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173
CHAPTER 7
Copyrighted Material
Effective, Intergranular, and
Total Stress
7.1 INTRODUCTION
The compressibility, deformation, and strength prop-erties
of a soil mass depend on the effort required to
distort or displace particles or groups of particles rel-ative
to each other. In most engineering materials,
resistance to deformation is provided by internal
chemical and physicochemical forces of interaction
that bond the atoms, molecules, and particles together.
Although such forces also play a role in the behavior
of soils, the compression and strength properties de-pend
primarily on the effects of gravity through self
weight and on the stresses applied to the soil mass.
The state of a given soil mass, as indicated, for ex-ample,
by its water content, structure, density, or void
ratio, reflects the influences of stresses applied in the
past, and this further distinguishes soils from most
other engineering materials, which, for practical pur-poses,
do not change density when loaded or unloaded.
Because of the stress dependencies of the state, a
given soil can exhibit a wide range of properties. For-tunately,
however, the stresses, the state, and the prop-erties
are not independent, and the relationships
between stress and volume change, stress and stiffness,
and stress and strength can be expressed in terms of
definable soil parameters such as compressibility and
friction angle. In soils with properties that are influ-enced
significantly by chemical and physicochemical
forces of interaction, other parameters such as cohe-sion
may be needed.
Most problems involving volume change, deforma-tion,
and strength require separate consideration of the
stress that is carried by the grain assemblage and that
carried by the fluid phases. This distinction is essential
because an assemblage of grains in contact can resist
both normal and shear stress, but the fluid and gas
phases (usually water and air) can carry normal stress
but not shear stress. Furthermore, whenever the total
head in the fluid phases within the soil mass differs
from that outside the soil mass, there will be fluid flow
into or out of the soil mass until total head equality is
reached.
In this chapter, the relationships between stresses in
a soil mass are examined with particular reference to
stress carried by the assemblage of soil particles and
stress carried by the pore fluid. Interparticle forces of
various types are examined, the nature of effective
stress is considered, and physicochemical effects on
pore pressure are analyzed.
7.2 PRINCIPLE OF EFFECTIVE STRESS
The principle of effective stress is the keystone of
modern soil mechanics. Development of this principle
was begun by Terzaghi about 1920 and extended for
several years (Skempton, 1960a). Historical accounts
of the development are described in Goodman (1999)
and de Boer (2000). A lucid statement of the principle
was given by Terzaghi (1936) at the First International
Conference on Soil Mechanics and Foundation Engi-neering.
He wrote:
The stresses in any point of a section through a mass of
soil can be computed from the total principal stresses, 1,
2, 3, which act in this point. If the voids of the soil are
filled with water under a stress u, the total principal
stresses consist of two parts. One part, u, acts in the water
and in the solid in every direction with equal intensity. It
is called the neutral stress (or the pore water pressure).
The balance 1 1 u, 2 2 u, and 3 3
u represents an excess over the neutral stress u, and it has
its seat exclusively in the solid phase of the soil.
Copyright © 2005 John Wiley Sons Retrieved from: www.knovel.com

174 7 EFFECTIVE, INTERGRANULAR, AND TOTAL STRESS
This fraction of the total principal stresses will be called
the effective principal stresses . . . . A change in the neutral
stress u produces practically no volume change and has
practically no influence on the stress conditions for failure
. . . . Porous materials (such as sand, clay, and concrete)
react to a change of u as if they were incompressible and
as if their internal friction were equal to zero. All the meas-urable
effects of a change of stress, such as compression,
distortion and a change of shearing resistance are exclu-sively
due to changes in the effective stresses 1, 2 and
. Hence every investigation of the stability of a saturated 3
body of soil requires the knowledge of both the total and
the neutral stresses.
In simplest terms, the principle of effective stress
asserts that (1) the effective stress controls stress–
strain, volume change, and strength, independent of the
magnitude of the pore pressure, and (2) the effective
stress is given by u for a saturated soil.1
There is ample experimental evidence to show that
these statements are essentially correct for soils. The
principle is essential to describe the consolidation of a
liquid-saturated deformable porous solid, as was done
for the one-dimensional case by Terzaghi and further
developed for the three-dimensional case by others
such as Biot (1941). It is also an essential concept for
the understanding of soil liquefaction behavior during
earthquakes.
The total stress can be directly measured or com-puted
using the external forces and the body force due
to weight of the soil–water mixture. A pore water pres-sure,
denoted herein by u0, can be measured at a point
remote from the interparticle zone. The actual pore wa-ter
pressure in the interparticle zone is u. We know
that at equilibrium the total potential or head of the
water at the two points must be equal, but this does
not mean that u u0, as discussed in Section 7.7. The
effective stress is a deduced quantity, which in practice
is taken as u0.
7.3 FORCE DISTRIBUTIONS IN A
PARTICULATE SYSTEM
The term intergranular stress has become synonymous
with effective stress. Whether or not the intergranular
stress is indeed equal to u cannot be ascertained i
without more detailed examination of all the interpar-
1The terms and are the principal total and effective stresses.
For general stress conditions, there are six stress components (11,
22, 33, 12, 23, and 31), where the first three are the normal stresses
and the latter three are the shear stresses. In this case, the effective
stresses are defined as 11 11 u, 22 22 u, 33 33
u, , , and . 12 12 23 23 31 31
ticle forces in a soil mass. Interparticle forces at the
microscale can be separated into the following three
categories (Santamarina, 2003):
1. Skeletal Forces Due to External Loading These
forces are transmitted through particles from the
forces applied externally [e.g., foundation load-ing)
(Fig. 7.1a)].
2. Particle Level Forces These include particle
weight force, buoyancy force when a particle is
submerged under fluid, and hydrodynamic forces
or seepage forces due to pore fluid moving
through the interconnected pore network (Fig.
7.1b).
3. Contact Level Forces These include electrical
forces, capillary forces when the soil becomes
unsaturated, and cementation-reactive forces (Fig.
7.1c).
When external forces are applied, both normal and
tangential forces develop at particle contacts. All par-ticles
do not share the forces or stresses applied at the
boundaries in equal manner. Each particle has different
skeletal forces depending on the position relative to the
neighboring particles in contact. The transfer of forces
through particle contacts from external stresses was
shown in Fig. 5.15 using a photoelastic model. Strong
particle force chains form in the direction of major
principal stress. The evolution and distribution of in-terparticle
skeletal forces in soils govern the macro-scopic
stress–strain behavior, volume change, and
strength. As the soil approaches failure, buckling of
particle force chains occurs and shear bands develop
due to localization of deformation. Further discussion
of microbehavior in relation to deformation and
strength is given in Chapter 11.
Particle weights act as body forces in dry soil and
contribute to skeletal forces, observed in the photo-elastic
model shown in Fig. 5.15. When the pores are
filled with fluids, the weight of the fluids adds to the
body force of the soil–fluids mixture. However, hydro-static
pressure results from the fluid weight, and the
uplift force due to buoyancy reduces the effective
weight of a fluid-filled soil. This leads to smaller skel-etal
forces for submerged soil compared to dry soil.
Seepage forces that result from additional fluid pres-sures
applied externally produce hydrodynamic forces
on particles and alter the skeletal forces.
7.4 INTERPARTICLE FORCES
Long-range particle interactions associated with elec-trical
double layers and van der Waals forces are dis-

INTERPARTICLE FORCES 175
External Load
Interparticle
Forces
Material
Copyrighted Interparticle
Forces
(a)
Body Force
Buoyancy Force
if Saturated
Viscous Drag by
Seepage Flow
Seepage
(b)
Capillary Force or
Cementation-reactive
Force
Electrical Forces
(c)
Figure 7.1 Interparticle forces at the particle level: (a) skeletal forces by external loading,
(b) particle level forces, and (c) contact level forces (after Santamarina, 2003).
cussed in Chapter 6. These interactions control the
flocculation–deflocculation behavior of clay particles
in suspension, and they are important in swelling soils
that contain expanding lattice clay minerals. In denser
soil masses, other forces of interaction become impor-tant
as well since they may influence the intergranular
stresses and control the strength at interparticle con-tacts,
which in turn controls resistance to compression
and strength. In a soil mass at equilibrium, there must
be a balance among all interparticle forces, the pres-sure
in the water, and the applied boundary stresses.
Interparticle Repulsive Forces
Electrostatic Forces Very high repulsion, the Born
repulsion, develops at contact points between particles.
It results from the overlap between electron clouds,
and it is sufficiently great to prevent the interpenetra-tion
of matter.
At separation distances beyond the region of direct
physical interference between adsorbed ions and hy-dration
water molecules, double-layer interactions pro-vide
the major source of interparticle repulsion. The
theory of these forces is given in Chapter 6. As noted
there, this repulsion is very sensitive to cation valence,
electrolyte concentration, and the dielectric properties
of the pore fluid.
Surface and Ion Hydration The hydration energy
of particle surfaces and interlayer cations causes large
repulsive forces at small separation distances between
unit layers (clear distance between surfaces up to about
2 nm). The net energy required to remove the last few
layers of water when clay plates are pressed together
may be 0.05 to 0.1 J/m2. The corresponding pressure
required to squeeze out one molecular layer of water
may be as much as 400 MPa (4000 atm) (van Olphen,
1977).
Thus, pressure alone is not likely to be sufficient to
squeeze out all the water between parallel particle sur-faces
in naturally occurring clays. Heat and/or high
vacuum are needed to remove all the water from a fine-grained
soil. This does not mean, however, that all the
water may not be squeezed from between interparticle
contacts. In the case of interacting particle corners,
edges, and faces of interacting asperities, the contact
stress may be several thousand atmospheres because
the interparticle contact area is only a very small pro-portion
( 1%) of the total soil cross-sectional area
in most cases. The exact nature of an interparticle con-tact
remains largely a matter for speculation; however,
there is evidence (Chapter 12) that it is effectively solid
to solid.
Hydration repulsions decay rapidly with separation
distance, varying inversely as the square of the dis-tance.
Interparticle Attractive Forces
Electrostatic Attractions When particle edges and
surfaces are oppositely charged, there is attraction due
to interactions between double layers of opposite sign.
Fine soil particles are often observed to adhere when
dry. Electrostatic attraction between surfaces at differ-ent
potentials has been suggested as a cause. When the

gap between parallel particle surfaces separated by dis-tance
d at potentials V1 and V2 is conductive, there is
an attractive force per unit area, or tensile strength,
given by (Ingles, 1962)
4.4 106 (V V )2 F 1 2 N/m2 (7.1) d2
where F is the tensile strength, d is in micrometers,
and V1 and V2 are in millivolts. This force is indepen-dent
of particle size and becomes significant (greater
than 7 kN/m2 or 1 psi) for separation distances less
than 2.5 nm.
Electromagnetic Attractions Electromagnetic at-tractions
caused by frequency-dependent dipole inter-actions
(van der Waals forces) are described in Section
6.12. Anandarajah and Chen (1997) proposed a method
to quantify the van der Waals force between particles
specifically for fine-grained soils with various geomet-ric
parameters such as particle length, thickness, ori-entation,
and spacing.
Primary Valence Bonding Chemical interactions
between particles and between the particles and adja-cent
liquid phase can only develop at short range. Co-valent
and ionic bonds occur at spacings less than 0.3
nm. Cementation involves chemical bonding and can
be considered as a short-range attraction.
Whether primary valence bonds, or possibly hydro-gen
bonds, can develop at interparticle contacts with-out
the presence of cementing agents is largely a
matter of speculation. Very high contact stresses be-tween
particles could squeeze out adsorbed water and
cations and cause mineral surfaces to come close to-gether,
perhaps providing opportunity for cold weld-ing.
The activation energy for soil deformation is high,
in the range characteristic for rupture of chemical
bonds, and strength behavior appears in reasonable
conformity with the adhesion theory of friction (Chap-ter
11). Thus, interatomic bonding between particles
seems possible. On the other hand, the absence of co-hesion
in overconsolidated silts and sands argues
against such pressure-induced bonding.
Cementation Cementation may develop naturally
from precipitation of calcite, silica, alumina, iron ox-ides,
and possibly other inorganic or organic com-pounds.
The addition of stabilizers such as cement and
lime to a soil also leads to interparticle cementation. If
two particles are not cemented, the interparticle force
cannot become tensile; they loose contact. However, if
a particle contact is cemented, it is possible for some
interparticle forces to become negative due to the ten-sile
resistance (or strength) of the cemented bonds.
There is also an increase in resistance to tangential
force at particle contacts. However, when the bond
breaks, the shear capacity at a contact reduces to that
of the uncemented contacts.
An analysis of the strength of cemented bonds
should consider three cases: (i) failure in the cement,
(ii) failure in the particle and (iii) failure at the ce-ment–
particle interface. The following equation can be
derived (Ingles, 1962) for the tensile strength T per
unit area of soil cross section:
1 n
Pk(7.2) T 1 e n
Ai
1
where P is the bond strength per contact zone, k is the
mean coordination number of a grain, e is the void
ratio, n is the number of grains in an ideal breakage
plane at right angles to the direction of T, and Ai is
the total surface area of the ith grain.
For a random and isotropic assembly of spheres of
diameter d, Eq. (7.2) becomes
Pk
(7.3) T d2(1 e)
For a random and isotropic assembly of rods of length
l and diameter d
Pk
(7.4) T d(l d/2)(1 e)
Bond strength P is evaluated in the following way (Fig.
7.2) for two cemented spheres of radius R. It may be
shown that
(R cos )
cosh R sin (7.5)

so for known , can be computed. Then, for cement
failure,
P 2 (7.6) c
where c is the tensile strength of the cement; for
sphere failure,
P ()2 (7.7) s
where R sin , and s is the tensile strength of
the sphere, and for failure at the interface

INTERPARTICLE FORCES 177
Figure 7.2 Contact zone failures for cemented spheres.
sin P 2R2(1 cos ) (7.8) 1
where 1 is the tensile strength of the interface bond.
In principle, Eq. (7.6), (7.7), or (7.8) can be used to
obtain a value for P in Eq. (7.2) enabling computation
of the tensile strength T of a cemented soil.
The behavior of cemented soils can depend on the
timing of cementation development. Artificially ce-mented
soils are often loaded after cementation has
developed, whereas cementation develops during or af-ter
overburden loading in natural soils. In the former
case, the particles and cementation bonding are loaded
together and contact forces can become negative de-pending
on the tensile resistance of cementation bond-ing.
The distribution and magnitude of skeletal forces
are therefore influenced by both geometric arrange-ment
of particles and the cementation bonding at the
particle contacts. In the latter case, on the other hand,
the contact forces induced by external loading are de-veloped
before cementation coats the already loaded
particles. In this case, it is possible that cementation
creates extra forces at particle contacts. In some ce-mented
natural materials, if the soil is unloaded from
high overburden stress, elastic rebound may disrupt ce-mented
bonds.
Cementation allows interparticle normal forces to
become negative, and, therefore, the distribution and
evolution of skeletal forces may be different than in
uncemented soils, even though the applied external
stresses are the same. Thus, the stiffness and strength
properties of a soil are likely to differ according to
when and how cementation was developed. How to
account for this in terms of effective stress is not yet
clear.
Capillary Stresses Because water is attracted to
soil particles and because water can develop surface
tension, suction develops inside the pore fluid when a
saturated soil mass begins to dry. This suction acts like
a vacuum and will directly contribute to the effective
stress or skeletal forces. The negative pore pressure is
usually considered responsible for apparent and tem-porary
cohesion in soils, whereas the other attractive
forces produce true cohesion.
When the soil continues to dry, air starts to invade
into the pores. The air entry pressure is related to the
pore size and can be estimate using the following equa-tion,
assuming a capillary tube as shown in Fig. 7.3a:
2 cos

aw ˆP (7.9) c rp
where is the capillary pressure at air entry, ˆc
P
is aw the air–water interfacial tension,

is contact angle de-fined
in Fig. 7.3, and rp is the tube radius. For pure
water and air, aw depends on temperature, for exam-ple,
it is 0.0756 N/m at 0C, 0.0728 N/m at 20C, and
0.0589 N/m at 100C. If the capillary pressure Pc
( ua uw, where ua and uw are the air and water
pressures, respectively) is larger than then air in- ˆP
, c
vades the pore.2 Since soil has pores with various sizes,
the water in the largest pores is displaced first followed
by smaller pores. This leads to a macroscopic model
of the soil–water characteristic curve (or the capillary
pressure–saturation relationship), as discussed in Sec-tion
7.11.
If the water surrounding the soil particles remains
continuous [termed the ‘‘funicular’’ regime by Bear
(1972)], the interparticle force acting on a particle with
radius r can be estimated from
2 It is often assumed that ua 0 (for gauge pressure) or 1 atm (for
absolute pressure). However, this may not be true in cases such as
rapid water infiltration when air in the pores cannot escape or the air
boundary is completely blocked.

Capillary Tube
Representing a Pore
2rp
dc
Material
Copyrighted θ
ua
uw
^ 2σaw cosθ
Pc = ρw gdc =
rp
(a) (b)
Figure 7.3 Capillary tube concept for air entry estimation: (a) capillary tube and (b) bundle
of capillary tubes to represent soil pores with different sizes.
2r2 cos

F r2 Pˆ aw (7.10) c c rp
where rp is the size of the pore into which the air has
entered. Since the fluid acts like a membrane with neg-ative
pressure, this force contributes directly to the
skeletal forces like the water pressure as shown in Fig.
7.4a.
As the soil continues to dry, the water phase be-comes
disconnected and remains in the form of me-nisci
or liquid bridges at the interparticle contacts
[termed the ‘‘pendular’’ regime by Bear (1972)]. The
curved air–water interface produces a pore water ten-sion,
which, in turn, generates interparticle compres-sive
forces. The force only acts at particle contacts in
contrast to the funicular regime, as shown in Fig. 7.4b.
The interparticle force generally depends on the sep-aration
between the two particles, the radius of the liq-uid
bridge, interfacial tension, and contact angle (Lian
et al., 1993). Once the water phase becomes discontin-uous,
evaporation and condensation are the primary
mechanisms of water transfer. Hence, the humidity of
the gas phase and the temperature affect the water va-por
pressure at the surface of water menisci, which in
turn influences the air pressure ua.
7.5 INTERGRANULAR PRESSURE
Several different interparticle forces were described in
the previous section. Quantitative expression of the in-teractions
of all these forces in a soil is beyond the
present state of knowledge. Nonetheless, their exis-tence
bears directly on the magnitude of intergranular
pressure and the relationship between intergranular
pressure and effective stress as defined by
u.
A simplified equation for the intergranular stress in
a soil may be developed in the following way. Figure
7.5 shows a horizontal surface through a soil at some
depth. Since the stress conditions at contact points,
rather than within particles, are of primary concern, a
wavy surface that passes through contact points (Fig.
7.5a) is of interest. The proportion of the total wavy
surface area that is comprised of intergrain contact area
is very small (Fig. 7.5c).
The two particles in Fig. 7.5 that contact at point A
are shown in Fig. 7.6, along with the forces that act in
a vertical direction. Complete saturation is assumed.
Vertical equilibrium across wavy surface x–x is con-sidered.
3 The effective area of interparticle contact is
ac; its average value along the wavy surface equals the
total mineral contact area along the surface divided by
the number of interparticle contacts. Define area a as
3Note that only vertical forces at the contact are considered in this
simplified analysis. It is evident, however, that applied boundary nor-mal
and shear stresses each induce both normal and shear forces at
interparticle contacts. These forces contribute both to the develop-ment
of soil strength and resistance to compression and to the slip-ping
and sliding of particles relative to each other. These interparticle
movements are central to compression, shear deformations, and creep
as discussed in Chapters 10, 11, and 12.

INTERGRANULAR PRESSURE 179
Material
(a)
Copyrighted Soil Particles
Continuous
Water Film
Negative pore pressure acts all
around the particles
Suction forces act only at particle
contacts and the magnitude of the
forces depends on the size of liquid
bridges.
(b)
Liquid
Bridges
Soil Particles
Interparticle
Forces
Pores of Radius
rp Filled with Air
Air
Figure 7.4 Microscopic water–soil interaction in unsaturated soils: (a) funicular regime and
(b) pendular regime.
Figure 7.6 Forces acting on interparticle contact A.
Figure 7.5 Surfaces through a soil mass.
the average total cross-sectional area along a horizontal
plane served by the contact. It equals the total hori-zontal
area divided by the number of interparticle con-tacts
along the wavy surface. The forces acting on area
a in Fig. 7.6 are:
1. a, the force transmitted by the applied stress ,
which includes externally applied forces and
body weight from the soil above.

2. u(a ac), the force carried by the hydrostatic
pressure u. Because a ac and ac is very small,
the force may be taken as ua. Long-range,
double-layer repulsions are included in ua.
3. A(a ac) Aa, the force caused by the long-range
attractive stress A, that is, van der Waals
and electrostatic attractions.
4. Aac, the force developed by the short-range at-tractive
stress A, resulting from primary valence
Material
Copyrighted (chemical) bonding and cementation.
5. Cac, the intergranular contact reaction that is gen-erated
by hydration and Born repulsion.
Vertical equilibrium of forces requires that
a Aa Aa ua Ca (7.11) c c
Division of all terms by a converts the forces to
stresses per unit area of cross section,
ac (C A) u A (7.12)
a
The term (C A)ac /a represents the net force across
the contact divided by the total cross-sectional area
(soil plus water) that is served by the contact. In other
words, it is the intergrain force divided by the gross
area or the intergranular pressure in common soil me-chanics
usage. Designation of this term by gives i
A u (7.13) i
Equations analogous to Eqs. (7.11), (7.12), and (7.13)
can be developed for the case of a partly saturated soil.
To do so requires consideration of the pressures in the
water uw and in the air ua and the proportions of area
a contributed by water aw and by air aa with the con-dition
that
a a a i.e., a → 0 w a c
The resulting equation is
aw A u (u u ) (7.14) i a a w a
In the absence of significant long-range attractions,
this equation is similar to that proposed by Bishop
(1960) for partially saturated soils
u (u u ) (7.15) i a a w
where aw/a. Although it is clear that for a dry soil
0, and for a saturated soil 1.0, the usefulness
of Eq. (7.15) has been limited in practice because of
uncertainties about for intermediate degrees of sat-uration.
Further discussion of the effective stress con-cept
for unsaturated soils is given in Section 7.12.
Limiting the discussion to saturated soils, two ques-tions
arise:
1. How does the intergranular pressure relate to i
the effective stress as defined for most analyses,
that is, u?
2. How does the intergranular pressure relate to i
the measured quantity, u0m , that is taken
as the effective stress, recalling (Section 7.2) that
pore pressure can only be measured at points out-side
the true interparticle zone?
Answers to these questions require a more detailed
consideration of the meaning of fluid pressures in soils.
7.6 WATER PRESSURES AND POTENTIALS
Pressures in the pore fluid of a soil can be expressed
in several ways, and the total pressure may involve
several contributions. In hydraulic engineering, prob-lems
are analyzed using Bernoulli’s equation for the
total heads and head losses associated with flow be-tween
two points, that is,
p v2 p v2 Z 1 1 Z 2 2
h (7.16) 1 2 1–2 2g 2g w w
where Z1 and Z2 are the elevations of points 1 and 2,
p1 and p2 are the hydrostatic pressures at points 1 and
2, v1 and v2 are the flow velocities at points 1 and 2,
w is the unit weight of water, g is the acceleration due
to gravity, and
h1–2 is the loss in head between points
1 and 2. The total head H (dimension L) is
p v2
H Z (7.17)
2g w
Flow results only from differences in total head;
conversely, if the total heads at two points are the
same, there can be no flow, even if Z1 Z2 and p1
p2. If there is no flow, there is no head loss and
h1–2
0.
The flow velocity through soils is low, and as a re-sult
v2/2g → 0, and in most cases it may be neglected.
Therefore, the relationship

WATER PRESSURE EQUILIBRIUM IN SOIL 181
p p Z 1 Z 2
h (7.18) 1 2 1–2 w w
is the basis for evaluation of pore pressures and anal-ysis
of seepage through soils and other porous media.
Although the absence of velocity terms is a factor
that seems to simplify the analysis of flows and pres-sures
in soils, there are other considerations that tend
to complicate the problem. These include:
1. The use of several terms to describe the status of
water in soils, for example, potential, pressure,
and head.
2. The possible existence of tensions in the pore wa-ter.
3. Compositional differences in the water from
point-to-point and adsorptive force fields from
particle surfaces.
4. Differences in interparticle forces and the energy
state of the pore fluid from point to point owing
to thermal, electrical, and chemical gradients.
Such gradients can cause fluid flows, deforma-tions,
and volume changes, as considered in more
detail in Chapter 9.
Some formalism in definition and terminology is
necessary to avoid confusion. The status of water in a
soil can be expressed in terms of the free energy rel-ative
to free, pure water (Aitchison, et al., 1965). The
free energy can be (and is) expressed in different ways,
including
1. Potential (dimensions—L2T2: J/kg)
2. Head (dimensions—L: m, cm, ft)
3. Pressure (dimensions—ML1 T2: kN/m2, dyn/
cm2, tons/m2, atm, bar, psi, psf)
If the free energy is less than that of pure water
under the ambient air pressure, the terms suction and
negative pore water pressure are used.
The total potential (head, pressure) of soil water is
the potential (head, pressure) in pure water that will
cause the same free energy at the same temperature as
in the soil water. An alternative definition of total po-tential
is the work per unit quantity to transport re-versibly
and isothermally an infinitesimal amount of
pure water from a pool at a specified elevation at at-mospheric
pressure to the point in soil water under
consideration.
The selection of the components of the total poten-tial
(total head H, total pressure P) is somewhat
arbitrary (Bolt and Miller, 1958); however, the follow-ing
have gained acceptance for geotechnical work
(Aitchison, et al., 1965):
1. Gravitational potential g (head Z, pressure pz)
corresponds to elevation head in normal hydrau-lic
usage.
2. Matrix or capillary potential m (head hm, pres-sure
p) is the work per unit quantity of water to
transport reversibly and isothermally an infinites-imal
quantity of water to the soil from a pool
containing a solution identical in composition to
the soil water at the same elevation and external
gas pressure as that of the point under consider-ation
in the soil. This component corresponds to
the pressure head in normal hydraulic usage. It
results from that part of the boundary stresses
that is transmitted to the water phase, from pres-sures
generated by capillarity menisci, and from
water adsorption forces exerted by particle sur-faces.
A piezometer measures the matrix poten-tial
if it contains fluid of the same composition
as the soil water.
3. Osmotic (or solute) potential s (head hs, pres-sure
ps) is the work per unit quantity of water to
transport reversibly and isothermally an infinites-imal
quantity of water from a pool of pure water
at a specified elevation and atmospheric pressure
to a pool containing a solution identical in com-position
to the soil water, but in all other respects
identical to the reference pool. This component
is, in effect, the osmotic pressure of the soil wa-ter,
and it depends on the composition and ability
of the soil particles to restrain the movement of
adsorbed cations. The osmotic potential is nega-tive,
that is, water tends to flow in the direction
of increasing concentration.
The total potential, head, and pressure then become
(7.19) g m s
H Z h h (7.20) m s
P p p p (7.21) z s
At equilibrium and no flow there can be no varia-tions
in , H, or P within the soil.
7.7 WATER PRESSURE EQUILIBRIUM IN SOIL
Consider a saturated soil mass as shown in Fig. 7.7.
Conditions at several points will be analyzed in terms
of heads for simplicity, although potential or pressure
could also be used with the same result. The system is
assumed at constant temperature throughout. At point
0, a point inside a piezometer introduced to measure

Material
Copyrighted Figure 7.7 Schematic representation of a saturated soil for analysis of pressure conditions.
pore pressure, Z 0, hm hm0, and hs0 0 if pure
water is used in the piezometer. Thus,
H 0 h 0 h 0 m0 m0
It follows that
P h u (7.22) 0 m0 w 0
the measured pore pressure.
Point 1 is at the same elevation as point 0, except it
is inside the soil mass and midway between two clay
particles. At this point, Z1 0, but hs 0 because the
electrolyte concentration is not zero. Thus,
H 0 h h 1 m1 s1
If no water is flowing, H1 H0, and
h h h m1 s1 m0
Also, because p1 p0 u0
u h h (7.23) 0 m1w s1 w
At point 2, which is between the same two clay par-ticles
as point 1 but closer to a particle surface, there
will be a different ion concentration than at 1. Thus,
at equilibrium, and assuming Z2 0,
u h h h h h 0 m2 s2 m1 s1 m0 w
A similar analysis could be applied to any point in the
system. If point 3 were midway between two clay par-ticles
spaced the same distance apart as the particles
on either side of point 1, then hs3 hs1, but Z3 0.
Thus,
u0 Z h h Z h h (7.24) 3 m3 s3 3 m3 s1 w
A partially saturated system can also be analyzed,
but the influences of curved air–water interfaces must
be taken into account in the development of the hm
terms.
The conclusions that result from the above analysis
of component potentials are:
1. As the osmotic and gravitational components
vary from point to point in a soil at equilibrium,

MEASUREMENT OF PORE PRESSURES IN SOILS 183
the matrix or capillary component must also vary
to maintain equal total potential. The concept that
hydrostatic pressure must vary with elevation to
maintain equilibrium is intuitive; however, the
idea that this pressure must vary also in response
to compositional differences is less easy to vi-sualize.
Nonetheless, this underlies the whole
concept of water flow by chemical osmosis.
2. The total potential, head, and pressure are meas-urable,
and separation into components is possi-ble
experimentally, although it is difficult.
3. A pore pressure measurement using a piezometer
containing pure water gives a pressure u0 wh,
where h is the pressure head at the piezometer.
When referred back to points between soil par-ticles,
u0 is seen to include contributions from
osmotic pressures as well as matrix pressures.
Since osmotic pressures are the cause of long-range
repulsions due to double-layer interactions,
measured pore water pressures may include con-tributions
from long-range interparticle repulsive
forces.
7.8 MEASUREMENT OF PORE PRESSURES IN
SOILS
Several techniques for the measurement of pore water
pressures are available. Some are best suited for lab-oratory
use, whereas others are intended for use in the
field. Some yield the pore pressure or suction by direct
measurement, while others require deduction of the
value using thermodynamic relationships.
1. Piezometers of Various Types Water in the pi-ezometer
communicates with the soil through a
porous stone or filter. Pressures are determined
from the water level in a standpipe, by a manom-eter,
by a pressure gauge, or by an electronic
pressure transducer. A piezometer used to mea-sure
pressures less than atmospheric is usually
termed a tensiometer.
2. Gypsum Block, Porous Ceramic, and Filter
Paper The electrical properties across a spe-cially
prepared gypsum block or porous ceramic
block are measured. The water held by the block
determines the resistance or permittivity, and the
moisture tension in the surrounding soil deter-mines
the amount of moisture in the block
(Whalley et al., 2001). The same principle can be
applied by placing a dry filter paper on a soil
specimen and allowing the soil moisture to ab-sorb
into the paper. When the suction in the filter
paper is equal to the suction in the soil, the two
reach equilibrium, and the suction can be deter-mined
by the water content of the filter paper.
These techniques are used for measurement of
pore pressures less than atmospheric.
3. Pressure-Membrane Devices An exposed soil
sample is placed on a membrane in a sealed
chamber. Air pressure in the chamber is used to
push water from the pores of the soil through the
membrane. The relationship between water con-tent
and pressure is used to establish the relation-ship
between soil suction and water content.
4. Consolidation Tests The consolidation pressure
on a sample at equilibrium is the soil water suc-tion.
If the consolidation pressure were instanta-neously
removed, then a negative water pressure
or suction of the same magnitude would be
needed to prevent water movement into the soil.
5. Vapor Pressure Methods The relationship be-tween
relative humidity and water content is used
to establish the relationship between suction and
water content.
6. Osmotic Pressure Methods Soil samples are
equilibrated with solutions of known osmotic
pressure to give a relationship between water
content and water suction.
7. Dielectric Sensors Such as Capacitance Probes
and Time Domain Reflectometry Soil moisture
can be indirectly determined by measuring the
dielectric properties of unsaturated soil samples.
With the knowledge of soil water characteristics
relationship (Section 7.11), the negative pore
pressure corresponding to the measured soil
moisture can be determined. The capacitance
probe measures change in frequency response of
the soil’s capacitance, which is related to dielec-tric
constants of soil particle, water, and air. The
capacitance is largely influenced by water con-tent,
as the dielectric constant of water is large
compared to the dielectric constants of soil
particle and air. Time domain reflectrometry
measures the travel time of a high-frequency,
electromagnetic pulse. The presence of water in
the soil slows down the speed of the electromag-netic
wave by the change in the dielectric prop-erties.
Volumetric water content can therefore be
indirectly measured from the travel time mea-surement.
Piezometer methods are used when positive pore
pressures are to be measured, as is usually the case in
dams, slopes, and foundations on soft clays. The other
methods are suitable for measurement of negative pore
pressures or suction. Pore pressures are often negative
in expansive and partly saturated soils. More detailed

descriptions and comparisons of these and other meth-ods
are given by Croney et al. (1952), Aitchison et al.
(1965), Richards and Peter (1987), and Ridley et al.
(2003).
7.9 EFFECTIVE AND INTERGRANULAR
PRESSURE
In Section 7.5, it was shown that the intergranular pres-sure
is given by
A u (7.25) i
where u is the hydrostatic pressure between particles
(or hm w in the terminology of Section 7.7). General-ized
forms of Eq. (7.24) are
u Z h h (7.26) 0 w mw sw
and
u h u Z h (7.27) m w 0 w sw
Thus, Eq. (7.25) becomes, for the case of no elevation
difference between a piezometer and the point in ques-tion
(i.e., Z 0),
A u h (7.28) i 0 s w
Because the quantity hs w is an osmotic pressure and
the salt concentration between particles will invariably
be greater than at points away from the soil (such as
in a piezometer), hs w will be negative. This pressure
reflects double-layer repulsions. It has been termed R
in some previous studies (Lambe, 1960; Mitchell,
1962). If hs w in Eq. (7.28) is replaced by the absolute
value of R, we obtain
A u R (7.29) i 0
From Eq. (7.25), it was seen that the intergranular
pressure was dependent on long-range interparticle at-tractions
A as well as on the applied stress and the
pore water pressure between particles u. Equation
(7.29) indicates that if intergranular pressure is to i
be expressed in terms of a measured pore pressure u0,
then the long-range repulsion R must also be taken into
account. The actual hydrostatic pressure between par-ticles
u u0 R includes the effects of long-range
repulsions as required by the condition of constant to-tal
potential for equilibrium.
In the general case, therefore, the true intergranular
pressure A u0 R and the conventionally i
defined effective stress u0 differ by the net
interparticle stress due to physicochemical contribu-tions,
A R (7.30) i
When A and R are both small, as would be true in
granular soils, silts, and clays of low plasticity, or in
cases where A R, the intergranular and effective
stress are approximately equal. Only in cases where
either A or R is large, or both are large but of signifi-cantly
different magnitude, would the intergranular and
effective stress be significantly different. Such a con-dition
appears not to be common, although it might be
of importance in a well-dispersed sodium montmoril-lonite,
where compression behavior can be accounted
for reasonably well in terms of double-layer repulsions
(Chapter 10).4
The derivation of Eq. (7.30) assumed vertical equi-librium,
with contributing forces parallel to each other,
that is, the intergranular stress is the sum of the i skeletal forces (defined as u0) and the elec-trochemical
stress (A R), as illustrated in Fig. 7.8a.
This implies that the deformation induced by the elec-trochemical
stress (A R) is equal to the deformation
induced by the skeletal forces at contacts [i.e., a ‘‘par-allel’’
model as described by Hueckel (1992)]. The
change in pore fluid chemistry at constant confinement
() leads to changes in intergranular stresses (), re- i
sulting in changes in shear strength, for example.
An alternative assumption can be made; the total
deformation of soil is the sum of the deformations of
the particles and in the double layers as illustrated in
Fig. 7.8b. The effective stress is then equal to the
electrochemical stress (R A):
R A u (7.31) i 0
This is called the ‘‘series’’ model (Hueckel, 1992), and
the model can be applicable for very fine soils at high
water content, in which particles are not actually in
contact with each other but are aligned in a parallel
arrangement. Increase in intergranular stress or ef- i
fective stress changes the interparticle spacing,
which may contribute to changes in strength properties
upon shearing.
4A detailed analysis of effective stress in clays is presented by Chat-topadhyay
(1972), which leads to similar conclusions, including Eq.
(7.29). was termed the true effective stress and it governed the i
volume change behavior of Na–montmorillonite.

ASSESSMENT OF TERZAGHI’S EQUATION 185
Electrochemical Force Electrochemical Force
Skeletal Force
σ = σ _ u0
Material
Copyrighted Skeletal Force
Electrochemical
Force A _ R
σi
σi
Deformation at
the Contact
σi = σ _ u0 + A _ R
(a)
Skeletal Force
σi
Skeletal Force
σ = σ _ u0
Total Deformation
at the Contact σi
σi = σ _ u0 = A _ R
(b)
Particle Deformation
by Skeletal Force
Electrochemical
Force A _ R
Electrochemical Force
Skeletal Force Electrochemical Force
Skeletal Force
Figure 7.8 Contribution of skeletal force ( u0) and electrochemical force (A R) to
intergranular force i: (a) parallel model and (b) series model.
Since the particles are arranged in parallel as well
as nonparallel manner, the chemomechanical coupling
behavior of actual soils can be far from the predictions
made by the above two models. In fact, Santamarina
(2003) argues that the impact of skeletal forces by ex-ternal
forces, particle-level forces, and contact-level
forces on soil behavior is different, and mixing both
types of forces in a single algebraic expression in terms
of effective stress can lead to incorrect prediction [e.g.,
Eq. (7.15) for unsaturated soils and Eq. (7.30) for soils
with measurable interparticle repulsive and attractive
forces].
7.10 ASSESSMENT OF TERZAGHI’S EQUATION
The preceding equations and discussion do not confirm
that Terzaghi’s simple equation is indeed the effective
stress that governs consolidation and strength behavior
of soils. However, its usefulness has been established
from the experience of many years of successful ap-plication
in practice. Skempton (1960b) showed that
the Terzaghi equation does not give the true effective
stress but gives an excellent approximation for the case
of saturated soils. Skempton proposed three possible
relationships for effective stress in saturated soils:
1. The true intergranular pressure for the case when
A R 0
(1 a )u (7.32) c
in which ac is the ratio of contact area to total
cross-sectional area.
2. The solid phase is treated as a real solid that has
compressibility Cs and shear strength given by
k tan (7.33) i
where is an intrinsic friction angle and k is a
true cohesion. The following relationships were
derived: For shear strength,
ac tan 1 u (7.34) tan
where is the effective stress angle of shearing
resistance. For volume change,

Table 7.1 Compressibility Values for Soil, Rock,
and Concrete
Material
Compressibilitya
per kN/m2 106
C Cs Cs /C
Quartzitic sandstone 0.059 0.027 0.46
Quincy granite (30 m deep) 0.076 0.019 0.25
Vermont marble 0.18 0.014 0.08
Concrete (approx.) 0.20 0.025 0.12
Dense sand 18 0.028 0.0015
Loose sand 92 0.028 0.0003
London clay (over cons.) 75 0.020 0.00025
Gosport clay (normally cons.) 600 0.020 0.00003
After Skempton (1960b).
aCompressibilities at p 98 kN/m2; water Cw 0.49
106 per kN/m2.
1 Csu (7.35) C
where C is the soil compressibility.
3. The solid phase is a perfect solid, so that 0
and Cs 0. This gives
u (7.36)
To test the three theories, available data were studied
to see which related to the volume change of a system
acted upon by both a total stress and a pore water
pressure according to

V
C
(7.37)
V
and also satisfied the Coulomb equation for drained
shear strength d :
c tan (7.38) d
when both a total stress and a pore pressure are acting.
It may be noted that this approach assumes that the
Coulomb strength equation is valid a priori.
The results of Skempton’s analysis showed that Eq.
(7.32) was not a valid representation of effective stress.
Equations (7.34) and (7.35) give the correct results for
soils, concrete, and rocks. Equation (7.36) accounts
well for the behavior of soils but not for concrete and
rock. The reason for this latter observation is that in
soils Cs /C and ac tan /tan approach zero, and,
thus, Eqs. (7.34) and (7.35) reduce to Eq. (7.36). In
rock and concrete, however, Cs /C and ac tan /tan
are too large to be neglected. The value of tan /tan
may range from 0.1 to 0.3, ac clearly is not negli-gible,
and Cs /C may range from 0.1 to 0.5 as indicated
in Table 7.1.
Effective stress equations of the form of Eqs. (7.32),
(7.34), (7.35), and (7.36) can be generalized to the gen-eral
form (Lade and de Boer, 1997):
u (7.39)
where is the fraction of the pore pressure that gives
the effective stress.5 Different expressions for pro-posed
by several researchers are listed in Table 7.2.
5A more general expression has been proposed as ij ijij u,
where ij is the tensor that accounts for the constitutive characteristics
of the solid such as complex kinematics associated with anisotropic
elastic materials (Carroll and Katsube, 1983; Coussy, 1995; Did-wania,
2002).
A more rigorous evaluation of the contribution of
soil particle compressibility to effective stress was
made by Lade and de Boer (1997) using a two-phase
mixture theory. The volume change of the soil skeleton
can be separated into that due to pore pressure incre-ment

u and that due to the change in confining pres-sure

( u) (or

u). The effective stress
increment
is defined as the stress that produces the
same volume change,
CV

V
V CV (

u) 0 sks sku 0
C V
u (7.40) u 0
where
Vsks is the volume change of soil skeleton due
to change in confining pressure,
Vsku is the volume
change of soil skeleton due to pore pressure change,
V0 is the initial volume, C is the compressibility of the
soil skeleton by confining pressure change, and Cu is
the compressibility of the soil skeleton by pore pres-sure
change. Rearranging Eq. (7.40) leads to
Cu

1
u (7.41) C
Lade and de Boer (1997) used this equation to de-rive
an effective stress equation for granular materials
under drained conditions. Consider a condition in
which the total confining pressure is constant [
(

ASSESSMENT OF TERZAGHI’S EQUATION 187
Table 7.2 Expressions for to Define Effective Stress
Pore Pressure Fraction Note Reference
1 Terzaghi (1925b)
n n porosity Biot (1955)
1 ac ac grain contact area per unit area of plane Skempton and Bishop (1954)
1 ac
tan
tan
Equation (7.34) Skempton (1960b)
1
CEquation (7.35); for isotropic elastic
s
C
deformation of a porous material; for solid
rock with small interconnected pores and
low porosity (Lade and de Boer, Material
1997)
Copyrighted Biot and Willis (1957), Skempton
(1960b), Nur and Byerlee (1971), Lade
and de Boer (1997)
1 (1 n)
Cs
C
Equation (7.43) Suklje (1969); Lade and de Boer (1997)
After Lade and de Boer (1997).
Figure 7.9 Variation of with stress for quartz sand and
gypsum sand (Lade and de Boer, 1997).
u) 0], but the pore pressure changes by
u.6 The
volume change of soil skeleton caused by change in
pore pressure (
Vsku) is attributed solely from the vol-umetric
compression of the solid grains (
Vgu). Hence,

V
C V
u C (1 n)V
u

V or sku u 0 s 0 gu
C C (1 n) u s (7.42)
where Cs is the compressibility of soil grains due to
pore pressure change and n is the porosity. Substituting
Eq. (7.42) into (7.41) gives

1 (1 n) Cs
u or C
Cs
1 (1 n) (7.43) C
Figure 7.9 shows the variations of with stress for
quartz sand and gypsum sand (Lade and de Boer,
1997). For a stress level less than 20 MPa, is essen-tially
one. Thus, Terzaghi’s effective stress equation,
while not rigorously correct, is again shown to be an
excellent approximation in almost all cases for satu-rated
soils (i.e., solid grains and pore fluid are consid-ered
to be incompressible compared to soil skeleton
compressibility).
6An example of this condition is a soil under a seabed, in which the
sea depth varies. This condition is often called the ‘‘unjacked con-dition.’’
Can the effective stress concept also be applied for
undrained conditions where drainage is prevented?
That is, when an isotropic total stress load of
iso is
applied, is
u equal to
iso? Using a two-phase mix-ture
theory, the total stress increment (
iso) is sepa-rated
into partial stress increments for the solid phase
(
s) and the fluid phase (
ƒ) (Oka, 1996). Consid-ering
that the macroscopic volumetric strains by two
phases are equal but of opposite sign for undrained

θ
Solid Surface
Material
θ
Copyrighted Air
Water
(reference fluid)
(a)
Air
Water
(reference fluid)
Solid surface
(b)
(c)
Air
Water
Solid
Figure 7.10 Wettability of two fluids (water and air) on a
solid surface: (a) contact angle less than 90, (b) contact an-gle
more than 90, and (c) unsaturated sand with water as the
wetting fluid and air as the nonwetting fluid.
conditions, Oka (1996) showed that the partial stresses
are related to the total stress as follows:
C Cs
ƒ
iso (C/n) (1 1/n)C C s l (7.44)
[(1/n) 1]C (C /n) C s l
s
iso (C/n) (1 1/n)C C s l
where n is the porosity, C is the compressibility of soil
skeleton, Cs is the compressibility of soil particles, and
Cl is the compressibility of pore fluid.
If the excess pore pressure generated by undrained
isotropic loading
is
u, the partial stress increment
for the fluid phase becomes (Oka, 1996)

n
u (7.45) ƒ
Combining Eqs. (7.45) and (7.46),
C Cs
u
(7.46) C C n(C C ) iso s l s
The multiplier in the right-hand side of the above
equation is in fact Bishop’s pore water pressure coef-ficient
B (Bishop and Eldin, 1950).7 For typical soils
(Cs 1.9 2.7 108 m2 /kN, Cl 4.9 109
m2 /kN, C 105 104 m2 /kN), so the values of B
are roughly equal to 1. Hence, it can be concluded that
Terzaghi’s effective stress equation is also applicable
for undrained conditions for most soils.
7.11 WATER–AIR INTERACTIONS IN SOILS
Wettability refers to the affinity of one fluid for a solid
surface in the presence of a second or third fluid or
gas. A measure of wettability is the contact angle,
which was introduced in Eq. (7.9). Figure 7.10 illus-trates
a drop of the reference liquid (water for Fig.
7.10a and air for Fig. 7.10b) resting on a solid surface
in the presence of another fluid (air for Fig. 7.10a and
water for 7.10b). The interface between the two fluids
meets the solid surface at a contact angle

. If the angle
is less than 90, the reference fluid is referred to as the
wetting fluid for a given solid surface. If the angle is
greater than 90, the reference liquid is referred to as
the nonwetting phase. The figure shows that water and
7A similar equation for B value has been proposed by Lade and de
Boer (1997).
air are the wetting and nonwetting fluid, respectively.8
The environmental SEM photos in Fig. 5.27 showed
that water can be either wetting or nonwetting fluid
depending soil mineralogy.
The contact angle is a property related to interac-tions
of solid and two fluids (water and air, in this
case).
as ws cos

(7.47)
aw
where as is the interfacial tension between air and
solid, ws is the interfacial tension between water
and solid, and aw is the interfacial tension between
8Some contaminated sites contain non-aqueous-phase liquids
(NAPLs). In general, NAPLS can be assumed to be nonwetting with
respect to water since the soil particles are in general primarily
strongly water-wet. Above the water table, it is usually appropriate
to assume that the water is the wetting fluid with respect to NAPL
and that NAPL is a wetting fluid with respect to air, implying that
the wettability order is water NAPL air. Below the water table,
water is the wetting fluid and NAPL is the nonwetting fluid.

WATER–AIR INTERACTIONS IN SOILS 189
106
105
104
Material
103
102
101
100
Copyrighted 7
5
3
2
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Volumetric Water Content θw
10-1
1
4
6
1 Dune Sand
2 Loamy Sand
3 Calcareous Fine Sandy Loam
4 Calcareous Loam
5 Silt Loam Derived from Loess
6 Young Oligotrophous Peat Soil
7 Marine Clay
Matric suction ua – uw (kPa)
Figure 7.11 Soil–water characteristic curves for some Dutch
soils (from Koorevaar et al., 1983; copied from Fredlund and
Rahardjo, 1993).
air and water. The microscopic scale distribution of
water and air is illustrated in Fig. 7.10c, whereby it is
assumed that water is wetting the grain surfaces.
The aforementioned discussion on wettability and
contact angle assumes static water drops on solid sur-faces.
It has been observed for movement of water rel-ative
to soil that the ‘‘dynamic’’ contact angle formed
by the receding edge of a water droplet is generally
less than the angle formed by its advancing edge.
Matric suction (or capillary pressure) refers to the
pressure discontinuity across a curved interface sepa-rating
two fluids. This pressure difference exists be-cause
of the interfacial tension present in the fluid–
fluid interface. Matric suction is a property that causes
porous media to draw in the wetting fluid and repel
the nonwetting fluid and is defined as the difference
between the nonwetting fluid pressure and the wetting
fluid pressure. For a two-phase system consisting of
water and air, the matric suction is
u u (7.48) n w
where un is the pressure of the nonwetting fluid (air)
and uw is the pressure of the wetting fluid (water).
Assuming that the soil pores have a cylindrical
shape, like a bundle of capillary tubes as illustrated in
Fig 7.3b, the interface between two liquids in each tube
forms a subsection of a sphere. The capillary pressure
is then related to the tube radius, contact angle, and
the interfacial tension between the two liquids. The
pressure drop across the interface is directly propor-tional
to the interfacial tension and inversely propor-tional
to the radius of curvature. It follows that higher
air pressure is required for air to enter water-saturated
fine-grained than coarse-grained materials.
Soil contains a range of different pore sizes, which
will drain at different capillary pressure values. This
leads to a soil–water characteristic relationship in
which the matric suction is plotted against the volu-metric
water content (or sometimes water saturation
ratio) such as shown in Fig. 7.11.9 The curves are often
determined during air invasion into a previously water-saturated
soil. As the volumetric water content de-creases,
as a result of drainage or evaporation, the
matric suction increases. When water infiltrates into
the soil (wetting or imbibition), the conditions reverse,
with the volumetric water content increasing and ma-tric
suction decreasing. Usually drainage and wetting
9The soil–water characteristic curve is referred to by a variety of
names depending on different disciplines. They include moisture re-tention,
soil–water retention, specific retention, and moisture char-acteristic.
processes do not follow the same curve and the volu-metric
water content versus matric suction curves ex-hibit
hysteresis during cycles of drainage and wetting
as shown in Fig. 7.12a. One cause of hysteresis is the
existence of ‘‘ink bottle neck’’ pores at the microscopic
scale as shown in Fig. 7.12b. Larger water-filled pores
can remain owing to the inability of water to escape
through smaller openings below in the case of drainage
or above in the case of evaporation. Another cause is
irreversible change in soil fabric and shrinkage during
drying.
The curves in Fig. 7.11 have two characteristic
points—the air entry pressure a and residual volu-metric
water content

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