This paper presents an alternative approach for estimating the Biot coefficient for the Grimsel granite, which appeals to the multi-phasic mineralogical composition of the rock. The modelling considers the transversely isotropic nature of the rock that is evident from both the visual appearance of the rock and determined from mechanical testing. Conventionally, estimation of the compressibility of the solid material is performed by fluid saturation of the pore space and pressurization. The drawback of this approach in terms of complicated experimentation and influences of the unsaturated pore space is alleviated by adopting the methods for estimating the solid material compressibility using developments in theories of multi-phase materials. The results of the proposed approach are compared with estimates available in the literature.

The classical theory of poroelasticity proposed by

The experimental procedure for determining the Biot coefficient

The Aar granitic rock (also referred to as Aare granitic rock) setting at Grimsel has been associated with initiatives related to the use of granitic rock formations as potential hosts for the creation of deep geologic repositories for the disposal of heat-emitting nuclear fuel waste. Detailed descriptions of the geological settings of the Aar Massif of the Central Alps are given by several authors including

During the geological evolution of the Aar Massif, the strata acquired different mineralogical compositions, and the studies by

Short petrographical descriptions of the rock samples analysed by

Geochemical descriptions of the rock samples across the Grimsel test site given by

Figure

The microstructure includes larger crystals of quartz (with dimensions of up to 8 mm), and this requires that a suitable representative volume element is considered, both in the mechanical testing and mineralogical property evaluations. Extensive geomechanical characterization studies have been performed on the Grimsel granite, and these are given in the references cited previously. Permeability studies are also reported by

The objective of this study is to employ the existing data on the mechanical characterization of the transversely isotropic granite to estimate the skeletal compressibility of the granite and to use XRD (X-ray diffraction) studies of the mineralogical composition of the Grimsel granite to estimate the compressibility of the solid phase composing the porous fabric.

The Grimsel granite sample taken from the PRP1 borehole, with a diameter of 110 mm and a length of 240 mm. The nominal planes of
stratification are inclined at about 50

The fabric of the Grimsel granite is indicative of a transversely isotropic material

We point out that the Poisson's ratio is generally defined as

Due to the isotropic behaviour in the

The skeletal bulk modulus for the transversely isotropic elastic material can be expressed in the form

In terms of the elasticity parameters that are applicable to the direction normal to the planes of stratification (

In the limit of material isotropy,

The estimation of the skeletal bulk modulus of the Grimsel granite can be attempted provided that the elasticity constants applicable to either an isotropic fabric or a transversely isotropic skeletal elastic behaviour can be identified. The geomechanical investigations of the granitic rocks at Grimsel have ranged from the estimation of the deformability and strength characteristics of the rock to the assessment of the in situ stress state. The interpretation of the available data for estimating the

Mineralogical composition of the Grimsel granodiorite (Gr-Gr)
(After

Mineralogical composition of the FEBEX granite
(after

The skeletal compressibility is also an important parameter in the interpretation of transient hydraulic pulse tests for estimating the fluid transport properties of low-permeability materials including granite and argillaceous limestones

The elasticity parameters were inferred through a computational back analysis of the overcoring technique; these authors also provide a comparison with the results obtained by

The majority of the studies focusing on the evaluation of the deformability characteristics of the Grimsel granite deal with isotropic elastic modelling. The possible influences of either elastic anisotropy or elastic transverse isotropy were addressed in the earlier study by

If a range of elastic behaviour can be clearly defined and if the elastic constants governing transverse isotropy can be determined, then, as shown by Eq. (

Elasticity properties for the Grimsel granite with the corresponding

The skeletal material of the Grimsel granite consists of a variety of mineral phases including quartz, biotite, anorthite, augite, microcline, and traces of pyrite and magnetite. The composition of these minerals was determined both at the XRD facilities at University of Montréal, QC, Canada and at the Department of Earth Sciences, Institute of Geology, ETH, Zurich

Mineralogical fractions of the Grimsel granite (data obtained by

Mineralogical fractions of the Grimsel granite (data obtained by the Earth Sciences Laboratory, University of Montréal).

In the multi-phasic approach, the objective is to determine the overall bulk modulus for the solid mineralogical phase by considering the bulk moduli for the separate mineral constituents and their volume fractions. Ideally this needs to be approached using a generalized theory of multi-phasic composites that can accommodate a mixture of any number of phases. Such a generalized theory is yet to be developed. The most widely used relationships are those by

The results given in

Using the mineralogical compositions obtained from XRD analyses given in Table

We consider the upper and lower bounds for a multi-phasic composite consisting of

Considering the multi-phasic data set given in Tables 6 and 7, respectively,

Considering the range of solid material compressibilities obtained from the two laboratory investigations, we can conclude that the lower (

The results for the skeletal compressibilities given in Table

Upper and lower limits for the Biot coefficient for the Grimsel granite;

Reduced data set for the upper and lower limits for the Biot coefficient for the Grimsel granite.

In theories developed for estimating the elasticity of multi-phasic materials, the most extensive studies relate to two-component elastic materials. Theories, however, have also been developed by several researchers to include a distribution of three elastic phases in the composite material. An early study in this area is by

A plausible alternate approach is to essentially reduce the components in Tables

As a guide, experimental results for the skeletal compressibility values that exceed the effective solid material compressibility of the minerals with the largest volume fractions should be disregarded. Therefore these results can be excluded without further comment. (Since the multi-phasic assessment of the compressibility of the solid material has a lower limit of approximately

The accurate estimation of the skeletal deformability characteristics of a porous rock is an essential prerequisite for estimating the Biot coefficient for a fluid-saturated poroelastic material. While the procedures for conducting either uniaxial or triaxial tests to estimate the skeletal deformability characteristics are well known, the exact procedure for estimating the elastic moduli, Poisson's ratio, etc., needs to be better documented so that the interpretations of experimental data can be consistent. The conventional procedure for the pressurization of a saturated sample of the rock and the measurement of the resulting sample strains when the externally applied cell pressure matches the pore fluid pressure is perhaps the best procedure for estimating the compressibility of the solid phases of the porous medium. This, however, is not a routine procedure for low-permeability materials, and substantial pressures need to be applied to ensure that volumetric strains of an accurately measurable value can be recorded.

Also, in such cases the strains could involve irreversible grain boundary frictional slip, and this needs to be excluded from the estimation of the solid material compressibility. Here, we advocate the use of a multi-phasic approach where the theories of composite materials can be used to estimate the compressibility of the solid material composing the porous skeleton. This is a relatively easy approach since XRD evaluations of the mineralogical phase composition are usually carried out to characterize the rock. In relation to the Grimsel granite, the analysis points to a Biot coefficient that has bounds rather than a specific value: i.e.

The data related to the research are all documented in the article. Additional information can be obtained from the lead author.

The concepts leading to the multi-phasic approach for estimating the Biot coefficient was proposed by APSS. The literature reviews and parameter estimations were performed by APSS, PAS, and MN. The experimental data related to the transversely isotropic modelling of the Grimsel granite were obtained by MN. The calculations of the Voigt–Reuss–Hill and Walpole bounds and their verifications were performed by PAS and APSS. The paper was written by APSS, PAS, and MN and formatted to LaTeX by MN.

The authors declare that they have no conflict of interest.

The work described in the paper was supported by a Discovery Research Grant awarded by the Natural Sciences and Engineering Research Council of Canada. This study is part of the In situ Stimulation and Circulation (ISC) project established by the Swiss Competence Center for Energy Research-Supply of Electricity (SCCER-SoE) with the support of Innosuisse. The third author would like to thank the financial support provided by the Swiss Innovation Agency Innosuisse, which is part of SCCER-SoE. The authors are also grateful to Eduardo Alonso (UPC, Spain), Lyesse Laloui (EPFL, Switzerland), Florian Amman (RWTH, Germany), Martin Mazurek (University of Bern, Switzerland), Stratis Vomvoris (NAGRA, Switzerland), Christian David (Université Cergy-Pontoise, France), Jonny Rutqvist (LBNL, USA), Farid Laouafa (INERIS, France), Son Nguyen (CNSC, Canada), Raphael Schneeberger (NAGRA, Switzerland) and Robert W. Zimmerman (Imperial College, UK) for drawing attention to the literature used in this study and for helpful comments. The authors gratefully acknowledge the comments made by Joseph Doetsch, Institute of Geophysics, Department of Earth Sciences ETH Zurich, which led to improvements in the presentation. The authors, however, are entirely responsible for the statements and conclusions presented in the paper.

This research has been supported by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2016-04676).

This paper was edited by David Healy and reviewed by two anonymous referees.