ABSTRACT: The elastic properties of the Copenhagen Limestone matrix material are spanning over more than two scales of magnitude. The sizes of rock constituents, the bulk density and the degree of induration of matrix material vary. The measured value of the elastic modulus depends on the inherent material properties, the size of the engaged volume and the measuring technique. The work presents a synthetic approach in determination of rock mass modulus by using theory of mixtures, thereby constructing a framework for interpretation of multi-scale investigations. Trend lines for mixtures observed on the results of acoustic tests are used to combine the UCS test results in order to obtain the modulus of the intact rock mass. Rock mass modulus relating to disturbed rock mass is derived by means of Hoek and Brown model (Hoek et al. 2002) and compared to the results of High Pressure Dilatometer Tests.

1 INTRODUCTION

Copenhagen Limestone (CL) is a carbonate rock with varying degree of induration, ranging from unlithified sediments to very strongly indurated rock. Determination of the rock mass modulus for CL presents a challenge, not only due to variation of induration, but also due to variable degree of fracturing. Variations occur on a centimetre scale.

A comprehensive ground investigation has been carried out in relation to Cityring project, comprising several hundreds of borings. Available measurements of elastic moduli include broad series of laboratory and field tests, including unconfined compression strength (UCS) and high pressure dilatometer tests (HPDT), as well as corresponding measurements of the sound velocities on laboratory specimens by means of piezo-crystals (PC) and vertical seismic profiling (VSP) in the field. The interpretation of these tests is aided by series of laboratory classification tests and geophysical tests in the field (porosity and density measurements). A comparative analysis of the results has been conducted and presented earlier (see e.g. Katić & Christensen 2014). For the sake of brevity, the evaluations presented herein are made for the average values covering the Cityring site as a whole.

CL is divided into 3 stratigraphic units, Upper, Middle and Lower (UCL; MCL and LCL, respectively), based on the results of geophysical and core logging, showing specific patterns of

EUROCK 2015 & 64th Geomechanics Colloquium. Schubert & Kluckner (ed.) © ÖGG

Composite elasticity of Copenhagen limestone

Nataša Katić Geo, Kgs. Lyngby, Denmark

Helle Foged Christensen Geo, Kgs. Lyngby, Denmark

limestone and flint forming the rock. The zone of interest for the tunnelling and related support structures such as station walls is mainly above LCL; hence, the evaluation focuses on CL above the LCL. Figure 1 shows typical samples from CL.

Figure 1. Sample cores from Upper Copenhagen Limestone (left) and Middle Copenhagen Limestone (right).

The top-most part of the CL is irregularly glacially and otherwise disturbed, which in general results with lower estimate of Geological Strength Index (GSI) in comparison with the lower CL. General range of GSI in CL is from 45 in top UCL to 55 in lower part.

CL is dominantly composed of calcite, with carbonate content reaching well above 95% in most of the material. CL is classified in terms of indurations H0 – H5, where indurations H0 and H1 represent unlithified material, while induration H5 is generally flint made of micro-crystalline quartz.

The flint in CL is irregularly distributed throughout the mass. In particular, MCL is characterized by the small nodules of flint. These are sometimes fully encapsulated within a sample, thus affecting the physical properties. Flint is more elastic and stronger material than calcite. Thus, assessed moduli in a sample from the Copenhagen Limestone may be higher than the moduli of pure calcite, which can be attributed to flint encapsulated within the sample.

Taking into account the scale of the structures of interest, CL is considered to be a homogeneous medium. None of the performed tests showed pronounced anisotropy with regards to strength and stiffness.

2 ELEMENTS OF ELASTICITY IN MIXED ROCK

2.1 Very small strain: Acoustic measurements

Rules of mixtures provide theoretical bounds for the stiffness of composite materials. Voigt and Reuss models relate to axial and transversal loading, thereby defining the uppermost and the lowermost bounds, respectively. The current work also considers other mixture models, in particular Raymer-Hunt Gardner and Wyllie time average curves (see e.g. Mavko et al. 1998).

In order to enable comparison with other calcite rocks, p-wave velocity (vp) and s-wave velocity (vs) from the available measurements on CL are presented on Figures 2 and 3, respectively. Comparison with the vp estimates for composite (porous) material is based on vp of pure calcite of about 6,600 m/s and full saturation by fresh water for a given porosity. The data is presented having in mind that the tests on UCS samples and VSP tests engage very different volumes of the rock as well as different confining stress levels. More so, while the VSPs are considered to be carried out in fully saturated medium, this condition may not be strictly fulfilled during the UCS testing.

It can be seen that the CL data plots close to or slightly below the Wyllie’s time average and Raymer-Hunt-Gardner curves. This is in agreement with other published data (e.g. Mavko et al. 1998, Eberli et al. 2003), whereas it is notable that the test results on samples with induration H2 guide the trend towards compacted mud samples presented by Eberli et al. (2003).

The VSP results follow the trend of the data on intact samples (see Figure 3). The results show that the rock mass response is similar to the response of samples with induration H2. This indicates a prevailing induration of the rock mass within the investigated profiles, but also includes an effect of the scale between the volume of UCS samples and volume involved in VSP response.

Figure 2. vp from PC tests on UCS samples and in situ VSP tests.

Figure 3. Left: vs from PC tests on UCS samples and VSP results; Right: comparison of measured vs and vp.

The data indicates that, for practical applications within the range of porosities observed in situ, the Raymer-Hunt-Gardner approximation is not likely to be exceeded, while Reuss model is a valid lower bound. The theoretical high bound curves, such as Voight or Hashin-Shtrikman, plot high above the data, and are of no technical interest. Therefore, they are not presented herein.

The plots show that all of the matrix constituents follow the same trend, hence that that the characteristic stiffness parameter (e.g. modulus of elasticity or shear modulus) can be estimated by determination of characteristic porosity or its proxies such as bulk density or water content. The trends also show that the stiffness contribution of the matrix constituents with induration lower than H2 is very small.

2.2 Small strain on intact rock: UCS tests

The modulus of the intact rock mass can be estimated from the LVDT measurements in UCS tests, by extrapolation of the evaluations of acoustic tests presented above.

For a core recovery less than 100%, two scenarios are derived. Namely, core loss occurs because the low induration part of the matrix is being washed out, in which case the core loss does not contribute to the stiffness of the mass. Alternatively, core loss can occur due to the clogging of the tube during drilling through high induration material, in which case, the contribution of the core loss

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is calculated as an assumed H5 contribution. The results of the UCS testing are summarized in the Table 1, together with the average induration count for the Cityring area based on the borehole logs.

Table 1. Average UCS results and induration count (CL) for the Cityring area based on the borehole logs.

Core loss H1 H2 H3 H4 H5 Porosity (UCS) [%] n/a 39 21 15 n/a σc (UCS) [MPa] n/a 4 16 33 n/a ELVDT (UCS) [MPa] (0)* 2,591 7,496 11,548 (40,000)*εx [%] n/a 0.17 0.21 0.28 n/a CL [%] 16 2 29 14 20 17

* Estimate adopted for evaluations, ref. Katić & Christensen 2014, cf. Hansen & Foged 2002 For the two scenarios, the Reuss (lower) bound of the elasticity modulus of the intact rock mass is estimated in the range 6,300 – 6,600 MPa. Assuming Poison’s ratio of 0.3, G of the intact rock mass is estimated in the range of 2,400 – 2,600 MPa. It should be noted that these values relate to the same level of strain as the UCS measurements, i.e. the small strain (<1%). In the same respect, Reuss estimate of the intact UCS strength is 6.1 – 6.2 MPa.

2.3 Large strain in disturbed rock mass: HPDT testing

High pressure dilatometer test (HPDT) is an in situ test, performed by expanding a cylindrical cavity of the boring (on average about 110 mm) by application of a certain pressure and measuring the cavity expansion in terms of the achieved shear strain. The shear moduli are thus obtained for initial expansion, and thereafter, for unload-reload branches performed at certain stress levels. It is assumed that no volumetric strain occurs; hence, there is no change in overall porosity during the test.

Due to the drilling technique, the bored pockets are not ideally smooth and some of the constituents become disturbed; typically, most of the material weaker than H2 is washed out from the interface zone, which slightly biases the results.

It is considered that the HPDT results reflect the deformability of the rock mass as a whole (the rock mass stiffness as per Hoek-Brown material model), including a certain level of damage due to the drilling technique. Overview of results of HPDT testing is presented in Table 2 and Figure 3.

Table 2. HPDT testing results overview.

Average Maximal pressure [MPa] 2 3 4 6 5 Maximal G [MPa] 552 740 963 1237 859 Max. expansion [mm] 9 (6.5*)

* 50% percentile

Initial moduli from HPDT are challenged by determination of the reference point for zero strain, which requires a totally unloaded cavity. In a rock material that is stress relived by drilling, the initial modulus is affected by drilling-induced disturbance and likely under-estimated. For the reloading branch, the pressures need to be above, and remain above, the overburden stresses to ensure that there are no effects of the drilling process. It is considered that all moduli obtained at pressures smaller than about 2 MPa are affected by the stress relief. At higher stress levels, the measured shear moduli are affected by the stress level. This is largely confirmed by the investigation, where only two tests showed shear moduli (obtained at 2 MPa) larger than the shear moduli obtained at higher pressures.

Figure 4. Cityring HPDT measurements. Left: Maximal shear moduli against applied pressure, with outlined lower and upper bounds; Right: Maximal shear moduli against maximal expansion, with outlined shear

moduli obtained at 3 MPa.

Figure 3 shows that the minimal shear modulus of the rock mass, at the level of disturbance imposed by drilling, approaches a limit of about 350 MPa. Here, the corresponding deformation can be approximated as one-half of the ratio of maximal expansion to the average diameter of the pocket, i.e. ½ x 9 mm / 110 mm, see Table 2. Although part of such a large deformation may still be caused by the recovery of the stress relief, it is clear that the deformation exceeds the threshold of large deformation. Based on the assumed Poisson’s ratio of 0.3, the corresponding deformation modulus of the disturbed rock mass is about 900 MPa.

3 COPENHAGEN LIMESTONE IN HOEK-BROWN MATERIAL FRAME

It is a common practice to estimate the rock mass deformation and strength parameters based on the Hoek and Brown (Hoek et al. 2002) model, which assumes that these properties are governed by sliding and rotation of intact blocks of rock defined by intersecting discontinuities (cf. Hoek et al. 2013). In this respect, the Hoek and Brown model provides a scale between the moduli obtained by averaging moduli based on theory of mixtures applied on UCS samples as intact moduli, and the moduli obtained by means of HPDT testing as rock mass moduli.

As the mechanical tests on the samples from the top of the CL showed no difference to the other parts of the CL, it is considered that the same input parameters can be used for the whole of the CL. Based on the presented average results of UCS tests on intact samples (Ei = 6,300 – 6,600 MPa and σci = 6.1 – 6.2 MPa) the modulus ratio, MR, is about 1,000 which is in agreement with the value suggested for micritic limestone (see Hoek & Diederichs 2006).

The variation of disturbance factor, D, between the intact and highly disturbed rock leads to a factor of 2-3 between the intact modulus and rock mass modulus. This parameter typically describes the blast damage, whereas for the excavation done by e.g. tunnel boring machine the parameter is typically lower. Given the drilling induced disturbances and very large strains developed during the HPDT testing, and the glacial disturbance of the limestone, it is considered that large damage factors are appropriate for application in CL.

The results of the Hoek and Brown model for calculating rock mass modulus Erm as function of D, obtained using RocLab 1.0 software are presented on Figure 5, together with the results of HPDT testing. It can be seen that the moduli obtained by HPDT and from Hoek and Brown model are in good agreement.

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4 CONCLUSION

The work presented challenges in a combined interpretation of laboratory and field testing of a highly variable Copenhagen Limestone. The framework for interpretation is based on multi-scale averaging of material parameters obtained in various tests.

Rules of mixtures observed on the results of acoustic tests are used to combine the UCS results on intact samples in order to obtain the intact modulus of the rock mass. Rock mass modulus relating to disturbed rock mass is derived by means of Hoek and Brown model (Hoek et al. 2002) and compared to the HPDT results. The obtained results are in agreement with the results obtained by in-situ HPDT tests and earlier suggested values (Katić & Christensen 2014).

ACKNOWLEDGEMENTS

The authors express their gratitude to the Metroselskabet and CMT for the permission to publish the data.

REFERENCES

Eberli, G. P., Beachle T.G., Anselmetti, F.S. & Incze M.L. 2003. Factors controlling elastic properties in carbonate sediments and rocks. The Leading Edge.pp.: 654-660 July 2003.

Foged, N., Jakobsen, L., Jackson P. & Erichsen L. 2007. Rock mass characterization for tunnels in the Copenhagen limestone. In: The Second Half Century of Rock Mechanics, Three Volume Set. 11th Congress of the International Society for Rock Mechanics, Olalla, C., Grossmann, N. & Ribeiro e Sousa, L. (eds.). CRC Press; Pap/Cdr edition (July 5, 2007).

Hansen, H.K. & Foged, N.N. 2002. Kalkens bjergmekaniske egenskaber. In: Ingeniørgeologiske forhold i København. dgf-Bulletin no. 19, Frederiksen J.K. (ed.) pp. 31-34. DGF, Danish Geotechnical Society.

Hoek, E. & Diederichs M.S. 2006. Empirical estimation of rock mass modulus. International Journal of Rock Mechanics & Mining Sciences 43 pp. 203–215.

Hoek, E., Carranza-Torres, C. & Corkum, B. 2002. Hoek-Brown Failure Criterion – 2002 Edition. In: Proceedings of NARMS-TAC 2002, Mining Innovation and Technology. Hammah R., Bawden, W., Curran, J. & Telesnicki M. (eds.), Toronto – 10 July 2002, pp. 267–273. University of Toronto.

Hoek, E., Carter, T.G. & Diedrichs, M.S. 2013. Quantification of the Geological Strength Index chart. 47th US Rock Mechanics / Geomechanics Symposium, San Francisco, CA, USA, June 23 - 26, 2013. ARMA, American Rock Mechanics Association 13-672.

Katić, N. & Christensen, H.F. 2014. Upscaling elastic moduli in Copenhagen Limestone. In Rock Engineering and Rock Mechanics: Structures in and on Rock Masses, Alejano, Perucho, Olalla & Jiménes (eds). 2014. pp. 235-240. Taylor & Francis Group, London.

Mavko, G., Mukerji, T. & Dvorkin, J. 1998. The Rock Physics Handbook. Tools for Seismic Analysis in Porous Media. Cambridge University Press.

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