## 5 Estimating UCS Using Linear and Parabolic Equations  
From a rock mechanics point of view, the UCS of a rock sample can be estimated based on the obtained strength at different confinements. As mentioned earlier, three levels of confinement are sufficient but five levels of confinement were chosen to be used in this research (2, 5, 10, 15, and 20 MPa). The purpose has been to produce more data points for a more accurate drawing of the peak strength envelope, and to have better observation of the rock behaviour. In this research, the estimation was performed using both linear and non-linear equations (or failure criteria) to fit the five obtained data points for each rock sample, plotting in the principal stresses domain. The linear equation is actually the Mohr-Coulomb failure criterion, and many non-linear equations/failure criteria have been presented and  
Figure 8: Sample C—test record of radial strain, axial stress, and cell pressure versus axial strain. “A” and “Bi” are peak strengths at the corresponding confinement. discussed by different authors, for example, [PERSON] [41], [PERSON] et al. [42], and [PERSON] et al. [43]. In this publication, based on their simplicity and common use, three failure criteria were selected for calculation: linear (Mohr-Coulomb), parabolic, and Hoek-Brown. The forms of the linear and parabolic equations are as follows:  
\[\sigma_{1}=k\sigma_{3}+\sigma_{ci} \tag{1}\]  
\[\sigma_{1}=a\sigma_{3}{}^{2}+b\sigma_{3}+c \tag{2}\]  
where  
* Equation (1) is actually the Mohr-Coulomb failure criterion written in the form of principal stresses;
* \(\sigma_{1}\) and \(\sigma_{3}\) are major and minor principal stresses, respectively;
* \(k\), \(\sigma_{ci}\), \(a\), \(b\), and \(c\) are the fitting parameters. The fitting parameters are defined by using the test results and a fitting procedure;
* \(k\) is related to the internal friction angle (\(\phi\)) of the rock with the equation as \(k=(1+\sin(\phi))/(1-\sin(\phi))\);
* \(\sigma_{ci}\) is also the UCS of the intact rock. The first option, linear fitting Equation (1), is actually the Mohr-Coulomb failure criterion. The second option is parabolic Equation (2), and the reason to choose a parabolic equation is due to the vast experience with Iddefjord granite at the NTNU/SINTEF laboratory. Based on a large number of triaxial tests being performed in the laboratory over the years, it can be concluded that the obtained strength envelope for this type of rock is curvilinear. The parabolic function fits better than a linear function to the results. The process of fitting the data points and the estimation of the UCS based on the triaxial test results with linear and non-linear options for each sample are presented in Figures 9-11. Figure 9: Sample A—estimation of UCS from triaxial test results using linear and parabolic equations. Figure 11: Sample C—estimation of UCS from triaxial test results using linear and parabolic equations. Figure 10: Sample B—estimation of UCS from triaxial test results using linear and parabolic equations. Results of the \"True-UCS\" and the \"Estimated-UCS\" from the tests are presented in Table 5.  
From the estimation of the UCS values (Figures 9-11), it can be seen that the coefficient of determination (R\({}^{2}\)) for the fitting equations is high for both the linear and non-linear equations. However, the R\({}^{2}\) for the non-linear fitting equation displays in general a better fit than the linear one for the tested samples. It means that a non-linear equation is a more appropriate fit for the obtained data than a linear equation for the Iddefjord granite used in this research. Further, the test results showed that the \"Estimated-UCS\" using the parabolic fitting equation is much closer to the \"True-UCS\" than using the linear fitting equation.