A.P.S. Selvadurai, S.M. Rezaei Niya

a Department of Civil Engineering and Applied Mechanics, McGill University, Montréal, QC, H3A 0C3, Canada

b Royal Melbourne Institute of Technology, Melbourne, Australia

ABSTRACT The Cobourg limestone is a very low porosity rock consisting of lighter nodular regions that are predominantly calcite and darker regions consisting of calcite, quartz, dolomite, and an appreciable clay fraction. This paper presents the application of the theory of multi-phasic composites to estimate the possible maximum effective thermal conductivity of the heterogeneous rocks. The thermal conductivity estimates are expected to be representative of the intact rock, without fractures or fissures that can influence the heat conduction process.The estimates are therefore indicative of the thermal properties of the rock in undisturbed regions unaffected by the influences of stress relief and excavation damage during construction of deep ground repositories for the disposal of heat-emitting nuclear waste.

Keywords:Thermal conductivity Cobourg limestone Multi-phasic approach Voigt, Reuss and Hill estimates Hashin and Shtrikman bounds Computational modeling

1. Introduction

The estimation of thermal properties of materials is one of the most widely investigated topics in the engineering sciences. Since the development of Fourier’s theory of heat conduction (Fourier,1822), the thermal properties of materials relevant to problems in engineering, physics and earth sciences have been extensively investigated.These studies deal with a variety of topics including(i)the basic thermal conductivity properties of crystals, (ii) thermal conductivity of multi-phasic composites and crystalline solids,(iii)influence of temperature and mass transfer on thermal conductivity, (iv) phase transformation during heating, (v) electrochemical and pore space topology alterations during heat conduction, (vi) development of laboratory-based experimental techniques for measurement of thermal properties, and (vii) field studies of thermal conductivity at the regional and continental scales. Clearly, the presentation of an all-encompassing review of heat-conduction, even restricted to geologic media, is a daunting task and certainly beyond the scope of this article.

Some noteworthy references related to this topic include the pioneering studies by Birch and Clark (1940a, b), Horai and Simmons (1969), Horai (1971), Kappelmeyer and Haenel (1974),Cermak and Rybach (1982), Robertson (1988), Brigaud and Vasseur (1989), Vasseur et al. (1995) and the recent article by Merriman et al. (2018) that specifically deals with the thermal conductivity of carbonate minerals and rocks. These articles themselves contain extensive references related to thermal conductivity properties of geologic media.

The deep geological disposal of nuclear waste in competent rock formations is being considered by many countries including Canada, Finland, France, Sweden, Switzerland, etc. (C?me et al.,1985;Laughton et al.,1986; Chapman and McKinley,1987; OECD,1988;Gnirk,1993; Gray,1993; Selvadurai and Nguyen,1996; Selvadurai,1997, 2017; IAEA, 2000; NIREX, 2004; Rutqvist et al., 2004, 2005;Alonso et al., 2005; Brady et al., 2009; Pusch et al., 2011;Selvadurai et al., 2011, 2015; NWMO, 2011a, b; Selvadurai and Jenner, 2013; OPG, 2016; Selvadurai and Najari, 2016, 2017;Yardley et al., 2016; Selvadurai and Glowacki, 2017, 2018). A recent review of international research efforts related to deep geological disposal of heat-emitting high-level nuclear fuel waste has also been compiled by Faybishenko et al. (2016).

This study examines an aspect of the potential of the Cobourg limestone for siting a deep ground repository (DGR) for high-level nuclear waste in the Cobourg formation(argillaceous limestone).In Canada, the heterogeneous Cobourg limestone (Fig. 1) and the surrounding geologic media were identified as a potential host rock for constructing a DGR to store low- and intermediate-levelnonheat emittingradioactive waste. A description of the geological setting of the rock formations containing the Cobourg limestone was given in NWMO (2011b).

Fig.1. The Cobourg limestone: (a) The heterogeneous features; and (b) The arrangement of the facies over a 100 mm × 120 mm area (Selvadurai, 2019a, b).

The investigation of the thermal properties of the Cobourg limestone,as a reference sedimentary rock from southern Ontario,Canada, was undertaken to supplement the research base necessary for understanding the suitability of sedimentary geologic formations for storage of high-level nuclear fuel waste.This highlevel waste is currently stored at the reactor sites and contributes to a growing environmental risk as the reactors age and approach their reliable life span. In the development of models to describe the coupled thermo-hydro-mechanical behavior of repository rocks(e.g.Selvadurai and Nguyen,1995;Stephansson et al.,1996;Selvadurai and Suvorov, 2016), the experimental evaluations of the parameters governing the coupled processes should address the issue of the representative volume element (RVE) needed to obtain meaningful effective properties of the heterogeneous Cobourg rock. Bulk mass density and uniaxial compressive strength measurements point to size-dependency effects(Selvadurai, 2017, 2019a) and suggest that sample sizes with dimensions greater than 75 mm will provide a suitable RVE to estimate the intact geomechanical properties. Large sample dimensions are needed to adequately capture the heterogeneity,but this carries with it the development of experimental facilities and methodologies that are non-routine. For example, the estimation of the poroelastic Biot coefficient (Biot, 1941) of the Cobourg limestone was discussed by Selvadurai (2019a). The extremely low permeability of the Cobourg limestone (i.e.permeabilityKin the range of 10-23-10-19m2from Vilks and Miller (2007) and Selvadurai (2019b)) means that an inordinate amount of time is required to saturate the rock and for the pressures applied at the boundary of the sample to reach and equalize at its interior. This is essential to accurately determine the elastic compressibility of the solid material composing the rock.To avoid this drawback, Selvadurai (2019a) used multi-phasic approaches to determine the solid material compressibility, providing a convenient method to estimate the Biot coefficient for very low permeability rocks. Selvadurai et al. (2019) used a similar approach to estimate the Biot coefficient of Grimsel granite.Selvadurai(2019b)showed that the multi-phasic approach can be used to estimate the effective permeability of a heterogeneous rock.The purpose of the present study is to apply the multi-phasic approach to estimate the effective thermal conductivity of the Cobourg limestone. The methodology follows from approaches adopted to estimate the thermal conductivity of rocks and other composites (Budiansky, 1970; Horai, 1971; Christensen, 1979;Brigaud and Vasseur,1989; Pribnow and Umsonst,1993; Vasseur et al.,1995; Revil, 2000).

2. The Cobourg mineralogy

The mineralogical compositions of the lighter and darker facies of the Cobourg limestone were obtained by Waber et al.(2007)(see also Koroleva et al.,2009)using the RIETVELD approach.The results obtained by Waber et al. (2007) are shown in Table 1. The mineralogical compositions were also obtained using the X-ray diffraction (XRD) facilities available in the McGill Institute for Advanced Materials (MIAM). A disc specimen measuring approximately 140 mm in diameter and 5 mm in thickness was used to obtain samples of the two regions of the limestone; other samples were also obtained without any preference for the facies that were present.The tests were performed using the Bruker D8 Discovery X-ray diffractometer and the mineralogical compositions were determined using the DIFFRAC.EVA software. The results are shown in Table 2 for comparison purposes. The estimations of the clay fractions in particular require experience in sample preparation and the interpretation of the XRD data. In this regard, the RIETVELD approach is expected to give better estimates of the clay fraction.

In consideration of the relative merits of the XRD interpretation approaches, we used the data obtained by Waber et al. (2007) for the mineralogical compositions of the lighter and darker facies(Table 1). Other features of the rock are given by Selvadurai (2017,2019a, b). The relative proportions of the lighter and darker facies were also estimated by dissecting an 80 mm×100 mm×300 mm cuboidal sample of the Cobourg limestone into 10 adjacent thick slabs measuring 80 mm×100 mm×8 mm.Photographic imaging of the larger surfaces of the slabs and their assembly was used to reconstruct the interior distribution of the lighter and darker facies.Details of the procedures used were fully documented by Selvadurai (2017). The reconstructed image of the plausible distribution of the lighter and darker species of the Cobourg limestone is shown in Fig. 2. The digitized images of the larger surface of the slabs were also used to reconstruct the interior distribution of the facies of the limestone.Both extrusion from the large surfaces and a solid modeling technique were used to reconstruct the speciesarrangements within the thick slab. The details of the procedures were documented in Selvadurai (2019b).

Table 1Mineralogical compositions of the lighter and darker facies of the Cobourg limestone. Data were obtained by Waber et al. (2007) and interpreted using the RIETVELD approach.It should be noted that(i)Smaller fractions of albite,K-feldspar and pyrites have been grouped with quartz;(ii)Small fractions of siderite were grouped with dolomite; and (iii) Chlorite is grouped with kaolinite.

Table 2Mineralogical compositions of the Cobourg limestone and its lighter and darker facies.Data were obtained from the McGill Institute for Advanced Materials(MIAM).XRD data were interpreted using the DIFFRAC.EVA software.

The volume fractions of the separate regions were estimated in previous research efforts (Selvadurai, 2017, 2019a, b); the average volume fraction of the darker species is estimated (Fig. 2) to beVDR?0.54. There can be a wide variability in estimation of the relative volume fractions, with estimates as low asVDR?0.35(Selvadurai, 2017, 2019a, b). For the purposes of illustrating the application of a multi-phasic approach to estimate the thermal conductivity,we selected the following volume proportions:VDR?0.54 andVLR?0.46,whereVLRis the volume fraction of the lighter facies.

3. Multi-phasic estimates of thermal conductivity

The two facies of the rock are composed of the basic minerals calcite, dolomite, quartz, clay and voids in varying proportions. To apply the multi-phasic approach, we required the thermal properties of these basic minerals. Extensive studies have been conducted by various investigators to estimate the thermal conductivities of what are largely crystals, with thermal conductivities that are anisotropic, i.e. the thermal conductivity is a second-order tensor with three principal values. For the minerals identified,the thermal conductivity is either isotropic,with a single thermal conductivity, or transversely isotropic such that the thermal conductivities can be identified either along or normal to the crystal planes.In order to assign a specific estimate for the effective thermal conductivity,the measure adopted is the geometric mean,given by

wherektis the thermal conductivity along the planes of stratification andknis the thermal conductivity normal to the planes of stratification. The geometric mean concept was used by Pribnow and Umsonst (1993) to estimate the thermal conductivity of quartzitic rocks(their estimate foras opposed to Eq.(1)) and in estimation of the fluid transport characteristics of heterogeneous porous media (Selvadurai and Selvadurai, 2010, 2014),where the permeabilities are non-zero and are roughly of the same order of magnitude in each of the three orthogonal directions.The estimation of the thermal conductivity of the minerals listed previously presents a challenge, since there is variability in the estimates given in the literature. The thermal conductivities also exhibit a temperature dependency,with a reduction in the thermal conductivities as the temperature increases. For consistency, we selected the data provided by Robertson (1988), who used estimates provided by Birch and Clarke (1940a, b) and the estimates given by Brigaud and Vasseur(1989)and Vasseur et al.(1995)(see Table 3).

4. Effective thermal conductivity of the Cobourg limestone

Assuming that the Cobourg limestone can be modeled as a medium where the facies are spatially distributed,which leads to a medium with isotropic effective properties, and provided that the phasic thermal conductivities do not lead to non-conducting thermal barriers, the effective thermal conductivity (k*) can be bounded by theupper boundVoigt (1928) and thelower boundReuss (1929) estimates in the form of (see also Wiener,1912):

where subscriptirepresents the calcite, dolomite, quartz, Montmorillonite and kaolinite;Viis the volume fraction indicated in Table 2; andkiis the respective thermal conductivity of the phase indicated in Table 3.

4.1. Thermal conductivity of the Cobourg limestone at 273 K

Considering volume fractions given in Table 2 and the thermal conductivity estimates for constituent minerals given in Table 3,and applicable to 273 K,the Voigt(1928)and Reuss(1929)bounds(Eq.(2))give the following range for the thermal conductivity of the lighter and darker facies in the Cobourg limestone:

Fig. 2. The lighter and darker facies within the Cobourg limestone(after Selvadurai,2019b). Area fraction = ADR/A, and volume fraction = VDR/V, where ADR and VDR refer to the area and volume fractions of the darker facies, respectively; and A and V refer to the total area and total volume, respectively.

Table 3Thermal conductivity of the basic minerals in the Cobourg limestone.The values for the clay minerals correspond to those applicable to their dry states.

Considering these results, the Voigt (1928) and Reuss (1929)bounds can be combined to provide the Hill (1952) estimates, i.e.

An improvement to the Hill (1952) average based on the Voigt(1928) and Reuss (1929) bounds can be obtained using the results given by Hashin and Shtrikman (1962) for a two-phase system. In this instance, the two constituents can be identified as the two separate facies withVDR?0.54 andVLR?0.46; this is an appropriate assumption that is consistent with the very low porosity of the lighter(nLR?0.006)and darker(nDR?0.01) facies.

Using the Hashin and Shtrikman (1962) bounds for the bulk thermal conductivity of the heterogeneous limestone,consisting of the lighter and the darker facies in the volume fractions indicated previously, we obtain

This result gives

The bounds give identical results and,in view of the near equal volume fractionsVDRandVLR, are close to the average thermal conductivity of the lighter and darker phases given by Eq. (5).

4.2. Thermal conductivity of the cobourg limestone at 373 K

The procedure can be extended to estimate the effective thermal conductivity of the Cobourg limestone at other temperatures.Considering the thermal conductivity estimates for the constituent minerals given in Table 3,and applicable to 373 K,the Voigt(1928)and Reuss (1929) bounds for the thermal conductivities of the lighter and darker facies are given by

The corresponding Hill (1952) estimates are

The Hashin and Shtrikman (1962) results give the following bounds:

Again, the bounds coincide and, considering the near equal proportions of the lighter and darker fractions,the results are close to the Hill (1952) averages,indicated by Eq. (10).

5. A fabric-based thermal conductivity estimation of the Cobourg limestone

Ideally,the mapping of the internal fabric of the limestone that contributes to the heterogeneity should be performed using noninvasive techniques such as X-ray tomography (XRT) imaging.This procedure was attempted on the smaller cylindrical samples but the sample dimensions could not provide accurate mapping of an RVE. As an alternative, a cuboidal block (80 mm ×100 mm × 300 mm) of the rock was dissected to recover 10 thick adjacent slabs measuring 80 mm×100 mm×8 mm.Both surfaces of each slab were photographed and digitally imaged, and a MATLAB software-based image processing technique was used to create black and white images. The images contained records of artefact such as fossils but these did not exceed 0.1% of the total area. The binarization of the photographs taken from the slabs was one of the challenging aspects of this research and similar efforts performed in connection with the study of permeability of the limestone were given in Selvadurai(2019a,b).First,all the photographs were taken with the same light intensity, background, and camera to have a consistent photographic basis. In this research program, three different approaches were employed to binarize the photographs:(i)Separation of the black and white regions manually by analyzing the photographs; (ii) Analyzing the photographs completely using the MATLAB code; and (iii) Using a combination of these two approaches, which is the basis for the binarization presented here.While it seems more consistent and accurate to use the approach(ii), the procedure will not necessarily result in a more accurate result, since the integrity of black and white regions is not preserved in this method. Also, since these regions will be imported into the finite element modeling software and remodeled, the regions need to be simplified in this step, which decreases the accuracy.Therefore,it was finally decided to analyze the photos using MATLAB and then remove the heterogeneities manually using CorelDRAW?. The threshold value was selected such that the volume and area fractions estimated during the previous research studies (Selvadurai, 2017, 2019a, b) were preserved.

The black and white separations of the slab faces are more of a qualitative classification rather than a distinct quantitative separation, and the photographic images contain transition regions rather than definite black/white regions. An approach that accounted for transition regions (three or more separate regions)proved to be unfeasible.The final decision concerning the binarized regions was considered to be the most pragmatic way in which the lighter and darker zones, defined according to the mineralogical compositions, could be delineated.

Fig. 3. Photographic and digitized images of the thick slab Section 2.

Fig. 4. Photographic and digitized images of the thick slab Section 4.

Fig. 5. Photographic and digitized images of the thick slab Section 6.

The black and white images were created using the CorelDRAW?Graphics Suite X4. Typical examples are shown in Figs. 3—6. In these figures, the designation B and T refer, respectively,to the lower and upper surfaces of a thin slab.These images were used to reconstruct plausible replicas of the lighter and darker facies within the 8 mm thickness of each slab. In order to obtain through-thickness replicas of the internal facies distributions of the Cobourg limestone slabs, we adopted simple techniques that allow extrapolation of the surface images of the slabs to their interiors.

Here we adopted two simple techniques to create plausible distributions of the facies within each slab. Attention will be focused on the generation of the darker regions with the understanding that, in view of the very low porosity of both facies, the remaining regions correspond to the lighter domains.The simplest approach is to ‘extrude’ the darker regions from either face to the mid-plane of the 8 mm thick slab.The second approach involves a solid modeling-type procedure that makes use of the LOFT command available in the software AutoCAD. The details of the reconstruction procedure were given by Selvadurai (2019b). For convenience of description, the geometry created using the LOFT command is referred to as theMorphedgeometry.Figs.7—10 show the typical replicas of the internal distribution of the lighter and darker facies in the thick slab specimen sections 2, 4, 6 and 8.Kriging techniques are an alternative procedure for estimating the internal distribution of properties using data available on the boundaries or surfaces of the domain (Selvadurai and Selvadurai,2010, 2014). This approach has merit when the thicknesses of the slabs are large in comparison to the dimensions of the internal fabric.

The internal geometry of the facies within each thick slab of the Cobourg limestone is discretized as tetrahedral finite elements to create a three-dimensional (3D) replica of the thick slab that can then be used to model the heat conduction process. The effective thermal conductivities within each thick slab alongthree orthogonal directionscan be determined by subjecting two opposite faces of the thick slab to constant temperatures (i.e. Dirichlet boundary conditions with constant values over each face but with a temperature differential between the two faces)and ensuring that the remaining adjacent faces are completely insulated (i.e. null Neumann boundary conditions).The effective thermal conductivities of the lighter and darker facies evaluated at 273 K and 373 K determined from the Hill (1952) estimates are used in the computational modeling, i.e. Eqs. (5) and (10).

Fig. 6. Photographic and digitized images of the thick slab Section 8.

The computational procedure can be applied to determine the effective thermal conductivities of each slab in three orthogonal directions.The computational model of each slab uses in excess of 210,000 elements and 280,000 degrees of freedom. It should be noted that by invoking null Neumann boundary conditions, the heat flow between slabs is suppressed. This is a limitation in the modeling approach;the alternative requires complete 3D modeling of all 10 assembled slabs with the appropriate boundary conditions to ensure one-dimensional heat flow through the assembly. This approach is computing intensive and was not pursued. The computational results derived from the modeling are summarized in Tables 4 and 5. For reference,the effective thermal conductivity of a slab normal to the plane area of the slab is denoted bykzand the effective thermal conductivities for the directions aligned with the edges of the thick slab perpendicular to thez-direction are denoted bykxandky.

The accuracy of the results presented in this study is controlled by the accuracy of the method employed and the accuracy level of the transport estimation in the lighter and darker regions. While the Hill (1952) estimate is used here to determine the thermal conductivity of lighter and darker regions, more accurate analysis can be performed by employing a range for the thermal conductivity of darker and lighter regions and determining the conductivity range of the overall sample.

To analyze the accuracy of the method, we performed another analysis,which is presented here.First,it should be mentioned that modeling the entire geometry resulting from combining all the slabs is not possible due to limitations of the finite element modeling software employed in this research. Multiple efforts to combine the geometries and mesh the overall geometry were unsuccessful due to limitations of the software. However, it was possible to combine four extruded slabs(slabs 7—10)and model the combined geometry;the results were compared with the weighted harmonic mean forkz, and the weighted mean for in-plane conductivity inx- andy-direction,kxandky, obtained from the data for the individual slabs. Considering the data for the temperature of 273 K, (i) the combined finite element model geometry giveskz=kx=ky=3.793 W/(m K),and(ii)the weighted mean and weighted harmonic mean estimates givekz=kx=ky=3.793 W/(m K).While this exact matching of quantities might be a coincidence,it suggests that the procedure for averaging the conductivities of the slabs to estimate the overall conductivity gives reasonably accurate estimates.

The results obtained from the finite element computations performed on the 10 thick slabs using the steady heat conduction analyses can now be combined to obtain the effective thermal conductivities for the assemblage.The weighted harmonic mean is used to estimate the effective thermal conductivity of the assembled cuboid of the Cobourg limestone in the direction normal to the plane of the plates and corresponds toand For theextrudedgeometry, we have

Fig. 7. Reconstructed domains identified for the finite element modeling of thick slab Section 2.

Fig. 8. Reconstructed domains identified for the finite element modeling of thick slab Section 4.

Fig. 9. Reconstructed domains identified for the finite element modeling of thick slab Section 6.

Fig.10. Reconstructed domains identified for the finite element modeling of thick slab Section 8.

Table 4Thermal conductivities of the individual slabs of the Cobourg limestone at 273 K derived from the COMSOL modeling 3.81 W/(m K)).

Table 4Thermal conductivities of the individual slabs of the Cobourg limestone at 273 K derived from the COMSOL modeling 3.81 W/(m K)).

Plate (i) Thermal conductivity (W/(m K))Extruded geometry Morphed geometry kiz kix kiy kiz kix kiy 1 3.791 3.791 3.791 3.791 3.791 3.791 2 3.793 3.793 3.793 3.793 3.793 3.793 3 3.792 3.792 3.792 3.792 3.792 3.792 4 3.791 3.791 3.791 3.791 3.791 3.791 5 3.791 3.791 3.791 3.79 3.789 3.789 6 3.791 3.791 3.791 3.791 3.791 3.791 7 3.791 3.791 3.791 3.790 3.79 3.79 8 3.792 3.792 3.792 3.791 3.791 3.791 9 3.794 3.794 3.794 3.791 3.791 3.791 10 3.795 3.795 3.795 3.794 3.794 3.794

Table 5Thermal conductivities of the individual slabs of the Cobourg limestone at 373 K derived from the COMSOL modeling 2.98 W/(m K)).

Table 5Thermal conductivities of the individual slabs of the Cobourg limestone at 373 K derived from the COMSOL modeling 2.98 W/(m K)).

Plate (i) Thermal conductivity (W/(m K))Extruded geometry Morphed geometry kiz kix kiy kiz kix kiy 1 2.955 2.955 2.955 2.955 2.955 2.955 2 2.957 2.957 2.957 2.957 2.957 2.957 3 2.956 2.956 2.956 2.956 2.956 2.956 4 2.955 2.955 2.955 2.955 2.955 2.955 5 2.955 2.955 2.955 2.953 2.953 2.953 6 2.955 2.955 2.955 2.954 2.954 2.954 7 2.955 2.955 2.955 2.954 2.954 2.954 8 2.956 2.956 2.955 2.955 2.955 2.955 9 2.959 2.959 2.958 2.955 2.955 2.955 10 2.96 2.96 2.96 2.959 2.959 2.959

wheretiis the plate thickness; andare the effective thermal conductivities of the Cobourg limestone slabs in direction normal to the plane of each slab for the extruded and morphed geometries, respectively. Similarly, the weighted mean can be used to estimate the effective thermal conductivity of the assembled cuboid in directions orthogonal to thez-axis,denoted byxandy:

The procedure can be repeated to determine the thermal conductivities applicable to 373 K. The results are as follows:

The results given in Eqs. (12)—(15) can be used to obtain the geometric mean for the effective thermal conductivities applicable to the temperatures of 273 K and 373 K, for the extruded and morphed fabric generations, i.e.

Similarly, the geometric mean estimates for the thermal conductivities applicable to the temperature of 373 K are given by

Considering the range of theoretical estimates for the effective thermal conductivities of the Cobourg limestone, it is concluded that the range of values are bounded by upper and lower limits:

Experimental evaluation of the thermal conductivity of Cobourg limestone is rare. Selvadurai and Najari (2017) used a value of thermal conductivity ofk?2.5 W/(m K)in their investigation of a saturated Cobourg limestone cylinder during boundary heating up to 70°C. The results of Pitts (2017) suggestk25°C?(3.04±0.9) W/(m K) andk100°C?(1.44±0.9) W/(m K); these results indicate that there is a trend in the reduction in the thermal conductivity with an increase in the test temperature but this is lower than the theoretical estimates presented in this study. It should also be noted that, although standard procedures are available for performing experiments to obtain the thermal conductivity of geologic materials, the details of the experimental procedures including contact thermal resistances at imperfect contacts and abnormally high temperature gradients (Selvadurai,1990, 2002) can influence the interpretation of experimental data when estimating the bulk thermal conductivity.

6. Concluding remarks

The paper presents alternative theoretical and computational procedures that can be used to estimate the thermal conductivity of a heterogeneous rock.The estimation is based on the mineralogical composition of the rock, which can be obtained quite accurately using conventional XRD tests. The accuracy and reliability of the theoretical evaluations rest on the accuracy with which (i) the mineralogical compositions and (ii) the thermal conductivities of the individual minerals can be estimated.In the case of the Cobourg limestone,there are visible lighter and darker heterogeneities that are both calcite-rich but contain varying proportions of dolomite,quartz and the clay minerals(illite,kaolinite and montmorillonite).This heterogeneity can lead to varying theoretical estimates based on the conventional Voigt(1928)and Reuss(1929)bounds and the Hill (1952) average and estimates based on the theoretical developments proposed by Hashin and Shtrikman (1962). These studies are complemented by computational models that are derived from 3D reconstructions of the lighter and darker regions of the rock present in 10 thick slabs of the limestone, measuring 80 mm × 100 mm × 8 mm, obtained by dissection of a cuboidal sample. These approaches have allowed the development of suitable bounds for the bulk thermal conductivity of the Cobourg limestone.

The reliability of the estimates for the bulk thermal conductivity of the Cobourg limestone ultimately depends on the accuracy with which the thermal conductivities of the minerals in the limestone can be determined.The estimation of the thermal properties of the minerals is not a routine exercise and requires access to sophisticated experimental procedures.The published literature,however,contains a wealth of information on thermal conductivity estimates for the minerals and these values are used in the development of the theoretical and computational estimates. The theoretical estimates for the bulk thermal conductivity of the Cobourg limestone are higher than those reported in the geotechnical literature,indicating that experimental procedures can introduce thermal resistances that can influence the experimentally-determined values of thermal conductivity. It should also be mentioned that the procedure presented in the paper is a convenient methodology that will be suitable when large test specimens are needed to capture RVEs in the scale of the heterogeneity.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interests associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The research support was provided by the Nuclear Waste Management Organization (NWMO), Toronto, Ontario, Canada, a Discovery Research Grant by the Natural Sciences and Engineering Research Council of Canada,and The James McGill Research Chairs Program. The authors are grateful to the Staff at Geoscience Research and Development at NWMO for their valuable comments on the drafts of the manuscript. The authors are grateful to Professor Martin Mazurek of the Rock—Water Interaction(RWI)Group,Institute of Geological Sciences,University of Bern,Switzerland,for drawing attention to their data on the mineralogical composition of the Cobourg limestone. A number of research assistants, graduate students and post-doctoral researchers in the Environmental Geomechanics Research group under the direction of the first author have contributed to the research program dealing with the Cobourg limestone. Their contributions are gratefully acknowledged.