Page 1
A. Verruijt,
Consolidation of soils,
Encyclopedia of Hydrological Sciences,
John Wiley & Sons, Ltd.,
Chichester, UK (2008).
DOI 10.1002/0470848944.hsa303.
Page 2
Consolidation of Soils
ARNOLD VERRUIJT
University of Technology, Delft, The Netherlands
The theory of consolidation is of major interest in the analysis of deformations of porous media, in soil and rock
mechanics, in nonsteady groundwater movement, and in the subsidence due to groundwater recovery and due
to the depletion of reservoirs of gas or oil. In this article, the basic theory of consolidation is presented, and
the theory is illustrated by some examples. The examples are taken from the fields of elementary soil mechanics,
nonsteady groundwater flow and the production of gas from a deep reservoir. The examples chosen all admit a
relatively simple analytical solution, using integral transformation methods.
INTRODUCTION
The theory of consolidation was originally developed by
Terzaghi (1925) in a study of the delay in the deformation
caused by the slow expulsion of water through the pores in a
material of low permeability under compressive loading, in
this case, a sample of clay. For the one-dimensional case, he
developed the mathematical description of the phenomenon,
on the basis of Darcy’s law for the flow of a fluid through
a porous medium, and his own concept of the effective
stress. He realized that in a soft soil, such as clay, the
deformations are caused by the effective stresses, defined
as the difference of the total stress and the pore pressure,
where the latter must be considered to act over the entire
surface of a cross section.
The theory was generalized to three dimensions and more
general materials, including porous rock, by Biot (1941),
and since then it has been applied to a large variety of
practical problems. A further generalization, to dynamic
problems, was made by De Josselin de Jong (1956) and
Biot (1956). One of the results from this generalization was
that, in general, there are two modes of compressive waves:
one in which the particles and the fluid move in phase and
another in which these two components move in opposite
directions. This last mode has been observed in laboratory
conditions, but it can be shown to be strongly damped (Van
der Grinten et al., 1987).
The original theory had been restricted to elastic defor-
mation behavior of the porous medium, but this restriction
was removed later, especially since the development of
modern numerical methods. Computer models are now
available that include more realistic models of soil behav-
ior, including plastic deformations and creep (Lewis and
Schrefler, 1998; Brinkgreve and Vermeer, 2002). In this
article, the study is restricted to the linear elastic version of
the theory.
The presentation of the basic equations of consolidation
theory in hydrology, soil mechanics, and rock mechanics
usually follows the lines established by Terzaghi and Biot,
which can be characterized as phenomenological. More
general presentations have been developed by De Boer
(2000) and Coussy (2004), but for the solution of engineer-
ing problems such more fundamental theories must usually
first be brought back to the form of Biot’s equations.
The combination of elastic deformations of a porous
matrix and the flow of a fluid through the pores leads
to the difficulty that in continuum mechanics stresses are
usually considered to be positive for tension, whereas in
hydrology the stress in the fluid is usually considered
positive for compression. To maintain Terzaghi’s principle
that the total stress equals the sum of the effective stress
and the pore pressure, in this article all stresses will be
considered positive for pressure. The price to be paid for
this convention is that in Hooke’s law, describing a linear
relation between stress and strain, a minus sign appears.
Much confusion also may be caused by the notations.
Each field of application has its own conventions for the
notation of the most important physical quantities, and
many authors have their own preferences, for instance, for
the elastic coefficients. In this article, it is attempted to
Encyclopedia of Hydrological Sciences. Edited by M G Anderson.
2005 John Wiley & Sons, Ltd.
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GROUNDWATER
follow the mainstream of soil mechanics and hydrology,
allowing for the use of both a compressibility coefficient C
and its inverse, the compression modulus K.
BASIC PRINCIPLES
Before deriving the basic equations of consolidation, it
is convenient to consider some of the basic principles
underlying the theory.
Undrained Compression
Consider an element of porous soil or rock, of porosity
n, saturated with a fluid. The element is loaded, under
undrained conditions, by an isotropic total stress of magni-
tude ?σ. The resulting pore water pressure is denoted by
?p. To determine the relation between ?p and ?σ, the load
is applied in two stages – an increment of pressure both in
the fluid and in the solid particles of magnitude ?p, and a
load on the soil, without any pore pressures, of magnitude
?σ − ?p. Compatibility of the two stages, requiring that
the total volume change is the sum of the volume changes
of the fluid and the solid particles is required only for the
combination of the two stages.
In the first stage, in which the stress in both fluid and
solid particles is increased by ?p, the volume change of
the pore fluid is
?Vf= −nCf?pV(1)
where Cfis the compressibility of the pore fluid (which may
include the compression of small amounts of isolated gas
bubbles) and V is the volume of the element considered.
The volume change of the solid particles is
?Vs= −(1 − n)Cs?pV(2)
where Cs is the compressibility of the solid material.
Assuming that all the solid particles have the same com-
pressibility, it follows that their uniform compression leads
to a volume change of the pore space as well (at this stage,
compatibility of the deformations of fluid and particles is
ignored) of the same magnitude. Thus the total volume
change of the porous medium is
?V = −Cs?pV(3)
In the second stage, the pressure in the fluid remains
unchanged, so that there is no volume change of the fluid,
?Vf= 0
(4)
The stress increment ?σ − ?p on the soil, at constant
pore pressure, leads to an average stress increment of
magnitude (?σ − ?p)/(1 − n) in the solid particles. The
resulting volume change of the particles is
?Vs= −Cs(?σ − ?p)V(5)
In this stage, the volume change of the porous medium as
a whole also involves the deformations due to sliding and
rolling at the points of contact of the particles. Assuming
that this is also a linear process, in a first approximation, it
follows that in this stage of loading
?V = −Cm(?σ − ?p)V(6)
where Cm is the compressibility of the porous medium.
It is to be expected that this is considerably larger than
the compressibilities of the two constituents: fluid and soil
particles, because the main cause of soil deformation is not
so much the compression of the fluid or of the particles,
but rather the deformation due to a rearrangement of the
particles, including sliding and rolling.
Owing to both these loadings, the volume changes are
?Vf= −nCf?pV
?Vs= −(1 − n)Cs?pV − Cs(?σ − ?p)V
?V = −Cs?pV − Cm(?σ − ?p)V
(7)
(8)
(9)
Because there is no drainage, by assumption, the total
volume change must be equal to the sum of the volume
changes of the fluid and the particles, ?V = ?Vf+ ?Vs.
This gives, with equations (7–9),
?p
?σ= B =
1
1 + n(Cf− Cs)/(Cm− Cs)
(10)
The derivation leading to this equation is due to
Bishop (1973), but similar equations were given earlier by
Gassmann (1951) and Geertsma (1957). The ratio ?p/?σ
under isotropic loading conditions is often denoted by B in
soil mechanics (Skempton, 1954). In early developments,
such as in Terzaghi’s publications, the compressibilities
of the fluid and of the soil particles were disregarded,
Cf= Cs= 0. In that case B = 1, which is often used as
a first approximation.
The Principle of Effective Stress
The effective stress, introduced by Terzaghi (1925), is
defined as that part of the total stresses that governs the
deformation of the soil (or rock). It is assumed that the total
stresses can be decomposed into the sum of the effective
stresses and the pore pressure by writing
σij= σ?
ij+ αpδij
(11)
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CONSOLIDATION OF SOILS
3
where σij are the components of total stress, σ?
components of effective stress, p is the pore pressure (the
pressure in the fluid in the pores), δijare Kronecker’s delta
symbols (δij= 1 if i = j, and δij= 0 otherwise), and α
is Biot’s coefficient, which is unknown at this stage. For
the isotropic parts of the stresses, it follows from equa-
tion (11) that
σ = σ?+ αp
In the case of an isotropic linear elastic porous material,
the relation between the volumetric strain ε and the
isotropic effective stress is of the form
ijare the
(12)
ε =?V
V
= −Cm?σ?= −Cm?σ − Cmα?p(13)
where, as before, Cm denotes the compressibility of the
porous material, the inverse of its compression modulus,
Cm= 1/K. This equation should be in agreement with
equation (9), which is the case only if
α = 1 −Cs
Cm
(14)
This expression for Biot’s coefficient is generally accept-
ed in rock mechanics (Biot and Willis, 1957) and in the
mechanics of other porous materials, such as bone or skin
(Coussy, 2004). For soft soils, the value of α is close to 1.
If the coefficient α is taken as 1, the effective stress
principle reduces to
α = 1 : σij= σ?
ij+ pδij
(15)
This is the form in which the effective stress principle is
often expressed in soil mechanics, on the basis of Terzaghi’s
(1925, 1943) original work. This is often justified because
soil mechanics practice usually deals with highly compress-
ible clays or sands, in which the compressibility of the solid
particles is very small compared to the compressibility of
the porous material as a whole. In this case, the effective
stress is also the average of the forces transmitted at the
isolated contact points between the particles.
THE BASIC EQUATIONS OF CONSOLIDATION
Using the principles presented above, and some general
principles of physics, the basic equations of consolidation
can be derived. This is first done for the simplified case of
a porous medium in which both the solid particles and the
fluid in the pores are incompressible.
Incompressible Fluid and Particles: Terzaghi’s
Theory
The basic equations of consolidation can be derived from
the principles of conservation of mass and momentum,
and a hypothesis on the movement of the fluid based on
Darcy’s law.
Conservation of mass of the fluid requires that
∂(nρf)
∂t
+ (nρfvi),i= 0
(16)
where ρf is the density of the fluid, and vi are the three
components of the (average) velocity of the fluid particles.
Partial differentiation of a variable a with respect to the
spatial coordinate xi is denoted as a,i and the summation
convention for repeated indices is applied. This means that
(nρfvi),i=∂(nρfv1)
∂x1
+∂(nρfv2)
∂x2
+∂(nρfv3)
∂x3
Similarly, conservation of mass of the solid particles
requires that
∂[(1 − n)ρs]
∂t
+ [(1 − n)ρswi],i= 0
(17)
where ρs is the density of the solid material and wi is
the (average) velocity of the solid particles. If the two
components are incompressible, as assumed in Terzaghi’s
theory, the two densities are constant, and then, after
elimination of the time derivative of the porosity from the
two equations, one obtains
wi,i+ [n(vi− wi)],i= 0
or
∂ε
∂t+ qi,i= 0
(18)
where qi= n(vi− wi) is the velocity of the fluid with
respect to the solids, considered as an average over an
elementary cross-sectional area of the porous medium as
a whole. This quantity is denoted as the specific discharge,
and it is this quantity that was measured by Darcy in his
famous experiments (Darcy, 1856). The idea that a relative
velocity should be used in the formulation of Darcy’s law
for deformable porous media was used implicitly in many
of the early descriptions of consolidation theory, including
those by Terzaghi, but it was first explicitly stated by
Gersevanov (1934).
Because ε = ?V/V, where V = Vp+ Vs, and the vol-
ume of the incompressible solids Vs is constant, so that
∂Vs/∂t = 0, it follows that
∂ε
∂t= (1 + e)∂e
∂t=
1
1 − n
∂n
∂t
(19)
in which e is the void ratio, e = Vp/Vs.
In the form (18), or an equivalent form using the
void ratio e or the porosity n, the basic equation of
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volume change can be found in many textbooks on soil
mechanics (Terzaghi, 1925; Terzaghi and Fr¨ ohlich, 1936;
Scott, 1963; Harr, 1966; Lambe and Whitman, 1969;
Verruijt, 1969; Craig, 1997, etc.), sometimes with an
additional term expressing the compression of the pore
fluid. The equation is often presented simply as an almost
self-evident expression of the deformation of saturated
soils, consisting of incompressible solid particles and an
incompressible fluid, in which the only possibility for a
volume change is the expulsion of pore fluid. It may be
noted that, in his first publications, Terzaghi expressed his
equations using a reduced (material) coordinate and using
the void ratio e as the variable in which the volume change
was expressed. This is perhaps somewhat complicated, but
it can be shown that his results are in complete agreement
with equation (18). And, furthermore, his presentation has
the advantage that it is also applicable for large strains of
the soil (Znidarcic and Schiffman, 1982). Unfortunately,
in a later publication (Terzaghi, 1943) the left-hand side
of equation (18) was expressed as ∂n/∂t, which differs
from the rate of volumetric deformation ∂ε/∂t by a factor
1 − n, and this results in an incorrect expression for the
final consolidation coefficient. Because this coefficient is
usually determined by comparing experimental results with
theoretical results, the mistake is of little consequence in
engineering practice. Also, the mistake has been generally
ignored in soil mechanics literature (e.g. Scott, 1963; Harr,
1966, Lambe and Whitman, 1969; Craig, 1997), even while
referring to Terzaghi’s (1943) book as the basic source, and
giving the correct form of equation (18), or an equivalent
form. Apparently, his successors have forgiven Terzaghi for
this mistake.
The specific discharge qimay be related to the pressure
gradient by Darcy’s law. For an isotropic material, this may
be formulated as
?∂p
where κ is the permeability of the porous medium
(expressed in square meters), µ is the fluid viscosity, and
gi is the gravity vector. In the case of a coordinate sys-
tem with the z axis pointing in the upward direction, the
only nonzero component is gz= −g, because gravity acts
in the downward direction. Assuming that the permeabil-
ity and the viscosity are constant (and remembering that ρf
has been assumed to be constant earlier), it follows from
equations (18) and (20) that
qi= −κ
µ ∂xi
+ ρfgi
?
(20)
∂ε
∂t=κ
µ∇2p =κ
µp,ii
(21)
One-dimensional Deformation
In general, a second basic equation must be obtained from
a consideration of deformation and equilibrium of the soil
mass. In Terzaghi’s original theory, it is assumed that
the only possible mode of deformation is in the vertical
direction, so that the volume change ε can be identified
with the vertical strain εzz, and, in the case of a linear
relationship between stress and strain, this vertical strain
can be expressed in terms of the vertical effective stress by
εzz= −mvσ?
zz
(22)
where mvis the vertical compressibility of the soil in the
case of lateral confinement, a soil property that can be
measured in an oedometer test (see Figure 1). Because the
effective stress in the case of an incompressible fluid and
particles equals the difference of the total stress and the
pore pressure, it follows that
∂ε
∂t=∂εzz
∂t
= −mv
?∂σzz
∂t
−∂p
∂t
?
(23)
Substitution of this result into equation (21) finally gives
∂p
∂t
=∂σzz
∂t
+ cv∇2p(24)
which is the basic equation of vertical consolidation in
an isotropic and homogeneous porous medium, assuming
an incompressible fluid and incompressible particles. The
coefficient cvis the consolidation coefficient, defined as
cv=
κ
mvµ=
k
mvρfg
(25)
where k is the hydraulic conductivity (or coefficient of
permeability), k = κρfg/µ. Equation (24) was first derived,
for the one-dimensional case of flow in the vertical direction
only, by Terzaghi (1925). It was also derived by Jacob
(1940), using somewhat different notations, for the case
of a compressible aquifer of thickness H, transmissivity
T = kH, and storativity S. The consolidation coefficient
can then be written as cv= T/S, which is equivalent to the
definition given in equation (25).
SOLUTION OF TERZAGHI’S PROBLEM
The simplest nontrivial application of the theory of consol-
idation is Terzaghi’s problem of vertical consolidation of a
clay sample confined in a stiff steel ring (see Figure 1). The
sample is loaded at a certain instant of time (considered as
t = 0) by a given load q, which is then kept constant. This
is usually called a one-dimensional compression test, or an
oedometer test.
Because the load remains constant, and the pore pressure
can vary in only in the vertical direction, the differential
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CONSOLIDATION OF SOILS
5
Figure 1
Terzaghi’s problem of a compression test
equation in this case, where the vertical coordinate is
denoted by z, is
t > 0 :∂p
∂t
= cv∂2p
∂z2
(26)
At the time of loading, no deformation could have taken
place, so that the effective stress remains zero, and the pore
pressure must equal the load,
t = 0 : p = q(27)
The boundary conditions are that the bottom of the
sample is impermeable, and that at the top of the sample,
the pore pressure is zero, assuming that the porous plate at
the top of the sample is very permeable,
z = 0 :∂p
z = h : p = 0
∂z= 0
(28)
(29)
The solution of this problem can be obtained by separa-
tion of variables, or by using the Laplace transformation.
The result is
p
q=4
π
∞
?
j=1
(−1)j−1
2j − 1
?
cos
?
(2j − 1)π
2
z
h
?
× exp
−(2j − 1)2π2
4
cvt
h2
?
(30)
This solution was given by Terzaghi (1925), using the
analogy with a heat conduction problem. It is illustrated
in Figure 2. It is important to note that the consolidation
process is practically finished when cvt/h2= 2. This also
means that if the consolidation process in the laboratory
takes 1h, for a sample of thickness 2cm, then the consol-
idation of a layer of the same material in the field, with a
thickness of 4 m, and with drainage on both sides, takes a
factor 1002as long, that is, 417days.
0
0
p/q
2
1
cvt
h2= 0.5
0.2
0.1
0.05
0.02
0.01
z/h
1
1
Figure 2
problem
Solution of the one-dimensional consolidation
THREE-DIMENSIONAL CONSOLIDATION
Basic Equations
In this section, the general theory of three-dimensional
consolidation is considered (Biot, 1941) for the general case
of a compressible fluid and compressible particles.
The compressibility of the fluid can most conveniently
be expressed by assuming that the constitutive equation of
the fluid is, in agreement with equation (7),
∂ρf
∂t
= ρfCf∂p
∂t
(31)
where Cf is the compressibility of the fluid. With equa-
tion (31), it follows from the equation of conservation of
mass of the fluid, equation (16), that
∂n
∂t+ nCf∂p
∂t+ (nvi),i= 0
(32)
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if a small term nCfvi∂p/∂xi, the product of a velocity and
a pressure gradient, which is small compared to the third
term of equation (32), is disregarded.
The constitutive equation of the solid material is taken
in agreement with equation (8),
∂ρs
∂t
=
ρsCs
1 − n
?∂σ
∂t
− n∂p
∂t
?
(33)
where Cs is the compressibility of the particle material.
Using equation (33) it follows from the equation of con-
servation of mass of the solid particles, equation (17),
that
?∂σ
where again, the terms expressing a product of the veloc-
ity and a stress gradient have been disregarded because
these are an order of magnitude smaller than the other
terms.
Elimination of the terms ∂n/∂t from equations (32) and
(33) gives
−∂n
∂t+ Cs
∂t
− n∂p
∂t
?
+ wi,i− (nwi),i= 0
(34)
wi,i+ n(vi− wi),i+ n(Cf− Cs)∂p
∂t+ Cs∂σ
∂t
= 0
(35)
or, noting that wi,i is the volumetric strain rate ∂ε/∂t,
and that the quantity n(vi− wi) is the relative velocity of
the fluid with respect to the particles, averaged over the
total cross section, which is usually denoted as the specific
discharge qi,
∂ε
∂t+ n(Cf− Cs)∂p
∂t+ Cs∂σ
∂t
= −qi,i
(36)
Because the isotropic total stress can be expressed as
σ = σ?+ αp (see equation 12), and the isotropic effective
stress can be related to the volume strain by σ?= −ε/Cm,
where Cmis the compressibility of the porous medium (the
soil) (see equation 13), it follows that equation (36) can
also be written as
α∂ε
∂t+ Sp∂p
∂t
= −qi,i
(37)
where Spis the elastic storativity of the pore space,
Sp= nCf+ (α − n)Cs
(38)
Equation (37) is the generalization of equation (18) for
a porous medium consisting of compressible particles and
a compressible fluid. The coefficient Sp is equivalent to
the inverse of Biot’s coefficient M. The special case of
incompressible particles (Cs= 0) is often used in soil
mechanics (Lambe and Whitman, 1969; Verruijt, 1969), but
this will not be considered further here.
Substitution of Darcy’s law, equation (20), into equa-
tion (37) gives
α∂ε
∂t+ Sp∂p
∂t
=
?κ
µp,i
?
,i
(39)
This is the generalized form of equation (21).
A second system of equations can be obtained by
considering equilibrium, compatibility, and deformation of
the porous medium. The equations of equilibrium can be
expressed as
σij,i+ fj= 0
where fj is a given body force, for instance expressing
gravity. Because of the effective stress principle, equa-
tion (11), the equations of equilibrium can be expressed
in terms of the effective stress and the pore pressure as
(40)
σ?
ij,i+ αp,j+ fj= 0
(41)
where α is Biot’s coefficient (see equation 14).
The effective stresses σ?
The simplest possible stress–strain relation occurs if it is
assumed that the deformations of the porous medium are
linear elastic. For an isotropic material, the relations are
then
ijdetermine the deformations.
σ?
ij= −?K−2
3G?εδij− G(ui,j+ uj,i)(42)
where K and G are the compression modulus and the
shear modulus of the porous material, and ui is the
displacement vector of the porous medium. The minus sign
is needed here because the stresses are considered positive
for compression. It may be noted that the volume strain ε
can be expressed in terms of the displacements by
ε = uk,k
(43)
It may further be noted that the quantity σ in previous
equations is the isotropic component of the total stress,
σ =1
σ?= −Kε
which expresses that K is the inverse of the compressibility
Cmintroduced above (see equation 13).
Substitution of equations (42) into the equations of
equilibrium (41) gives
??K −2
Equations (39) and (45) form a system of four partial
differential equations in the four basic unknown variables:
3σkk. For isotropic compression, equation (42) gives
(44)
3G?ε?
,j+ [G(ui,j+ uj,i)],j− (αp),j= fj
(45)
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CONSOLIDATION OF SOILS
7
the three displacement components uiand the pore pressure
p. Subject to suitable initial and boundary conditions, this
system of equations may be solvable. The equations were
first presented, using somewhat different parameters, by
Biot (1941).
For a homogeneous material, the equations (45) can be
written in full as
?K +1
?K +1
?K +1
3G? ∂ε
3G?∂ε
3G?∂ε
∂x+ G∇2ux− α∂p
∂y+ G∇2uy− α∂p
∂x= fx
(46)
∂y= fy
(47)
∂z+ G∇2uz− α∂p
∂z= fz
(48)
The fluid conservation equation (39) can now be written
as
α∂ε
∂t+ Sp∂p
∂t
=κ
µ∇2p(49)
These are the basic equations of the theory of consolida-
tion, for a homogeneous and isotropic porous medium.
Derivation Using Mixture Theory
It may be interesting to compare the derivation of the basic
equations with a derivation using mixture theory (Bowen,
1980, 1982; Coussy, 2004; Morland et al., 2004). In this
approach, the basic equations are considered for the two
components of the mixture separately. For conservation of
mass, this was done above, leading to equation (36),
wi,i+ n(vi− wi) + n(Cf− Cs)∂p
∂t+ Cs∂σ
∂t
= 0
(50)
The second set of basic equations consists of the equa-
tions of conservation of momentum for the fluid and for the
particles,
∂(nρfvj)
∂t
∂[(1 − n)ρswj]
∂t
+ (np),j+ nρfgj+ Fj= 0
(51)
+ (σij− npδij),i+ (1 − n)ρsgj− Fj= 0
(52)
where Fjis the interaction force between fluid and particles,
and where σijis the total stress. The interaction force is now
postulated to consist of two parts,
Fj=n2µ(vj− wj)
κ
+ pn,j=nµqj
κ
+ pn,j
(53)
where, as before, µ is the viscosity of the pore fluid,
and κ is the permeability of the porous medium. The first
term of equation (53) represents the friction of the moving
fluid through the system of particles, and the second term
represents the interaction due to the change of shape of
the pore space. This term corresponds to the interaction
between the fluid and the container in a fluid-filled container
in hydrostatics, as first considered by Stevin (1586) (see
Dijksterhuis, 1955). This interaction is the solution of
the hydrostatic paradox (Sears et al., 1976). That this
term is really necessary can be seen by substitution into
equation (52) for the quasi-static case, when the inertia
terms are disregarded. This then gives
qj= −κ
µ(p,j+ ρfgj) =
k
ρfg(p,j+ ρfgj)(54)
where k = κρfg/µ is the hydraulic conductivity. Equa-
tion (50) is Darcy’s law, see also equation (20), which is
generally accepted.
Adding the two equations (51) and (52) gives, if the
inertia terms are again disregarded, considering quasi-
statics only, as is usual in hydrology,
σij,i+ ρgj= 0
(55)
where ρ = nρf+ (1 − n)ρs, the total density of the porous
medium. Equation (55) is equivalent to equation (40), the
equation of equilibrium of the porous medium as a whole,
except that now the body force is specified to be gravity. It
may be concluded that the traditional approach, following
Terzaghi and Biot, leads to the same equations as the
approach using mixture theory. It should also be noted that
it is essential that the deformations of the porous medium
are expressed in terms of the effective stress, and this
requires the application of the effective stress principle, in
the form of equation (11), or in its simplified form with
α = 1, for incompressible particles.
Undrained Response
For the analysis of many problems, it is necessary to
formulate an initial condition. If this involves a sudden
(stepwise) loading along part of the boundary of the porous
medium, it can be expected that pore pressures will be
generated at the instant of loading, which will gradually
dissipate, until a final steady state is reached. The initial
deformations and pore pressures can often be calculated by
considering that at the instant of loading no pore fluid could
have been drained from the porous medium. It now follows
from equation (37), by integrating over a very small time
step ?t, and assuming that the divergence of the specific
discharge vector (qi,i) is finite, that
αε0+ Spp0= 0
(56)
where the subscript 0 denotes the (incremental) value
immediately after application of the load. It follows that
Page 9
8
GROUNDWATER
the initial pore pressure can be related to the initial volume
strain by
p0= −αε0
Sp
(57)
Using
stress–strain relation given by equation (41), the total stress
can now be related to the strains by
the effective stressequation (11)and the
t = 0 : σij= σ?
× εδij− G(ui,j+ uj,i)
ij+ αpδij= −?Ku−2
3G?
(58)
where
Ku= K +α2
Sp
(59)
the undrained compression modulus of the porous medium.
In the special case of an incompressible fluid and incom-
pressible particles (Cf= Cs= 0, so that Sp= 0), the
undrained compression modulus becomes infinitely large.
This corresponds to the notion in early soil mechan-
ics that saturated soft soils, such as clays, are initially
incompressible.
It can be concluded that the response of an elastic porous
material at the instant of loading can be determined by
solving a purely elastic problem with shear modulus G and
compression modulus Ku, using the boundary conditions for
the displacements and the total stresses. If this problem has
been solved, the initial pore pressures can be determined
using equation (57), or, better still, by the relation
t = 0 : p0=
ασ0
α2+ KSp
(60)
which can be derived from equations (53) and (54) by
noting that in the undrained analysis ε0= −σ0/Ku. Equa-
tion (60) has the advantage that it has a finite limit in
the case of incompressible constituents, whereas in equa-
tion (57) both the numerator and the denominator then are
zero. Actually, in the case of incompressible constituents
(Cf= Cs= 0), it appears that p0= σ0, which corresponds
to the basic notion of consolidation theory that a saturated
soil is initially incompressible, so that the initial isotropic
effective stress σ?
pressible fluid but incompressible particles, equation (60)
reduces to
0must be zero. In the case of a com-
t = 0, Cs= 0 : p0=
σ0
(1 + nCfK)
(61)
which is a well-known result in soil mechanics (De Josselin
de Jong, 1963; Verruijt, 1969).
Cryer’s Problem
As an example of a three-dimensional problem in which
the coupling of the groundwater flow and the deformation
of the porous medium plays an essential role, a problem
first presented by Cryer (1963) is considered. The problem
concerns a spherical clay sample (see Figure 3) that is
surrounded by a layer of filter material and an impermeable
membrane around the filter layer. The sample is loaded by
an all round compressive load at time t = 0, which causes
an initial pressure in the water. In the drain around the
sample the pore pressure remains zero, by a connection to
an external container.
The problem is defined by two basic equations, the first of
which is the equation of equilibrium in the radial direction,
∂σ?
∂r
rr
+ 2σ?
rr− σ?
r
tt
+ α∂p
∂r= 0
(62)
where σ?
of effective stress. The second basic equation is the equation
of continuity of the pore fluid, equation (39), in spherical
coordinates,
rrand σ?
ttare the radial and tangential components
α∂ε
∂t+ C0∂p
∂t
=κ
µ
?
∂2p
∂r2+2
r
∂p
∂r
?
(63)
where C0= nCf+ (α − n)Cs, which represents the com-
bined effect of the compressibilities of the fluid and the
solid particles.
Using Hooke’s law for the spherical components of
stress, equation (62) can be expressed in terms of the radial
displacement u and the volume strain ε. This leads to the
relation
?K +4
The problem is further defined by the four boundary con-
ditions. At the center of the sphere, the radial displacement
and the pressure gradient must be zero, and at the outer
boundary of the sphere (r = a), the pore pressure must be
3G?∂ε
∂r= α∂p
∂r
(64)
Figure 3
Cryer’s problem of a clay sphere
Page 10
CONSOLIDATION OF SOILS
9
zero, and the radial total stress must be equal to the given
pressure q, for t > 0.
The problem can be solved using the Laplace trans-
formation (Churchill, 1972). This leads to the following
expression for the pore pressure in the center:
pc
p0
= 2m
∞
?
j=1
sinξj− ξj
mξjcosξj+ (2m − 1)sinξj
exp
?−ξ2
jcvt
βa2
?
(65)
where the coefficients ξj are the positive roots of the
equation
(1 − mξ2
In this solution, the parameters cv,m, and β have the
following meaning:
cv=κ?K +4
µ
?K +4
The coefficient cvis the usual coefficient of consolida-
tion. In the case of an incompressible fluid and incom-
pressible solid particles, the Biot coefficient α = 1, and the
pore space storativity Sp= 0. It then follows that β = 1
and m =?K +4
Figure 4 shows the results for three values of Poisson’s
ratio, for the case of an incompressible fluid and incom-
pressible particles. It is interesting to note that for all values
of ν < 0.5, the pore pressure in the center of the sphere ini-
tially increases before it is finally reduced to zero. This is
caused by the drainage, which starts at the outer shell of the
sphere, and which leads to a tendency for shrinkage of that
outer shell. This leads to an additional compressive stress on
j)sinξj− ξjcosξj= 0
(66)
3G?
3G?(1 + KSp/α2)
; β = α2+?K +4
3G?Sp;
m =
4G
(67)
3G?/4G. For this case, the solution (65)
reduces to the solution given by Cryer (1963).
the practically incompressible core of the sphere, so that an
additional pore pressure is generated. The effect can also be
considered as a consequence of the rapid (immediate) trans-
mission of static stresses, and the gradual progress of the
diffusive process of groundwater flow. An initial increase of
the pore pressure was also obtained by Mandel (1953) for
a sample compressed between two rigid plates, with lateral
drainage. The effect is usually called the Mandel–Cryer
effect. It has been verified experimentally for the case of
the sphere by Gibson et al. (1963) and Verruijt (1965).
The problem of the sudden loading of a cylinder has
been solved by De Leeuw (1964) for the case of an
incompressible fluid and incompressible particles. The
solution involves Bessel functions, but proceeds in the same
way as for the sphere. As can be expected, it leads to similar
results, with a somewhat less pronounced Mandel–Cryer
effect at the center of the cylinder. The solution of the
generalized problem, including the compressibility of the
fluid and the particles, has been presented by Detournay
and Cheng (1993); see also Wang (2000).
Uncoupled Consolidation
The problems of Mandel and Cryer illustrate the strong
coupling between the flow of the fluid and the deformation
of the porous medium. This is in contrast to the mathe-
matically similar theory of thermoelasticity, in which the
magnitude of the coefficients is such that the coupling is
not so pronounced (Boley and Weiner, 1960), and engi-
neering problems can usually be solved by first considering
the heat conduction problem, and then studying the effect
of temperature changes on the deformation. In the theory
of consolidation, such a procedure is possible only in cer-
tain approximations. Some of these are considered in this
section.
A first possibility of uncoupling occurs when it is
assumed that the isotropic total stress remains constant in
0.00010.0010.01 0.11.010.0
0
1
2
pc/p0
ct/a2
v = 0.5
v = 0
v = 0.25
Figure 4
Pore pressure in the center
Page 11
10
GROUNDWATER
time (Rendulic, 1936). Because, in general, for an isotropic
material, σ = −Kε + αp, it now follows that
∂ε
∂t= −α
K
∂p
∂t
(68)
This means that the volume strain ε can be eliminated
from the basic equation (39), which leads to
∂p
∂t
= cc∇2p(69)
where cc is a modified consolidation coefficient, for
isotropic compression,
c =
κK
µ[α2+ KSp]
(70)
Another case of uncoupling, rather popular in ground-
water hydrology, occurs when it is assumed that the ver-
tical deformations are so much larger than the horizontal
deformations that the latter can be disregarded. The basic
equation (39) can then be written as
α∂εzz
∂t
+ Sp∂p
∂t
=κ
µ∇2p(71)
where it has been assumed that the soil is homogeneous, so
that κ/µ is constant. If it is now also assumed that the ver-
tical total stress remains unchanged during consolidation (a
reasonable assumption for many problems in groundwater
hydrology), the vertical strain rate may be related to the
pore pressure as follows:
∂εzz
∂t
= −mv∂(σzz− αp)
∂t
= αmv∂p
∂t
(72)
where mv is the vertical compressibility of the porous
medium, which can be related to the elastic moduli
by mv= 1/?K +4
3G?. Substitution of equation (72) into
equation (71) gives
∂p
∂t
= c∇2p(73)
where c is now a generalized consolidation coefficient, for
vertical deformations only,
c =
κ?K +4
3G?
µ?α2+?K +4
It may be noted that for soft soils, when the compress-
ibilities of the pore fluid and the solid particles can be
disregarded, the factor between square brackets in equation
(74) is equal to 1.
3G?Sp
? =
k
γwmv[α2+ Sp/mv]
(74)
The difference between the two equations (69) and
(73) is only in the value of the consolidation coefficient.
As this parameter is difficult to determine accurately by
calculating it from its definition, which involves several
other parameters that are difficult to determine, it is
often more convenient to simply assume the validity of
an equation of the form (73), ignoring the background
of the consolidation coefficient, and then to determine
its value from the results of a field test. In reservoir
engineering, dealing with the effects of the production
of gas or oil from a deep reservoir, a similar procedure
is often used (Gambolati et al., 2000). The value of the
consolidation coefficient of a reservoir may be predicted
from laboratory tests on core samples, but can more
accurately be determined from the results of monitoring
the decrease of the fluid pressure in the reservoir and the
corresponding deformation of the rock (Mobach and Te
Gussinklo, 1994; Ba´ u et al., 2002).
The practical applicability of a two-stage approach, in
which the pore pressures are determined from a diffusion
equation using an appropriate total storativity, has been
demonstrated in an extensive and careful study by Gambo-
lati et al. (2000), in which the results of such an uncoupled
analysis are compared with those of a fully coupled anal-
ysis, and the differences appear to be minor. This appears
to be a conclusion that applies to most problems of hydrol-
ogy and reservoir operation, in which the consolidation is a
result of fluid withdrawal. It does not apply to many prob-
lems of soil mechanics, in which the consolidation often is
a result of a time-dependent loading of the soil. This usually
requires a fully coupled approach.
Theis’ Problem
A classical problem in hydrology is the case of unsteady
flow to a well in a confined aquifer of infinite extent (Theis,
1935; Jacob, 1940); see Figure 5.
Using polar coordinates, the basic equation can be
written as
∂p
= c
∂t
?
∂2p
∂r2+1
r
∂p
∂r
?
(75)
For a well of constant discharge Q0, and infinitely small
radius, in an infinite aquifer of thickness H, the solution is
p = −Q0µ
4πκHE1
?
r2
4ct
?
(76)
where E1(x) is the exponential integral (Abramowitz and
Stegun, 1964). The solution can easily be obtained using
the Laplace transformation.
The vertical displacement of the soil surface (the subsi-
dence) is of particular interest. It can easily be calculated
by noting that, because the vertical total stress has been
Page 12
CONSOLIDATION OF SOILS
11
H
Q0
Figure 5
Theis’ problem of a well in a confined aquifer
assumed to be constant, any decrease in pore pressure
immediately leads to an increase of the vertical effective
stress, and thus to a vertical strain. It follows that in this
case
εzz= −Q0mvµ
4πκHE1
?
r2
4ct
?
(77)
where mv is the vertical compressibility of the soil.
Expressed in terms of the elastic coefficients, this parameter
can be written as mv= 1/?K +4
3G?. The total subsidence
?
4ct
of a layer of thickness H, which is denoted by w, is
w =
Q0µ
4πκ?K +4
3G?E1
r2
?
(78)
For many more problems of unsteady flow in confined
or semiconfined (leaky) aquifers, solutions have been
obtained, and can be found in the literature (see, e.g.
Hantush, 1964; Bear, 1979).
Horizontal Displacements
From a theoretical viewpoint, the classical approach to
problems of unsteady groundwater flow in aquifers is
not consistent. On the one hand, the flow is considered
to be strictly or mainly horizontal, and, on the other
hand, the horizontal deformations are ignored. This is
unsatisfactory because the horizontal flow of the water
exerts a horizontal friction force on the soil, which can
be expected to result in horizontal displacements. In the
case of a water-producing well, it can be expected that
there will be horizontal displacements directed toward
the well. Such horizontal displacements have indeed been
measured in practice (Wolff, 1970). A simple model to
analyze horizontal as well as vertical deformations in a
pumped aquifer was suggested by Verruijt (1969) and
elaborated by Bear and Corapcioglu (1981). In this model,
the aquifer is assumed to be deforming under constant
vertical total stress and with zero shear stresses along
its upper and lower boundaries. Using this schematization
in the three-dimensional consolidation theory, the pore
pressure distribution for the case of a well of constant
discharge Q0is found to be
p = −Q0µ
4πκHE1
?
r2
4c?t
?
(79)
where c?is a modified consolidation coefficient,
c?=
κ?K +1
3G?
µ?α2+?K +1
3G?S0
?
(80)
Because the soil in this model is less stiff than in the
model considered before, as can be seen by comparing
equations (74) and (80), the consolidation process will be
somewhat slower than in the classical solution, but the
shape of the drawdown curve is the same.
The displacements are found to be
u = −1
2w0
r
H
?
E1(r2/4c?t) +4c?t
r2
?
1 − exp
?
−r2
4c?t
???
(81)
w = w0E1
?
r2
4c?t
?
(82)
where
w0=
αQ0µ
8πκ?K +1
3G?
(83)
Comparison of equation (82) with the classical result
(equation 78) shows that the three-dimensional analysis
leads to a subsidence that is about 50% smaller. Equa-
tions (81) and (82) were derived by Bear and Corapcioglu
(1981) for the case of incompressible constituents.
Figure 6 shows the horizontal and vertical displacements,
as a function of r/H, for three values of the dimensionless
time parameter c?t/H2. It appears that there are indeed
horizontal displacements of the same order of magnitude
as the vertical displacements near the well, thus confirming
the superiority of a three-dimensional model.
Page 13
12
GROUNDWATER
c′t/H 2 = 25
c′t/H 2 = 50
c′t/H 2 = 100
0
0
1020304050
r/H
10 w0
u
w
Figure 6
Horizontal and vertical displacements
It may be noted that the difference between the origi-
nal model with vertical deformations only and the three-
dimensional model may be interesting from a theoretical
viewpoint, but may not be important in engineering prac-
tice, as the values of the consolidation coefficient and the
compressibility can most conveniently be determined from
the results of a field test, and then the basic physical nature
of the parameters is not relevant.
Subsidence by a Point Well in a Half Space
An interesting example of the effect of steady groundwater
flow on the displacements in a deep soil body is the case of
a point well in a half space, bounded by an upper boundary
along which the water supply is sufficient to maintain a
constant pressure (Booker and Carter, 1986), as illustrated
in Figure 7.
z
x
y
Figure 7
Well in half space
The solution for the pore water pressures in the lower
half space z > 0 is
p = −Q0µ
4πκ
?1
R1
−
1
R2
?
(84)
where R1is the distance to the sink, located at a depth h
below the surface, and R2is the distance to an imaginary
source of equal strength, located at the image point at a
height h above the surface. The flow of the groundwater
causes the porous medium to deform, and these deforma-
tions can be determined using the equations of equilibrium,
as expressed by equations (46–48). In this case, it is more
convenient to first investigate the deformations and stresses
caused by a single sink or source in an infinite medium, and
then superimpose the effect of the sink and the source. In
this symmetric setting, the result is that the normal stresses
at the surface z = 0 are zero, but the shear stresses σzrare
not zero. This can be corrected by adding a third problem:
the elastic problem of a half space loaded by a given shear
stress distribution. Because for such problems an effective
method has been developed, using the Hankel transforma-
tion (Sneddon, 1951) this is an elementary mathematical
exercise. The final result is that the displacements are
ur= −
αQ0µ
8πκ?K +4
R1
3G?
+ 4(1 − ν)(R2− z − h)h
×
?r
−
r
R2
rR2
−2rzh
R23
?
(85)
Page 14
CONSOLIDATION OF SOILS
13
0
0
12345678
r/h
2
1
z/h
Figure 8
Displacements of a mesh of squares
uz=
αQ0µ
8πκ?K +4
×
R1
3G?
+h + z
?h − z
R2
+ 2(1 − 2ν)h
R2
+2z(h + z)
R23
?
(86)
The subsidence of the surface z = 0 is of particular inter-
est. If this is denoted by w, it is found from equation (86)
that
z = 0 : w =
(1 − ν)αQ0µ
2πκ?K +4
3G?
h
?
h2+ r2
(87)
As already noted by Booker and Carter (1986), the
maximum subsidence, just above the sink, is independent
of the depth of the sink, although the radius of the area of
influence is proportional to that depth. The displacements
of a mesh of small squares are illustrated in Figure 8, for
the case ν = 0.
Subsidence Caused by the Depletion of a Gas Field
Although the theory of consolidation applies equally well
to a porous medium saturated with water as to a porous
rock filled with oil or gas, and contributions to the the-
ory have come from both fields of application, the types of
problems are often quite different. Perhaps the main rea-
son for this difference is that soils near the surface of the
earth surface are usually well permeable, even though the
permeability of a clay layer may be much smaller than
the permeability of a sand layer, and groundwater flows
through the soil up to great depths, whereas oil and gas
accumulate only in reservoirs when these are hydraulically
isolated from the overlying layers. Thus the assumption of a
homogeneous porous medium is an acceptable proposition
in hydrology, but is an implausibility in reservoir engineer-
ing. For example, the problem considered just before, of
a producing well in a homogeneous porous medium up to
the surface, is very well imaginable in groundwater hydrol-
ogy, but is not relevant to the deformations caused by the
depletion of a reservoir of gas or oil. Such a reservoir may
consist of a deformable porous medium, but is bounded by
layers of impermeable rock (this is one of the conditions
for the preservation of the reservoir). As the fluid pres-
sure in the reservoir is reduced, the effective stresses will
increase and the reservoir will be compacted. This results
in a deformation in the surrounding rock and soil, which is
experienced as a subsidence of the soil surface. To predict
this subsidence is not so much a problem of coupled con-
solidation, but a problem of continuum mechanics, or, in its
simplest form, a problem of elastic deformation, in which
the reduced fluid pressure in the reservoir is the cause of
the deformation. These fluid pressures may be predicted
by a reservoir simulator, and are often measured in situ
during production of the reservoir. A powerful prediction
model for the analysis of the deformations was developed
by Geertsma (1973). A simplified version of this model
will be presented here. The basic element in this model is a
nucleus of compression at a depth h in a semi-infinite elas-
tic solid (Mindlin and Cheng, 1950; Sen, 1950). For this
case, the vertical displacement of the free surface is
w =
(1 − ν)?Vh
π(h2+ r2)3/2
(88)
where ?V is the volume of material extracted at a
small spherical nucleus. On the basis of this solution, the
subsidence caused by a disk-shaped reservoir (see Figure 9)
Page 15
14
GROUNDWATER
x
y
z
Figure 9
Disk-shaped reservoir in half space
can be determined by integration over the volume of the
reservoir. For a circular reservoir of radius a, and thickness
d, in which the vertical strain is ε0, the subsidence in the
point x = x0, y = y0, z = 0, is
1
2π
00
?h2+ (r cosθ − x0)2
where w0is the maximum subsidence, in the origin,
?
The integral has been expressed as tabulated functions
(integrals of Bessel functions) by Geertsma (1973). For a
reservoir of arbitrary shape, the integral must be evaluated
numerically. Results for a circular reservoir, obtained by
numerical integration, are shown in Figure 10, for the case
of a very large reservoir, with a/h = 20 and ν = 0.5. It
is interesting to note that the subsidence curve for such
a large reservoir is relatively flat in the region above the
disk. It is also interesting to note that in the basic solution of
equation (88) the power in the denominator is3
a much faster reduction to zero in radial direction than in the
w
w0
=
?2π
dθ
?a
r
+(r sinθ − y0)2
?3
2
dr(89)
w0= 2(1 − ν)ε0d1 −
1
(1 + a2/h2)1/2
?
(90)
2, indicating
0
0
1
1
2
w/e0d
r/a
Figure 10
a/h = 20 and ν = 0.5
Subsidence for a disk-shaped reservoir, for
case considered in the previous example, shown in Figure 8,
where the power in the denominator was1
2.
Further Examples
Many more examples of solutions of the theory of con-
solidation can be found in the literature; see, for instance,
Zaretsky (1967), Rice and Cleary (1976), Detournay and
Cheng (1993), Wang (2000), and Coussy (2004). Complex
practical problems are usually solved numerically, using the
finite element method; see, for instance, Sandhu and Wilson
(1969), Verruijt (1995), Lewis and Schrefler (1998), Gam-
bolati et al. (2000), Brinkgreve and Vermeer (2002), and
Gambolati et al. (2005).
CONCLUSION
The theory of consolidation has been derived, for soils
consisting of incompressible or slightly compressible solid
particles, and saturated with an incompressible or slightly
compressible pore fluid. Some elementary analytical solu-
tions have been presented, to illustrate some of the main
features of the consolidation phenomenon, such as the retar-
dation of the pore pressure development because of a pos-
sible deformation of the solid skeleton, and the relevance
of three-dimensional deformations in many applications.
Although a fully coupled three-dimensional analysis may
be preferable from a theoretical viewpoint, in problems
of hydrology and reservoir operation sufficient accuracy
can usually be obtained by calculating the pore pressures
from a diffusion-type equation, and then calculating the
deformations using the distribution of pore pressures as a
given body force.
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