The paper examines the coupled thermo-hydro-mechanical (THM) processes that develop in a fractured rock region within a fluid-saturated rock mass due to loads imposed by an advancing glacier. This scenario needs to be examined in order to assess the suitability of potential sites for the location of deep geologic repositories for the storage of high-level nuclear waste. The THM processes are examined using a computational multiphysics approach that takes into account thermo-poroelasticity of the intact geological formation and the presence of a system of sessile but hydraulically interacting fractures (fracture zones). The modelling considers coupled thermo-hydro-mechanical effects in both the intact rock and the fracture zones due to contact normal stresses and fluid pressure at the base of the advancing glacier. Computational modelling provides an assessment of the role of fractures in modifying the pore pressure generation within the entire rock mass.

The longevity constraints with regard to the safety of deep geologic
sequestration of high level radioactive waste suggest that the conventional
scientific approaches to the investigations that involve laboratory and
field studies must be complemented by approaches that will allow for predictions
on timescales that are beyond conventional geological engineering
activities involving underground facilities (Laughton et al., 1986; Chapman
and McKinley, 1987; Selvadurai and Nguyen, 1997; Rutqvist et al., 2005;
Alonso et al., 2005). Current concepts for deep geologic storage require an
assessment of the geological setting containing a repository to account for
geomorphological processes that can occur over timescales of several
thousands of years. The major geomorphological process that is identified
over this timescale is glaciation. Our attention is restricted to a
geologic setting that has been investigated in connection with the DECOVALEX
project, and the domain of interest incorporates a system of fractures that
are hydraulically connected but mechanically sessile under the influence of
the glaciation loading. The domain of interest, which contains the set of fractures, is
derived from the international DECOVALEX III project (Chan and Stanchell, 2008).
The highly idealized geosphere model is based on data from the Whiteshell
Research Area (WRA) in Manitoba, Canada. The studies by Chan and Stanchell
(2005, 2008) provide the database of material properties in the simulations
conducted on stationary aspects of glacial loads with specific emphasis on
the hydro-mechanical modelling that uses the continental-scale model of the
Laurentide Ice Sheet developed by Boulton et al. (2004). The dominant
fracture network closely resembles the simplified version of an actual
fracture network found in a small (approx.

The purpose of this paper is to examine coupled thermo-hydro-mechanical (THM) problems for a fluid-saturated rock mass that contains fracture zones and is subjected to advancing glacial loads. The mechanical response of a fractured rock mass to a glacial cycle was studied in the papers by Selvadurai and Nguyen (1995, 1997, 1999) and Steffen et al. (2014); it was shown that fracture zones can lose stability or slip during or after the deglaciation period of the cycle. The glacial cycle scenario was also included in the DECOVALEX studies (Chan et al., 2005) of a fractured rock mass and the current study extends this work to include THM effects and a more accurate treatment of the glacial loading. A secondary objective of the study is to assess the capabilities of a computational multiphysics finite element code COMSOL Multiphysics® in examining a model domain that consists of an intact rock mass and fracture zones, both of which can exhibit THM processes.

The formulation of a coupled hydro-mechanical (HM) problem for fully saturated geological materials was presented by Biot (1941) and reformulated by several investigators including Rice and Cleary (1976) and Hamiel et al. (2005). The HM coupling gives rise to the Mandel–Cryer effect (Selvadurai and Yue, 1994; Selvadurai and Shirazi, 2004), which can explain the momentary increase in the pressure head in an aquifer when the ground water is drained upwards along a fault (Stanislavsky and Garven, 2003) and can predict changes in the fluid pressure induced by the slip of geological faults (Hamiel et al., 2005). Thermo-hydro-mechanical effects have also been of interest to many geoenvironmental processes including the geologic disposal of heat-emitting nuclear waste, frictional heating of faults (Lachenburch, 1980; Rice, 2006), geothermal energy extraction (Dickson and Fanelli, 1995), and ground freezing resulting from buried chilled-gas pipelines (Selvadurai et al., 1999a, b). Solutions to pure fluid flow in heterogeneous formations and coupled thermo-poroelastic problems for fluid-saturated geological materials have been obtained by several researchers (Booker and Savvidou, 1985; McTigue, 1986; Selvadurai and Nguyen, 1995, 1997; Nguyen and Selvadurai, 1995; Selvadurai, 1996; Khalili and Selvadurai, 2003; Selvadurai and Selvadurai, 2010; Selvadurai and Suvorov, 2012, 2014). Recent mathematical and computational studies of THM behaviour of fluid-saturated media with both elastic and elasto-plastic skeletal behaviour are given by Selvadurai and Suvorov (2012, 2014) and experimental manifestations of the thermo-poroelastic Mandel–Cryer effect are also given by Najari and Selvadurai (2014).

The behaviour of a rock mass can be influenced by the presence of fractures and faults. Coupled THM behaviour of fractured rocks has been studied by several investigators and extensive references to this work can be found in areas related to geosciences and geomechanics (Noorishad et al., 1984; Selvadurai and Nguyen, 1995; Nguyen et al., 2005; Rutqvist et al., 2002; Guvanasen and Chan, 2000; Chan and Stanchell, 2008; Tsang et al., 2009). Thermo-hydro-mechanical processes in fractured rock formations can be analysed using two approaches: in the first approach, by modelling discrete fractures and specifying their locations within a host rock, which is modelled as a fracture-free medium; and in the second approach by introducing the influence of fractures implicitly through the derived overall constitutive equations for the fractured medium.

Within a discrete fracture modelling approach, three distinct finite element formulations can be identified: special interface or joint elements (Selvadurai and Nguyen, 1995, 1999; Ng and Small, 1997; Nguyen and Selvadurai, 1998; Guvanasen and Chan, 2000; Steffen et al., 2014), the embedded manifold approach (Guvanasen and Chan, 2000; Juanes et al., 2002; Graf and Therrien, 2008; Erhel et al., 2009), and the conventional or direct approach, in which the fractures are modelled with the finite elements of the same spatial order; e.g. 3-D fractures are modelled with 3-D finite elements (Stanislavsky and Garven, 2003; Chan et al., 2005; Chan and Stanchell, 2005; Sykes et al., 2011). The multiphysics finite element code COMSOL Multiphysics® used in this study has an efficient 3-D mesh generator, well suited for discretizing complex geometries containing distinct narrow regions, such as fractures. Thus, in this paper, the THM problems investigated in connection with glacial loading will be examined using the conventional or direct approach.

In this study the rock mass is assumed to be an isotropic thermo-poroelastic
domain with a network of dominant fractures that is integral with the
surrounding intact rock. The intact rock is assumed to contain small-scale joints, minor
fractures, pores and voids, and thus they are not explicitly included into
the model. The dominant fracture network represents large-scale faults and closely resembles
the simplified version of the real fracture network found in a small
(approx.

In this work attention is restricted to a glacier of very large length which
is initially at rest and then starts to move along the bedrock surface. The
glacier motion or advance is assumed to be very small compared to the
glacier length. Since the glacier loading is highly non-uniform near the
glacier front and very flat near its centre, the major change in the
response of the underlying bedrock caused by a small glacier advance is
expected to occur near the initial location of the glacier front. Therefore,
we will place the rock mass model domain containing fractures near the
glacier front initial position and observe the response of the rock and
fractures there. The size of the model domain is also chosen to be very
small compared to the glacier length although larger sizes would naturally
provide for better accuracy of the solution. Since the nonlinear effects are

The paper is organized as follows. Mathematical description of the thermo-poroelasticity problem is given in Sect. 2 whereas the finite element model is described in detail in Sect. 3. A few numerical tests that validate the model are presented in Sect. 4 and Sect. 5 contains the main computational results that illustrate the influence of glacial loading and temperature change on the development of fluid pressure, velocity, temperature, displacement and mean effective stress in the entire rock mass and within the individual fractures.

For a linear elastic isotropic fluid-saturated rock, the total stress tensor

For an isotropic fluid-saturated porous medium, Darcy's law can be written:

We assume that the heat transfer in the system is through conduction only
and the fluid flow velocity both in the pore space of the intact rock and
through the fractures is slow enough so that the convective heat transfer
terms can be neglected. It is also assumed that the deformations of the
intact rock and the fractures along with the fluid flow processes do not
result in generation of heat. Therefore, heat transfer in the entire rock
mass is described by the Fourier law of heat conduction:

Stress equilibrium equations written in terms of displacements

From the fluid mass conservation law and Darcy's law (Eq. 3) the fluid flow
equation can be derived as

From the thermal energy balance equation and Fourier's law (Eq. 4) the heat
conduction equation can be obtained as

Geometry of the fractured rock mass subjected to loading by the advancing glacier.

Glacial loads on the surface of a rock mass arise due to the accumulation of
snow, ice and water (Aschwanden and Blatter, 2009). The water content within
a glacier is generally small (approximately 5

Consider a model of a rock mass in the form of a parallelepiped with upper
surface

In order to examine the response of a poroelastic rock mass subjected to
glacial loads, it is important to adequately take into account the
interaction of the glacier with the underlying bedrock. On the temporal
scale of interest to the glacial loading of a rock mass containing fracture
zones, a simpler representation of the glacier can be adopted. Firstly,
mechanical loading due to the weight of the ice sheet needs to be prescribed
(Bangtsson and Lund, 2008; Read, 2008). The mechanical loading corresponds to the changing ice-sheet
thickness

On the lower surface of the rock mass,

Initial fields within the rock mass correspond to the fields existing at the
onset of glacier motion and thus should include stresses due to the weight
of the glacier at its initial position, geostatic stresses, hydrostatic
fluid pressure, the temperature distribution due to the geothermal heat
flux, etc. In this work we focus our attention only on the

Figure 1 shows geometry of the fractured rock mass model used in the present
study. The region of the rock mass is a parallelepiped with in-plane
dimensions

Each fracture zone is defined by its surface which constitutes a planar
quadrilateral formed by the four vertices lying in the same plane. To
facilitate construction of the fracture surface, a 2-D workplane is
constructed for each fracture. Orientation of the workplane is defined by
specifying coordinates of the vector normal to the fracture surface. The
fracture contour is drawn in the workplane by specifying coordinates of the
four vertices of the fracture surface, transformed to the local coordinate
system of the workplane. Subsequently, the fracture surface is extruded at a
distance equal to the thickness of the fracture zone (10

If a fracture originates at the upper surface of the rock, the upper edge of the fracture zone is expected to coincide with the upper surface of the rock, which is horizontal. However, after the extrusion process described above, if the fracture zone is not strictly vertical, the upper edge of the fracture (being orthogonal to the fracture surface) will be inclined with respect to the horizontal surface of the rock. Such a configuration may cause problems later during mesh generation since the edge of the fracture lies too close to the surface of the rock. To correct this geometry, the fractures' surfaces in their respective workplanes can be extended in such a way that, after extrusion and embedding, the fracture intersects the upper surface of the rock and extends beyond it. Then we simply cut off the portion of the fracture zone above the upper surface of the rock by using, for example, the subtraction operation. Similar corrections can be performed with respect to other edges of fractures that lie too close to the lower surface of the rock or to the perimeter surfaces. It should be noted that the fracture obtained in this way is a prismatoid and no longer an extruded feature.

Typically, it is desirable to have the capability of inserting into the 3-D
geometry as many fractures as possible. On the other hand, if the geometry
becomes too complex, the mesh may not be generated. Problems with mesh
generation may arise when (a) the fractures are not parallel, i.e. have
various orientations in space, (b) the number of intersections between
fractures is too large, (c) the distance between the neighbouring fractures is
too small, i.e. they are too close to each other, and (d) the thickness of
fractures is too small. These conditions naturally place a limit on the
number and the type of fractures that can be successfully inserted. We
succeeded in inserting only 21 fracture zones into the geometry of the given
rock mass as shown in Fig. 1. The basal rigid stratum will have an
influence on the THM processes that are investigated. The depth of 1.7

The COMSOL Multiphysics® finite element software has a
powerful mesh generator engine. For 3-D analysis only tetrahedral elements
can be used and by default the order of the elements is quadratic. There are
several predefined element sizes such as “Extremely Coarse”, “Extra Coarse”, “Coarser”, “Coarse”, “Normal”, “Fine”,
“Finer”, “Extra Fine”, “Extremely Fine” and, in addition, the user can
define the element size, “Custom” mesh size. Each mesh is characterized by
the following parameters: maximum element size, minimum element size,
maximum element growth, resolution of curvature and resolution of narrow
regions. The values of these parameters are geometry-dependent. For example,
for the given geometry of the rock mass, a parallelepiped with in-plane dimensions of

If the model geometry is too complex, it might be impossible to construct
the mesh with the predefined mesh size due to the error messages. For
example, for the present geometry of the fractured rock mass,
COMSOL Multiphysics® failed to construct the “Normal” mesh, but
managed to construct the “Extra Fine” mesh consisting of 3 321 534 elements and a
“Fine” mesh with 223 530 elements. To reduce the number of elements even further
we define the “Custom” mesh size for which we assign the following values to the
parameters: maximum element size 1690

If the error message appears during mesh generation, it is recommended that
the mesh size parameters be modified; even a small change in a parameter
value can sometimes allow COMSOL Multiphysics® to successfully
construct the mesh. For example, for the given geometry with 21 fractures, one
could construct the “Custom” mesh if the minimum element size is set to 152–154, 157,
159, 161, 167, 168, 170, 172, 186 or 189

Finite element discretization of the rock mass with a system of interconnecting fracture zones. A discretization of 158 728 tetrahedral elements is used.

The primary emphasis of this work is to examine the THM response of the fractured rock mass when it is subjected to advancing glacial loads. In general, advance or retreat of the glacier can be caused by (a) sliding of the glacier along the bedrock and (b) a change in the accumulation and/or ablation rates. In the first case the shape of the glacier does not change, while in the latter case the shape of the glacier, i.e. its length and height, evolves. In the following we consider the glacier advance caused only by sliding, i.e. case (a).

Figure 1 shows the glacier profile at time

The thickness of the glacier is non-uniform and can be described by the
function

The velocity of the glacier

As mentioned, the initial or in situ state is not modelled in the present
study and therefore the loading applied to the surface of the rock mass
only corresponds to the difference between the current position of the
glacier, at time

In the following we focus our attention on the response of the rock mass over a short time (e.g. 5900 years), for a small glacier advance, when the glacier front is still close to the fractured region of the rock and the glacier centre remains at a very large distance from the fractured region. Thus, we do not compute the results corresponding to the maximum value of the glacier loads, when the glacier centre itself overruns the model domain. The choice of the time frame can be explained by the highly non-uniform distribution of the glacier load near the front of the glacier; our intention is to study the effect of this non-uniform load. In addition, we believe that the glacial sliding is limited to small distances compared to the overall glacier length.

The response of the rock mass was obtained by using the transient finite
element (FE) solver in COMSOL Multiphysics®. The numerical
integration in time was implicit by default, which implies that a matrix
factorization is required at each time step. Selecting the proper FE solver
is of outmost importance because of the size of memory required,
computational time limitations and issues related to convergence. In order
to solve the HM problem (Eqs. 5–6) for the “Custom” mesh, a direct
solver, such as the SPOOLES (SParse Object Oriented Linear Equations Solver) solver, requires a very large memory (larger
than 30

Distribution of the glacier thickness in the ice sheet marginal zone. The exact glacier shape is shown with a solid line and the approximation to the exact shape, used in the computational simulations, is shown with a dashed line.

The fracture zones have THM properties substantially different from those of the intact rock. In particular, the permeability of the fracture zones is taken to be about six orders of magnitude greater than the permeability of the intact rock, whereas Young's modulus is assumed to be approximately 10 times smaller.

The properties of the intact rock and the fractures used in our
computational simulations are the following: Young's modulus

In this section we assess our choice of the user-defined “Custom” mesh, consisting
of 158 728 tetrahedral elements (Fig. 2), and show that this mesh is
suitable for solving the given problem of the bedrock response if the
glacier speed is as low as 1.27

Vertical fluid velocity [

The first evaluation procedure involves analysis of the HM solutions for the homogeneous rock, i.e. the rock in which properties of the fractures are the same as those of the intact part. Here we examine how the presence of the densely refined regions in the mesh, in close proximity to the “fracture” zones, may affect the solution. To do so, in addition to the “Custom” mesh we consider the regular mesh with only 5786 elements that does not account for the presence of “fracture” zones.

Figure 4 shows the distribution of vertical fluid velocity

The second evaluation procedure employed involves a comparison of the solutions for two meshes: the user-defined “Custom” mesh with 158 728 elements and the pre-defined “Extra Fine” mesh with 3 321 534 elements. Taking into account the fact that the “Extra fine” mesh has a massive number of elements, which would require a considerable amount of memory, we could solve only the “pure” hydraulic problem (Eq. 6) in which the deformations of the porous skeleton are suppressed, thus reducing the number of unknowns.

Figure 5 shows the distribution of vertical fluid velocity

Vertical fluid velocity [

Figures 6–15 show the hydro-mechanical response of the fractured rock
subjected to glacial loading, described previously, at time 5900 years. The
time-dependent fluid pressure and vertical compressive stress induced by the
glacial load are applied to the upper surface of the rock mass as prescribed
by Eqs. (8) and (9). The solution is obtained by solving the hydro-mechanical
problem (Eqs. 5, 6) on the user-defined “Custom” mesh. For more accurate
results, the model domain, shown in Figs. 1 and 2, was augmented along the

Figure 6 shows the typical fluid pressure distribution at the interior, fractured region of the rock and
exterior, homogeneous region at time 5900 years. To capture the response in the fractured region,
the fluid pressure was plotted in the cross section where the

Distribution of fluid pressure [

Variation of fluid pressure [

Figure 7 shows the fluid pressure variation at a depth of 200

Figure 8 shows the vertical fluid velocity

Figure 9 shows variation of the vertical fluid velocity in the intact
part of the rock mass with fractures in the cross sections

Distribution of the vertical fluid velocity [

Vertical fluid velocity [

Distribution of the vertical fluid velocity [

Variation of the vertical fluid velocity [

Vertical displacement [

Variation of the vertical displacement [

Figure 10 illustrates the vertical fluid velocity

Figure 11 shows variation of the vertical fluid velocity in the largest
fracture zone along its length, at a depth of 200

Figure 12 shows the vertical displacement or deflection in the rock mass
with fracture zones at time 5900 years. The displacement is negative under
the glacier, which is indicative of the mechanical compression of the rock,
and positive outside the glacier, which is indicative of the conventional
heave associated with glacial loading (Walcott, 1970, 1976; McNutt and
Menard, 1978; Selvadurai, 1979, 2009b, 2012; Selvadurai and Nguyen, 1995).
The maximum negative displacement (

Figure 13 shows variation of the vertical displacement in the rock mass with
fracture zones in the cross sections of the fractured and fracture-free
regions

Distribution of the mean effective stress [

Figure 14 shows the mean effective stress

Figure 15 shows the mean effective stress variation in the intact part of
the rock with fracture zones in the cross sections

Variation of the mean effective stress [

In the thermo-hydro-mechanical problem the temperature change caused by
glacier advance or growth must be prescribed at the upper surface of the
rock. Chan et al. (2005) examined the change in the ground surface
temperature through the entire glacial cycle using climate/temperature
simulation. From this simulation it was clear that before the arrival of the
glacier, the ground surface temperature decreases significantly from

In Figs. 16–19 we illustrate the response of the rock mass with fracture
zones subjected to the glacial load described previously and, in addition,
to the effects of cooling associated with the advance of the glacier. The
time-dependent fluid pressure, the vertical compressive stress and the
temperature change are applied to the upper surface of the rock mass as
prescribed by Eqs. (8), (9) and (11). As indicated previously, the initial or
in situ state is not taken into account in the present study and, therefore,
only the surface loading and temperature that correspond to the difference
between the current position of the glacier, at time

Temperature change distribution [

Figure 16 shows the temperature change generated within the rock 5900 years
after initiation of the glacier motion. The temperature change on the upper
surface is prescribed according to Eq. (14). The temperature change under the
glacier reaches the maximum of

Figure 17 shows the variation of the temperature change at the upper surface
of the rock mass and at a depth of 800

Temperature change [

Figure 18 shows the mean effective stress in the intact part of the rock
with fractures at time 5900 years. The rock mass is subjected to the
compressive stress, the fluid pressure induced by the glacial load and the
temperature field (Eq. 14) associated with the glacial advance. We note that the
compressive mean stress at the lower surface of the rock mass is almost
equal to that in the rock subjected only to the mechanical load and fluid
pressure (see Fig. 14). However, the tensile mean stress, which occurs in
the fractured region in the vicinity of the upper surface, is increased from
0.15 to 0.28

Figure 19 shows variation of the mean effective stress in the intact part of
the rock mass with fractures at a depth of 200

Distribution of the mean effective stress [

In the present work, the behaviour of a fractured rock mass subjected to
advancing glacial loads was analysed. The glacial load was taken into
account by prescribing a spatially and time-dependent compressive stress and
fluid pressure on the upper surface of the rock mass. Resulting fluid
pressures, vertical fluid velocities, displacements, and mean stress
distribution diagrams were obtained through computational simulations that
incorporated THM processes. Results were plotted for the specific time of
5900 years during which the glacier advanced across the model domain by a
distance of approximately 7500

The computational results show that there is a difference between the fluid pressures at the upper and lower surfaces of the rock mass due to the low permeability of the intact rock. This pressure difference creates a vertical pressure gradient that contributes to the vertical fluid velocity component, which is a major feature of the fluid flow in the present problem. We also observed a difference in the fluid pressure distributions within the fractured region and the surrounding homogeneous region of the rock. In fact, the fluid pressure within the fracture zones and the intact rock in close proximity to the fractures is larger than that in the homogeneous region, located remotely from the fracture zones. This can be attributed to the large permeability of the fractures.

Variation of the mean effective stress [

The distribution of the vertical fluid velocity within the intact part of the fractured rock is complex. In general, the direction of the fluid velocity is negative (downward) beneath the glacier and positive (upward) beyond the glacier margin. The magnitude of the downward velocity is larger than the upward velocity, which is consistent with the direction associated with the meltwater flow. As was already noted, between the subvertical fractures and near horizontal fractures the vertical fluid velocity in the intact rock is generally smaller than that in the homogeneous region. The vertical fluid velocity within the fractures of the rock is about 50 000 times larger than the velocity in the intact rock. The direction of the fluid velocity in the fractures is mainly negative but large velocities in the positive direction can also occur, especially in the subvertical fractures that are not parallel to the ice sheet front. The role of the fracture zones and their dominant influence on the fluid flow in the region during a glaciation event is thus confirmed by the present study.

The mean effective stress in the intact part of the rock mass is mostly
compressive, but tensile stresses do occur between and around the fractures,
at close proximity to the upper surface of the rock. The maximum compressive
stress is at the lower surface of the rock at the initial position of the
ice sheet front. The glacial loading induces compression of the rock, i.e.
negative strain in the direction of the rock thickness. For the situations
examined in the paper, this axial compression is relatively uninfluenced by
the presence of fracture zones. In addition to the compressive stress and
fluid pressure, the given rock mass was subjected to the cooling associated
with a glacier advance. Since the present study considers a small glacier
advance of only 7500

This study is closely related to the work performed earlier by Chan et al. (2005)
and Chan and Stanchell (2005, 2008) in that it also examines the
effect of the glacial loading on the behaviour of fractured bedrock. In their
work the rock was modelled as a thermo-poroelastic medium and contained 3-D
(but narrow) fracture regions, which were also assumed thermo-poroelastic.
The novelty of the present model is the use of a quite complicated fracture
network consisting of 21 fractures whereas only 5 fractures were included
into the models described in the papers by Chan et al. (2005) and Chan and
Stanchell (2005). The NWMO report by Chan and Stanchell (2008) deals with a
more complex fracture network consisting of 19 fractures but the fracture
thickness is set to 20

Computational modelling of fractured rock regions capable of accounting for THM processes is an essential tool for examining the response of rock masses at timescales relevant to glaciation scenarios. The computational approach allows for the examination of the relative importance of components in the THM model and the study of their influence on both the intact rock and the fracture zones at timescales relevant to glaciation scenarios and ultimately to deep geologic disposal concepts. In the modelling considered in this paper, both the fracture zones network and the intact rock are assumed to be sessile. The modelling can be extended to include activation of faults, e.g. in a combined analysis with large-scale models of GIA; natural hydraulic heterogeneities of the geologic media (Selvadurai and Selvadurai, 2010, 2014), particularly, at close proximity to the fracture surfaces; and stress-induced alterations of intact permeability. These effects can have an important influence on groundwater flow patterns in the domain. The information on the influences of stress states on permeability evolution in fractures is particularly scant and in a detailed analysis relevant to a deep geologic repository setting this aspect needs to be considered. Also implicit in the study is the absence of both radiogenic and geothermal heating effects; the influences of the latter could be important on the timescale of a glaciation event.

The source files used in the computations performed in connection with this
paper can be found at the following links:

The work described in this paper was initiated through the research support
provided by the Nuclear Waste Management Organization, Ontario, and through a
Discovery research grant awarded to A. P. S. Selvadurai by the Natural Sciences and
Engineering Research Council (NSERC) of Canada. A co-author (P. A. Selvadurai) is
grateful to NSERC for a post-graduate scholarship, which enabled him to
participate in the research. The opinions expressed in the paper are those
of the authors and not of the research sponsors. The authors are grateful to
the topical editor Lutz Gross and two anonymous reviewers for their
valuable comments, which led to significant improvements in the
presentation.