# Viscoelastic Strains of Palaeozoic Shales under the Burger's Model Description
## 3 Laboratory Rheological Analyses of Paleozoic Rocks
### Initial Creep Analysis in Paleozoic Shales on 1D Models  
Rock creep is often described by 1D models, which in the case of both uniaxial and triaxial stresses show the dependency of axial strain on time. Such analyses allow the definition of the character of creep and the determination of the correctness of the initial assumptions in the creep model for the analyzed rocks. Figure 2 shows an example of creep strain obtained for a shale sample from the Pelplin Formation (W260A, see Table 1). The sample was analyzed on three levels of differential load applied at 25, 50, and 75% of (\(\sigma_{1}-\sigma_{3}\))\({}_{max}\). As shown in the case of sample W260A for all loading ranges, creep has classical characteristics, from initial (instantaneous) elastic strain, through primary creep, and ending with secondary creep (Figure 3) [35, 36, 37]. Notable is that the largest loading, assessed at the level of 75% (\(\sigma_{1}-\sigma_{3}\))\({}_{max}\) before the experiments, did not cause a distinct transition to the third progressive creep in sample W260A. A similar case was observed in most tests, where creep ended at a constant creep velocity (secondary creep). Applying the most popular 1D creep models, i.e., standard, [PERSON]'s and exponential models (Figure 4) for creep description, it can be initially assessed, which one corresponds well to the creep in Paleozoic rocks from the Baltic Basin in the analyzed loading range. Figure 1: (a) Preparation procedure of the tested samples; (b) triaxial cell with a mounted rock sample; (c) triaxial loading path CTC applied in the creep experiments. Figure 2: Sample W260A: (**a**) creep strains for the whole loading path; (**b**) axial and lateral creep strains for loading levels applied at 25%, 50%, and 75% (\(\sigma_{1}-\sigma_{3}\))\({}_{max}\). The standard model has a restricted application for the description of creep in the analyzed shales because creep in this model approaches a constant strain value, which usually is not observed in rock creep. This is also the case in the analyzed shales (Figure 2). In [PERSON]'s model, creep approaches an asymptote described by the following equation:  
\[\sigma_{0}\bigg{[}\frac{1}{E_{1}}+\frac{1}{E_{2}}+\frac{t}{\eta_{1}}\bigg{]}, \tag{1}\]  
In advanced creep at a constant velocity--secondary creep--the inclination of the creep strain can be determined from the relationship \(\frac{\sigma_{0}}{\eta_{1}}\). This phenomenon is observed in most medium and high strength rocks [34; 39; 40]. It is also noted in analyzed shales (Figure 2). The advantage of the presented models is the rather simple physical interpretation of their constants, which may be used directly in the construction of constitutive relations of viscoelastic media. The exponential model used in rock creep description [37] is the most universal regarding the accuracy of creep description. Despite the fact that constants \(B\) and \(n\) may have a specific physical interpretation [27], they do not directly describe the constants of viscosity or elasticity as structural models (e.g., [PERSON]'s). Due to good match to the obtained results, the exponential model is often used to determine the creep function \(J(t)\) and creep compliance below [15; 27; 37]:  
\[J(t)=Bt^{n}, \tag{2}\]  
Figure 4: The most commonly applied 1D creep models: (**a**) standard, (**b**) Burgers, (**c**) Norton–Bailey [38]. Figure 3: Typical creep characteristics comprising three stages: primary, secondary, and tertiary creep. where \(B\) and \(n\) are constants. When analyzing the characteristics of creep in the representative sample W260A on three loading levels (Figure 2b), it can be assumed that both the [PERSON]'s and [PERSON]'s models should correctly describe creep of the shale sample. Owing to the fact that the model constants can be interpreted physically, the Burger's model was selected for further analyses. Axial and transverse shale creep was analyzed in the loading range of 25-75% (\(\sigma_{1}-\sigma_{3}\))\({}_{max}\).