A Multiscale Theory of Swelling Porous Media II. Dual Porosity Models for Consolidation of Clays
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Transcript of 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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72 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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74 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 75
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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76 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 77
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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78 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 79
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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80 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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)
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 81
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
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82 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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]).
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 83
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)
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84 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 85
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
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86 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 87
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
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88 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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)
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 89
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
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90 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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
:
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92 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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
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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 93
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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94 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 95
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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96 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
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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A MULTISCALE THEORY OF SWELLING POROUS MEDIA 97
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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98 MÁRCIO A. MURAD AND JOHN H. CUSHMAN
periodicity assumption to get
g divx
v0f