a multiscale theory of swelling porous media ii. dual porosity models for consolidation of clays

8/19/2019 A Multiscale Theory of Swelling Porous Media II. Dual Porosity Models for Consolidation of Clays

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Transport in Porous Media 28: 69–108, 1997. 69c

1997 Kluwer Academic Publishers. Printed in the Netherlands.

A Multiscale Theory of Swelling Porous Media:II. Dual Porosity Models for Consolidation of

Clays Incorporating Physicochemical Effects

MÁRCIO A. MURAD1 and JOHN H. CUSHMAN21 Laborat ́ orio Nacional de Computaç ˜ ao Cient ́ ifica, LNCC/CNPq, Rua Lauro Muller 455, 22290 – Rio de Janeiro, Brazil2Center for Applied Math, Math Sciences Building, Purdue University, W. Lafayette, IN 47907 U.S.A. e-mail: [email protected]

(Received: 13 August 1996; in final form: 21 February 1997)

Abstract. A three-scale theory of swelling clay soils is developed which incorporates physico-chemical effects and delayed adsorbed water flow during secondary consolidation. Following earlierwork, at the microscale the clay platelets and adsorbed water (water between the platelets) areconsidered as distinct nonoverlaying continua. At the intermediate (meso) scale the clay platelets andthe adsorbed water are homogenized in the spirit of hybrid mixture theory, so that, at the mesoscalethey may be thought of as two overlaying continua, each having a well defined mass density. Withinthis framework the swelling pressure is defined thermodynamically and it is shown to govern theeffect of physico-chemicalforces in a modified Terzaghi’s effective stress principle.A homogenizationprocedure is used to upscale the mesoscale mixture of clay particles and bulk water (water next tothe swelling mesoscale particles) to the macroscale. The resultant model is of dual porosity typewhere the clay particles act as sources/sinks of water to the macroscale bulk phase flow. The dualporosity model can be reduced to a single porosity model with long term memory by using Green’sfunctions. The resultant theory provides a rational basis for some viscoelastic models of secondaryconsolidation.

Key words: swelling clay soil, mixture theory, homogenization, consolidation, swelling pressure,disjoining pressure, dual porosity.

1. Introduction

Swelling clay soils consisting of an assemblage of clusters of hydrous alluminium

and magnesium silicates with an expanding layer lattice are widely distributed in

the earth’s crust. Their behavior is of paramount importance in almost all aspects

of life, where they are responsible for many reactions and processes. For example,

compacted clays such as bentonite have been extensively used to impede the move-

ment of water through cracks and fissures. They play a critical role in various waste

isolation scenerios such as barriers for commercial land fills. In the context of oil

and gas production, drilling muds play a critical role ([45, 75]). Swelling clays also

play a critical role in the consolidation and failure of foundations, highways and

runways. For example, the foundation engineers face problems when a foundation


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70 MÁRCIO A. MURAD AND JOHN H. CUSHMAN

is built on expansive soils, since as the moisture content increases, the soil swellsand heaves upward, and as the moisture decreases, the soil compacts and the ground

surface recedes and pulls away from the foundation walls [76]. An accurate model

capable of capturing the swelling of clays will have important consequences incivil and petroleum engineering, hydrology, geology and soil science.

Since Terzaghi [72] and Biot [16,17] first proposed linear poroelastic models

for deformable media, the criterion for rupture and failure of soils has been based

on the concept of effective stress. Effective stresses are defined as the difference

between total applied stresses and bulk water pressure. Classically these stresses are

interpreted as contact stresses, i.e. transmitted between points of the intergranular

contact. It has been experimentally verified that the classical Terzaghi effective

stress principle describes accurately coarse-grained soils such as sands, silts and

low and medium plastic clays such as kaolinite or illite. However, its classical

form has been found to be inadequate for explaining deformation of swelling

clays, particularly active plastic clays such as bentonite and montmorillonite. The

reason is that the classical effective stress principle assumes that no other forcesexcept the effective stress and pore pressure are present. The existence of physico-

chemical forces within and between the clay particles are excluded. Interparticle

forces arising from physico-chemical mechanisms have been demonstrated to be

of paramount importance for active clays. Researchers have heuristically modified

Terzaghi’s effective stress principle to account for physico-chemical forces and

as a consequence different mechanistic pictures of the various stresses have been

derived (see Sridharan and Rao [70], Sridharan [68], Lambe [48], Morgensten and

Balasubramonian [60]). A comparison between thesedifferent mechanisticpictures

can be found in, e.g., Hueckel [39, 40] or Graham et al. [31].

The nature of physico-chemical forces remains controversial. In contrast to the

effective stress, net attractive(A)-repulsive(R) forces between the clay particles donot depend upon direct contact. They have at least three components: the Van der

Wals attraction, electrostatic (or osmotic) repulsion and surface hydration (a struc-tural component). This latter component arises due to the hydrophilic structure of

the platelets which manifest short range bonding forces between the minerals and

water. These forces are usually referred to as ‘hydration forces’ (Israelachvili [41]).

The complex mechanisms underlying theconstitutive behavior of a hydrophilic clay

soil are a consequence of its complicated microstructure. Due to their tremendous

specific surface area and their charged character, clusters of clay platelets when

hydrated form ‘particles’ consisting of an assemblage of platelets and adsorbed

water. These particles swell under hydration and shrink under desication. The

platelet-water hydration forces cause the macroscale behavior of clays to signifi-

cantly differ from granular nonswelling media. The hydration forces modify thethermodynamical properties of the water in the interlamellar spaces and conse-

quently its properties vary with the proximity to the solid surface (Low [54–56],

Grim [32]). The interlamellar water is termed adsorbed water to distinguish it

from its bulk or free-phase counterpart (i.e. water free of any adsorptive force).


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Both the heuristic modifications of Terzaghi’s effective stress principle andthe ad-hoc viscoelastic models for secondary consolidation need to be rigorously

justified by upscaling the microscale behavior. Theoretical approaches which have

been used to develop models of porous media include mixture theory and othermethods which propagate microscopic governing equations to the larger scale

(e.g. homogenization, volume averaging, etc.). Hybrid mixture theory, HMT (see

Hassanizadeh and Gray [33, 34]) consists of classical mixture theory in the sense

of Bowen [18] applied to a multiphase system with volume averaged balance

equations. HMT is applicable to a multi-phase mixture in which the characteristic

length of each phase is ‘small’ relative to the extent of the mixture. An average

value for each phase property is established at every point in the mixture, forming

M

coexisting continua at each point. Macroscale dependent variables are defined to

be as consistent as possible with their microscale counterparts, so that an analogue

of classical Gibbsian thermodynamics can be developed. Variables such as swelling

pressure must often be defined in a somewhat nonintuitive fashion. Constitutive

equations are developed on the averaged scale and are subject to constraints placedby the entropy inequality (Coleman and Noll [21]). HMT has been used extensively

to improve our understanding of flow and deformation in nonswelling porousmedia

(see Hassanizadeh and Gray [35–37]). More recently, the theory has been clarified

and extended to derive remarkable results for two scale single porosity swelling

systems such as smectitic clay pastes (Achanta et al. [1, 2]). This framework has

since been extended to three-scale swelling systems (i.e. porous systems composed

of swelling porous particles and bulk fluid filled voids or cracks), by Murad et al.

[61], Bennethum and Cushman [12], and Murad and Cushman [62].

A three-scale model (micro, meso and macro) of a porous matrix consisting of

porous swelling particles is depicted in Figure 1. The particles are in contact with

one another and bulk water. Each particle consists of clay colloids and adsorbedwater. In Murad et al. [61], Bennethum and Cushman [12, 13] and Murad and

Cushman [62] the adsorbed water is treated as a separate phase from the bulk

water. At the microscale the model has two phases, the disjoint clay platelets and

the adsorbed water. At the mesoscale (the homogenized microscale) the model

consists of the clay particles and the bulk water. The macroscale consists of the

bulk water homogenized with the mesoscale particles. To propagate information

between scales, several types of upscaling methods can be used. For example, one

can upscale the microscale to the mesoscale using methods such as, e.g., homo-

genization or volume averaging. Since the microscopic adsorbed water is viewed

as a thin film coating the clay minerals, this method of upscaling would be very

complex as it would involve averaging the thin film governing equations coupled

with those governing the large deformations of the solid phase (e.g. elasticity,viscoelasticity). Alernatively, one can perform such upscaling by adopting the HMT

of Hassanizadeh and Gray [34, 35] together with a proper theory of constitution

including appropriate internal variables needed to capture the swelling character of

the system. The mesoscopic internal constitutive variable which captures particle


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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 73

swelling is the volume fraction. Hence one can perform a simpler upscaling bypursuing the framework of Achanta et al. [2] and adopting a volume fraction

hybrid mixture theory in the sense of Bedford and Drumheller [11].

This type of upscaling was adopted in Murad et al. [61] who used the hybridmixture theory of Achanta et al. [2] to upscale the microscale to the mesoscale. Then

assuming Stokesian slow bulk-water movement, the homogenization procedure

(Bensoussan et al. [15], Sanchez-Palencia [64]) is used to upscale the mesoscopic

governing equations to the macroscale. The extension of this approach to coupled

water flow and solid deformation with disconnected (entrapped) bulk phase fluid

and clay particles was developed in Murad and Cushman [62]. In their framework,

macroscopicgoverningequations for flow and deformation were rigorously derived

by upscaling the microstructure. A different approach was adopted in Bennethum

and Cushman [12, 13]. Here information was propagated to the mesoscale by

averaging the microscale balance laws, but a constitutive theory was not developed

on the mesoscale. Rather, the mesoscopic equations were again averaged directly to

the macroscale and a constitutive theory developed at this latter scale by exploitingthe entropy inequality in the sense of Coleman and Noll [21]. The fundamental

difference between these two approachesis the developmentof a constitutive theory

on the mesoscale in the former and not in the latter. Each approach has advantages

and disadvantages which are governed by the type of experiments one wishes to

run. The upscaling technique from the meso to the macroscale pursued in Murad

and Cushman [62] was a straightforward homogenization of the entire hydrophilic

swelling clay soil. It yielded a macroscopic Darcy’s law in which the velocity is

a superposition of the adsorbed and bulk water velocities. This type of model has

been referred to as a ‘parallel flow type model’ (Showalter [66, 67]) because the

secondary mesoscopic adsorbed/bulk water flow is neglected and the geometry of

the cells is suppressed in the upscaling procedure. One of the disadvantages of thisapproach is that both the particles and bulk phase fluid have the same time scale and

therefore the model cannot incorporate delayed adsorbed water flow. This lattertype of flow is crucial for explaining secondary consolidation.

A notable consequence of the HMT framework of Murad and Cushman [62]

is the appearance of a new stress component, the ‘hydration stress tensor’, which

accounts for physico-chemical effects. However, experimental validation for these

stresses and their relation with other measurable physicochemical quantities such

as swelling pressure still needs to be clarified. Our first goal is to provide insight into

the physical interpretation of this physicochemical quantity. We then compare the

constitutive equations of Murad and Cushman [62] with Low’s swelling pressure

concept and then we provide an alternative way of measuring hydration forces in

active clays. In addition, we show that hydration forces lead to the appearance of a new thermodynamic quantity which governs the excess in pressure of the clay

particles relative to the classical pore pressure of Biot [16, 17] for nonswelling

granular media. We shall illustrate that, although Low’s swelling pressure has


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Derjaguin’s disjoining pressure as a microscale counterpart, this new excess inpore pressure does not.

Furthermore, to derive three-scale macroscopic governing equations we adopt

the homogenization procedure to upscale the mesoscale governing equations of the clay particles together with the Stokesian slow bulk-water movement. We aim

at deriving macroscopic equations that incorporate the delayed flow of adsorbed

water during secondary consolidation. Within the framework of the homogenization

procedure this can be achieved naturally by simply adopting a different scaling

law for the mesoscopic conductivity for the clay particles than that of Murad

and Cushman [62]. This scaling captures the secondary mesoscopic adsorbed-

bulk water flow. This yields a macroscopic picture which exhibits an additional

capacity term to account for the momentum interchange between the particles and

surrounding bulk fluid and leads to a dual porosity or distributed microstructure

model for swelling clay soils. This technique has been successfully used to model

naturally fractured reservoirs in which the system of fractures plays the role of

the bulk system (where the macroscopic flow takes place) and the matrix blocksbehave analogous to the clay particles and are treated as sources/sinks to the bulk

phase (see, e.g., Arbogast, Douglas and Hornung [4, 5, 27] and references therein).

The macroscopic bulk phase flow is influenced at the mesoscale through distributed

source/sinks of momentum which govern the creep constitutive equations for the

macroscale effective stress tensor. In the linear case the dual porosity model can

be solved by using a Green’s function and then reducing it to a single integro-

differential equation of Volterra type in which the kernel appears related to the

geometry of the clay particles. The integro-differential equation can then be related

to some viscoelastic models proposed for secondary consolidation. We remark that

the derivation of the memory effects (due to the delayed adsorbed water secondary

flow) within the HMT three-scale framework of Bennethum and Cushman [12]would require a more general exploitation of the entropy inequality involving

history dependent constitutive variables. Hence, we adopt the homogenizationprocedure.

In the theory developed herein we will assume that the exchangeable cations

are concentrated on the clay surface, such that the platelets negative surface charge

is effectively screened. In other words, as in Low [57, 58], we will consider that

surface hydration is the dominant component of the swelling pressure.

2. Constitutive Equations for the Mesoscale Swelling Clay Particles

We begin by reviewing the main mesoscale results of the constitutive theory devel-

oped by Murad and Cushman [62] for a system composed of clay-platelets andadsorbed-water (the clay particle). Then we re-examine the constitutive theory to

obtain a better feel for the hydration and swelling stresses.


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Figure 1. Three-scale model for clay (from Murad et al. [61]).

2.1. NONEQUILIBRIUM RESULTS

Consider the clay particles as a mixture of two phases (the solid clay platelets and

liquid adsorbed water) viewed as coexisting continua, which undergo independent

motions x = x

X

; t ; = l ; s with respect to each reference configuration (here

x denotes the spatial position of the particle of the -phase at time t with respect

to a reference position X

). Let the subscript = l ; s denote the adsorbed liquid

and solid phase respectively and let

,t

,

and A

denote the averaged density,

symmetric particle stress tensor, volume fraction and intensive Helmholtz potential

of phase . Further, let T denote temperature (assumed equal in both phases) and

let the average mesoscopic strain tensor of the solid phase Es

be given as

Es =

12

Cs

I ;

(2.1)

where Cs

= F T s

Fs

with Fs

= grad xs

denoting the deformation gradient of the solid

phase (with grad denoting the differentiation with respect to a material particle on

the mesoscale).

Assume that on the mesoscale the solid and fluid phases are incompressible,

nonheat conducting, and that the adsorbed water is nonviscous. By postulating

constitutive dependenceof the free energies in the form A

= A

T ; Es

= l ; s

and using the Coleman and Noll method of exploiting the entropy inequality [21],

Murad and Cushman [62] obtained the following constitutive equations for the

stress tensors t

l

tl

=

l

p

̀

I; (2.2)

s

ts

=

s

p

s

I+

t e s

+

t ls

; (2.3)

where the tensors t e s

and t ls

are defined by

te s

=

s

s

Fs

@ A

s

@

Es

F T s

;

tls

=

l

l

Fs

@ A

l

@

Es

FT s

(2.4)


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and p

is the thermodynamic pressure of the -phase. For incompressible media,the Coleman–Noll method is applied to a modified entropy inequality obtained by

adding to the original entropy inequality the continuity equations premultiplied by

Lagrangian multipliers as constraints (see [53, 61, 62] for details). The thermo-dynamic pressures p

turn out to be identified with the Lagrange multipliers. In

the compressible case it is postulated A

= A

T ;

; Es

and application of

the Coleman–Noll method yields the same constitutive equations (2.2) and (2.3)

with the exception that the thermodynamic pressures have their classical definition

p

=

2 @ A

= @

.

In the constitutive theory of Murad and Cushman [62] it was postulated that

A

l

depends on Es

and r Es

was included in the list of independent variables (see

also Bennethum and Cushman [12]). This implies the mesoscale thermodynamics

of the adsorbed water differ from that of a bulk phase fluid and also, unlike

non-swelling granular media, leads to the appearance of the new tensor t ls

. The

inclusion of r Es

allows for strain induced flow of the adsorbed water at the

mesoscale. This dependence of A l

on the mesoscale solid strain is more generalthan that of Achanta etal. [2] and Murad et al. [61], who postulated A

l

T ;

l

. Their

constitutive theory was based on the experimental observationsof Low [56] relating

the behavior of the adsorbed water to the platelet separation h . The assumption

thatA

l

depends on

l

describes accurately swelling particles with interlayer spaces

between 25 and 100 Å . In this range the adsorbed fluid can withstand the hydrostatic

swelling pressure but not shear stress. The additional dependence of A l

on shear

deformations is motivated by the fact that for small platelet separation h (less

than 10 molecular diameters or 25 Å ), the adsorbed fluid molecules become more

ordered and arrange themselves in layers parallel to the surface, the film becomes

structured, inhomogeneous, anisotropic, its effective viscosity rises dramatically

and is able to sustain a shear stress even at equilibrium (see, e.g., Israelachviliet al. [42], Schoen et al. [65], Cushman [22]).

The physical interpretation of (2.3) can be obtained by comparing it with the

analogous results of Hassanizadeh and Gray [35] for nonswelling granular media.

As we shall illustrate next, the difference between these two types of media is

the stress tls

for swelling media which arises due to the additional constitutive

assumption A l

= A

l

Es

. We shall illustrate that t ls

governs physico-chemical

stresses within the clay particles and may also be identified with the swelling

pressure.

2.2. PHYSICAL INTERPRETATION OF THE NEW TENSORt

l

s

FOR SWELLING MEDIA

Equation (2.3) is crucial in the present formulation since it contains importantinformation on the constitutive behavior of the solid phase stress tensor for the

swelling particles. To exploit its physical significance let us introduce the total

particle stress tensor t and the particle thermodynamic pressure p as

t=

s

ts

+

l

tl

; p =

l

p

l

+

s

p

s

: (2.5)


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By adding (2.2) and (2.3) and using (2.5) we obtain

t+ p

I=

te s

+

t ls

:

(2.6)

The above result gives important insight into the stress in swelling particles. To

elucidate this consider a fixed solid strain Es

and define the bulk phase B to be fluid

unaffected by the solid phase (e.g. nonswelling granular media). By definition, the

free energy of a bulk fluid A B

does not change with the proximity of the solid and

therefore is independent of Es

. Therefore t ls

= 0 and noting t e s

only depends on the

fixed solid strain Es

, (2.6) reduces to

tB

+ p

B

I = t e s

; (2.7)

where now the subscript B is used to denote the corresponding property for a

non-swelling granular medium. Equation (2.7) has been derived by Hassanizadeh

and Gray [35] within the context of hybrid mixture theory applied to nonswellinggranular media. In classical soil mechanics the above result resembles in form

Terzaghi’s effective stress principle at the mesoscale for nonswelling media with

p

B

and t e s

normally referred to as pore pressure (or bulk phase pressure) and

effective stress tensor. In classical stress analysis of nonswelling media the pore

pressure p B

has a similar definition to p in (2.5) except that it is assumed equal to

both thermodynamic fluid and solid pressures, i.e. (see, e.g., [16, 35])

p

B

=

s

p

s

+

l

p

l

= p

l

= p

s

; (for a nonswelling medium) (2.8)

The effective stress tensor te s

measures stresses induced by mineral to mineral

contact and primarily controls the deformation of nonswelling systems such as

sands, silts, and low and medium plastic clays such as kaolinite or illite. Themodified effective stress principle (2.6) for swelling media has the additional term,

tls

. Unlike coarse-grained soils, whose stress mechanisms are primarily controlled

by the contact stresses te s

, swelling clays such as montmorillonite contain the

additional stress component tls

which governs the deformation of the swelling

particles. Clearly this additional intra-particle stress results from the presence of

adsorbed water within the particles. It is of physico-chemical nature and can be

viewed as a stress structural component arising from surface hydration. Whence, as

in Murad and Cushman [62], we henceforth denote t ls

the hydration stress tensor .

An important consequence of (2.6) is the partition of the total particle stress tensor

into its platelet t e s

and adsorbed water tls

components. Consequently we can

overcome some limitations in the works of Lambe [48] and Hueckel [40] where it

is assumed that only one stress exists in the platelets and adsorbed water, which

is measured as the difference between the total macroscopic stress and bulk phase

pressure.We next relate the hydration stress tensor tl

s

to the swelling pressure which

has for many years been used to study swelling clays (see Low [56, 57]). Figure 2


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Figure 2. Swelling experiment (from Low [57]).

depicts a classical reverse osmosis swelling pressure experiment performed by Low

wherein bulk water is separated from a well ordered parallel clay platelet-adsorbed

water mixture by a semipermeable membrane. Due to the hydrophilic interaction

between the adsorbed water and the clay minerals the clay tends to swell as water

penetrates the region between its superimposed layers and forces them apart. Inthis experiment an overburden pressure P is applied normally to the clay-water

mixture and the average interlayer separation,h

, of the platelets is measured. The

excess in the overburden pressure relative to the bulk pressure p B

is the swelling

pressure , at equilibrium

P p

B

:

(2.9)

Considering p B

= p atm, with p atm denoting the atmospheric pressure, Low exam-

ined the equilibrium swelling pressure of different montmorillonites clays satur-

ated with adsorbed water. For incompressible fluid he found that the dimensionless

swelling pressure = p atm satisfies the empirical relation

+

1=

exp

1

e

1

e

= B

exp

e

;

(2.10)

where e = l

= 1 l

is the void fraction, e is the void fraction when = 0

(i.e. when P = p B

), is a constant that is related to the specific surface area

and the cation exchange capacity,B =

exp = e

, and the notation

for the

dimensionless swelling pressure has been maintained.

In what follows we pursue a more general definition of the mesoscale swelling

pressure which retains the same physical interpretation as (2.9) under the equilib-

rium conditions of the swelling experiment depicted in Figure 2. For simplicity

we assume incompressibility and first note that in Low’s swelling pressure experi-

ment, the reference bulk phase pressure p B

is defined in the domain occupied by the

bulk water. Unfortunately the generalization of Low’s definition (2.9) to the casewhere particles undergo nonequilibrium processes requires a pointwise definition

for x; t . We thus pursue a local definition for relative to a reference virtual


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bulk fluid which shall be locally constructed. To this end begin by defining thechemical potential density (Gibbs free energy) of the the adsorbed and bulk fluids

G

A

+

1 p

; = l ; B :

(2.11)

where, for simplicity the notation =

l

=

B

has been used when the fluids are

assumed incompressible. We then invoke a classical Gibbsian result which states

that at equilibrium, the chemical potentials of a single constituent coexisting in

two phases are equal (see, e.g., Callen [20]). Using the maximum entropy principle

Callen [20] illustrated this result in a classical example of osmotic water-pressure

difference across a semipermeable membrane and showed that the chemical poten-

tial of the water is constant in this experiment. The same idea can easily be applied

to Low’s swelling experiment to show that at equilibrium the chemical potential of

the adsorbed water is equal to that of the bulk water, i.e. G l

= G

B

. We make use

of this result to characterize the virtual local reference bulk fluid (denoted by the

subscript B ) where x ; t 0 : This reference bulk water is constructed at instan-taneous equilibrium with the adsorbed water such that their chemical potentials are

equal, and the swelling pressure

x; t

locally represents a pressure excess due to

the interaction of the water with the clay. In other words, would be zero if the

properties of the water were unaffected by the interaction with the solid phase, as

in the case of a bulk fluid. If we denote the free energy of the reference bulk fluid

by A B

, the postulate G l

= G

B

together with (2.11) gives

A

l

+

1 p

l

= A

B

+

1 p

B

: (2.12)

The above resultprovides a partial relation between thethermodynamicalproperties

of the adsorbed water and reference bulk fluid. To complete the characterization

of this local reference state recall that, in the absence of thermal effects, A l

onlydepends on

l

in Low’s experiment [58]. Denote

l

= e

=

1 e

as the volume

fraction defined in Low’s relation (2.10) for which = 0 with l

=

l

. At l

adsorbed water behaves as a bulk fluid and, hence, A l

l

= A

B

. This, combinedwith (2.12) yields

p

B

= p

l

+ A

l

A

B

= p

l

Z

l

l

@ A

l

@ s

d s : (2.13)

Since A l

is a function of l

, the above result furnishes a definition for the reference

bulk phase pressure p B

in terms of f p l

;

l

;

l

g . Together with (2.7) this provides

the local characterization of p

B

and tB

for a fixed solid deformation. We now

redefine the swelling pressure locally relative to p B

. Begin by noting that definition

(2.9) is restricted to an equilibrium well ordered parallel platelet arrangement, in

which there is no mineral to mineral contact effective stresses. One may generalizethe swelling pressure concept to particles of curved shape, to incorporate particle

shear stress and nonequilibrium viscous effects. To do so we introduce a vectorial


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definition for , namely the swelling stress vector which we shall henceforth denotein boldface. For a mesoscopic surface of unit normal n, for t

B

given as in (2.7),

and p B

defined in (2.13), define locally as

x ; n; t t tB

n : (2.14)

This mesoscopic definition is consistent with the microscale vectorial definition

for the disjoining pressure proposed by Kralchevsky and Ivanov [47] and Ivanov

and Kralchevsky [43] for curved thin films undergoing nonequilibrium processes

where viscous effects are important. In addition, (2.14) reduces to Low’s swelling

pressure definition (2.9) in the swelling experiment at equilibrium. To show this

recall that in the swelling pressure experiment of Figure 2, t= P

I ( P denotes

the overburden pressure) and the effective component vanishes for the arrangement

of parallel platelets. Using this in (2.7) we get tB

= p

B

I and (2.14) reduces tothe classical swelling pressure relation = n with = P p

B

. Note that in

general t tB

may have off-diagonal components and consequently may also

have a tangential component to the mesoscale surface.

An open question is the role the excess in thermodynamic pressure of the

adsorbed fluid, relative to the local reference bulk phase pressure p l

p

B

, plays

during particle consolidation. Unlike Low’s swelling pressure

, which has Der-

jaguin’s disjoining pressure as a microscopic counterpart, the excess p l

p

B

does

not exhibit any microscale analogy. The reason is the different thermodynamic

representation adopted at the mesoscale (e.g. A l

= A

l

T ;

l

; Es

for compressible

media) than that of thin films. This latter microscopic formulation usually adopts a

different Legendre transformation in which the Helmholtz free energy is replaced

by the Gibbs energy as a thermodynamic potential (see Derjaguin et al. [25]), or

even in the free energy representation the microscopic film density is eliminatedfrom the list of independent variables (see e.g. Li [51, 52]).

Further, note that since p l

affects the total particle thermodynamic pressure, p ,

through (2.5), we are led to also quantify an excess in the total particle pressure p

relative to the bulk phase p B

. Henceforth, we shall refer to this difference as the

excess in pore pressure,

B

, i.e.

B

p p

B

:

(2.15)

In analogy to the swelling pressure, the above definition reflects locally the excess

in pore pressure due to the interaction between the adsorbed water and the clay

minerals. In other words, B

would be zero if the properties of the water were

unaffected by the interaction with the solid phase.

Next we proceed to derive a relation between the hydration stress tensor t ls

and

the swelling stress vector and the excess in pore pressure B

. To this end define

0

x; t ;

n

B

x; t

n

x; t ;

n : (2.16)


13/40


Applying (2.6) to a mesoscale surface of normal n, and using definitions (2.7),(2.14) and (2.15) we find

tls

n=

t

t e s

+ p

I

n

=

t

t e s

+ p

B

+ p p

B

I

n

=

t

tB

+ p p

B

I

n

=

B

n =

0

: (2.17)

Hence, projection of t ls

onto the normal to a mesoscale surface may be interpreted

as the difference between the excess in pore pressure and swelling stress vector.The above result leads to an alternative way of writing the modified Terzaghi’s

effective principle (2.6) in terms of the reference bulk phase pressure p

B

, rather

than the particle thermodynamical pressure p . Define the tensor

t

=

tls

B

I (2.18)

and combine with (2.17) to get

t

n = : (2.19)

We may think of t

as a swelling stress tensor , since in analogy to the classical

Cauchy argument, the projection of t

onto a mesoscale surface of unit normal n

gives the swelling stress vector .

Further combining (2.6) with (2.15) and (2.18) we get

t= p

I+

t e s

+

t ls

= p

B

I+

t e s

+

t ls

B

I

= p

B

I+

te s

+

t

:

(2.20)

This result is an alternative form of t which expresses the mesoscopic modified

effective stress principle (2.6) with p replaced by p B

. Physico-chemical forces

in (2.20) are measured by the swelling stress tensor t

. This alternative way of

expressing the modified Terzaghi’s principle resembles in form some heuristic

modified effective stress principles for clays discussed in, e.g., Sridharan and Rao

[70] or Lambe [48]. Historically, physico-chemical forces have heuristically been

modeled at the macroscale through the addition of a term to Terzaghi’s principle

which measures the effect of net repulsive ( R I) and attractive ( A I) forces between

particles. This stress is commonly denoted by R A I. Denoting the intra-particle

mesoscopic counterpart of these stresses as

r

a

, Equation (2.20) is a first rational

attempt at a rigorous derivation of the modified Terzaghi’s principle. From (2.20)

we have r a = t

which shows that the net attractive-repulsive intra particle

forces arising from surface hydration are governed by the swelling stress tensort

. Hence, we can reproduce the basic mechanical models for stress partitioning

between solid and fluid phasesdiscussed for example in Sridharan and Rao [70] and


14/40


Lambe [48]. The effective stress concept for soils with physico-chemical stressesand the role of hydration forces in carrying the total load has been controversial

and consequently more than one definition for effective stress has been proposed.

For example, Lambe [48] defined effective stresses as the difference between totalstress and pore pressure. Using Lambe’s definition in (2.20) the effective stress is

the sum of the mineral contact and swelling stress components. On the other hand

Sridharan and Rao [70] argued against Lambe’s definition and defined effective

stress as the mineral-mineral contact stress as it controls the resistance against

failure. Hence, if we define the mesoscopic effective stress tensors tL

and tS R

in

the sense of Lambe and Sridharan and Rao respectively by

tS R

=

te s

;

tL

=

te s

+

t

:

Then the modified effective stress principle at the mesoscale is

t= p

B I+

t L = p

B I+

t S R +

t = p

B I+

tS R +

r

a:

(2.21)

As we shall see further in Section 6, this result has an macroscopic analogy.

Consequently some controversial aspects in stress analysis in cohesive soils are

clarified within the current approach. Note that the deviatoric part of t

does

not necessarily vanish. This suggests that in general r

a is a full rank tensor.

The presence of off-diagonal components in t

may be important at low moisture

contents where according to Schoen et al. [65], Israelachvili [42] and Cushman[22]

the adsorbed fluid and the solid surface may support shear forces at equilibrium,and consequently the swelling stress vector may also have a tangential component

at the mesoscopic surface. As we shall show next, if we exclude the range of

moisture where the adsorbed fluid can support shear stresses, then r a reduces to

a multiple of the identity, i.e. r a = r a I and the swelling stress vector actsnormal to the surface, i.e.

=

n.

2.3. SWELLING AND HYDRATION STRESSES AT MODERATE MOISTURE CONTENT

A simplified scalar concept of hydration and swelling stresses can be obtained

by considering a moderate moisture content greater than that occupied by 10 fluid

monolayers. In this range the adsorbed water can not withstand shear at equilibrium

and therefore a well ordered particle composed of parallel platelets as depicted in

Low’s experiment can only compress or expand and thus support no shear forces.

As we will show, in the range of moderate moisture content, the only term in the

right hand side of (2.6) and (2.20) with nonzero off-diagonal components is the

effective stress tensor t e s

. The hydration and swelling stress tensors t ls

and t

reduce

to multiples of the identity. The assumption of moderate moisture content can be

easily imposed by postulating that A l

does not depend on the deviatoric part of thesolid strain E

s

. On the other hand, unlike the bulk liquid, dependence of A

l

upon

the volumetric strain is still retained (see Achanta et al. [2] or Murad et al. [61]).


15/40


Thus, let J s

= det Fs

be the Jacobian of the incompressible solid motion whosematerial derivative satisfies [35, 62]

D

s

J

s

s

D t

=

0:

Denote the volume fraction of the reference configuration by s

=

s

Xs

. After

integration

J

s

s

=

s

: (2.22)

Hence J s

governs the volumetric mesoscopic deformation of the solid phase. In the

range of moderate moisture content, as in [2, 61], we assume that the free energy

of the adsorbed fluid depends on volumetric strains by postulating A l

= A

l

J

s

or

A

l

= A

l

l

since they are coupled by (2.22). Using (2.1) in (2.4) together with

the identity @ J 2s

= @ Cs

Cs

= J

2s

I (Eringen [28]) and (2.22) we get

tls

= 2 l

Fs

@ A

l

@ Cs

F T s

= 2 l

@ A

l

@ J

2s

@ J

2s

@ Cs

Cs

=

2

l

J

2s

@ A

l

@ J

2s

I=

2

l

J

2s

@ A

l

@

s

@

s

@ J

2s

I

=

l

s

J

s

@ A

l

@

s

I=

l

s

@ A

l

@

l

I (2.23)

Together with (2.18) this shows that in the range of moderate moisture contents t ls

and t

reduce to multiples of the identity. Hence, we are led to introduce the scalar

components of tl

s

and t

, namely the hydration pressure p

and swelling pressure , as

p

=

@ A

l

@

l

; =

13

tr t

: (2.24)

Using the above definition for p

in (2.23), the hydration stress tensor is given by

tls

= p

l

s

I= p

l

1

l

I:

(2.25)

Since t

is a multiple of the identity, using definition (2.24) in (2.19) we have

= n with

t = I = p l s B I (2.26)

where (2.18) and (2.25) have also been used in the last equality. In addition, using

(2.25) in (2.3) and (2.6), respectively, along with definition (2.15) yields

ts

= p

s

+ p

l

I+

s

1t e s

(2.27)


16/40


t

t e s

= p + p

l

s

I= p

B

+ p

l

s

B

I

= p

B

+ I; (2.28)

where (2.26) has also been used in the last equality in (2.28). The above result isour modified mesoscopic Terzaghi’s effective principle in the range of moderate

moisture content. When comparing with (2.21) we now have r a = r a I =

I which shows that intra-particle net attractive-repulsive forces are governed by

the swelling pressure. Moreover, in applying (2.28) to Low’s swelling experiment

and recalling that t= P

I and te s

=

0 because the platelets are ordered, we

reproduce the classical swelling pressure definition = P p B

.

2.4. EQUILIBRIUM RESULTS

Our aim in this subsection is to obtain a relation between hydration forces and

swelling pressure at equilibrium. Then we can make use of Low’s experimental

relation (2.10) to derive the dependence of the hydration pressure on the volume

fraction, i.e. the relation p

= p

l

. Following Truesdell and Toupin [73], it is

postulated that at equilibrium entropy is a maximum and entropy generation is

a minimum. Application of these conditions to the entropy inequality yields at

equilibrium (see Murad and Cushman [62] for details)

p

l

= p

s

= p : (2.29)

The above result states that at equilibrium, the thermodynamic pressures of the

solid and adsorbed fluid phases are equal. Recall that this reproduces (2.8) which

is a result that has been extensively used in the theory of granular nonswelling

media even at nonequilibrium (see, e.g., [17, 35]). As we shall see in the nextsubsection the equality between p l

and p s

may not necessarily hold in swelling

systems away from equilibrium. Moreover, using (2.29) in (2.15) and combining

with (2.13) yields

B

= p

l

p

B

=

Z

l

l

p

s

ds :

(2.30)

If we combine the above result with (2.26) we get

=

B

l

s

p

=

Z

l

l

p

s ds p

l

1 l

: (2.31)

Since p

has the thermodynamical definition (2.24), the above result provides

an alternative thermodynamic definition for the swelling pressure at equilibrium.

Using Low’s relationship (2.10) in (2.31), one can determine the relation

p

= p

l

and therefore Low’s experimental result for

may provide alternative


17/40


ways of measuring hydration forces in clays. This procedure is somewhat differentfrom that of Achanta et al. [2] and Murad et al. [61] where Low’s result was

reproduced by neglecting the stresses in the solid phase. Differentiating (2.31) with

respect to

l

we get

l

d p

d l

+ 2 p

+

1

1 l

d

d l

= 0 :

Whence

d 2l

p

d l

+

l

1 l

d

d l

= 0 ;

which after integration and using p

l

=

l

= 0 yields

2l

p

l

=

Z

l

l

s

1 s

d s

d s ds :

Using Low’s result for

in the right-hand side we can derive a relation for p

l

.

Consider for simplicity = B = 1 in (2.10) and denote by Ei x the exponential-

integral function defined as

Ei X

Z

X

1

exp

d = ln X + 1

X

n = 1

X

n

nn!:

We then have

2l

p

=

Z

l

l

s

s

2 1 s exp

1 s

s

ds

=

Z

l

l

e exp

1

e

d 1= e = Ei

1

e

l

l

= Ei Ei

1 l

l

;

where Ei = Ei 1 = e = Ei 1 l

=

l

. The above relation is a first attempt

to develop constitutive relations for hydration forces in swelling clay particles as

they appear measured by the relation p

= p

l

.

2.5. NEAR-EQUILIBRIUM RESULTS

We begin by presenting the near equilibrium results of Murad and Cushman [62] inthe range of moderate moisture content. These results were derived by linearizing


18/40


the entropy inequality about equilibrium. In particular, when linearizing aboutf v

l ; s

; D

s

l

= D t g , where vl ; s

vl

vs

and D s

= D t @ = @ t + vs

r denote the

velocity of the adsorbed water relative to the solid phase and material derivative

following the solid phase respectively, the following results were obtained

l

vl ; s

=

Kl

( r p l

+ p

r

l

) ; (2.32)

p

l

p

s

=

D

s

l

D t

; (2.33)

where Kl

and

are material coefficients and for simplicity gravity has been

neglected. Equation (2.32) is the mesoscopic Darcy’s law for the adsorbed water

with Kl

=

Kl

l

denoting the permeability tensor of the clay particles. In addition

to a pressure gradient, the above form of Darcy’s law contains a gradient of a

generalized interaction potential which accounts for swelling. The appearance

of this additional term is consistent with the fact that volume fraction gradients

provide a potential for adsorbed water flow in a swelling medium. From (2.32) wecan overcome the limitations of the works of Ma and Hueckel [59], and Hueckel

[40] where the adsorbed water is often termed ‘immobile water’ and consideredpart of the solid phase. Further, note that using definitions (2.11) and (2.24) in

(2.32) we have by the chain rule

l

vl ; s

=

Kl

( r p l

+ r A

l

) = Kl

r G

l

: (2.34)

The above reproduces the well known result that the gradient of the chemical

potential provides the generalized force for flow of matter, i.e. matter tends to

flow from regions of high chemical potential to regions of low chemical potential.Alternatively, recallfrom the classical thermodynamics of Stokesian fluids(Eringen

[28]) that in the absence of thermal effects,A

B

is constant for an incompressiblebulk fluid. We can then make use of (2.12) and rewrite Darcy’s law in its classical

form in terms of the gradient of reference bulk fluid as follows

l

vl ; s

=

Kl

(r p

B

+ r A

B

)=

Kl

r p

B

:

(2.35)

Equations (2.32), (2.34) and (2.35) consist of alternative forms of writing Darcy’s

law for the adsorbed water flow. As we shall illustrate in the next sections the

adoption of a particular form is somewhat related to the choice of primary variables

in the set of governing equations for the particles.

We now turn to the physical interpretation of (2.33).Using (2.33) in (2.5) we

also have

p = p

l

s

D

s

l

D t

: (2.36)

Using (2.33) and (2.36) in (2.27) and (2.28) respectively yields

ts

=

p

l

+ p

l

+

D

s

l

D t

I+

s

1te s


19/40


t t e s

=

p

l

+ p

l

s

+

s

D

s

l

D t

I: (2.37)

Equation (2.36) tells us that near equilibrium, the thermodynamic pressure of the

adsorbed fluid and solid phases are not necessarily equal. Thus, the commonlyassumed equality between p

l

and p s

for granular nonswelling media (2.8) maynot necessarily hold, especially for swelling systems. The coefficient

may be

thought of as a retardation factor which among other effects, accounts for the re-

ordering of the adsorbed water molecules as they are disturbed, i.e. an entropic

effect (see Bennethum et al. [14]). If this is the only source of retardation, then it

follows that for a granular medium,

0, since there is very little ordering of the

bulk liquid phase in such a medium. The evaluation of

requires experimental

study. In a different fashion, some information on this coefficient can be obtained

by averaging the constitutive relations for the nonequilibrium disjoining pressure

of microscopic thin liquid films (see [47, 43]). To this end use (2.36) in (2.15) along

with (2.13), and obtain the following near equilibrium relation for

B

B

= p p

B

= p

l

p

B

s

D

s

l

D t

=

Z

l

l

p

s d s s

D

s

l

D t

: (2.38)

When combined with (2.26) this yields for the swelling pressure

=

Z

l

l

p

s d s p

l

s

s

D

s

l

D t

: (2.39)

Hence, we may think of

as composed of two parts. A static (equilibrium) com-ponent e q

measured by the first two terms in the right-hand side and a viscous

(non-equilibrium) component neq measured by the last term. The motivation for

this decomposition is based on a similar microscopic result proposed Kralchevsky

and Ivanov [47] and Ivanov and Kralchevsky [43] for the viscous disjoining pres-

sure of thin films away from equilibrium. After neglecting convective effects and

using conservation of mass @ l

= @ t +

l

div vl

= 0 we have

eq =

Z

l

l

p

s d s p

l

s

neq=

s

@

l

@ t

=

s

l

div vl

:

Kralchevsky and Ivanov [47] have advocated that the microscopic counterpart

of the purely viscous nonequilibrium component neq accounts for the excess inthe viscosities of the thin film relative to the bulk phase. Thus, one can extend this

argument to the mesoscale and possibly identify the coefficient s

l

with the


20/40


difference between the averaged mesoscopic volumetric viscosity of adsorbed andbulk water. This averaged excess in viscosity, which we shall denote by

l ; B

, was

also measured by Low [54] who experimentally obtained the following analogous

relation to (2.10)

l ; B

=

B

exp

e

=

B

exp

1 l

l

;

where is another characteristic constant that depends on the nature of the mont-

morillonite. Thus

l ; B

=

l

s

and the above provides a first attempt to

measure the non-equilibrium coefficient

of the viscous disjoining pressure neqin the average sense. Of course, this claim is subject to experimental validation.

3. Linearized Governing Equations for Clay Particles and Bulk Water

The infinitesimal theory for the clay particles is obtained following the standard

linearization procedure: Assume that particles are initially homogeneous, isotropic

and at equilibrium. Expand A

( = l ; s ) in a Taylor series about equilibrium and

retain quadratic terms in A

and linear terms in the set of governing equations. In

particular, if we assume thatA

s

is an isotropic function of Es

, depending only on

its invariants to fulfill the usual objectivity requirements (Eringen [28]), then the

linearization procedure is exactly analogous to that of the classical linear isotropic

elasticity theory [28]. Let us consider that the clay particles are initially at an

equilibrium state given by Es

=

0,

l

=

l

and

s

=

s

s

=

1

l

and letA

l

=

A

l

l

denote the free energy of the adsorbed fluid at the reference configuration.For simplicity assume initially a well ordered parallel platelet arrangement within

each particle such that the reference configuration is free of effective stresses.

Let p = p l , K l I = K l l and = s l and let the strain tensor beidentified with its linearized form

Es

= r

s us

; (3.1)

wherer

s us

=

1=

2 r

us

+ r

uT s

, with us

denoting the displacement of the solid

phase. Let f s

;

s

g denote the pair of Lame coefficients of the platelet matrix, and

let denote a material coefficient of the adsorbed water. Postulate the quadratic

expansions

s

s

A

s

=

s

2

tr Es

2+

s

tr E2s

;

A

l

= A

l

+ p

l

l

+

2

l

l

2:

Then the linearized forms of (2.4) and (2.24) are

te s

=

s

tr Es

I+ 2

s

Es

; p

= p

+

l

l

: (3.2)


21/40


Equation (3.2) is nothing but the mesoscopic version of the classical Biot’s linear

elastic constitutive equation for the effective stresses. Denote by f B

; ;

0

g the

values of f

B

; ;

0

g

at the equilibrium reference state obtained by setting

l

=

l

in (2.30) and (2.31) together with definition (2.16). We then have

B

=

Z

l

l

p

s

ds ; =

B

p

l

s

;

0

=

B

= p

l

s

: (3.3)

Introduce the functionsf

l

andg

l

as

g

l

=

l

s

d p

d l

l

=

l

+ p

s

l

; f

l

= g

l

+ p

:

The linearized forms of (2.38), (2.39) and (2.16) are

B

=

B

p

l

l

D

s

l

D t

;

=

B

p

l

s

g + p

l

l

D

s

l

D t

= f

l

l

D

s

l

D t

; (3.4)

0

= p

l

s

= p

l

s

+ g

l

l

:

(3.5)

We are now ready for our mesoscopic linearized governing equation in the clay

particle domain. By neglecting all inertial and convective effects, the linearized

mass balances for the solid and fluid phases reduce to

@

l

@ t

+

l

div vl

= 0 ;@

s

@ t

+

s

div@ u

s

@ t

= 0 :

After adding them up and using the constraint

s

+

l

=

1, the above can be

rewritten in terms of the percolation velocity ql

l

vl ; s

as

div ql

+ div@

us

@ t

= 0 ;@

l

@ t

+

s

div ql

= 0 :

From the constitutive equations (2.32), (2.37), (3.2) and (3.5) together with thebalance laws, for E

s

as in (3.1), our system of linearized equations governing the

swelling clay particles written in terms of the unknowns f us

;

ql

;

te s

;

l

;

0

; p

l

;

tg is


22/40


Mass Conservation of the Adsorbed Water

@

l

@ t

+

s

div ql

= 0:

Total Mass Conservation

div ql

+ div@

us

@ t

= 0 :

Total Momentum Balance

div t=

0:

Total Particle Stress Constitutive Equation

t = p l

I + te

s

+

0

+

@

l

@ t

I :

Linearized Effective Stress Constitutive Relation

te s

=

s

div us

I + 2 s

r

s us

:

Linearized Hydration Stress Constitutive Relation

0

= p

l

s

+ g

l

l

:

Modified Darcy’s Law for the Adsorbed Water

ql

= Kl

r p

l

+ p

r

l

:

4. Mesoscopic Problem for Clay Particles and Bulk Water

Let l

and f

denote the clay particle and bulk water domains respectively, and

let be the interface between them. For given l

and s

and a set of coefficientsf K

l

;

;

s

;

s

; p

; g g at the initial equilibrium state, the above system of linearized

equations governs the swelling of the particles in

l

. In addition, following earlier

work, [61, 62], the slow Newtonian movement of the bulk phase is governed by

the classical Stokes problem

div tf

= 0 in f

;

tf

= p

f

I+

2

f

r

s vf

in

f

;

div vf

= 0 in f

:


23/40


24/40


Using (2.28), (2.35) and (3.4) the alternative mesoscopic formulation in terms of the primary unknowns f u

s

; ql

; t e s

;

l

; ; p

B

; t g and f tf

; vf

; p

f

g is given by

div tf

=

0 in

f

;

tf

= p

f

I+

2

f

r

s vf

in

f

;

div vf

= 0 in f

;

div t = 0 in l

;

t= p

B

I+

t e s

I in

l

;

te s

=

s

div us

I + 2 s

r

s us

in l

;

= f

l

l

@

l

@ t

in

l

;

div ql

+ div@

us

@ t

= 0 in l

;

@

l

@ t

+

s

div ql

= 0 in l

;

ql

= K

l

r p

B

in l

;

tn=

tf

n on ;

ql

n=

vf ; s

n on ;

p

B

= p

f

on ;

l

=

l

in l

; t = 0 ;

div us

= 0 in l

; t = 0 :

After obtaining p B

and l

within this formulation, p l

can be evaluated in a post-

processing approach using (5.1).

6. Macroscale Behavior: Two Scale Asymptotic Expansions

In this section we use the homogenization procedure to upscale the mesoscopic

results derived in the previous section to the macroscale. Our swelling clay at

the macroscale is idealized as a bounded domain " with a periodic structure.Following the general framework of the homogenization procedure, described,

for example, in Bensoussan et al. [15] and Sanchez-Palencia [64], we introduce


25/40


Figure 3. Two elements of equivalent clay soils.

Figure 4. The reference cell Q and its distribution over the homogenized soil.

mesoscopic and macroscopic lengths, denoted by l and L , which characterize

the mesoscopic size of the period and the field respectively. Their ratio " l = L .

Consider " as the union of disjoint parallelepiped cells, Q " , congruentto a standard

Q consisting of the union of several clay particles Q l

completely surrounded by

a connected bulk water domain Q f

. Let the systems of bulk phase water and clay

particles in "

be denoted by "

f and "

l , respectively. The " -model on "

consistsof the mesoscopic governing equations of Section 5 on each subdomain "

f

and

"

l

. Our starting point, " = 1, corresponds to our mesoscopic model. For " 1

a swelling clay soil is posited wherein the centers of the bulk phase channels are

located " -times the reference distance apart, though congruent to the reference

cell (Figure 3). The homogenized model for the macroscopic clay soil is obtained

by letting " ! 0 while the lattice extends to infinity. As we shall show next

the limit model " !

0

consists of a distributed model with microstructure in

which the macroscopic swelling clay soil is viewed as two coexisting systems: one

representing the clay particles and the other representing the bulk water. The picture

corresponding to the limiting model is depicted in Figure 4, where a mesoscopic

cellQ

is assigned to each point x of the macroscopic bulk phase domain.

The approach developed next is similar to that proposed by Arbogast and co-

workers [3–5, 27] for flow in naturally fractured reservoirs. For simplicity we

consider the limiting case of the clay geometry wherein the clay particle systemis disconnected. Following the terminology of fissured media this geometry is

termed totally fissured medium (TFM). Since particles are completely isolated


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from each other by the bulk phase fluid there is no direct mass and momentumtransfer from particle to particle. Instead the adsorbed water first flows into the

bulk phase where it passes into another particle or remains in the bulk phase. As

a consequence, most of the flow passes through the bulk phase, while the storageof the fluid takes place in the system of clay particles. This picture corresponds

to an idealized clay soil wherein clay particles are highly ordered so that there is

no solid-solid contact. In reality, the clay is not well ordered and there is some

particle-particle contact. A porous medium which exhibits interaction between

particles through their interfaces is termed partially fissured medium (PFM). In

PFM particles are connected to neighboring particles, so that a percentage of the

water flow passes through particle interconnections and therefore particles are not

only coupled indirectly through the bulk phase system. The modeling of PFM

requires an additional coexisting system that governs vicinal water flow from

particle to particle. The homogenization tools for upscaling such media requires

more complexity (see Douglas et al. [26] and Showalter [67]) and will be saved for

a latter occasion.A crucial point in the analysis of dual porosity models is the proper scaling of

the coefficients by appropriate powers of " . The idea is to conserve flow in some

sense and consequently avoid degeneration of the governing equations as " ! 0.

Following Arbogast and co-workers [5, 27], this is done by considering the scaling

law K "

l

= K

l

"

2. This scaling has the effect of making the particles progressively

less permeable as " ! 0 and consequently preserves the secondary particle bulk

phase flux. In addition, recalling the standard homogenization procedure of the

Stokes problem, the bulk water viscosity coefficient f

is also rescaled by " 2 (see

Auriault [7], Sanchez-Palencia [64]).

The upscaling is achieved by considering every property to be of the form f x; y

(where x and y denote the macroscopic and mesoscopic coordinates, respectively,with y = " 1x) and then postulating two scale asymptotic expansions for the

set u " consisting of primary unknowns. We then expand our set of unknowns

f

us

;

ql

;

t e s

;

l

; ; p

B

;

tg and f t

f

;

vf

; p

f

g in terms of the perturbation parameter "

u" =

u0+ "

u1+ "

2u2+

with the coefficients ui , -periodic in y. Inserting the above expansions into the set

of mesoscopic governing equations with the differential operator @ = @ x

replaced by

@ = @

x

+ "

1@ = @

y

we obtain, after a formal matching of the powers of " , successive

cell problems. For the bulk water we have

divy

t0f

= 0 ; (6.1)

divx

t0f

+ divy

t1f

= 0 ; (6.2)

t0f

= p

0f

I; (6.3)


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t1f

= p

1f

I+ 2

f

r

s

y

v0f

; (6.4)

divy

v0f

=

0;

(6.5)

divx

v0f

+ divy

v1f

= 0 ; (6.6)

and for the clay particles

s

4

y y

u0s

+

s

+

s

r

y

divy

u0s

= 0 ; (6.7)

divy

t0= 0 ; (6.8)

divx

t0 + divy

t1 = 0 ; (6.9)

t0= p

0B

+

0

I+

te 0s

; (6.10)

t1= p

1B

+

1

I+

te 1s

; (6.11)

te 0s

=

s

divx

u0s

+ divy

u1s

I + 2 s

r

s

x

u0s

+ r

s

y

u1s

; (6.12)

0= f

0l

l

@

0l

@ t

; (6.13)

divy

q1l

+ divx

@

u0s

@ t

+ divy

@

u1s

@ t

= 0 ; (6.14)

@

0l

@ t

+

s divy q1l

=

0;

(6.15)

q0l

= 0 ; (6.16)

q1l

= K

l

r

y

p

0B

; (6.17)

along with the boundary conditions

v0f

@

u0s

@ t

n = 0 on ; (6.18)

q1l n =

v1f @

u1s

@ t

n on ; (6.19)

2 s

r

s

y

u0s

+

s

divy

u0s

I n = 0 on ; (6.20)

t0

t0f

n= 0 on ; (6.21)


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t1

t1f

n= 0 on ; (6.22)

p

0B

= p

0f

on

(6.23)

and initial conditions

0l

=

l

; in l

; t = 0 ; (6.24)

divx

u0s

+ divy

u1s

= 0; in l

; t = 0 : (6.25)

Next we formally collect our homogenized results. Recall that within the above

alternative formulation p l

was replaced by p B

and therefore a post-processing is

still required for evaluation of p 0l

6.1. DARCY’S LAW FOR THE BULK WATER FLOW

Using (6.1) in 6 : 3 we have t0f

= p

0f

x ; t I. In addition, noting that u0s

satisfiesthe Neumann problem given by (6.7) and boundary condition (6.20), we have

u0s

=

u0s

x; t . The macroscopic Darcy’s law for the bulk water relative to the solid

phase follows from the well-known upscaling of the Stokes problem (6.2)–(6.5)

together with boundary condition (6.18) (see, e.g., Auriault [7], Sanchez-Palencia

[64]). Introducing the mean value operator

e

= j j

1Z

d

i

; i = l ; f

and defining the macroscopic volume fractions of the particles and bulk phase,

respectively, by n

= j

j = j j ; = l ; f , and the averaged bulk phase velocity

relative to the solid phase by f qf

0

f vf

0 n

f

@ u0s

= @ t , we have

f qf

0= K

f

r p

0f

;

which is the classical Darcy’s lawgoverning the macroscopic bulk water movement.

If we assume that the swelling medium is isotropic at the macroscale then the scalar

K

f

denotes the macroscopic hydraulic conductivity for the bulk phase flow.

6.2. MODIFIED TERZAGHI’S EFFECTIVE STRESS

To derive the macroscopic modified Terzaghi’s effective stress we apply the meanvalue operator to (6.9)–(6.11), use the boundary condition (6.22) together with

(6.2), (6.3), the divergence theorem, and periodicity assumptions to obtain


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divx

e t0

= divx

e te s

0 r

x

f

p

B

0 r

x

e

0

= j j

1

Z

t e 1s

1+ p

1B

I

n d

= j j

1Z

t1f

n d

= j j

1Z

f

divy

t1f

d f

= j j

1Z

f

divx

t0f

d f

= n

f

r

x

p

0f

:

Defining the total macroscopic stress tensor and bulk phase pressure as

T=

tf

in f

;

t in l

;

P

f

=

p

f

in f

;

p

B

in l

;

the above result together with boundary conditions (6.21) and (6.23) yield

divx

e T0

=

0;

e T0

=

e t0

n

f

p

0f

I=

f

P

f

0+

e

0

I+

e te s

0;

(6.26)

where the averaged form of (6.10) has also been used. This is similar in form to

the modified effective stress principle of Sridharan and Rao [70]

T = P f

I + t e s

+ R A I

with R A I = e

0I. This shows that, in analogy with the mesoscale results, if we assume that swelling is governed by surface hydration, then the net attractive-

repulsive forces between the clay particles are governed by the macroscopic

swelling pressure. The approach presented herein provides a rational attempt to

model constitutive responses associated with physico-chemical forces in clays.

According to Sridharan [68, 69, 70], the magnitude of the two last components

of the right-hand side of (6.26) varies with the type of clay considered and the

moisture content. For example, for coarse-grained soils such as sands, silts and

low and non-expansive medium plastic clays such as kaolinite or illite, the stress

mechanisms are primarily controlled by the contact stresses e t e s

0. While for active

plastic smectitic clays such as bentonite and montmorillonite the stress mechanisms

appear to be governed by the swelling stress componente

0

I.

6.3. MASS BALANCE

Finally, we derive the macroscopic mass balance by averaging (6.6) and (6.14)

using the boundary condition (6.19) along with the divergence theorem and the


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periodicity assumption to get

g divx

v0f

a multiscale theory of swelling porous media ii. dual porosity models for consolidation of clays

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