Validation and Practical Application of a Data Reduction Software for the Analysis of Data from Stress Relief Tests

Carrying out safe rock excavations (open-pit or underground), for civil or military purposes, must always deal with the nature of the materials crossed and with their ability to self-sustain. An essential condition for carrying out rock excavations is therefore the knowledge of the strength and deformability characteristics of the materials being excavated, as well as their on-site stress conditions.

The methods for assessing the state of stress in rocks on-site, based on the physical reference principle adopted, are generally classified into four groups: stress release, hydraulic fracturing, compression of rock samples, and propagation of fractures induced by drilling. Compared to the methods of hydraulic fracturing and fracture propagation that require decidedly more relevant equipment and financial resources for their execution, the first method is certainly the most widespread, both for its adaptability to different case studies and for the relative simplicity of the experimental determination. In fact, the technique is based on sensors of different shapes, which are able to evaluate, according to different directions, the endured strains from small rock surfaces, consequent to their separation from the boundary rock. Together with the development of procedures and equipment for the experimental determination of the natural and/or induced stresses acting on site, methods for interpreting experimental data have been developed by different authors, through the application of mathematical techniques. In this context, specific analytical and numerical solutions have been proposed for the data collected thanks to different geotechnical instruments, such as: the "USBM borehole deformation gage" (Leeman 1959; Obert et al. 1962; Panek 1966; Crouch & Fairhurst 1967; Merril 1967; Niwa et al. 1971; Suzuki 1966, 1971); the "biaxial confinement device" (Fitzpatrick 1962); the "flat jack" (Merril et al. 1964; Rocha et al. 1966); the "CSIR-doorstopper" and "triaxial strain cell" or "hollow inclusion cell" (Mohr 1956; Leeman & Hayes 1966; Leeman 1964, 1968, 1969, 1971; Hiramatsu & Oka 1968; Oka 1979; Rocha et al. 1974; Worotnicki & Walton 1976; "borehole slotting" (Becker & Werner 1994; Bock & Foruria 1983); the "strains at the hemi-spherical borehole end" (Sugawara et al. 1984; Sugawara & Obara 1986); and the "strains at the conical end" (Kobayashi et al. 1991). This innovative drive has been followed, in the last twenty years, exclusively by the evolution of existing measurement methods which are the result of technology transfer, such as: automatic data acquisition by means of miniaturised PCs directly connected to the measurement device and transmission of the data acquired and stored during the measurement via Wi-Fi (Gullì et al. 2006; Iabichino & Cravero 2010).

In the context of excavations, both for mining purposes and for the construction of infrastructures, the most widespread techniques for determining the state of natural and/or induced stress in the rocks on site, are those which, based on "stress relief", allow the determination of the complete stress tensor, with a single set of measurements taken from a single borehole. Among these techniques, due to the relative simplicity of execution and, above all, to the wide availability and consistency of studies on data interpretation methods, the one most used is the technique proposed by the Commonwealth Scientific and Industrial Research Organisation (CSIRO), also called the Hollow Inclusion cell, or HI cell.

The measuring cell (HI cell) is a two-component epoxy resin cylinder with a length of approximately 100 mm, an internal diameter of approximately 32 mm and a thickness of approximately 3 mm, equipped with three (3) rectangular strain gauge rosettes, each consisting of three (3) or four (4) resistive electrical strain gauges, arranged at 120° along its inner median circumference (Fig. 1). For the execution of the experimental test, the external surface of the HI cell is made integral, by means of a gluing operation, to the internal wall of a pilot hole with a nominal diameter of 38 mm, drilled into the rock to reach the desired depth. Once the polymerisation of the glue is completed, the stresses are released by making a circular notch (over-coring) coaxially to the pilot hole equipped with the HI cell, using a thin-walled core barrel or a double-core barrel with a minimum internal diameter of about 120 mm.

The strain intensities, evaluated for each direction of measurement, are obtained by the difference between those acquired at the end of the over-coring operation and those acquired at the start (Fig. 2) and are related to the stress tensor through the elastic properties of the rock, which must be known. Given the wide choice of both the methods to be adopted for the determination of the pseudo-elastic constants of the rock being excavated, and the number of samples to be tested for each rock involved in the project, it follows that the choice of adopting greater/lesser accuracy in the characterisation of the rock mass is directly linked to the importance of the project. In any case, the best approximation that can be reached in determining the pseudo-elastic characteristics of the rock has a direct reflection in the best estimate of the stress state of the rock itself. In this regard, it should be noted that according to Hakala (2006), when performing experimental tests with the HI cell, dispersions between 1.0 MPa and 3.8 MPa are acceptable, due to the intensity of the main stresses; and between 8° and 15° for the direction of the stresses.

As previously shown, different mathematical solutions are proposed in the literature, all quite complex, to determine the state of stress acting in rocks for which a continuous, homogeneous, linear elastic behaviour, both isotropic and anisotropic, has been hypothesised. The objectives of this study do not refer to the analytical developments to determine the stresses acting in a rock, as 9 or 12 strains are known that are independent of each other, so the fundamentals of the most widely used data reduction software for the purpose, those of the program to be validated, as well as the difference between the calculation schemes themselves will be summarised.

The most commonly used softwares for the interpretation of data coming from stress relief tests (Smith 1982; Larson 1992; Worotnicki 1993) evaluate the acting stress tensor, using displacements and strains determined at the boundary of a circular hole of infinite length, made in a material considered to be continuous, linearly elastic and isotropic, with known geotechnical characteristics. A further hypothesis adopted in the numerical calculations is that of "plane deformation", which considers the strains detected, developed exclusively in the survey plane and in the plane orthogonal to the axis of the hole. The stresses acting around the hole are obtained using the classical Kirsch elastic solution (1898) or its generalised version for non-linearly dependent (non-parallel) holes provided by Hiramatsu and Oka (1962, 1968), Fairhurst (1968a, ba, b, 1968a, ba, b), Leeman (1968), and Fama and Pender (1980). The quoted solutions, albeit with the limitations implicit in their basic hypotheses (the component of the axial stress exclusively dependent on the Poisson's ratio, strength and deformability characteristics of the sensor adopted for the strain/displacement measurements, considered negligible), are widely used in construction site practice. This is largely attributable to the simplicity of the geomechanical characterisation required by the numerical calculations for the rock under study. In fact, the determination of the pseudo-elastic modules characterising the rock under excavation, although conducted on the basis of different assumptions made in the design field (Amadei 1983; Hirashima & Hamano 1987; Amadei & Stephansson 1997), is generally carried out by hypothesising, for the on-site rock, an isotropic, linearly elastic behaviour, and adopting, for each of the three necessary characteristics E (Young's modulus), ν (Poisson's ratio), and G (transverse stiffness), an average value derived from specifically normed tests. These modules/coefficients can be evaluated in greater detail both in the laboratory and on site using established procedures, such as those listed by Fitzpatrick (1962) or the ISRM (1979).

The software for the interpretation of data from stress release tests (Cravero et al. 2005, 2016) is called "Berni1". It was originally developed by Amadei (1983), then modified to be used on computers operating at 32/64 bit and updated with different numerical methods and mathematical routines (taken from the Fortran IMSL libraries), and equipped with specific routines for the evaluation of the main stress tensor. This software is aimed at determining the state of stress in the boundary of an elastic inclusion, and has an annular section, integral to the walls of a circular hole made in a continuous, elastic, linear and anisotropic medium. The analytical formulation of the state of stress at the boundary of holes made in an anisotropic medium was given by Lekhnitskii (1963), while Amadei (1983) contributed the generalisation applicable to the rock mass, both of the Lekhnitskii formulations and of the hypothesis of "plane deformation" adopted, to calculate the distribution of stresses around arbitrarily oriented holes in equally arbitrarily arranged anisotropic rock masses. In greater detail, the Berni1/BM2000 software addresses and provides analytical solutions to two problems: the calculation of the deformations induced around a hole however oriented in an anisotropic medium, since the stresses acting at infinity are known (direct problem); the calculation of the stresses acting around a hole in any orientation, made in an anisotropic medium, known as the induced strains (inverse problem). Although the analytical formulations proposed by Amadei (1997) for the interpretation of stress release data are widely recognised as "rigorous", as they take into account the influence both of the stiffness of the measuring instrument and of the adhesive used for making the sensor itself integral with the rock, their application to practical cases is rather rare. One of the reasons for this lack of application is certainly the difficulty in the determination of the geotechnical characteristics of the rock, when considered transversely isotropic (the case of metamorphic rocks). In fact, to calculate the stresses acting around a hole made in a transversely isotropic medium using the strains resulting from the stress release, knowledge of five independent parameters is required, as already pointed out: E₁, E₂, ν₁, ν₂ and G_1,2, which cannot simply be assessed through laboratory or on-site tests.

A simple method for determining the pseudo-elastic modules is offered by Nunes (1997, 2002) who, on the basis of the transformations of the elastic and anisotropic constants developed by Lekhnitskii (1963) during his studies on the distribution of stresses in superimposed flat plates crossed by holes of different shapes, developed an original analytical method: through specific formulations, this allows the data acquired to be used by carrying out simple confinement tests on hollow cylindrical specimens equipped with a "Hollow Inclusion cell", obtaining the 5 pseudo-elastic deformability modules (E_1,2, ν_1,2, G_1,2), together with the orientation parameters of the rock's isotropy planes (α, β). The proposed method is valid for transversely isotropic rocks with an anisotropy ratio between the elastic modules (E₁/E₂) in the range of 1 ~ 2.

The validation of the BM2000 software was first required to build two sets of data relating to the geomechanical characterisation of the rocks to be examined. The results of the stress state analysis in the case of rock with linear elastic behaviour, both isotropic and anisotropic (transversely isotropic), were evaluated separately.

In the first case, with the same data of "relieving” from over-coring to be evaluated, the results of the analyses carried out through BM2000 were compared with those from Berni1, and from the most common software available in the literature and widely used in the sector (Smith82, STRESsOUT).

In the second case, again with the same data to analyse, the results obtained and shown by Amadei (1983) through Berni1 were compared with the similar results from BM2000.

In order to make the data required by the various softwares homogeneous, a first analysis was carried out regarding the sign conventions and the reference systems adopted by them, as well as the geometry of the measuring device (Hollow Inclusion cell).

In this regard, it has to be pointed out that the Smith82 and STRESsOUT software represent the strain and the compressive stress with the positive sign, whereas Berni1 and BM2000 adopt the negative sign for the same parameters. In any case, it has to be kept in mind that the over-coring of a portion of rock on-site has the local consequence of tension "relief", so that the stresses resulting from the analyses have a sign opposite to the strains used at the beginning of the calculation (positive input strains correspond to negative output stresses).

The reference systems adopted by the different software used for validation are generally three (3): two sets of axes orthogonal to each other and three Cartesian two-dimensional coordinates lying in a known position on planes tangential to the measurement hole. The first orthogonal triad generally coincides with the NS and EW axes of the geographical reference, completed by the Zenith axis. The second one has an axis coinciding with the axis of the hole used for the execution of the experimental test and the other two axes lying in the plane normal to the axis of the hole. The third reference system identifies the position of the measuring elements (strain gauge) by means of two angles: the first (circumferential angle) identifies the position of the centre of the single strain gauge along the circumference of the section of the hole; the second (rotational angle) identifies the inclination of the axis of the single strain gauge with respect to the horizontal axis of the two-dimensional reference system associated with the plane tangential to the measurement hole at the point of application of the strain gauge rosette. For example, this latter angle is evaluated in an anticlockwise direction according to Berni1 and BM2000, thus assuming a value of 90° if the axis of the strain gauge considered has the same direction as the axis of the measuring hole, and a value of 0° if the axis of the strain gauge is arranged according to the measurement section. For the different software used, Table 1 presents the individual reference systems adopted to identify the arrangement of the strain gauges in the hole in order to highlight the differences. Table 2 shows the orientations of the electrical strain gauges along the circumference of the measuring instrument, depending on the specific software analysed. Again, with the aim of highlighting the differences among the software used, Table 3 shows the arrangement of the orthogonal reference triad associated with the measurement hole.

Table 4 presents the "transformations" necessary to make the inputs of a generic data reduction software "compatible" with those of another data reduction software selected for the validation of BM2000.

Further preliminary adaptations that were necessary to compare the results of the software chosen to validate BM2000 on the basis of "homogeneous" data were: the adaptation of the data relating to the geometry of the instrument used and the choice of the four (4) "K_i" stress concentration factors related to the calculation conditions (plane deformation), as well as the dimensions and deformability values of the HI cell used. In detail, respectively E = 2000 MPa and ν = 0.3 (-) for the Young's modulus and the Poisson's ratio of inclusion were adopted, as well as the symbols ("a") for the internal radius of the measuring cell; ("b") for the radius of the pilot hole used to install the cell and ("r") for the radius of the circumference where the electrical strain gauges are positioned. The data on the radii necessary for the calculation are expressed as a ratio, where a/b = 2 (-), r/a = 0.75 (-). The K_i coefficients used in the Smith82 and STRESsOUT software are calculated based on the solution developed by Savin (1961) in the case of a ring integral with a hole made in an isotropic and linear elastic material (Fama & Pender 1980; Jalkanen 1982).

The results obtained show that the BM2000 software developed at the IGG-CNR Geomechanics Laboratory, applied to rock with linear elastic behaviour, both isotropic and transversely isotropic, is fully satisfactory, provided that the strain characteristics of the medium and the geometry of the measuring instrument adopted are known.

In fact, observing the results in the isotropic case in the comparison software (Berni1, Smith82, STRESsOUT), after having correctly homogenised the input data, the differences are in the order of a few hundredths of MPa for the stresses and a few degrees for the angles of the stresses with respect to the reference axes.

Most likely, the differences detected are due to the greater calculation precision obtained thanks to the use of different IMSL subroutines within the numerical methods and to the inaccuracies due to the positioning of the individual sensors (strain gauges) along the circumference of the measurement section of the pilot hole. The software, which has been successfully validated, allows, once the deformability characteristics of the rock are known, the complete state of stress to be determined even in the case of rocks with transversely isotropic behaviour.

In this respect, the state of natural and/or induced stress in open pit or underground activity of dimension stones with more or less evident planes of weakness, where the ratio between the elastic modules in orthogonal directions is > 1, can be examined more closely. This, in agreement with Hakala (2006), implies that the strain anisotropy (E₁/E₂) between the intensities of 1.14 MPa and 1.33 MPa has a noticeable systematic effect on the interpretation of the stress state on-site and, for this purpose, the elastic parameters of the rock at the measuring point should be defined as accurately as possible.

In this regard, the required experimental determination can be carried out directly on-site with a circumferential press capable of containing the over-cored rock sample with the hollow inclusion cell still inside it, using the method proposed by Nunes (1997, 2002). Another experimental method, still being tested, for defining the deformability characteristics of the on-site rock with transversely isotropic linear elastic behaviour is based on the use of a circumferential press capable of containing the over-cored rock sample with the hollow inclusion cell still inside it, and a numerical process of minimisation implemented by means of the BM2000 software.