Page 1

A. Verruijt,

Consolidation of soils,

Encyclopedia of Hydrological Sciences,

John Wiley & Sons, Ltd.,

Chichester, UK (2008).

DOI 10.1002/0470848944.hsa303.

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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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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

theeffective 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

0.00010.0010.010.1 1.010.0

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

10 20304050

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.

REFERENCES

Abramowitz M.A. and Stegun I.A. (1964) Handbook of

Mathematical Functions, National Bureau

Washington.

Ba´ u D., Ferronato M., Gambolati G. and Teatini P. (2002) Basin-

scale compressibility of the northern Adriatic by the radioactive

marker technique. G´ eotechnique, 52, 605–616.

Bear J. (1979) Hydraulics of Groundwater, John Wiley & Sons

Ltd: New York.

Bear J. and Corapcioglu M.Y. (1981) Mathematical model for

regional land subsidence due to pumping. 2. Integrated aquifer

subsidence equations for vertical and horizontal displacements.

Water Resources Research, 17, 947–958.

BiotM.A. (1941)General

consolidation. Journal of Applied Physics, 12, 155–164.

Biot M.A. (1956) Theory of propagation of elastic waves in

a fluid-saturated porous solid. The Journal of the Acoustical

Society of America, 28, 168–191.

Biot M.A. and Willis D.G. (1957) The elastic coefficients of

the theory of consolidation. Journal of Applied Mechanics, 24,

594–601.

of Standards:

theory ofthree-dimensional