reading, Mass., p. 323.)

vapor in the capillary tube is increased over the normal pressure, p0, of the air and its water vapor. An 

air bubble is formed at the lower end of the tube. When 

the bubble becomes unstable and is about to break 

away, its shape is very nearly that of a hemisphere of 

radius R. If A is the surface area of the hemisphere,

(3)

If the radius is increased to R+dR, the surface is 

increased by

(4)

Substituting this into equation 1, the work done in 

creating this additional surface is

Let us now find the work done by the pressure difference, p p0 . The net force against a small element of 

the hemispherical surface of area dS is

(P Po)dS (6)

This assumes that the increase of hydrostatic pressure 

from the water surface to the bottom of the bubble is 

negligible. In increasing the size of the bubble, this 

element of surface moves out radially through a distance dR. The work on this small element of the 

surface is the product of force and distance, or

(P po)dSdR (7)

When this work is integrated over all the bubblesurface elements (see equation 3), the total work done 

by the pressure forces is

dW=(p-p0)2*R*dR (8)

Equations 5 and 8 may be equated and simplified. 

Furthermore R, the radius of the hemispherical bubble, is very nearly the same as r, the radius of the 

tube. Therefore, the pressure difference between the 

concave and convex sides of the liquid interface is

2<r 2<r (9)

As shown by Adam (1930, p. 14), for any curved 

surface (shaped like a blowout patch, for example) the 

pressure difference is

(10)

where RI and R2 are the principal radii of curvature. 

RI and R2 are here taken as positive for a meniscus 

whose interface is concave to the vapor. However in 

normal usage R is taken as positive when the interface 

is convex to the vapor, as for a water bubble in air or 

a mercury meniscus. If normal usage were followed, 

it would be necessary to have minus signs in equations 

9 and 10 to have the pressure over the meniscus come 

out greater than the pressure below the meniscus, 

using the negative value of radius. It is largely to
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D5

avoid the use of these minus signs that R is taken as 

positive for the case of a meniscus concave to the vapor.

ANGLE OF CONTACT

The molecules in a liquid-vapor interface near a solid 

are affected by forces of cohesion (attractive forces of 

p. D2) exerted upon them by other molecules of the 

liquid and by forces of adhesion exerted upon them bv 

the molecules of the solid

The dot in figure 3 represents a molecule in the 

surface layer of a liquid, and the vectors Fa and Fe

FIGURE 3. Contact angle of a liquid-vapor surface near a solid. Fa is the adhesion 

force; Fe, the cohesion force, F, the resultant force, and 0, the angle of contact, 

(After Sears, F. W., 1950, Mechanics, heat, and sound: 2d ed., Addison-Wesley 

Pub. Co., Inc., Reading, Mass., p. 325.)

represent the adhesive and cohesive forces acting on 

that molecule.

If the magnitudes of Fa and Fc are as shown in figure 

3A, the resultant force is F. The liquid surface will 

curve upward near its contact with the solid because 

a liquid can be in equilibrium only when the force on 

its surface is perpendicular to the surface. The surface 

must be perpendicular to F and the angle of contact, 

0, is acute. If the forces of adhesion and cohesion have 

the relative magnitudes shown in figure 3JB, the liquid 

surface curves down near the solid contact, and 6 is 

obtuse.

643855 O 62 2

When the forces of adhesion and cohesion are such 

that the angle of contact is zero, the liquid tends to 

spread over a solid surface, or to "wet" the surface. 

When the forces are such that the angle is large, the 

liquid tends to bunch up in beads or droplets on the 

solid surface and is said to be nonwetting.

RISE OF WATER IN A CAPILLARY TUBE

Figure 4 shows a glass capillary inserted in water. If 

the radius of the tube and the forces of cohesion and

FIGURE 4. Rise of water in a capillary tube, po is the atmospheric pressure, p', 

the pressure under the meniscus, z, the height of capillary rise, R, the radius of 

curvature of the meniscus, r, the radius of the tube, and S, the angle of contact. 

(After Sears, F. W., 1950, Mechanics, heat and sound: 2d ed., Addison-Wesley 

Pub. Co., Inc., Reading, Mass., p. 326.)

adhesion are such as to give a meniscus that is curved 

concave upward, the water will rise in the tube. The 

following explanation of this capillary rise is adapted 

from Sears (1950, p. 327-328).

Let po be the total pressure of the atmosphere plus 

its water vapor above the flat surface outside the capillary tube. Under the flat surface, the hydrostatic 

pressure also is p0 . Neglecting the small variation of 

water-vapor pressure and air pressure over the small 

height, the total pressure above the meniscus also is p0 . 

Finally, for hydrostatic equilibrium the pressure at the 

level Q in the capillary tube also must equal p0 .
D6 FLUID MOVEMENT IN EARTH MATERIALS

In figure 4, R is the radius of curvature of the curved 

surface, r is the radius of the capillary tube, and 6 is the 

angle of contact. Therefore (Adam, 1930, p. 15):

R=

cos 6

(ID

where the tube is so narrow that the meniscus may be 

taken as spherical. Equation 9 gives the pressure 

difference across a meniscus. In the narrow tube, p0, 

atmospheric pressure, is on the concave side of the 

meniscus. Thus it is the greater pressure. Let p' be 

the lesser pressure on the convex side just below the 

meniscus. Then using this terminology in equation 9,

Substituting the value of R from equation 11,

. 2o- cos e P=P° 

From hydrostatics, the pressure at Q is

(12)

(13)

(14)

where z is the height of capillary rise, p is the density 

of liquid water, and g is the acceleration of gravity. 

Substituting the value of p' from equation 13 into 

equation 14,

2o- cos 0 . P»=PO - \-pgz (15)

Rearranging, the final equation for the rise of water in 

the capillary tube is

2o- cos e

(16) z=-

pgr

The angle of contact between clean water and clean 

glass is nearly zero. Therefore, as a convenient 

approximation, equation 16 is often written as

or as

pgr

2(T P9Z= 

(17)

(18)

These equations state in effect that the liquid ' 'rises or 

falls in the tube, until the height of the meniscus is 

such that the weight of the column of liquid adjusts 

the pressure inside and outside the tube to equalitv" 

(Adam, 1930, p. 295).

For a noncircular capillary interstice, equation 10 

instead of equation 9 would be used in 12, and the final 

form would be

(19)

Therefore, other things being equal, the height of capillary rise increases as the radius of the capillary tube 

decreases.

"Most organic liquids and water form zero angles 

with clean glass and silica; exceptions to this rule may 

occur if the glass has become dry and the liquid is 

advancing over the surface" (Adam, 1930, p. 180). 

The contact angle of water with a mineral surface 

varies with the state and cleanliness of the mineral 

surface.

"The angle assumed when a liquid advances to a 

position of rest over a dry solid surface is larger than 

the angle when the liquid recedes to a position of rest 

over a wetted surface, and this was called by Sulman 

(1919) the hysteresis of contact angle; he suggested that 

the advancing or retreating contact angles are unstable 

forms of a definite equilibrium contact angle (static 

angle)" (Partington, 1951, p. 166). Edser (1922, p. 

290), citing Sulman (1919), gives values for water from 

13° to 58° receding and from 62° to 95° advancing for 

different mineral surfaces.

Partington (1951, p. 167) credits Langmuir and Bartell 

and co-workers for stating that, with proper precautions, equilibrium contact angles can be obtained and 

measured. Wark and Cox (1932), reviewing more recent 

work than that of Edser, say that really clean surfaces 

of most minerals have very small contact angles. 

Throughout this report, it will be assumed that the 

contact angles between water and the mineral grains 

of the soil are small, so that their cosines may be taken 

as unity and equations 18 and 19 are applicable.

WATER VAPOR IX UNSATUBATED SOILS

Important phenomena in unsaturated soils result 

from the transfer of water molecules across liquidvapor surfaces and the diffusion of water vapor.

MOLECULAR CAUSES OF VAPOR PRESSURE

When a liquid evaporates, or vaporizes, molecules 

fly off against the forces of molecular attraction into 

the overlying vapor. If the vapor is confined over the 

liquid, some vapor molecules return to the surface and 

become liquid again (condense). Eventually, when the 

density of the vapor is such that the average number 

of molecules returning to the liquid per unit time is 

equal to the average number leaving per unit time, 

the vapor is in equilibrium with the liquid. A vapor 

in equilibrium with its liquid is said to be saturated, 

and the equilibrium pressure is called the vapor tension 

or vapor pressure.

VAPOR PRESSURE AND TEMPERATURE

The pressure of a vapor in equilibrium with its liquid 

depends largely upon the temperature, and this depend-
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D7

ence is described by the Clausius-Clapeyron equation. 

As written by Fermi (1937, p. 66), the equation is

dp _ X

dT~T(Vv -Vl}

(20)

where

dp=change in vapor pressure 

dT= change in absolute temperature 

X=heat of vaporization of the liquid at constant

temperature 

T= absolute temperature at which the change of

state occurs

V,=specific volume of the vapor 

F{=specific volume of the liquid

If it is assumed that V{ is negligible as compared 

with Vv, that the vapor satisfies the equation of state 

of an ideal gas, and that the heat of vaporization, X, 

is constant over a wide range of temperatures, a simple 

approximate formula shows the manner in which vapor 

pressure is related to temperature. As written by 

Fermi (1937, p. 67), it is

_XM

p=-Ce ~« T (2V 

where

C= a constant 

R = the gas constant 

M=the molecular weight of the vapor 

e=the exponential base, 2.71828 . . .

Thus, vapor pressure increases with temperature.

VAPOR PRESSURE AND AMOUNT OF DISSOLVED SOLUTE

Let p0 be the vapor pressure of a free pure solvent. 

Add an amount of dissolved material whose value is 

given by the mole fraction b. The vapor pressure of 

the solvent will be lowered to p according to the 

equation,

p=p(>e-t> (22)

as written by Edlefsen and Anderson (1943, p. 129).

VAPOR PRESSURE AND HYDROSTATIC PRESSURE IN THE

LIQUID

According to Edlefsen and Anderson (1943, p. 136), 

the relation

(23)

is true where dp is the change in vapor pressure caused 

by a change of hydrostatic pressure, dP. The subscript T denotes that these changes occur at constant 

temperature. v v and v t are the specific volumes of the 

vapor and liquid, respectively.

The fraction Vi/v, is in general very small. Therefore, the effect of a change in the hydrostatic pressure

upon the pressure of the vapor in equilibrium with a 

liquid is small.

Edlefsen and Anderson (1943, p. 138) describe an 

equation that gives the increase in vapor pressure of 

water, Ap, in atmospheres, corresponding to the application of a hydrostatic pressure of P atmospheres. It 

is assumed that the normal vapor pressure p0, in 

atmospheres, of free water at the temperature T, is 

known. The equation is

1.002(P 2X105P2 ) 

= 4.548r

Po

(24)

According to Edlefsen and Anderson this formula 

shows that, if a hydrostatic pressure of 20 atmospheres 

is applied to water at 27°C, the vapor pressure of the 

water is increased from 0.035 to approximately 0.03551 

atmosphere. Thus, a relatively large change in hydrostatic pressure causes a relatively small change in vapor 

pressure. Similarly, if a tension of 20 atmospheres is 

placed on the water, the vapor pressure of the water 

will be decreased by 0.00051 atmosphere, and the 

humidity will be 98.5 percent of the humidity over a 

free water surface at the same temperature. A soil in 

this condition is close to the permanent wilting percentage, which is commonly taken at a suction of 15 

atmospheres. Thus, as a soil dries, the vapor pressure 

and the humidity relative to that over a free water surface are decreased by only a small amount.

BAROMETRIC EQUATION

The barometric equation, or Laplace law of atmospheric pressure, gives the variation of gas pressure with 

height in a uniform gravitational field. It is usually 

used to determine the variation of air pressure with 

height. According to Edlefsen and Anderson (1943, 

p. 129), "It applies equally well, however, to the variation of vapor pressure with height above a free liquid 

surface, as, for example, water, when the whole system 

has come to equilibrium at the same temperature, in 

a uniform gravitational field." Because of the importance of the Laplace law in this development, the 

following derivation is included.

Consider a vertical column of gas of unit crosssectional area (fig. 5).

According to the ideal-gas law,

pv=RT (25) 

where

p=the pressure, in dynes per square centimeter,

at a distance z above the reference point 

v=specific volume of the gas 

T= absolute temperature 

R=gas constant, in ergs per degree gram
D8 FLUID MOVEMENT IN EARTH MATERIALS

FIGURE 5. Column of vapor in a gravitational field, pit is the vapor pressure at the 

free water surface and p, vapor pressure at a height, z above the free water surface. 

(After Edlefsen and Anderson, 1943, p. 129.)

IR is equal to the universal gas constant, 83,140,000 

ergs per degree gram mole divided by the weight of a 

gram mole. The molecular weight, or the weight of a 

gram mole of water vapor, is 18.02 grams. For water 

vapor, IR would therefore be about 4,620,000 ergs per 

degree gram.

Rewriting equation 25, and remembering that specific volume is the inverse of density,

, p (26)

where 7 is the density of the gas, in grams per cubic 

centimeter.

The vapor pressure of the gas in the column in figure 

5 changes by the amount dp in going from A to B. 

This decrease in pressure is equal to the weight of the 

gas or vapor in the length of column represented by dz. 

The weight is ygdz. Thus,

-dp=ygdz (27) 

Eliminating 7 by means of equation 26 and rearranging,

(28)

Integrating from 2=0 at the free water surface where 

p=p0 , to z=z where p=p,

P __~ 9L 

and

(Edlefsen and Anderson, 1943, p. 130). 

Equation 30 can be put into the final form

(29)

(30)

__£2 

i /n_ ^ R * (31)

Thus, if p0 is the vapor pressure at the free water surface, 

where 2=0, equation 31 determines the pressure p 

at a height z. It is interesting that the rate of change 

of vapor pressure with height outside a capillary tube 

is much smaller as predicted by equation 31 than the 

rate of change of hydrostatic pressure with height 

within the tube, pgz. This is as predicted by equation 23.

KELVIN EQUATION

DERIVATION

The vapor pressure over a concave water meniscus 

is less than the vapor pressure over a free flat water 

surface. Thus, a vapor-pressure deficiency exists 

over a meniscus in a capillary tube when compared 

with the vapor pressure over a free flat water surface. 

The relation between the vapor pressure and the radius 

of curvature of the meniscus is described by the Kelvin 

equation (Thompson, 1871). Because of the importance of this equation, it is now derived for a capillary 

surface connected to a flat surface following the derivation of W. O. Smith (1936, p. 228-230).

Let the system in figure 6 be isolated so that there 

is no air pressure. Then pQ is the water-vapor pressure 

over the flat surface. The hydrostatic pressure in 

the liquid just under the flat surface also is equal 

to po and is due to the weight of the overlying water 

vapor. Let pa be the water-vapor pressure just over 

an element, dS, of capillary surface, and let /> ' be the 

hydrostatic pressure just under the element, dS, of 

capillary surface. The hydrostatic pressure, pa , under 

the element of curved surface equals the vapor pressure, 

pa, minus the pressure difference across the curved 

surface due to surface tension.

If the system in figure 6 were open to the air, all 

the pressures would be increased by the atmospheric 

pressure. Furthermore, the change in atmospheric 

pressure with height would have to be considered, as 

well as the change in water-vapor pressure with height. 

Therefore when suitable adjustments are made the
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D9

FIGURE 6. A curved liquid-vapor surface! n contact with a free flat liquid surface, 

po is the water-vapor pressure over the flat surface, pa , the water-vapor pressure 

just over and p£, the hydrostatic pressure just under an element, dS, of capillary 

surface at a height, above the flat surface. (After Smith, W. O., 1936, p. 229).

following derivation is applicable regardless of the 

type of gas hi the system:

The pressure in the water in the capillary tube at 

the same level as the flat surface is

Po=Pa+pgz (32)

Similarly, the pressure in the vapor outside the capillary 

tube just over the flat surface is

po=pa+ I vgdz

Jo

(33)

where y is the density of the vapor. Equating equations 32 and 33,

Pa+ I ygdz=pa + pgz (34) Jo 

This can be rearranged to

Pa~Pa = pgz ygdz (35) 

Jo

The value for the difference of pressure across the 

curved surface can now be substituted from equation 

10:

2 (36)

taking the radii of curvature as positive.

The two terms on the right side of equation 36 give 

the pressure differences between the flat and curved 

water surfaces in the water and vapor columns, respectively. To make them more usable, everything 

will be put in terms of pressure. This can be done 

by starting with the simple relationship between 

pressure and height in the vapor, equation 27.

dp= ygdz

Integrating between 0, p0 and z, p,

fp PZ

dp= ygd, 

J PO Jo

(p po) = 

Jo

and

(37)

(38)

(39)

Equation 39 may be substituted for the second term 

on the right side of equation 36, giving

(40)

To put the first term on the right side of equation 

40 into terms of pressure, equation 37 is rewritten as

79

(41)

Setting up the integral between 0, p0 and z, p,

pd, p* (42)

Jo Jp0 79 

Integrate the left side, and

' ^± (43)

We cannot integrate the right side of equation 43 

because y, the density of the vapor, varies with pressure. To eliminate y from equation 43, consider Boyle's 

law:

(44)

To Po

or

Po

(45)

Putting this value of y into equation 43, it becomes

z=-£*- f'& (46) 

Wo J Pn P

Integrating,

po

(47)
DIG FLUID MOVEMENT IN EARTH MATERIALS

Substituting into equation 40, the Kelvin equation is

^> (48)

Equation 40 can be simplified if it is assumed that 

(p Po) is negligible and that Ri=R2 =r, as is approximately true for a capillary tube of circular cross section. Then 48 becomes

Po r pp0

From the ideal gas law,

(49)

(50)

where v v0 is the specific volume of the vapor at p0. 

Then,

R77

(51)

Substituting for p0 in equation 49 and remembering 

that p, the density of water, is the inverse of v t, the 

specific volume of water, we arrive at the following 

forms of the Kelvin equation:

__ 

r RT

and

P=p0 e r

(52)

(53)

The minus is in equation 53 because the values of R1} 

R2 , and r were assumed as positive if the meniscus is 

concave to the vapor. If positive values of r are 

substituted into equation 53, the pressure over the 

meniscus comes out less than that over a flat surface. 

Using the form shown in 53, «the radius of curvature is 

negative when the meniscus is convex to the vapor, as 

in a raindrop. If a negative value of r is inserted in the 

equation, the vapor pressure over a raindrop comes out 

greater than that over a flat surface.

Equation 53 applies where the capillary surface is in 

liquid continuity with the flat water surface. The 

equation may be derived, from thermodynamic considerations (Edlefsen and Anderson, 1943, p. 142-146), 

for a curved liquid-vapor interface that is separated 

from the flat liquid surface.

For the discontinuous-liquid system, we again take 

the radius of curvature of the interface to be positive 

for a liquid surface concave to the vapor and negative 

for a surface convex to the vapor. From the equation, 

the smaller the radius of curvature of a capillary 

meniscus, or the greater the curvature of the meniscus,

the greater is the vapor-pressure deficiency above the 

meniscus.

SIGNIFICANCE

Equating the exponential terms in equation 53, the 

Kelvin equation, and equation 31, the barometric 

equation, rearranging terms, and substituting density 

for the inverse of the specific volume of the water, we 

arrive back at equation 18

pgz=-

2er (54)

Equation 54 does not contain any of the properties of 

the gas. Thus, the height of capillary rise of a given 

liquid in a tube or interstice of given radius is the same 

whether the surrounding atmosphere consists of water 

vapor alone or water vapor and air.

The Kelvin equation may be looked at in several 

ways. If suction is applied to a wet sand, the vapor 

pressure is decreased. Therefore, the radii of curvature 

of the equilibrium minisci are greatly reduced, and the 

menisci are drawn deeper into the narrowing necks of 

the soil pores. The hydrostatic pressure in the water 

on the convex sides of the menisci also is decreased 

because of the pressure differentials across such curved 

surfaces.

Adam (1941, p. 13-14) gives a physical explanation 

of these phenomena:

The vapor pressure over a convex surface is greater than that 

over a plane; and over a concave surface it is less. The difference 

depends on the fact that condensation of vapor on a small 

convex drop of a liquid increases its surface area, so that the 

surface tension tends to oppose the condensation and to increase 

the vapor pressure. On a plane surface condensation does not 

alter the surface area, and oh a concave surface the surface area 

is diminished by condensation of more vapor, so that the surface 

tension aids condensation in this case.

Adam (1930, p. 22) gives solutions of the Kelvin 

equation for the increase of vapor pressure, p/po, over a 

drop of water of radius r, at 20° C, with p0 equal to 

17.5 mm of mercury:

r (cm)

-10-*

-10-5

-10-7

P/Po

1.001

1.011

1.114

2.95

Thus, a small change in vapor pressure greatly changes 

the radius of the equilibrium meniscus.

To demonstrate the use of these equations, assume 

that a suction equivalent to 15 atmospheres is applied 

to a soil. This value of negative hydrostatic pressure 

is commonly taken for the permanent wilting 

percentage. Let us compute the radius of the circular 

pore in which the menisci will be lodged at equilibrium.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY Dll

For this case,

Vi l cm3 per gram 

v»= 57,800 cm3 per gram 

dP=l5 atmospheres =15,194,000 dynes per cm2

Therefore, the vapor-pressure deficiency can be computed from equation 23. It is dp=263 dynes per cm2 . 

Thus, even at the permanent wilting percentage, the 

vapor-pressure deficiency in the soil is very small. 

(See also p. D7.)

To determine the equilibrium meniscus, or the pore 

that has a sufficiently small radius to support the equilibrium meniscus, we refer to the Kelvin equation. For 

a temperature of 20°C, p0, the vapor pressure over a 

flat water surface, may be taken as 17.5 mm of 

mercury.

2>o=17.5 mm of mercury=23,300 dynes per cm2

<r=73.5 dynes per cm for water 

Vi = 1 cm3 per gram

T=293° Kelvin

______ergs

= 83,100,000

degree gram-mole 

ergs

am

=4'620'000 degree gram f°r Wat"r VapOT 

p=po dp=23,100 dynes per cm2

Substituting into equation 52 or 53, the radius of the 

equilibrium meniscus is about 0.00001 cm, or 0.0001 

mm.

The same result can be found by substituting into 

equation 54 a value of z equivalent to 15,500 cm. 

This is the height of a hanging column of water that 

would give a tension of 15 atmospheres if the hanging 

column of water could support such tensions.

As will be discussed later, other forces are generally 

more important than surface tension in pores of this 

small size. Therefore, this computation is more of a 

demonstration than an accurate portrayal of the meniscus size in a soil at the permanent wilting percentage.

Because the Kelvin and barometric equations apply 

whether or not a liquid continuum exists, equation 54 

also applies to both cases. Whether or not a pore is 

in liquid continuity with the water table, the radius of 

curvature of the meniscus at equilibrium can be computed from the pore height above the water table by 

equation 54.

It may seem surprising that equation 54 applies 

equally when there is a liquid continuum, as in the 

capillary fringe, and when there is not, as above the 

capillary fringe. However, as discussed later, both 

regions fall within the same gravitational field. If the 

value of g is assumed to be constant the gradient of

gravitational potential is constant throughout both 

regions.

Equation 54, in the case of a liquid continuum, may 

be looked upon as a statement of the equivalence of the 

pressures exerted by surface tension and by the weight 

of a hanging column of water in continuity with the 

water table. If a break occurred halfway up this column, equation 54 would apply to the lower half of the 

column and would predict the radius of curvature of 

the meniscus at the top of the half column. The value 

of z in equation 54 would be smaller for the meniscus 

at the top of the half column than for the meniscus at 

the top of the whole column. Therefore, the radius 

of curvature would be larger for the meniscus at the top 

of the half column. This radius of curvature would be 

greater than the radius of the tube and would develop 

on the cutoff lip at the top of the lower half-column 

tube. The meniscus at the bottom of the upper half 

column would be in vapor equilibrium with the meniscus 

at the top of the bottom half column and would have 

the same radius of curvature. Therefore, the forces 

at the midpoint of the column would be the same as if 

there were no break in the column, and equation 54 

would apply to the entire column. The same reasoning 

can be projected to cases where there are progressively 

increasing numbers of breaks in the column. Eventually, as the limit is approached, we arrive at the case 

of an isolated pore separated from the water table by 

vapor-filled pores.

Equations 31, 53, and 54 may be looked at in another 

way. Figure 7 shows a capillary tube immersed in 

water. The pressure of the water vapor relative to that 

at the water table decreases with height according to 

equation 31.

Because the capillary tube has a fixed radius, only 

one radius of curvature is possible for the meniscus. 

A given vapor-pressure deficiency is associated with that 

meniscus curvature according to equation 53. At 

equilibrium, the meniscus must be located so that the 

vapor-pressure deficiency above it is the same as the 

pressure deficiency in the adjacent exterior vapor as 

given by equation 31.

Assume that the radius of the tube is such that the 

vapor-pressure deficiency above the meniscus is 10 

dynes per cm2. Assume also that water is run into the 

tube until it is filled to level A. Then the vaporpressure deficiency over the meniscus would be less 

than that in the adjacent exterior vapor, and a condition of unbalance would exist. Water would drain 

from the tube immediately, and the meniscus would 

fall until it stablized at position B. The vapor-pressure 

deficiency over the meniscus would then be the same 

as that in the adjacent exterior vapor, and the system 

would be in equilibrium.
D12 FLUID MOVEMENT IN EARTH MATERIALS

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FIGURE 7. Position of the meniscus in a capillary tube and adjacent vapor pressure*.

An analogous situation exists in the wedge-shaped 

pore in figure 8. Because the wedge is fixed in space 

relative to the water table, the vapor-pressure deficiency over it must be the same as that in the adjacent 

exterior vapor, 20 dynes per cm2 , at equilibrium. If 

the wedge is filled to the dashed line, where the meniscus 

curvature is smaller than that required for equilibrium, 

the vapor-pressure deficiency might be only 10 dynes 

per cm2 . Water would evaporate and diffuse away 

from the meniscus, which would be drawn into the 

narrower part of the wedge. At equilibrium, the curvature would be increased so that the pressure deficiency 

over the meniscus would be 20 dynes per cm2 .

For a capillary tube immersed in mercury, the meniscus would be convex to the vapor. According to the 

Kelvin equation, the vapor pressure over the meniscus 

in the capillary tube would be greater than that over 

a flat surface. Therefore, the meniscus would have to 

be depressed below the flat surface to a level where 

the increase of hydrostatic pressure in the exterior 

liquid would equal the increase in pressure over the 

meniscus.

The foregoing development and equation 54 apply 

where surface-tension forces are dominant. In clay 

soils, in very small interstices, and very close to the 

particle surfaces, other forces become more important.

Then equation 54 no longer applies, as is discussed 

later.

As a result of analyses similar to the foregoing it is 

sometimes assumed that the water in an unsaturated 

soil is under considerable tension. Although under 

certain conditions water has considerable tensile 

strength, the tensile strength is very small when the 

water contains dissolved gases. It is small also if the 

absolute pressure approaches the vapor pressure of the 

water, and bubbles of water vapor form and disrupt 

the liquid phase. It is likely "that the fraction of 

water which is actually in tension is never large except 

perhaps at very high water contents and small tension. 

This conclusion was also reached by Edlefsen and 

Anderson. We regard apparently large values of 

soil-moisture tension or soil suction to be artifacts 

accounted for by the influence of the electrical double 

layer" (Bolt and Miller, 1958, p. 928). (See p. D20.)

DIFFUSION

Water vapor moves by diffusion. The distribution 

of solutes throughout a liquid also occurs by diffusion. 

Finally, osmotic effects are due to diffusion. Therefore, the nature of the diffusion process will be 

considered in this report.

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Free water surface

FIGURE 8. Position of the meniscus in a wedge-shaped pore and adjacent vapor

pressures.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D13

Gas released in an evacuated vessel distributes itself 

uniformly throughout the vessel, as explained by Meyer 

and Anderson (1939, p. 83-92) in their elementary 

discussion of diffusion. The distribution is accomplished by the thermal activity of the gas molecules and 

is an example of diffusion.

If the gas is released in a vessel containing air, it 

takes longer for the gas to move to all parts of the 

vessel. The freedom of movement of the gas molecules 

is impeded by the air molecules, and the diffusion is 

retarded. If the pressure of air in the jar is increased 

by reducing its volume (which is equivalent to increasing the concentration of air molecules), the rate of 

diffusion is still less.

Two gases in separate vessels mix by diffusion when 

the vessels are put in communication. Liquids also 

diffuse into each other, though more slowly than gases 

because of the closeness of the molecules. Solids also 

exhibit diffusion (Duff, 1937, p. 104).

If the temperature remains constant, the partial or 

diffusion pressure of a gas, solvent, or liquid is directly 

proportional to its own concentration, or the number of 

molecules per unit volume. Dalton's law of partial 

pressures states that the total pressure exerted by a 

mixture of real gases is the sum of the pressures that 

the gases would exert if each occupied the whole space 

alone. To visualize this, assume that the molecules of 

a gas are far apart. Molecules of other gases may be 

slipped in among them without perceptibly altering 

the frequency with which they collide with the vessel 

wall and without altering the pressure that the first 

gas exerts (Deming, 1944, p. 85). Hildebrand (1947, 

p. 50) explains this by pointing out that the molecules 

of one gas species would "maintain the same average 

kinetic energy at the same temperature regardless of 

the presence of any other species of molecules, and 

therefore the part of the pressure which is due to the 

impacts on the vessel walls of one species, called the 

partial pressure of that gas, would be the same no 

matter what other gases are present."

Diffusion occurs only when the concentration or 

partial pressure of the diffusing substance is not uniform throughout the system. It is different from mass 

movements such as wind, current, or convection. In 

mass movement the moving units are not single molecules, but generally extensive assemblages of molecules.

Each substance in a system diffuses in a direction 

and at a rate that depends on its own gradient of 

diffusion pressure (partial pressure) or on the gradient 

of concentration. The diffusion-pressure gradient 

equals the difference in diffusion pressure between two 

regions, divided by the distance through which the 

diffusing molecules travel. According to the principle 

of independent diffusion, the direction in which any

643855 O 62 3

substance will diffuse is unaffected by the gradients or 

the directions or rates of diffusion of other substances 

in the system. In any system several substances may 

diffuse in different directions at the same time. In 

fact a substance can diffuse into a region of smaller 

partial pressure but greater total pressure, and the 

total pressure in the region will then be even greater 

than the total pressure in the area from which diffusion 

occurred.

The rate of diffusion of a gas varies directly with its 

diffusion gradient, directly with its temperature, inversely with the square root of its density, and inversely 

with the concentration of the medium through which 

it is diffusing. The diffusion of a solute through a 

solvent is much slower than the diffusion of gases 

because of the densely packed molecules of the liquid 

which impede the diffusion of the dissolved molecules 

or ions. The diffusion of solute particles is governed 

by principles very similar to those which control the 

diffusion of gases. Although the freedom of movement 

of liquid molecules is to some extent restrained by 

internal cohesive forces, the molecules also possess 

thermal energy. Like gases and solutes, liquids exhibit 

diffusion phenomena. If water is brought into contact 

with another liquid, such as ether, with which it is 

only slightly miscible, a slow diffusion of the molecules 

of the ether takes place into the water and of water 

into the ether, and the process continues until the two 

liquids are mutually saturated.

Diffusion is described by the well-known onedirection, nonlinear, concentration-dependent diffusivity equation (see, for example, Bruce and Klute, 1956, 

p. 458^59):

where

fi=the concentration of ions or molecules

x = distance

is the diffusivity which is a function of the 

concentration, fi. This equation states that the change

in concentration with time, ^rr> is a function of the 

rate of change with distance, ^r-» of the product of the

da diffusivity, D(Q), and the concentration gradient,-^-

OSMOSIS

Suppose a vessel containing a sucrose solution is 

placed in contact with one containing plain water. 

There will be a diffusion-pressure gradient for sucrose
D14 FLUID MOVEMENT IN EARTH MATERIALS

from the solution toward the water and a diffusionpressure gradient for water from the water toward the 

solution. According to the principle of independent 

diffusion, water molecules and sucrose molecules will 

diffuse in opposite directions until the solute concentration and the diffusion pressure become uniform.

Suppose that the two vessels are connected through 

a differentially permeable or semipermeable membrane 

that is permeable to the solvent (water) but impermeable 

to the solute (sucrose). Water will diffuse into the 

sucrose solution but the sucrose will not be able to 

penetrate the membrane and diffuse into the pure water. 

This is called osmosis. As a result of the net movement 

of solvent into the solution, the total pressure in the 

sohition becomes greater than that in the pure solvent 

at equilibrium. We say net movement because solvent 

actually moves in both directions across the membrane. 

However, at equilibrium, the movements in both directions are equal. Thus, osmosis may be considered as 

the building up of a differential total pressure in one 

part of a system through the restricting of diffusion in 

one direction and the allowing of diffusion in the other 

direction by means of a differentially permeable membrane.

"Osmotic pressure may be defined as the maximum 

pressure which can be developed in a solution when 

separated from pure water by a rigid membrane permeable only to water" (Meyer and Anderson, 1939, 

p. 94). The osmotic pressure of a solution can be 

determined by enclosing it in a vessel formed of a rigid 

membrane permeable only to water, immersing this 

vessel in pure water, and exerting just enough pressure 

on the solution to prevent any increase in its volume due 

to the entrance of water. The osmotic pressure of the 

solution is quantitatively equal to the imposed hydrostatic pressure (Meyer and Anderson, 1939, p. 94-95).

It is common practice to speak of solutions possessing 

osmotic pressures whether or not they are under such 

conditions that a hydrostatic pressure can develop within them. In other words, the term "osmotic pressure" 

is commonly used to denote the potential maximum 

hydrostatic pressure which would develop in a solution 

were it placed under the necessary conditions. It will 

be used in this way in this report.

Osmotic pressure is used sometimes to designate also 

the actual hydrostatic pressures or turgor pressures 

developed as a result of osmosis. Actual hydrostatic 

(turgor) pressures developed during osmosis seldom 

equal the osmotic, or potential maximum, pressure. 

For example, assume that a solution having an osmotic, 

or potential maximum, pressure of 12 atmospheres is 

enclosed in a stoppered vessel formed of a membrane 

permeable only to water. Assume further that the 

vessel in turn is immersed in a solution having an

osmotic pressure of 8 atmospheres. Water will diffuse 

inward until, at equilibrium, the actual hydrostatic 

(turgor) pressure developed in the internal solution will 

be 4 atmospheres. Its osmotic, or potential maximum, 

pressure will still be nearly 12 atmospheres. Even if 

the external liquid is pure water, the actual hydrostatic 

(turgor) pressure developed in the internal solution will 

not be equal to its original osmotic pressure unless the 

membrane is completely inelastic (Meyer and Anderson, 

1939, p. 94-95).

Because osmotic pressures are defined in terms of 

final hydrostatic (turgor) pressures developed by 

osmosis, flow occurs in the direction of greater osmotic 

pressure. For example, flow occurs from pure water 

(lower osmotic pressure) into a solution (higher osmotic 

pressure). However, flow is always from the region 

of greater to the region of lesser partial pressure of the 

diffusing substance. This is because the region of 

greater osmotic pressure is the region of lesser partial 

pressure of the diffusing substance.

When a substance is dissolved in water, the diffusion 

pressure of the water in the resulting solution is decreased and its osmotic pressure increased as compared 

with that of pure water at the same temperature and 

pressure. The diminution of diffusion pressure is proportional, within a wide range of solution concentrations, to the number of solute particles in a given 

volume of the solvent.

The osmotic pressure of a solution is a measure of the 

diffusion-pressure deficit of the water in that solution. 

Suppose that solution A has an osmotic pressure of 20 

atmospheres and solution B an osmotic pressure of 12 

atmospheres. Then the diffusion-pressure deficit of A 

is 20 atmospheres, and that of B is 12 atmospheres. 

Water moves toward solution A where the diffusionpressure deficit, or the osmotic pressure, is greater. 

After the water movement, solution A will exert an 

actual hydrostatic (turgor) pressure of 8 atmospheres 

at equilibrium, and the membrane will exert a wall 

pressure of 8 atmospheres to contain it. Because the 

actual pressure is greater in A at equilibrium, the 

diffusion pressure of water in A will increase, and the 

diffusion-pressure deficit of the water in A will no 

longer be 20 atmospheres, but will be 12 atmospheres. 

Thus at equilibrium the increase in actual hydrostatic, 

or turgor, pressure inside the solution raises the diffusion pressure of the water in the solution to equal that 

in the outside water (Meyer and Anderson, 1939, p. 97).

FORCE, WORK, ENERGY, POTENTIAL, AND GRADIENT

The following sections of this report are devoted 

largely to discussions of potentials and forces in soil 

water. Therefore, it is advisable to discuss first the 

meaning of force, work, energy, potential, and gradient.
REVIEW OF SOME ELEME1STTS OF SOIL-MOISTURE THEORY D15

Force is commonly defined as a push or pull (Sears, 

1950, p. 1). It may be thought of as the cause of 

acceleration or the cause of a change in the state of rest 

or state of uniform motion of a body. The relation 

between the force on a body and its mass and acceleration is described by Newton's second law of motion 

(Sears, 1950, p. 78)

F=ma (56) 

where

m=mass, in grams

a= acceleration, in centimeters per second squared 

F= force, in gram centimeters per second squared 

= dynes

When a force acts on a body the product of the force 

and the distance the body moves in the direction of the 

force is the work performed by the force. Mathematically,

W=Fx cos a (57) 

where

F= the force causing the motion, in dynes 

x=ihe distance that the body is moved by the

force, in centimeters 

a=the angle between the direction that the force

is acting and the direction that the body

is moving 

W=the work done, in dyne centimeters, or ergs

Energy is the capacity for doing work. A moving 

body possesses kinetic energy equal to the work it can 

do before it is brought to rest. By definition it is equal 

to the product of one-half the mass and the square of 

the velocity of the body.

Potential energy is the capacity that a body or 

system of bodies has for doing work by virtue of its 

position or configuration (Smith, A. W., 1948, p. 45). 

For example, a spring expanded beyond its unstretched 

length has potential energy because it can do work 

when it contracts. Similarly, a body at some elevation 

above a gravitational datum has potential energy with 

respect to that datum because it can do work as it/falls 

from the higher elevation to the datum. The amount 

of potential energy it has is proportional to the height 

above the datum. In other words, the higher it is, the 

more work it can do upon falling to the datum. Because 

energy is the capacity to do work, it has the same 

dimensional units as work, ergs.

A scalar "is any quantity which although having 

magnitude does not involve direction. For example, 

mass, density, temperature, energy, quantity of heat, 

electric charge, potential, ocean depths, rainfall, 

numerical statistics such as birth rates, mortality or

population, are all scalar quantities" (Coffin, 1911, p. 1). 

Any quantity which involves both magnitude and 

direction is a vector quantity. "Any vector quantity 

may be represented graphically by an arrow" (Coffin, 

1911, p. 1).

According to d'Abro (1951, p. 71), the name" field" 

is given to the continuous distribution of some "condition" prevailing throughout a continuum. When the 

condition is adequately described at each point of space 

by a scalar, it is known as a scalar field. Temperature 

is such a condition and the temperature distribution 

throughout a volume is a physical illustration of a 

scalar field. In many cases the condition at each point 

of space has a direction as well as a magnitude, and 

the field is a vector field. A field of force or the field 

defined by the instantaneous velocities of the different 

points of fluid in motion are illustrations of vector 

fields.

Consider a number of points in space. Suppose 

that with each of those points there is an associated 

value of potential. Potential is the work required to 

take a unit mass or volume from a datum to the point 

in question against the field forces. The potential 

might be due to gravitational, mechanical, or electrical 

causes. The potential at a point may be thought of as 

the potential energy of a unit body at the point. Because energy has no direction, the potential at the 

point is a scalar property or a scalar point function. 

Its value depends only upon the values of the coordinates of the point. Suppose further that the scalar 

point functions together with their first space derivatives are continuous and singly valued functions. 

Then the ensemble of points in the given region together 

with the corresponding values of the scalar point 

function "constitute what is called the Field of the 

Scalar Point Function" (Wills, 1931, p. 75). Examples 

of such fields are gravitational or electrical fields.

A level or equipotential surface of a scalar point 

function is a surface for all points of which the function 

has the same value (Wills, 1931, p. 76). For example, 

all points at the same elevation have the same gravitational potential and can be connected by such an 

equipotential surface. For another example, the 

equipotential surfaces around an isolated electrostatic 

point charge would consist of concentric shells.

Consider next an ensemble of points each of which 

is associated with a vector function the value of which 

depends only upon the coordinates of the point. Examples would be the gravitational force field associated 

with a field of gravitational potential, the velocity of a 

moving fluid associated with a field of pressure potential, 

or an electric force field due to a distribution of electric 

charges. The ensemble of points in the given region, 

together with the corresponding values of the vector
D16 FLUID MOVEMENT IN EARTH MATERIALS

point function, constitute "the Field of the Vector 

Point Function in the region" (Wills, 1931, p. 75). 

The potential gradient at a point is the rate of change 

of potential with distance measured in the direction in 

which this variation is a maximum (Gray, 1958, p. 219). 

As shown in figure 9, the ground-water gradient would

U

FIGURE 9. Ground-water gradient= .

be -^ where As is the difference in elevation of the

water table at two wells a horizontal distance Aa: cos 0 

apart, where 0 is the angle between the inclination of 

the water table and the horizontal. 

"The symbol V, read 'del,' defined by writing:

T d , - d . 7 d

^ ^~+J 5: \-k -^~ 

ox J by ds (58)

is an operator which, acting upon the scalar point 

function u, produces the gradient of this function" 

(Wills, 1931, p. 77). x, y, and z are the rectangular 

coordinates of the system in which u is defined. ~i, ], 

and k are three mutually perpendicular unit vectors, 

which are present in the operator because the gradient 

of the scalar function is a vector. A bar over a term 

indicates that the term is a vector quantity. Therefore, 

the gradient of u is

=-

Ox

. 7\-k 5 (59)

where u is a scalar point function such as potential. 

"Thus, the gradient of a scalar field is a vector field, 

the vector at any point having a magnitude equal to 

the most rapid rate of increase of * * * the scalar 

point-function * * * at the point and in the direction 

of this fastest rate of increase, i.e., perpendicular to the 

level surface at the point" (Hague, 1939, p. 35).

The total change of the value of u in a distance dr, is 

du. This is the scalar product of the gradient and the 

distance, or

du=d~r Vu (60)

where the dot indicates the scalar product (Wills, 1931, 

p. 77).

The gradient of a scalar field is a vector field as 

shown by equations 58 and 59. Furthermore, because 

the direction of maximum change is perpendicular to 

the equipotential surfaces, the gradient of a scalar field 

denotes a vector field such that the vectors are normal 

to the equipotential surfaces. Such fields, where the 

equipotential surface and the vectors are perpendicular, 

are said to be orthogonal. In effect, for hydraulics this 

simply expresses the fact that water flowing down a hill 

follows the steepest grade. It is often valuable to map 

fields of associated scalar and vector point functions. 

An example is given in figure 10. In studies of graviFIGURE 10. Approximate flow net in the vicinity of an irrigation pond. The dashed 

lines are equipotential lines, or lines of equal hydraulic head, in feet. The solid 

lines are flow lines which enclose areas of equal ground-water flow.

tation, the lines or surfaces of equal potential are lines 

or surfaces of equal gravitational potential. The 

orthogonal vectors are lines of equal gravitational force. 

In hydraulics, a map of scalar and vector point 

functions is called a flow net (fig. 10). The scalar field 

consists of distributions of values of pressure or head. 

Equipotential surfaces are surfaces of constant pressure 

or head. The lines orthogonal to the scalar field map 

the vector field and are streamlines which indicate 

directions of flow.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D17

In hydraulics, a velocity potential is often defined as 

a scalar function of space and time such that its space 

rate of change with respect to a given direction, or its 

gradient in that direction, is the fluid velocity in that 

direction. It is analogous to the force potential, whose 

gradient in a given direction is the force intensity in 

that direction. In such a flow net, as in figure 10, flow 

lines, or streamlines or flow tubes, include elements of 

equal flow. Therefore they are closer together in areas 

of greater velocity.

Rewriting equation 57, and changing the notation so 

that F stands for force, which is always a vector,

W=Fx cos a (61)

Because potential energy is the ability a body has to 

do work by virtue of its position or configuration, W 

is the potential energy of the body after it is moved a 

distance from a datum. The negative of the work done 

by a conservative field of force is the potential u, and 

according to Joos (1934, p. 81)

dW=Fdx=-du (62)

The minus sign is necessary because the potential decreases in the direction of the force or flow.

Suppose we want the force. It is simply obtained 

from

F= -Vu (63)

Thus, "every conservative force may be represented as 

the negative of the gradient of a scalar point function, 

the potential" (Joos, 1934, p. 81).

POTENTIALS IN SOIL WATER

In soil moisture, the

total potential is defined as the minimum energy per gram of 

water which must be expended in order to transport an infinitesimal test body of water from a specified reference state to 

any point within the liquid phase of a soil-water system which 

is in a state of rest. The reference state is commonly taken to 

be a pool of pure water at the same temperature with a flat 

surface exposed to atmospheric pressure at a known elevation. 

Under conditions in which the soil will spontaneously adsorb 

water from the reference state, the potential is a negative 

quantity. In any soil-water system in a state of rest, the total 

potential does not vary from point to point in the system (Bolt 

and Miller, 1958, p. 918).

The total potential depends upon the individual 

potential fields and upon individual force fields which 

affect the test body of water as the transfer is made. 

In unsaturated soils these potentials are presumed to be 

due to (a) gravitational attraction, (b) hydrostaticpressure difference, or gradient of pressure potential, 

(c) osmotic-pressure differences due to differences of 

content both of soluble salts and of anions and cations

in Gouy double layers associated with the solid surfaces, 

(d) adhesion of water to the solid surfaces, (e) temperature gradients, and (f) gradients of chemical potential 

due to changes in solute concentration. If the last 

two of these are considered to be negligible, the total 

potential at any point is the scalar sum of the potentials 

due to the first four effects (Childs and Collis-George, 

1948, p. 78). Thermodynamic considerations and the 

use of the free-energy concept are necessary to include 

the thermal and chemical potentials (Edlefsen and 

Anderson, 1943).

GRAVITATIONAL POTENTIAL

All terrestrial water lies within the earth's gravitational field. Gravitational potential is the energy 

required to move an infinitesimal mass or volume of 

water from a datum where gravitational potential is 

arbitrarily taken as zero, to a given position in the field 

against the attraction of gravity. According to this 

definition of gravitational potential,

Q=F.z (64)

where

6?= the gravitational potential, or gravitational 

energy, at an elevation z above the datum, 

where the gravitational potential is arbitrarily taken as zero. It is the work done, 

in ergs, in moving an infinitesimal mass or 

volume of water from the datum to the 

height z, in centimeters.

Fg=the force of the gravitational field against 

which the water is moved, in dynes.

The relation between Q and z is positive because the 

gravitational potential increases as the given mass or 

volume of water is moved in the direction of increasing 

z, or vertically upward.

According to Newton's second law of motion,

(65)

where

w=the mass of water moved, in grams.

a=g the acceleration of gravity, in centimeters 

per second squared. The acceleration is 

assumed to be constant within the relatively 

small range of values of z normally encountered in soil-moisture work.

Combining equations 64 and 65, the energy, in ergs, 

to move any mass of water, m, from the datum to the 

height z centimeters is the scalar quantity

G=myz (66)
D18 FLUID MOVEMENT IN EARTH MATERIALS

To find the gravitational potential in ergs per gram of 

water, Gm, divide equation 66 by m. Then

G

(67)

To find Gv, the gravitational potential, in ergs per 

cubic centimeter of water, divide equation 66 by v, the 

volume of the test mass of water, and

__U_mgz_

'"""» v ' (68)

where p=density of the liquid water.

The gravitational potential, in ergs per gram of 

water, or in ergs per cubic centimeter of water, depends only upon the height within the field above the 

datum. Furthermore, it increases at a uniform rate 

with increasing 2, assuming g to be constant. The 

gravitational potential is not affected by the contents 

of the field, whether liquid, vapor, or solid. It depends only upon the energy required to move vertically a unit test mass or volume from a datum to 

any position in the field.

To determine the gravitational force on a mass m of 

water at elevation z in the field one applies the operator 

V, equation 58, and differentiates the potential. For 

equation 66 operated upon by 58,

(69)

The x and y terms present in equation 58 do not exist 

because the gravitational force operates only in the 

vertical, or 2, direction. The minus sign shows that the 

force is directed downward while height, z, increases 

upward and the potential increases with z. In other 

words, the force is directed in a direction opposite to 

that of increasing potential.

Differentiating, the gravitational force on the mass 

of water, m, is

(70)

where Fg is in dynes. Changing the notation to represent the magnitudes of the quantities only, we arrive 

back at equation 65,

Fg=mg (71)

Similarly, if Fg is force per gram of water,

(72)

Thus, the gravitational force, in dynes per gram of 

water, is numerically equal to the acceleration of

gravity. Finally, if Fg is the gravitational force per 

unit volume of water,

F.= pg (73)

The force on a unit mass or volume at any point in the 

field is constant because of the assumption that g is 

constant. This independence of the gravitational 

potential of the contents of the field explains why 

equation 54 is equally applicable whether or not there 

is a liquid continuum. It explains why equation 54 

applies equally to the continuous liquid phase below 

the capillary fringe and to the discontinuous liquid 

phase above the capillary fringe.

The acceleration of gravity, g, may be taken as 980 

cm per sec2 . Therefore, according to equation 67, 

the change in gravitational potential per gram equals 

980 ergs per gram per centimeter of change in height. 

The density of water, p, is 1 gram per cm3 . Therefore, 

according to equation 68, the change in gravitational 

potential per cubic centimeter of water is 980 ergs per 

cm3 of water per centimeter of change in height.

The datum for gravitational potential is usually 

taken as the water table. At this point 2=0 and 

G=Q. Therefore z is the height above the water table 

and z is the depth below the water table. Below the 

water table the gravitational potential is negative by 

an amount equal to 980 ergs per gram of water per 

centimeter depth below the water table, or 980 ergs per 

cm3 of water per centimeter depth below the water 

table, tunes the depth below the water table. Above 

the water table, it is equal to 980 ergs per gram of 

water per centimeter height above the water table, or 

980 ergs per cm3 of water per centimeter height above 

the water table, times the height above the water table, 

regardless of whether the liquid phase is continuous or 

discontinuous.

According to equations 72 and 73, the force on the 

water is equal to 980 dynes per gram of water, or 980 

dynes per cm3 of water throughout the field.

Although the gravitational potential and force at any 

point in the gravitational field are independent of the 

material filling the field, this is not true for the pressure 

within the field. The variation of pressure with height 

is directly proportional to the density of the phase filling 

the field. In equation 73 the force on each cubic 

centimeter of water is constant. However, when the 

water is in a liquid state, there are more of these cubic 

centimeters of water per unit change in height than it 

the water were in a vapor state. Therefore, the forces 

add up more quickly in the liquid phase, and the changes 

in pressure are greater than hi the vapor phase. Looked 

at in another way, a column of liquid water weighs 

more than a column of water vapor, and the pressure
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D19

is a function of the weight of the overlying column of 

water. According to elementary hydrostatics,

where

(74)

&p=change in pressure between two points a

vertical distance z apart 

D density of the liquid or vapor

The soil is saturated for some distance above the 

water table. For a somewhat greater distance above 

the water table, some interconnecting systems of pores 

are in liquid continuity with the water table. Because 

the density of water is 1 gram per cm3 , the pressure in 

the water above the water table changes at the rate of 

980 dynes per cm2 per centimeter of change in height, 

according to equation 74.

In the unsaturated soil outside the interconnecting 

systems of liquid-filled pores, the pores are partly filled 

with water vapor. Taking the density of saturated 

water vapor, at a temperature of 20 °C and at atmospheric pressure, as 0.000017 grams per cm3, equation 74 shows that the change in the pressure in a 

vapor-filled continuum is about 0.02 dyne per cm2 per 

centimeter of change in height. For greater accuracy, 

the compressibility of water vapor should be considered 

by using the barometric equation.

HYDROSTATIC-PRESSURE POTENTIAL

According to Childs (1957, p. 14), a suction must be 

applied in order to withdraw water from a soil or to 

prevent the soil from imbibing water. The greater the 

applied suction, the more water is withdrawn, and the 

lower will be the moisture content when the soil has 

reached equilibrium at the applied suction.

If water is withdrawn from a soil that does not 

shrink upon drying, air must enter the pore space and 

air-water interfaces must be present in the pore space. 

Such curved interfaces can be maintained only by 

capillary forces. Hence, surface tension acting in the 

interfaces provides a mechanism of soil-water retention 

against externally applied suctions (Childs, 1957, 

p. 14). This is one of several types of force that retain 

water against externally applied suctions. Furthermore, this is the mechanism whereby a negative hydrostatic pressure and pressure potential are developed 

within the soil.

As in the case of gravitational potential, hydrostaticpressure potential may be taken as zero at the water 

table. Of course the absolute pressure at the water 

table is atmospheric pressure, equivalent (at sea level) 

to that of a column of mercury 76 cm high, or 1,013,000 

dynes per cm2 . Because this pressure is taken as zero,

all pressures are relative to atmospheric pressure or are 

gage pressures. Below the water table at equilibrium 

hydrostatic pressuress are positive because the hydrostatic-pressure potential increases with increasing depth. 

Above the water table at equilibrium the water is under 

a negative hydrostatic pressure, or tension, because the 

hydrostatic-pressure potential becomes increasingly 

negative with increasing height above the water table 

(Baver, 1948, p. 204).

At the water table, then, the gravitational potential 

and the hydrostatic-pressure potential are arbitrarily 

taken as zero. In the absence of other forces the total 

potential at the water table is therefore zero. Under 

equilibrium conditions the total potential must also be 

equal to zero at all points above and below the water 

table. Thus, at equilibrium and where other forces 

are not present, the gravitational potential and the 

hydrostatic-pressure potential must everywhere be 

numerically equal (Baver, 1948, p. 204; Childs and 

Collis-George, 1950b, p. 243).

According to equation 67, the gravitational potential 

per gram of water decreases at the rate gz with increasing depth below the water table. Therefore at equilibrium the hydrostatic-pressure potential per gram of 

water must increase at the same rate with increasing 

depth below the water table, or

t=-gz (75)

where ^=the hydrostatic-pressure potential. It is the 

energy, in ergs per gram of water, to move a gram of 

water from the water table to a depth z below the 

water table against the hydrostatic-pressure forces. 

The minus sign is present because the pressure potential 

increases downward, whereas z is measured upward. 

In similar fashion from equation 68,

. fnc\ \l/= pgz ( *>)

\f/ is now the hydrostatic-pressure potential, in ergs per 

cubic centimeter of water. It must be remembered 

that below the water table z is negative, and according 

to equations 75 and 76 the hydrostatic-pressure potential, ^, is therefore positive.

A liquid continuum extends to a height above the 

water table that is determined by the pore sizes and by 

equation 54, as will be discussed later. This is the 

same capillary rise that occurs when a capillary tube 

is immersed in water. This region, where some or all 

of the pores are filled with water that is continuous with 

water in the zone of saturation, is the capillary fringe. 

At equilibrium, and in the absence of other potentials, 

the hydrostatic-pressure potential equals the gravitational potential in the capillary fringe, and its value is 

given by equations 75 and 76. In this case z is positive,
D20 FLUID MOVEMENT IN EARTH MATERIALS

and the hydrostatic-pressure potential is negative. 

This situation corresponds to the fact that in the zone 

of aeration the soil exhibits a negative pressure, or 

hydrostatic-pressure deficiency, or suction, or tension.

The liquid-water system is discontinuous above the 

capillary fringe. It consists of isolated capillary interstices, or groups of such interstices, that are filled with 

water and separated from similarly filled interstices by 

others that are filled with soil air and water vapor. In 

this part of the zone of aeration, equations 75 and 76 

also hold at equilibrium and give negative values of 

hydrostatic-pressure potential for positive values of z. 

Curved liquid-vapor interfaces are present in many of 

the pores and equation 54 also applies. The actual 

pressures in the water vapor in this belt can be determined from the barometric equation 31. The pressures 

in the water on the convex sides of the menisci can be 

determined from equations 9 or 10.

Above the water table the hydrostatic pressure in 

the liquid is negative relative to atmospheric pressure. 

The greater the height above the water table, the greater 

will be the curvature of the equilibrium liquid-vapor 

interfaces. The interfaces are drawn deeper into the 

smaller interstices, and the larger interstices are 

emptied of water, giving larger hydrostatic-pressure 

deficiencies in the pore water. However at a sufficiently 

great height above the water table, or at sufficiently 

great values of externally applied suction, only very 

small pores remain full of water. Other forces then 

become more important than surface tension as discussed below. Thus extremely high negative hydrostatic pressures, or tensions, do not exist in the soil 

water. In fact, as shown by Edlefsen and Anderson 

(1943, p. 205), much of the water is actually under 

compression, even though it is in equilibrium with an 

extremely large applied suction.

OSMOTIC-PRESSURE POTENTIAL

A third type of potential found in soil water is 

osmotic-pressure potential. It is due to differences in 

content of both soluble salts and anions and cations in 

Gouy double layers associated with the solid surfaces 

of soil particles (Childs and Collis-George, 1948).

According to the concept of Gouy (Adam, 1941, p. 

342), cations, or positively charged particles, frequently 

become dissociated from soil particles of clay size 

(Childs, 1957, p. 7-8). The surface of the soil particle 

then has a negative charge. Such particles are called 

micelles. Particles having similar electrical charges 

repel each other and particles having opposite electrical 

charges attract each other. Therefore, in the vicinity of 

negatively charged soil-particle surfaces, there is an increased concentration of cations (positively charged

ions) and a decreased concentration of anions (negatively charged ions), whereas cation and anion concentrations are equal in parts of the solution that are more 

remote from the solid-liquid interface. The arrangement of charges at the surface of the micelles is known 

as an electric, or Gouy, double layer. Because the total 

ion concentration increases as the solid-liquid interface 

is approached, the osmotic pressure also increases.

Because the negatively charged soil micelles attract 

cations from the soil solution, the net concentration of 

ions in the space between any two given micelles is 

greater than that in the solution remote from the micelles. Because of this difference in concentration, an 

osmotic force tends to force water into the space between the micelles, thereby tending to force them apart. 

Thus the hydrostatic pressure between the micelles is 

in excess of the hydrostatic pressure in the remote solution. If suction is applied to a soil that exhibits this 

phenomenon, removal of water as a result of application 

of the external suction relieves the hydrostatic pressure 

on the space between the micelles. Therefore, the soil 

contracts upon drying (Childs, 1957, p. 23-28).

The ultimate ionic concentration at equilibrium in 

the neighborhood of a Gouy double layer at the surface 

of a clay-size particle represents a balance between the 

segregating forces and the diffusing forces. The segregating, or ion-concentrating, forces are those due to the 

mutual electrical attraction between the micelle and 

oppositely charged ions and to adsorptive forces. The 

diffusing, or ion-diluting, forces include the normal 

ionic diffusion that would occur from a region of higher 

ion concentration to one of lower ion concentration and 

the diffusion of water in the direction of the higher concentration. Corresponding to this equilibrium, there is 

an osmotic-pressure gradient which is a function of distance from the solid surface. The osmotic pressure at 

any point corresponds to a negative hydraulic-pressure 

potential. It must be regarded as a hydrostatic-pressure deficiency because it tries to force water in the 

direction of greater ion concentration that is, toward 

the particle and into the electric double layers. This 

contribution to total potential might be appreciable 

within as much as 50 A (A=1 Angstrom unit = 10~8cm) 

from the dissociating surface (Childs and Collis-George, 

1948, p. 79).

Osmotic-pressure potential due to differences in ion 

concentration is important in soils that shrink upon 

drying. Such soils contain clay minerals and humus. 

Childs (1957, p. 23) refers to water retention in such 

soils as the mechanism of "water retention by particle 

repulsion." It is the main source of the negative hydrostatic pressure that is exerted against an externally 

applied suction applied to a soil that shrinks upon drying. It could be of importance also in extremely dry
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D21

sandy soil, where the remaining water is retained in very 

small wedges or close to the particle surfaces.

ADSORPTION POTENTIAL

"Attractive (adsorptive) forces between solids and 

water can be divided in two categories, short-range 

forces and long-range forces" (Bolt and Miller, 1958, 

p. 918). The short-range forces are effective in a range 

less than 100 A from a solid surface, whereas the longrange forces may extend beyond 100 A.

Of the two types of short-range forces, one consists of 

chemical forces, which are "caused by localized interactions between the electron clouds of surface atoms 

and water molecules" (Bolt and Miller, 1958, p. 918- 

919). These forces are localized close to the surfaces of 

the particles and have little effect on adsorption.

The second type of short-range force consists of 

Van der Waals forces. They have their origin in the 

electrostatic attraction of the nucleus of one molecule 

for the electrons of another. Because of the geometry 

of the molecule this attraction is largely but not 

completely compensated for by the repulsion of electrons by electrons and of nuclei by nuclei (Pauling, 

1958, p. 321). These short-range

forces, although not as localized as the chemical forces, are in 

the case of interaction between water and a solid surface still 

of short range as a result of the destructive interference with 

fields from other atoms as the distance between the atoms 

under consideration increases. Although such forces account 

for the cohesive strength of the water and the phenomenon of 

surface tension, it seems unlikely that the effective range with 

regard to the adsorption of water on solid surfaces extends 

beyond the first few molecular layers adjacent to the interface 

(Bolt and Miller, 1958, p. 919).

In general, the short-range forces may be neglected 

except at low water contents usually encountered 

only in the laboratory. It must be noted, however, 

that other workers ascribe the cohesive strength of 

water and the phenomenon of surface tension to the 

molecular hydrogen bonding, or dipole nature of the 

molecule (Hendricks, 1955, p. 11-12).

To understand the long-range forces, consider the 

nature of the water molecule. "* * * The water 

molecule may be supposed to have a localized distribution of electrical charge in its outer parts. The concentrations of charge are located at the apexes of 

a tetrahedron, two being positively charged with two 

negatively. The molecule is thus a dipole; i.e., it is, 

as a whole, electrically neutral but the 'center of 

gravity' of the negative charge is separated from that 

of the positive" (Childs, 1957, p. 9). The long-range 

adhesive forces are due to the attraction between the 

positive ends of the water dipoles and the electrostatic 

field emanating from the negatively charged soil 

particles. The relatively small separation between

the oppositely charged ends of the dipole becomes less 

important the farther away the dipole is from the soil 

particle. At some distance the dipole acts as a neutral 

particle because, for all practical purposes, the two 

opposite charges can be considered to coincide. Therefore, even these "long-range" adhesive forces must be 

of rather short range even though they may attain 

large values near the soil particle. This attractive 

or negative force may extend to a maximum distance 

from the soil particle of more than 100 A (Bolt and 

Miller, 1958, p. 919).

The adsorptive forces strongly retain water near the 

solid surfaces, and large forces may be required to 

remove this water. However, these adsorptive forces 

become prominent only when soils are dried far beyond 

the state of dryness that is normally found in nature.

THERMAL POTENTIAL

According to Edlefsen and Anderson (1943), the 

free energy of soil moisture, or any other fluid, increases 

with the temperature. Furthermore, temperature and 

other thermodynamic variables of state help determine 

phase changes in the soil fluids. Therefore, thermal 

potential must be accounted for in determining the 

total potential of soil water. Unfortunately, "There 

can be little doubt that the most complex and least 

understood area in the field of soil-water relationships 

is that of the effect of temperature gradients applied 

to moist soils" (Winterkorn, 1958b, p. 113).

Moisture movement by vapor diffusion or by a 

combination of vapor diffusion and capillarity can 

be appreciable when thermal gradients are present 

(Smith, W. O., 1943; Gurr- and others, 1952; Taylor 

and Cavazza, 1954). However, thermal transfer of 

water occurs in great quantity only when relatively 

large thermal gradients are present in the soil water. 

Furthermore, evidence suggests that moisture flow 

in response to temperature gradients occurs mainly 

or completely in the vapor phase (Hutcheon, 1958, 

p. 114; Kuzmak and Sereda, 1958, p. 146). It is most 

important near the land surface where the largest 

thermal gradients are found. At depth, thermal 

gradients are generally small (Baver, 1948). The 

moisture transfer due to these small thermal gradients 

at depth may be quantitatively substantial over a 

period of time and for a given area. However, it is 

relatively small when compared with the moisture 

transfer due to the other gradients. As stated by 

Richards and Richards (1957, p. 53), "There is evidence 

that, except very near the soil surface and then mainly 

for longtime effects involving weeks or months, vapor 

transfer of water in soil in the root zone of growing 

crops is not very significant agriculturally." Therefore,
D22 FLUID MOVEMENT IN EARTH MATERIALS

it will be assumed that thermal gradients are negligible 

and that moisture movement occurs isothermally.

CHEMICAL POTENTIAL

By chemical potential is meant the potential due 

to the osmotic energy of ions free in the aqueous 

solution. This is distinct from the potential due to 

the osmotic energy of ions held in the electrical double 

layers on the solid particle surfaces (Winterkorn, 

1958a, p. 330).

If free salts are absent or are uniformly distributed, 

chemical potential may be ignored (Low, 1958, p. 56). 

In addition, it "is usually omitted in considering water 

movement because the solutes generally move more 

readily than the water with respect to the soil" 

(Gardner, 1958, p. 78).

TOTAL POTENTIAL AND ITS MEASUREMENT

The advantages of using potentials in the soil water 

rather than the forces on the soil water now become 

apparent. Assuming isothermal conditions and uniform solute concentration, the total force on the soil 

water would be the sum of the forces due to gravity, 

hydrostatic pressure, osmosis, and adsorption. Because forces are vectors and have direction, a vectorial 

addition would be required at every point. On the 

other hand, potential is a scalar property, and the 

total potential at any point in an unsaturated soil is 

the scalar sum of the four component potentials discussed above:

0=p02+(^-e-£) (77) 

where

0=total potential, in ergs per cubic centimeter 

pgz=gravitational potential, in ergs per cubic 

centimeter

$=hydrostatic-pressure potential, in ergs per 

cubic centimeter

c=osmotic-pressure potential, in ergs per cubic 

centimeter

|= adsorption potential, in ergs per cubic centimeter 

(Childs and Collis-George, 1948, p. 79.)

The total force can be determined from the gradient 

of the total potential.

The potential of soil water can be measured by 

bringing the soil water into equilibrium with a manometer. Because all unsaturated soils have some continuous air-filled pore space, a manometer would read 

zero, or atmospheric pressure, in contact with an unsaturated soil. Therefore, a porous membrane made of 

sintered glass spheres, unglazed ceramic, cellophane, 

porvic, or sausage casing is interposed between the

soil on one side and an external body of water which is 

in contact with the manometer. The membrane to be 

used should have pores sufficiently small to remain full 

of water at all values of negative pressure to be encountered in an experiment. Because these pores 

remain full of water, air cannot enter them and put 

the manometer out of operation. On the other hand, 

the water in the soil is continuous with the water in 

the membrane, with the body of water on the manometer 

side of the membrane, and with the manometer. 

Thus, pressures are transmitted from the soil water 

to the manometer.

In unsaturated soils a negative pressure or suction 

must be maintained on the manometer side of the 

porous membrane for equilibrium to be achieved with 

the soil water. One commonly speaks of measuring 

soil-water suction because one measures "the suction 

prevailing in an external body of water which is in 

equilibrium with the soil. In some cases, as in sand, 

we shall see that we may safely infer the internal soil 

water suction from such measurements, while in others, 

as in clays, we may not; but in the latter case the true 

internal suction is almost meaningless because [of its] 

varying from point to point within wide limits, while 

the equilibrium suction of an external water body is 

definable and significant" (Childs, 1957, p. 16). The 

"internal soil water suction" refers to the actual tension 

or hydrostatic-pressure deficiency in the water.

When a sand or other soil that does not shrink upon 

drying is brought into equilibrium with such a manometer, the suction in the external water body comes to 

equilibrium with the resultant of the forces due to 

gravity and to hydrostatic-pressure deficiency from 

surface-tension effects. In these nonshrinking soils 

the osmotic effect is unimportant. The adsorptive 

effects are also unimportant at normal moisture contents. In this case, the external suction is a measure of 

the hydrostatic-pressure deficiency resulting from the 

surface tension at the curved air-water interfaces.

When a clay or other soil that does shrink upon 

drying is brought into equilibrium with such a manometer, the suction in the external water probably comes 

to equilibrium with the resultant of the forces due to 

gravity, to hydrostatic-pressure deficiency from surfacetension effects, and to osmotic pressure (or pressure 

involved in water retention by particle repulsion). 

Where the shrinkage effect is completely dominant, 

the necessary external suction required to bring about 

equilibrium is equal to the osmotic repulsive pressure 

(Childs, 1957, p. 27).

The relation between the external suction applied 

to a soil and the amount of water that the soil retains 

against that dewatering suction (the moisture content) 

is the moisture characteristic. A typical moisture
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D23

10.000

o

OC 1000

Iz

I- *

o

z

Z 100

o

Io

v>

10

\

10 20 

MOISTURE, IN PERCENT BY VOLUME

FIGURE 11. Moisture characteristic of a fine sand. (Courtesy of N. A. Willits, 

assistant professor of soils, Rutgers University.)

characteristic is shown in figure 11. The moisture 

characteristic of a nonshrinking soil is related to the 

withdrawal of moisture from pores against the forces 

of surface tension. Therefore it is an indication of the 

pore-size distribution of the soil. For soils that do 

shrink upon drying, it is related to the osmotic pressure 

and to the separation of the clay micelles (which depends upon the amount of water between the micelles). 

Thus, when shrinkage experiments show that water 

loss is not accompanied fully by the entry of air into 

the pore space, the soil suction may be interpreted 

either as repulsive pressure between the micelles due 

to osmosis or as hydrostatic-pressure deficiencies due 

to surface tension.

In nonshrinking soils the contribution of osmotic 

potential, e, and adsorption potential, £, to total 

potential, <£, can be ignored. The moisture characteristic can be used as if it were a curve of moisture content, 

/3, versus negative hydrostatic-pressure potential due 

to surface tension, \f/. Thus it can be used to determine 

the pore-size distribution. Childs and Collis-George 

(1950a, p. 396) use this derived pore-size distribution 

to determine the hydraulic, or capillary, conductivity. 

For clays, where the suction removes water by drawing 

the grains closer together, by orientation of clay-mineral 

plates into lower potential energy positions or by the

interpenetration of double layers, the moisture characteristic cannot be used to determine the pore-size 

distribution.

The term (^ e £) in equation 77 is expressed as a 

pressure deficiency because it is determined from the 

suction in an external body of liquid at equilibrium. 

However, this does not mean that there are large 

hydrostatic-pressure deficiencies in the soil. As discussed above, it is doubtful that water can exist in a 

state of tension greater than 1 atmosphere under normal 

conditions. The true internal hydrostatic-pressure 

deficiency of the water, in a clay especially, varies 

greatly from point to point. In fact the water is 

probably under great compression in many parts of the 

system.

Consider equation 77 when moving from point to 

point in either a shrinking soil or close to the particles 

in a nonshrinking soil at a fixed elevation. At the 

fixed elevation pgz is constant. If the system is in 

equilibrium the total potential, <j>, must be zero, and 

the term in parentheses must everywhere be the 

negative equivalent of pgz. Near a solid surface or in 

a clay micelle e and £ may have extremely large 

negative values. To keep the term in parentheses 

constant and equal to pgz, ^ would have a very large 

positive value. Thus the water would be under 

considerable positive hydrostatic pressure that is, 

under compression near the solid surface.

Moisture-characteristic curves relating external suction and moisture content exhibit hysteresis. The 

hysteresis in nonshrinking soils can be explained by 

considering a small capillary interstice that is hydraulically connected to the rest of the system through a 

larger interstice. If a relatively large suction is applied 

during drying, both interstices will empty. During a 

subsequent wetting the overall suction might decrease 

to some intermediate value. Assume that the smaller 

interstice would normally fill with water at this intermediate suction but the larger interstice would not. 

Because water cannot fill the larger interstice, it cannot 

reach and fill the smaller, and both interstices remain 

empty. Therefore the moisture content may be lower 

at a given suction during wetting than at the same 

suction during drying. In soils that shrink upon drying, 

the hysteresis might reflect the irreversible orientation 

of the clay plates during drying and wetting cycles, 

respectively (Childs, 1957, p. 28).

Several instruments measure potentials in soil 

water by bringing the soil water into equilibrium with 

a manometer as described above. The most important 

are the tensiometer, the asbestos tension table, the 

porous-plate pressure apparatus, and the pressure 

membrane apparatus (Baver, 1948, p. 211; Childs and 

Collis-George, 1950b, p. 239). Another method con-
D24 FLUID MOVEMENT IN EARTH MATERIALS

sists of measuring the pressure of vapor in equilibrium 

with the soil vapor when the soil vapor is at equilibrium 

with the soil liquid. A variation of this method consists 

of measuring the freezing-point depression, because the 

freezing point and vapor pressure are lowered with 

increasing hydrostatic-pressure deficiency or increasing 

osmotic-pressure deficiency (Edlefsen and Anderson, 

1943, p. 113).

Caution is necessary in using the vapor-pressure or 

the freezing-point-depression methods because they 

measure the total free energy of the soil water (Edlefsen 

and Anderson, 1943, p. 201). The total free energy 

is the potential shown by equation 77 plus energy 

components due to temperature and to the presence of 

free solutes. The energy potential due to the presence 

of free solutes is distinct from that due to the osmotic 

pressure attributable to the ions held in the Gouy 

double layer. However it is also commonly called an 

osmotic potential and is attributable to the energy 

changes stemming from the changes in solute concentration (Schofield, 1948, p. 129). This free-solute 

component of the total free energy does not contribute 

to the pressure deficiency in the water and does not cause 

water movement. Therefore it must be subtracted 

from the total free energy as measured by the vaporpressure or freezing-point methods, to get the actual 

pressure deficiency in the water due to the surface 

tension or osmotic forces discussed above.

By means discussed above, the total potential, 0, 

can be measured. The gravitational potential, pgz, 

also can be determined. Under equilibrium conditions 

the total potential at the water table and elsewhere 

in the system is zero, and the gravitational potential is 

everywhere equal to the sum of the terms in parentheses 

in equation 77. Even under nonequilibrium conditions, 

if the total potential and the gravitational potential 

are known, the term in parentheses can be evaluated. 

However it is usually not possible to resolve the term 

in parentheses into its component potentials. Therefore 

equation 77 is most commonly used in the form

(78)

(79)

where

and S is the capillary potential. Its interpretation for 

shrinking and for nonshrinking soils is outlined above. 

In most situations, as discussed above, it equals the 

suction potential. The gradient of the sum of the 

gravitational and capillary potentials gives the watermoving forces in unsaturated soils.

EQUILIBRIUM DISTRIBUTION OF POTENTIAL AND 

WATER IN SOIL

The distribution of component potentials within an 

unsaturated soil is discussed by Bolt and Miller (1958).

The distribution of water and potentials at equilibrium 

in a nonshrinking sand is discussed below.

In a nonshrinking sand the capillary potential, S, is 

equal to the hydrostatic-pressure potential, ^, because 

osmotic and adsorption potentials are negligible. 

At equilibrium the total potential, <£, is everywhere 

equal to zero because there are no gradients of total 

potential at equilibrium, and the total potential is 

equal to zero at the water table. Therefore, letting 

<£ equal zero in equations 77 and 78, the gravitational 

potential, pgz, is everywhere equal to the negative 

value of the capillary potential ( 2), or to the negative value of the hydrostatic-pressure potential ( \f/). 

Furthermore, equation 54 also applies; pgz in 54 may 

be interpreted as the gravitational potential, in ergs 

per cubic centimeter, and 2afr may be interpreted a? 

the negative of the hydrostatic-pressure potential, in 

ergs per cubic centimeter.

DISTRIBUTION IN THE ZONE OP SATURATION

The datum for gravitational potential and for 

capillary potential is usually taken as the water table. 

In the zone of saturation below (the zone filled with 

water under atmospheric or greater pressure), the 

gravitational potential decreases at the rate of 980 

ergs per cm3 of water per centimeter of increase in 

depth below the water table according to equation 

68. Therefore the capillary potential S, in this case 

equivalent to the hydrostatic-pressure potential ^, 

increases at the rate of 980 ergs' per cm3 of 

water per centimeter of increase in depth below the water 

table. This corresponds to the rate of increase of 

hydrostatic pressure with depth in a body of water, 

980 dynes per cm2 per centimeter of increase in depth.

DISTRIBUTION IN THE CAPILLARY FRINGE

The capillary fringe is an irregular belt of saturation 

extending above the water table. The water in this 

belt is at less than atmospheric pressure. Interconnected systems of saturated pores exist, making a 

continuum of saturated interstices extending to the 

water table. Here, as elsewhere in the system, the 

gravitational potential increases at the rate of 980 ergs 

per cm3 of water per centimeter of increase in height 

above the water table. Therefore the negative value 

of the hydrostatic-pressure potential, or the capillary 

potential, increases at the rate of 980 ergs per cm3 of 

water per centimeter of increase in height above the 

water table. The hydrostatic pressure in the liquid 

continuum decreases at the rate of 980 dynes per cm2 

per centimeter of increase in height above the water 

table.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D25

DISTRIBUTION IN PINCHING AND SWELLING TUBES 

WITHOUT HORIZONTAL CONNECTIONS

The following discussion describes the capillary rise 

of water in pinching and swelling tubes of circular 

cross section. It is greatly simplified and is designed 

only to illustrate the general type of occurrence of 

water in such a system. A more rigorous description 

of capillary rise in a somewhat more realistic soil 

model is presented by W. O. Smith and others (1931).

Equation 54 describes the capillary rise in a tube of 

uniform cross section, the angle of contact being assumed equal to zero. It may be rewritten as

pgr (80)

The pinching and swelling tubes have cross sections 

that vary with height. However, assuming that the 

cross sections are circular, equation 80 must be satisfied at any possible meniscus position. For purposes 

of illustration, it is implied also that the contact angle is 

always zero regardless of the pore geometry.

Taking

(7=73.5 dynes per cm for water at 20°C 

p=l gram per cm3 for water 

#=980 cm per sec2

equation 80 becomes

r2=0.15 cm2 (81)

Figure 12 is a plot of equation 81 showing the possible 

meniscus positions, or possible heights of capillary 

rise, that correspond to different pore sizes under the 

simplifying assumption.

Before a meniscus can lodge in a pore of radius r at a 

height z above the water table, another condition besides that indicated by equation 81 must be satisfied. 

A pore of the proper size must exist at the corresponding 

distance above the water table. To demonstrate this 

point, consider a hypothetical distribution of pores.

Assume for example that the pores are distributed in 

vertical sequences that are completely isolated from 

adjacent vertical sequences. Assume further that the 

radii of the pores vary in the vertical direction according to a sine-curve distribution. Thus, we are replacing the pores by vertical tubes that pinch and swell 

like a sine curve. Let

a=the minimum neck radius

6=the height from one neck to the next or from

one swell to the next 

c=the difference between the maximum and

minimum neck radii.

(See fig. 13.)

0.10 .20

r, IN CENTIMETERS

FIGURE 12. Plot of the capillary-rise equation for water, rz=0.1S cm*, z Is the 

height of capillary rise, in centimeters, for a pore of givej| circular radius (r) in 

centimeters.

The neck radius at any height, z, is

. . trZ r a-\-c sm -?- 

o

(83)

where ir=3.14159 ....

The vertical lines around the trigonometric term indicate that the absolute value of that term is to be used. 

Consider the hypothetical case where 

a=0.01 cm 

6=0.1 cm 

c=0.1 cm

Equation 82 becomes

r=0.01+0.1sin (83)

which is expanded in figure 13.

Equation 81 and figure 12 show the equilibrium 

hydraulic relationship between 2 and r. Equation 83 

and figure 13 show the actual geometrical relationship 

between 2 and r. When both are reproduced on the 

same scale and one is superimposed on the other, the 

points of intersection of the two curves satisfy both 

equations 81 and 83. They indicate the possible
D26 FLUID MOVEMENT EN EARTH MATERIALS

0.05 0.10 0.15

r, IN CENTIMETERS

FIGURE 13. Hypothetical variation of pore radius (r) with height above the water 

table, z, according to the relation r=0.01+0.1|sin 10irz|.

meniscus positidhs, or the possible heights of capillary 

rise. However, because the menisci can occupy only 

the lower halves of the pores, the intersections with 

the upper parts of the pores must be ignored. This 

principle can be expressed mathematically by saying 

that, in equation 83, z is a number of the form

z=djo

where d, /, and o are digits or groups of digits such that

d=0 to n 

/=0.0 to 0.9 

o=0.00 to 0.05

Figure 14 shows equation 81 plotted on the same 

scale as figure 13, between 2=2.00 and 2=2.25. When 

overlaid on figure 13, the three points of intersection 

are 2=2.022 and r=0.074, 2=2.121 and r=0.071, and 

2=2.220 and r=0.68. These are possible solutions of 

the two equations and show possible heights of equilibrium menisci and the radii of the interstices required 

at those heights above the water table.

Substituting equation 80 into equation 82, the possible values of pore size that satisfy the two equations 

are expressed by

i 

r=a+csm pgrb

,_ , .

(84)

2.25

2.20

CO

tr ui

H 

Ul

UI

o

2J5

2.10

2.00

0.05 OJO OJ5 

r, IN CENTIMETERS

0.20

FIGURE 14. Plot of the capillary-rise equation, rz=0.1,5 cm^, on the same scale used 

in figure 13.

Or, substituting equation 81 into equation 82,

0.157T r=a-\- csm

rb (85)

Any possible pore radius in the given pore-size distribution that can support a water column in equilibrium 

with gravity must satisfy this equation subject to the 

limitations of the simplifying assumptions.

Applying equation 85 to the hypothetical example,

r=0.01+0.1 sin (86)

Solving by trial and error, the largest radius that 

satisfies the relation is r=0.1042 cm. Substituting 

this value of r into equation 81, the minimum height of 

capillary rise is 1.44 cm.

The largest radius in the hypothetical distribution of 

pinching and swelling tubes is 0.11 cm. This would 

hold a meniscus at a height of 1.36 cm according to 

equation 81. According to equation 83 the pore extends 

from 1.30 to 1.40 cm. However, the part of the pore 

that has a radius of 0.11 cm (the widest part of the pore) 

is at a height of 1.35 cm above the water table and not 

at a height of 1.36 cm as required by equation 81. At 

both 1.35 cm and 1.36 cm above the water table the
REVIEW OF SOMifi ELEMENTS OF SOIL-MOISTURE THEORY D27

radii are smaller than those required for equilibrium. 

Because this situation pertains throughout this pore, 

the water would rise above the pore into the next pore. 

The equations indicate that it would rise to a height of 

1.44 cm into the next pore, which goes from 1.40 to 1.50 

cm, and reach stability where the radius is 0.1042 cm. 

This would be the height of "minimum capillary rise" 

as it is called by some investigators (Smith, W. O., and 

others, 1931, p. 18).

The maximum height of capillary rise is determined 

in similar fashion. The smallest pore radius is 0.01 cm, 

and this can hold a meniscus at a height of 15 cm above 

the water table according to equation 81. A neck of 

this radius is located at exactly 15 cm according to 

equation 83, and this is the maximum height of capillary 

rise. In this case, there can be no capillary rise above 

this height because there is no pore small enough to 

support an equilibrium meniscus above this height.

Other possible positions of capillary rise exist between 

the maximum and minimum heights. In fact there is 

a possible meniscus position in each of the intervening 

pores. Consider which of these possible positions the 

top of the capillary fringe will assume.

Assume that the top of the capillary fringe is at the 

height of minimum capillary rise, 1.44 cm above the 

water table. If the water table then slowly rises, the 

meniscus moves upward toward the center of the pore 

that extends from 1.40 to 1.50 cm above the water table. 

As discussed above, when the center of the pore is 1.36 

cm above the water table the meniscus is lodged in that 

pore center where the radius is 0.11 cm. If the water 

table continues to rise, the meniscus rapidly passes 

through the upper part of the pore and lodges at an 

equilibrium position in the next pore. Thus, for a 

rising water table, the top of the capillary fringe in this 

hypothetical distribution of pores remains a distance 

above the water table about equal to the minimum 

height of capillary rise.

Suppose that the water table after rising reaches 

equilibrium where the top of the capillary meniscus is 

1.36 cm above the water table and the pore radius is 

0.11 cm. If the water table then falls, the meniscus is 

gradually drawn downward into the lower half of the 

pore in order to develop a greater curvature. When the 

water table is 15 cm below the top of the capillary fringe 

the meniscus is in the narrowest part of the pore, the 

neck, where the radius is 0.01 cm. If the water table 

continues to fall, the meniscus is pulled rapidly through 

the neck into the pore below, where it lodges close to 

the neck of the lower pore. Thus, for a falling water 

table, the top of the capillary fringe in this hypothetical 

distribution of pores remains a distance above the water 

table about equal to the maximum height of capillary 

rise.

The most common positions of the capillary fringe 

are at the maximum and minimum positions of capillary 

rise. However, other positions are possible. Examples 

were given above where intermediate positions were 

assumed as the water table changed from a rising to a 

falling or from a falling to a rising condition.

As the water table falls in this hypothetical simplified 

model, the meniscus is pulled through successive pores, 

and the pores left above the meniscus are empty. 

Thus, the moisture content changes abruptly from 

saturation to near dryness, at the top of the capillary 

fringe.

The illustration above is a simple example of soilmoisture hysteresis. The moisture content of the 

region between the positions of maximum and minimum 

capillary rise depends upon the previous history of 

water-table fluctuation. In this extreme case of pinching and swelling tubes a region may be saturated when 

the water table falls, and it may be dry when the water 

table rises. Similarly, actual soils exhibit hysteresis 

and are wetter on dewatering than on rewetting, even 

though the suctions applied are the same.

DISTRIBUTION IN TIOHTLY PACKED SPHERES

Consider perfect spheres in the most compact, or 

rhombohedral, packing as shown in figures 15 and 16 

(Slichter, 1899, p. 309-310). The arrangement of 

the pores differs in several ways from that of the 

nonuniform vertical tubes postulated previously.

1. The interstices no longer have smooth walls. They 

are complicated wedge-shaped openings having 

wedge-shaped reentrants.

2. The interstices are no longer isolated horizontally 

but are connected in a continuous system. It 

is as if lateral capillary tubes connect the vertical 

capillary tubes.

3. Although the individual pores are all similar, a 

horizontal plane through the system intersects 

different pore widths and shapes if the unit 

rhombohedron is tilted.

These differences from the previous model have only 

minor effects on the height and shape of the capillary 

fringe. Instead of being at a uniform elevation, the 

top of the capillary fringe rises and falls as it goes 

from pore sections of smaller radius to those of larger 

radius. However, because all the pores are similar, 

the undulations of the top of the capillary fringe are 

minor and do not exceed one sphere diameter in height. 

Another difference between the two models is that the 

interstices can no longer be assumed to be circular, and 

equation 87 would have to be used instead of equation 

80 for a more rigorous solution. It is

(87)
D28 FLUID MOVEMENT IN EARTH MATERIALS

FIGURE IS. Unit rhombohedron formed by passing surfaces through the centers 

of eight contiguous spheres in the most compact packing of a mass of spheres. 

(After Slichter, 1899, p. 309.)

where RI and R2 are the principal radii of curvature 

of the meniscus.

The main differences between the two models are 

shown by the dewatering process. When the water 

table falls in rhomobohedrally packed spheres or in 

an actual soil, the interstices are not completely dewatered. As the meniscus is pulled down through 

the main part of a given pore, little wedges of water 

remain in the reentrants of the pore.

Figure 17 shows a pore having horizontally inclined 

reentrant wedges of gradually decreasing cross section 

leading to adjacent pores. Assume that the capillary 

fringe is in the position of maximum capillary rise 

with the meniscus in the neck at the top of the large 

pore, thereby satisfying equation 80 or 87. The 

reentrant wedges will be completely full of water, as 

shown in figure 17A

As the water table falls, the meniscus is drawn 

through the main pore into the neck below. The 

continuous water film is separated, and there are separate menisci in the pore and in the reentrant wedges,

FIGURE 16. Unit element of the pore space in a mass of spheres packed in the most 

compact manner possible, a plaster cast of the interior of the rhombohedron in 

figure 15. The spheres are not quite in contact; their surfaces are separated about 

0.5 cm. Ufter Slichter, 1899, p. 310.)

as shown in figure 175. The curvature of the menisci 

in the pore and in the wedges is governed by equations 

80 and 87. As the water table continues to fall, the 

meniscus at the top of the capillary fringe is pulled 

into the underlying pore. However, little isolated 

wedges of water remain in the reentrants.

DISTRIBUTION ABOVE THE CAPILLARY FRINGE

DISTRIBUTION IN TIGHTLY PACKED SPHERES

Consider the little isolated wedges of water in the 

pore reentrants that are now above the liquid continuum. The water wedges must come to equilibrium 

with the vapor that fills the pores at the same height 

above the water table. Transfer of water into and 

out of the reentrant w.edges occurs by vapor movement 

until equation 87 is satisfied.

Suppose that the water table stabilizes at a depth 

of 100 cm below the reentrant pore. If circular cross 

sections are assumed for illustrative purposes, according to equations 80 or 81 the equilibrium pore 

size would be 0.0015 cm. Water would evaporate and

FIGURE 17. Dewatering of a main pore and of the reentrants.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D29

diffuse away from the liquid surfaces in the wedgeshaped reentrants until the menisci were drawn into 

the wedges to a point where their radii were 0.0015 cm. 

If the smallest parts of the reentrants had radii greater 

than 0.0015 cm, water would evaporate and diffuse 

away from the menisci until the reentrants were completely dewatered. Only those reentrants having 

radii of 0.0015 cm or smaller would retain any water 

that is in equilibrium with the gravitational potential 

at that level. Of course this assumes that water is 

retained in the pores only by the surface-tension forces 

and that adsorptive and other forces are negligible.

Equation 78 provides another way of looking at 

this problem. At equilibrium, <f> is zero and S must 

equal the gravitational potential. The required value 

of negative capillary potential, or hydrostatic-pressure 

potential, can develop only in a pore 0.0015 cm in 

radius. If there is no pore that small, sufficient retaining capillary or hydrostatic-pressure force cannot be 

developed to balance the dewatering gravitational force. 

Equilibrium is not possible, and the interstice is dewatered by gravity flow.

At 100 cm above a rising water table, water vapor 

would diffuse to any meniscus in a pore less than 

0.0015 cm in radius. The vapor would condense on 

the meniscus, reducing its curvature and energy of 

water retention until the equilibrium meniscus size 

of 0.0015 cm was reached.

DISTRIBUTION IN RANDOMLY DISTRIBUTED PORES

Consider next the hypothetical extreme case where 

there is a random distribution of pore sizes. Furthermore, assume that the soil is of infinite horizontal 

extent. As a final simplification, assume circular pores 

so that equation 81 may be used.

Somewhere in the system, a pore having a radius 

greater than that required by equation 81 will be 

located close above the water table. At that point, 

there will be no capillary fringe and no capillary rise. 

Elsewhere, it will be possible to find a continuum of 

small pores extending to a great height above the 

water table. Furthermore, each pore radius will be 

equal to or smaller than the radius required by equation 

81 at that height above the water table. There will 

be a saturated continuum filling these pores and thus 

a very large height of capillary rise. Finally, all 

heights of capillary rise between these two extremes 

will occur, and the top of the capillary fringe will be a 

very uneven surface. Of course the exact height and 

shape of the capillary fringe will depend upon whether 

the soil is in a dewatering or rewatering phase. Above 

a falling water table, for example, very large pores can 

remain full of water provided that the required meniscus

is maintained in an overlying pore of sufficiently small 

radius.

In this random distribution of pore sizes, occasional 

isolated water-filled pores will be surrounded by vaporfilled pores. Similarly, occasional isolated vapor-filled 

pores will be surrounded by water-filled pores. In 

general, however, at a given level there will be different 

combinations or groups of water-filled pores whose 

menisci are sufficiently small to satisfy equation 81. 

Here again hysteresis is important. For example, 

consider a region containing a group of relatively 

large pores completely surrounded by pores small 

enough to retain water at a particular elevation and 

to satisfy equation 81. At a given stage in a rewatering 

phase, water might diffuse toward and condense in 

the smaller pores until they were filled with water. 

However the larger internal pores might not be filled 

at that stage of the rewatering phase, and menisci 

might face inward toward the larger pores. In dewatering from a state of saturation, on the other hand, 

the situation would be similar to that at the top of the 

capillary fringe for a falling water table. The surrounding smaller pores might reach equilibrium according to equation 81 and retain the water in their own 

pores and in the larger interior pores as well. Therefore the entire region might remain saturated even 

though the larger internal pores would normally be 

dewatered at that elevation. An analogous hysteresis 

effect would prevail in a region consisting of small pores 

surrounded by larger pores.

Two extremes have been presented. One consists 

of uniform interstices having a capillary fringe of 

virtually uniform height and individual reentrant 

wedges retaining water above the capillary fringe. 

The other consists of a random distribution of pores 

having a capillary fringe the top of which is extremely 

uneven. Water would be retained erratically above 

the capillary fringe in all combinations from single 

filled pores to very large webs of water filling great 

numbers of pores. The actual situation at equilibrium 

in the field lies between these extremes. Most soils 

have some sorting and horizontal stratification. Therefore, in most cases, the water distribution at equilibrium 

should be closer to the first case than to the second. 

In extremely variable material the actual water distribution at equilibrium might approximate more ^losely 

the random situation.

MOVEMENT OF SOIL MOISTURE 

PHYSICAL PICTURE

Soil moisture can exist and move in three ways as 

liquid water, as water vapor, and as adsorbed water. 

"It seems probable that, in soils, moisture transfer in
D30 FLUID MOVEMENT IN EARTH MATERIALS

the adsorbed phase can be significant only under rather 

special conditions, such as where the soil is very dry 

and possesses a large specific surface" (Philip, 1958, 

p. 157). Thus in the following discussion it is assumed 

that movement of adsorbed water is negligible and that 

the water moves chiefly in the liquid and vapor phases. 

Soil moisture moves in response to gravitational, 

hydrostatic-pressure, adsorption, osmotic, temperature, 

and chemical potential gradients. As discussed previously, temperature and solute concentration are taken 

as constant throughout the system. In addition, the 

discussion is restricted to a sand, where the osmotic and 

adsorption potentials are negligible. Moisture movement is then governed by the gradient of the total 

potential, which according to equation 78 is

(88)

Here, S is the capillary potential of a nonshrinking soil. 

It is virtually equivalent to ^.

PORE UNDERLAIN BY A CAVITY

Assume that a given pore has the general shape 

shown in figure 18 and is underlain by a cavity. Figure 

ISA shows the pore neck partly filled with water and at

FIOUEE 18. Pore underlain by a cavity.

equilibrium. Therefore the upward and downward 

forces on the water are equal.

The downward forces are the pull of the lower 

meniscus and the weight of the water in the pore, 

pgz, where z is now the height of the water in the neck. 

If the pore is circular, this pull is denoted by equation 

9, which is

(p-Po)=~ (89)

The upward force is the pull of the upper meniscus, 

also denoted by equation 9. The upward pull by the 

upper meniscus must exceed the downward pull by the

lower meniscus by an amount equal to pgz. Consequently the radius of the upper meniscus must be 

smaller according to equation 9, and it will be lodged in 

a smaller part of the neck than will the lower meniscus.

Suppose water is added to the system until the soil 

overlying the cavity becomes saturated. Eventually 

the upward meniscus will not be able to develop a small 

enough radius to support the growing weight of water 

in the pores in addition to the pull of the lower meniscus. 

Therefore the lower meniscus will move to the bottom 

of the pore and become convex toward the cavity, as 

shown in figure 18B.

The pressure in the cavity is atmospheric pressure. 

At equilibrium, the pressure on the upper side of the 

lower meniscus (concave to the liquid) is greater than 

atmospheric pressure according to equation 9. Thus 

the upper and lower menisci act in the same direction 

to support the weight of water in the pore.

Positive hydrostatic pressure due to the weight of the 

overlying liquid is required to overcome the pressure 

differentials across the menisci before water can flow 

into the cavity. Thus the soil above a cavity must be 

saturated and under sufficient head to overcome the 

pressure differences across the interfaces before water 

can move into the cavity. If the pore radius is large, 

it will take very little positive pressure to overcome the 

pressure difference, according to equation 89. If the 

pore is small, it might take considerable hydrostatic 

pressure a considerable zone of saturation over the 

cavity to overcome the larger interfacial pressure 

differences associated with pores of smaller radius.

Assume that the pore is in the moisture condition 

shown in either A or B of figure 18 and that the cavity 

is underlain by pores having a lower moisture content. 

No liquid movement can occur across the cavity as 

discussed above. However, equilibrium does not exist 

because the pores are drier below the cavity than above 

it. The water menisci below the cavity are drawn 

farther into the pore necks and their radii of curvature 

are smaller than above the cavity. According to the 

Kelvin equation, vapor pressures are therefore smaller 

below the cavity than above it. A vapor-pressure 

gradient would be established across the cavity, and 

water would evaporate from the upper menisci, diffuse 

across the cavity, and condense on the lower menisci. 

Movement would cease at equilibrium when all the 

moisture contents and menisci are such that equation 80 

is satisfied everywhere in the system (assuming the 

validity of the simplifying assumptions).

Once the upper pore has sufficient positive hydrostatic head for liquid flow to occur, it occurs very 

quickly. On the other hand, the diffusion of water 

vapor is very slow. Thus water moves at greatly 

different rates under these different conditions.
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D31

FIGURE 19. Epre underlain by a pore of similar size.

PORE UNDERLAIN BY PORES OF SIMILAR SIZE

Figure 19 shows two similar necks separated by a 

pore. There is less water in the lower neck than in the 

upper. The balance of forces in each neck is similar to 

that described for the case of the cavity. Tn each neck 

the capillary pull of the lower meniscus plus the weight 

of the water in the neck equals the capillary pull of the 

upper meniscus; the upper meniscus exerts a greater 

pull than does the lower meniscus and must be lodged 

in a narrower part of the neck. As in the case of the 

cavity, water cannot flow as a liquid across the pore, 

and moisture equilibration occurs by slow evaporation 

at the upper neck, diffusion to the lower neck, and 

condensation at the lower neck.

Assume that entrapped air can be bled from the pores 

by lateral connections. As water is added to the upper 

neck in figure 19-4, meniscus b moves into the upper 

pore and meniscus a moves into the lower pore. Meniscus b is always lodged in a narrower neck section than 

meniscus a, however, because it must balance the weight 

of the water in the neck as well as the pull of meniscus a.

If insufficient water is added to move meniscus a to 

the midpoint of the lower pore, liquid flow does not 

occur. Instead, relatively slow downward movement 

of water occurs across the pore by evaporation, diffusion, and condensation. Because the radius of curvature 

of meniscus a in figure 19 B is larger than in figure 19 

A, vapor pressure at this meniscus is greater and a 

diffusion occurs more rapidly than in figure 19 A.

If sufficient water is added to move meniscus a 

past the midpoint of the pore, water will flow rapidly

FIGURE 20. Pore underlain by a smaller pore.

to the lower part of the pore. At equilibrium the 

amount of water in the lower neck will be slightly 

greater than in the upper neck in order to satisfy 

equation 80.

If sufficient water is added to fill all the pores and 

entrapped air is removed, surface tension ceases to be 

a factor. Water movement through the pores then 

follows the rules of saturated flow.

PORE UNDERLAIN BY PORES OF DISSIMILAR SIZE

The upper pore in figure 20 is underlain by a smaller 

one. Meniscus a will reach the midppint of the lower 

pore while meniscus b is still in the neck of the larger 

pore. After meniscus a reaches the midpoint of the 

smaller pore, water will flow rapidly to the lower part 

of the small pore. Thus liquid flow will occur at a 

moisture content which is lower than that required for 

liquid flow where the upper pore is underlain by a 

pore of equal or larger size. In fact, if the lower pore 

is sufficiently smaller than the upper, it can fill with 

water while the upper pore remains almost empty.

If the lower pore is larger than the upper, the reverse situation is present. In figure 21, for example, 

it would be necessary for the upper pore to be full
D32 FLUID MOVEMENT IN EARTH MATERIALS

FIGURE 21. Pore underlain by a larger pore.

and to have sufficient hydrostatic head before the 

water could move as a liquid into the larger pore 

below. Vapor diffusion would occur across the underlying large pore as long as there was any difference 

in the vapor pressures above and below. This is 

similar to the case of the pore underlain by a cavity.

RANDOMLY DISTRIBUTED PORES

Several conclusions are possible from the considerations discussed above. For a given pore size and distribution, there is a moisture content below which 

liquid movement through the pores does not occur. 

Movement occurs exclusively by vapor transfer or 

else by vapor transfer from neck to neck and by liquid 

transfer through the necks. At higher moisture 

contents, the liquid movement occurs as a series of 

discontinuous jumps from neck to neck as discussed 

above. At sufficiently high moisture contents, a 

series of pores might be filled so that the flow through 

them follows the laws of saturated flow. For some 

types of soil geometry there might be a continuous 

vapor bypassing liquid-filled wedges. Similarly there 

might be liquid-filled passages bypassing vapor-filled 

pore centers.

The following description pertains to a soil containing 

a random distribution of pores. Some passageways are 

filled only with water vapor, and water moves through 

them slowly only as a vapor. Some passageways 

contain vapor-filled pores and liquid-filled necks. The 

movement through these channels occurs by vapor

transfer through the pores and liquid transfer through 

the necks. Some passageways are in a similar condition 

except that there is enough water for the menisci to 

jump across the pores. Water movement then occurs 

relatively rapidly by the mechanism of liquid jumps. 

Finally, some passageways are completely full of water 

and movement occurs rapidly by saturated flow.

A given arrangement of pores at a given moisture 

content transmits water at a given velocity under a 

given head difference. The ability to transmit water 

under a unit head difference is known as the capillary 

conductivity, or unsaturated conductivity, of the soil 

at the given moisture content. Capillary conductivity 

describes the sum of the movement by the different 

mechanisms described above. It includes movement 

by vapor flow, by vapor flow in the pores and liquid 

flow in the necks, by liquid jumps, and by liquid flow. 

As moisture content increases, more systems of pores 

contain more water. Therefore more systems of pores 

transmit water by the liquid-flow mechanisms, and 

capillary conductivity increases. Abundant experimental data show that capillary conductivity increases 

with increasing moisture content and with decreasing 

suction.

Poiseuille's law

(90)

gives the discharge rate through a capillary tube under 

conditions of viscous flow, where

Q=the discharge rate

r=the radius of the tube

77=the absolute or dynamic coefficient of viscosity

Z=the length of the tube 

PI and P2 =the pressures at the ends of the tube

The largest pores empty first as soil moisture decreases. Because of the fourth-power relationship 

between Q and r in equation 90, saturated flow through 

a soil decreases very rapidly as the radii of the saturated 

pores diminish in size. In addition, as moisture content decreases " * * * the chance of water occurring 

in pores or wedges isolated from the general threedimensional network of water films and channels 

increases. Once continuity fails, there can be no 

flow in the liquid phase, apart from flow through 

liquid 'islands' in series-parallel with the vapor system * * *" (Philip, 1958, p. 153).

Most of the conductivity of a wet soil stems from 

pores in material of silt size and larger. When a soil 

is dry the only pores that are filled and capable of 

liquid flow are the smaller pores. Liquid movement
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D33

through these pores is very small. The larger pores, 

substantially dewatered, transmit water only by the 

slow mechanisms of (1) vapor transfer, and (2) vapor 

transfer through the pores and liquid flow through 

the necks. At such low moisture contents the conductivity is very small.

In field soils, drainage and water transmission occur 

rapidly at high moisture contents. At a critical range 

of moisture contents, where pores of silt size and larger 

are being virtually dewatered, the rate of drainage 

becomes very small. For some soils this critical moisture content corresponds to field capacity. Field 

capacity, or field moisture capacity, is the " amount 

of water remaining in a well-drained soil when the 

velocity of downward flow into unsaturated soil has 

become small" (Soil Sci. Soc. America, 1956, p. 433).

Applied water distributes itself uniformly throughout a soil only if it is sufficient in quantity to bring 

the entire soil to a certain critical moisture content 

or range of moisture contents. These critical moisture 

contents correspond to those at which values of capillary conductivity become small during drainage. The 

applied water percolates downward as a belt, whose 

moisture content is between the critical one and saturation. It leaves the overlying soil near the critical 

moisture content. After the applied water moistens 

a certain depth to this moisture content, movement 

practically ceases. More water must be applied to 

obtain further rapid penetration as a liquid (Veihmeyer, 

1939, p. 544).

According to Bodman and Colman (1944, p. 117), 

downward moving water advances behind a "wettingfront," or sharp differential in moisture content 

(Remson and others, 1960, p. 153). If only enough 

water is applied to wet a certain depth of soil to the 

critical moisture content, movement of the wetting 

front practically ceases. Further moisture movement 

occurs only by vapor transfer and other slow processes 

discussed above. More water must be applied to 

obtain further penetration as a liquid. Of course, all 

movement finally stops at static equilibrium, when 

only enough moisture has been left behind so that the 

capillary forces equal the gravitational forces.

This behavior results because a certain degree of 

saturation is necessary before any given distribution 

of pore sizes can begin to transmit water rapidly as a 

liquid. At lower moisture contents, water movement 

through the larger of these pores depends upon slow 

vapor diffusion. Water movement through the smaller 

of these pores is negligible even when the pores are 

filled. Therefore rapid water transmission can occur 

only at moisture contents where series of larger pores 

are sufficiently wet for water to move through them 

as a liquid. Below these moisture contents, water

movement as a vapor through the larger pores and as 

a liquid through the smaller pores is very slow, and 

the wetting front remains practically stable.

DIFFERENTIAL EQUATION FOB UNSATUBATED FLOW

HISTORICAL BACKGROUND

In order to understand the derivation and meaning 

of the unsaturated-flow equation, it is helpful to consider first the derivation and meaning of the similar 

equation for saturated flow. This is because "the 

simplest type of porous flow problem deals with 

'saturated' media in which all of the pores are completely filled with one homogeneous liquid" (Miller 

and Miller, 1956, p. 324). Historically, the equation 

was first derived for the saturated case and later 

adapted to the unsaturated case.

Henry Darcy discovered in 1856 an empirical proportionality between macroscopic flow rate and driving 

force (Darcy, 1856). By adding a conservation-of-matter condition for steady flow to Darcy's law written in 

differential form, Slichter in 1899 obtained an equation 

identical in form with the Laplace equation (Slichter, 

1899, p. 330). Finally, by adding a conservation-ofmatter condition for unsteady flow to Darcy's law written in differential form, Theis obtained an equation 

identical in form with the heat-flow equation (Theis, 

1935). "This has since formed the basis for the successful development of saturated flow technology" 

(Miller and Miller, 1956, p. 324). This type of equation 

is known as the equation of heat conduction in studies 

of heat conduction, as the diffusion equation in studies 

of chemical diffusion, and as the nonequilibrium equation in studies of ground water.

Several difficulties delayed the development of the 

analogous equation for unsaturated flow. The first difficulty stems from the nature of the potentials and the 

driving forces in unsaturated systems. In saturated 

systems the potentials are relatively simple, involving 

only position and pressure, and are easily measured by 

means of water wells and piezometers. The potentials 

in unsaturated flow systems are much more complicated 

and involve components of gravitational, hydrostaticpressure, osmotic-pressure, adsorption, temperature, 

and chemical potentials. Unsaturated-flow potentials 

also vary in a complicated way with moisture content, 

and hysteresis enters into the relationship between 

them. Furthermore, measurements of potential and 

moisture content in soils are difficult and tedious to 

make. An additional difficulty not encountered in saturated flow is due to the fact that the transmission constant, or capillary conductivity, is dependent upon the 

moisture content and thus is variable.
D34 FLUID MOVEMENT IN EARTH MATERIALS

Buckingham (1907) recognized the analogy between 

potentials in unsaturated-flow systems and in other 

flow systems and introduced the concept of capillary 

potential. Richards (1931, p. 323-324) was able to 

adapt the heat-flow or diffusion type of equation to 

unsaturated flow by writing the moisture content and 

capillary conductivity as unspecified independent functions of the capillary potential or the suction. In this 

way, he arrived at the type of equation that forms the 

basis for unsaturated-flow studies. Richards assumed 

that Darcy's law holds for unsaturated systems and 

subsequent work has shown the validity of this assumption (Philip, 1958, p. 153).

As has been discussed, unsaturated flow involves 

several different mechanisms. The particular mechanism that dominates varies with the moisture content. 

It may seem surprising that a single equation can describe flow under these different conditions, but it can 

be done because of the fortunate circumstance that the 

equation for saturated flow is of the same form as the 

equation governing the diffusion of water vapor. Thus 

by proper specification of the transmission coefficient as 

a variable to cover the sum of the different mechanisms 

operating at different moisture contents a single equation can be used (Philip, 1958, p. 158).

It is recognized by some investigators (Edlefsen and 

Anderson, 1943) that the thermodynamic specification 

of the liquid and vapor phases is incomplete in current 

use of the equation. However, no solution to this 

difficulty is currently available.

The best answer available for the hysteresis problem 

has been to confine the solution of the equation to 

periods of either soil drying or soil wetting so that moisture content and capillary conductivity can be treated 

as single-valued functions of the capillary potential.

Because of the importance of the unsaturated-flow 

equation, its derivation is presented. It is derived from 

the equation of continuity and the Darcy equation, 

which are discussed first.

EQUATION OF CONTINUITY

The equation of continuity is chiefly a statement of 

the law of conservation of matter. It "states that 

the fluid mass in any closed system can be neither 

created nor destroyed" (Muskat, 1937, p. 121). It 

can be derived from the fact that the change of mass in 

a small unit rectangular parallelepiped equals the difference between the mass entering and the mass leaving.

Figure 22 shows a unit rectangular parallelepiped 

with center at x, y, z. The mass flow of moisture 

through the parallelepiped is to be computed first. 

This is the net flow of moisture out of the volume element per unit time, in grams of water per square 

centimeter per second.

FIGURE 22. Unit rectangular parallelepiped.

Let Vx equal the mass flow of moisture per unit time 

in the x direction, through a unit cross-sectional area 

at a distance x from the yz plane. Vx is measured in 

grams per square centimeter per second.

Then, the expression

da;

equals the change in the mass flow of moisture per unit 

time per unit cross-sectional area in the x direction, 

with distance from the yz plane.

The mass flow through an elemental plane of the 

parallelepiped parallel to the yz plane and cutting the 

point x, y, z is Vxdydz, in grams per second. The mass

dx flow into the side dydz at a distance x from the yz

plane is

^ -jdydz

or

(91)

(92)

Similarly, tfye flow out of the side dydz at a distance 

x-\-dx from the yz plane is 

2

dx (93)

The net mass flow in the x direction out of the unit 

rectangular parallelepiped per unit time is the flow out,
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D35

equation 93, minus the flow in, equation 92. Making 

the subtraction, it is

, , , dxdydz (94)

Similarly, let

Fj,=the mass flow of moisture per unit time in the 

y direction, through a unit cross-sectional 

area at a distance y from the xz plane.

Fz =the mass flow of moisture per unit time in the 

z direction, through a unit cross-sectional 

area at a distance z from the xy plane.

Then the net mass flow in the y direction out of the unit 

parallelepiped per unit time is

dxdydz (95)

The net mass flow in the z direction out of the unit 

parallelepiped per unit time is

dxdydz (96)

The total mass flow out of the unit parallelepiped per 

unit time, in grams per second, is the sum of equations 

94, 95, and 96. Adding and simplifying, it is

dy dxdydz (97)

The loss of mass of water in the unit parallelepiped 

per unit time is to be computed next. Let

per = the bulk density of the medium or soil, in grams

per cubic centimeter. 

/3=the moisture content on a dry- weight basis.

This is a decimal fraction obtained by

dividing the mass of water in grams by the

mass of dried soil in grams.

The mass of water per volume of soil, in grams per 

cubic centimeter, is

P*P (98)

The mass of water in the unit parallelepiped of volume 

dxdydz is

(99)

The loss of mass of water in the unit parallelepiped per 

unit time, in grams per second, is

^ dxdydz (100)

where £=time, in seconds.

According to the law of conservation of matter, the 

mass loss of water in the parallelepiped equation 100 

must equal the flow out of the parallelepiped equation 

97, or

dt dxdydz- ^1 dxdydz (101) 

oz j

Dividing by dxdydz,

dt (102)

which is the equation of continuity.

Let V be a vector giving the mass flow of moisture, 

in grams per second per square centimeter, in the 

direction of the line of flow. Then,

V=iVx+]Vy+kV, (103)

Using the vector operator defined in equation 58 on 

equation 103,

But, 

Therefore,

Comparing equations 102 and 106

_d(p^) v

(105)

(106)

(107)

V is the vector operator known as del or nabla. It 

will be recalled also that the product of this operator and 

a scalar such as potential is a vector known as the 

gradient. The scalar product of this operator and a 

vector, such as velocity above, is known as the divergence (Brand, 1947, p. 183).

The divergence operator applied to a vector function 

gives at each point the rate per unit volume at which 

the physical entity is issuing from that point. If the 

divergence is positive, as in equation 107, there must 

be a source of water located at the point, or else water 

must be leaving the point. If water is leaving the 

point the storage of water at that point must be decreasing as shown by the negative time derivative on 

the left side of equation 107. If the divergence is zero 

and there are no sources or sinks, equation 107 shows 

that the rate of change of storage also is zero and that 

steady-state conditions pertain.
D36 FLUID MOVEMENT IN EARTH MATERIALS

DARCY'S LAW

As stated by Richards (1931, p. 323), "Darcy, 

working with mediums under saturated conditions, 

found that the flow of water through a column of 

soil is directly porportional to the pressure difference 

and inversely proportional to the length of the column. 

For low pressure gradients it has been found by 

numerous investigators (Stearns, 1927; King, 1899) 

that this law is in exact agreement with experiment 

and it is entirely analogous to the well-known law 

of Poiseuille for the flow of liquids through capillary 

tubes. However, both of these laws fail to hold 

for high pressure gradients. The limits within which 

they are true and the modifications which a second 

approximation requires can be determined only by 

exhaustive experiments on a wide range of materials. 

In view of the experimental data now available it is 

assumed that Darcy's law holds for the low velocities 

and pressure gradients dealt with in this paper."

Darcy's law may be written as

where

(108)

VV =SL vector giving the volume flow of moisture, 

in cubic centimeters per square centimeter per second

-fiT=the coefficient of capillary conductivity 

 V0=the negative gradient of the total potential

If 0 is in ergs per gram, K is in seconds.

FLOW EQUATION

The mass flux is the product of the fluid density 

and the volume flux, or

V=P V. (109)

where p=the density of the water, in grams per cubic 

centimeter.

Therefore, equation 107 can be changed to

(HO)

where Vv is the vector giving the volume flow of 

moisture.

The value of VV) the volume flow, can be replaced 

by means of Darcy's law, equation 108. Then the 

equation of continuity, equation 110, becomes

(HI)

This is the general equation for unsaturated flow.

For a system in which flow is occurring in the vertical 

direction only, and for which pa is constant, equation 

111 becomes

d|8 d f

>rr~>r ot oz |_ >r oz

Equation 78 may be rewritten as

0:

where

(112)

(113)

0=the total potential, in ergs per gram 

2=the vertical height above the datum 

#2=the gravitational potential, in ergs per gram 

S=the capillary potential, in ergs per gram

Substituting this into equation 112,

(114)

Assuming g and p to be constant and dividing by pa , 

we have

If |3 and S may be considered to be related by a singlevalued function, equation 115 can be rewritten as

if

the equation for vertical flow is

(117)

(118)

Similarly, for unidirectional horizontal flow, the flow 

equation is

K is known as the diffusivity. The reciprocal of the 

term ^ is analogous to specific heat in the theory of

heat flow, and Klute (1952a, p. 106) proposes the name 

specific moisture capacity for it.

Under conditions of steady-state flow the mass loss 

of water, or the change in moisture content with time,

>TT> is zero. Therefore, the left sides of equations

102, 107, 111, 118, and 119 are equal to zero. This 

was discussed previously in regard to equation 107, the 

continuity equation. It was pointed out that if the
REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D37

divergence is zero and there are no sources or sinks, 

the rate of change of storage of water also is zero and 

steady-state conditions pertain.

Under steady-state conditions the general equation 

for unsaturated flow, equation 111, becomes

0=v(p-KW>) (120)

When the density of water, p, and the coefficient of 

capillary conductivity, K, are constant, equation 120 

can be rewritten as

VV=0 (121) 

or

^+^+^4=0 (122)

This is the well-known Laplace equation. Similarly, 

for steady-state conditions, the equation for vertical 

flow of soil moisture, equation 118, becomes

(123) V

Finally, for steady-state conditions the equation for 

unidirectional horizontal flow, equation 119, becomes

(124) dx\ da

These equations are difficult to use because the 

capillary conductivity, K, the specific moisture capacity,

^> the total potential, <j>, the diffusivity, K, and the

capillary potential all depend on the moisture content. 

To solve the equations, the functional dependence of 

capillary conductivity, specific moisture capacity, 

potential, and diffusivity on moisture content must be 

determined by empirical or other means. Exact 

solutions of equation 123 under given boundary conditions are available from Richards (1931, p. 329), 

Remson and Fox (1955, p. 308), Wind (1955), and 

Gardner (1958a).

In the past few years, numerical, or iterative, methods 

have been devised for the solution of the flow equation 

for the nonequilibrium or transient case under given 

boundary conditions (Klute, 1952a, Klute, 1952b; 

Luthin and Day, 1955; Day and Luthin, 1956, p. 445; 

Philip, 1957; Youngs, 1957). It is hoped that a large 

number of solutions applicable to soil-moisture flow 

under different conditions will be available within a 

few years.

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