Modeling Brazilian Tensile Strength Tests on a Brittle Rock Using Deterministic, Semi-deterministic, and Voronoi Bonded Block Models

Abstract

This study aims to numerically investigate how various common simplifications of grain structure representation in bonded block models affect simulations of rock mechanical behavior. Specimens of Wausau granite were characterized mechanically through Brazilian tensile strength tests for this work. The samples were also characterized petrographically using thin section microscopy, scanning electron microscopy-based automated mineralogy, and visual inspection. Four types of representations of the Wausau granite samples were developed, including 6 detailed manually developed deterministic models, 6 semi-deterministic models, and 120 randomly generated representations (Voronoi models). First, a calibrated set of micro-properties was determined using the deterministic representations to simulate the Brazilian tensile strength measurements. Next, the study examined the ability of different Voronoi tessellations to adequately represent the grain structure for the purposes of accurate tensile strength simulation. This was evaluated by comparing Voronoi model results to the deterministic grain structure model results and laboratory test results. The findings of the study show that the four types of models used in this study can all provide realistic representations of the mechanical behavior of rock. The study confirms that standard Voronoi approximations of grain structures can be reasonably used in lieu of less practical, manually developed representations of the grain structure. Specifically, Voronoi models can properly replicate the geometric heterogeneity within the grain structure, even though they simplify some of its geometric attributes.

Highlights

• Four types of models of granite specimens were generated, each type representing the specimen grain structure with a different degree of realism.

• Brazilian tensile strength simulation results obtained using deterministic, semi-deterministic, and two different Voronoi structures were compared.

• Validity of randomly generated Voronoi models to adequately approximate the geometric heterogeneity within the grain structure was investigated.

• Voronoi models provided strengths nearly equivalent to those obtained from the more complex deterministic and semi-deterministic models.

Appendices

Appendix A: Brazilian Tensile Tests

Three BTS tests on Wausau Granite were conducted following the ASTM International D3967-95a (2001) procedures in the Earth Mechanics Institute laboratory at the Colorado School of Mines. The specimens were loaded at 110 N/sec until failure occurred via formation of diametral/axial fractures. Figure 

Fig. 9

Images of the specimens before and after completion of the tests

9 shows the three specimens before and after completion of the tests. The specimen geometrical details are as follows: (1) BTS-1—diameter = 51.4 mm, thickness = 25.1 mm; (2) BTS-2—diameter = 51.4 mm, thickness = 25.2 mm; (2) BTS-3—diameter = 51.4 mm, thickness = 24.7 mm. The load recorded by the hydraulic press was converted to indirect tensile strength using the following equation (International Society for Rock Mechanics 1978):

$${\sigma }_{t}=\frac{2P}{\pi Dt}$$

where P is the load at failure, D is the diameter of the specimen, and t is the thickness of the specimen.

Appendix B

The results in Sect. 3.2 correspond to BBMs that have four different grain types and 10 corresponding mineral–mineral associations, with properties listed in Tables 7 and 8, respectively. To ensure that the observed behaviors are not a function of the specific set of micro-properties or material heterogeneity (i.e., mismatch in grain elastic and contact properties) and more generally depict the role of geometric heterogeneity, additional models were run with different micro-properties.

• (1) Effect of contact tensile strength

Contact tensile strength directly controls the emergent macroscopic tensile strength of BBMs (Kazerani and Zhao 2010; Fabjan et al. 2015). To understand its influence, two sets of models were run, where the contact tensile strengths were varied by ± 25% (with respect to those listed in Table 8). Each set consisted of 72 models: 6 deterministic (3 specimens, 2 faces each), 6 semi-deterministic (3 specimens, 2 faces each), 30 Voronoi HGS (3 specimens, 2 faces each, 5 realizations per face), and 30 Voronoi UGS (3 specimens, 2 faces each, 5 realizations per face). Model results and overall statistics are presented in Fig. 

Fig. 10

Comparison of laboratory results (LAB) against corresponding deterministic (DET), semi-deterministic (SEM), Voronoi HGS (VO-1), and Voronoi UGS (VO-2) BTS. a Comparison with 25% decrease in contact tensile strength, b Comparison with 25% increase in contact tensile strength, and c Comparison with no change in contact tensile strength, i.e., calibrated properties

10 and Table

Table 12 Summary of the average BTS generated using each type of model

12, respectively. For ease of comparison, model strengths corresponding to the calibrated set of micro-properties are added to Fig. 10 and Table 12.

As expected, the BTS increased and decreased for all four model types with an increase or decrease in contact tensile strength, respectively. The change in simulated BTS is less than 25% in the majority of the models because of the complex non-linear relationship between contact strength and local fracture development. With respect to the calibrated case, the average mean strengths changed by similar amounts (2–2.7 MPa; Table 12), such that the strengths are similar across the four model types in both the + 25% and − 25% cases. This supports the general observation in Sect. 3.2 that there is a limited effect of geometric heterogeneity on modeled BTS.

The variability in BTS follows the same trend as the change introduced in the contact tensile strength, i.e., standard deviation (STD) of − 25% case < STD calibrated < STD + 25% case. This is explained by the greater absolute heterogeneity in the contact tensile strength in the + 25% case in comparison with the − 25% case. For example, if the tensile strengths of two contact types are 2 MPa and 3 MPa, then the difference between them changes to 0.75 MPa (lower heterogeneity) and 1.25 MPa (higher heterogeneity) with − 25% and + 25% change, respectively. It follows that the − 25% model is more homogeneous in an absolute sense than the + 25% model, and hence demonstrates lower variability in BTS. This also means that contact strength heterogeneity has some influence on BTS, but the influence is similar across the different grain structures considered.

• (B) Effect of material heterogeneity

For this analysis, two different types of parameter homogenization were considered (Sinha and Walton 2020).

Grain property homogenization: 6 sets of grain elastic properties were computed for the 6 faces (BTS-1A, BTS-1B, BTS-2A, BTS-2B, BTS-3A, BTS-3B) by weight averaging the properties listed in Table 7 with the mineral proportions in Table 1. All grains in the model were assigned 1 set (out of 6 sets) of elastic property depending on which face it represented. No homogenization of the contact property was performed.

Complete homogenization: In addition to the grain property homogenization, contact properties were also homogenized by weight averaging the properties listed in Table 8 with the associated total contact length in every model. Since the contact lengths for the 10 mineral–mineral associations varied across the models, the weighted average calculation had to be performed individually for each model. Simply stated, in these models, a single set of grain and contact properties was assigned to all the grains and contacts.

Similar to the previous analysis considering variations in tensile strength, 72 models were run for each case—6 deterministic (3 specimens, 2 faces each), 6 semi-deterministic (3 specimens, 2 faces each), 30 Voronoi HGS (3 specimens, 2 faces each, 5 realizations per face), and 30 Voronoi UGS (3 specimens, 2 faces each, 5 realizations per face). Results are presented in Fig. 

Fig. 11

Comparison of laboratory results (LAB) against corresponding deterministic (DET), semi-deterministic (SEM), Voronoi HGS (VO-1), and Voronoi UGS (VO-2) BTS. a Comparison with grain property homogenization, b Comparison with grain and contact property homogenization

11 and Table

Table 13 Summary of the average BTS generated using each type of model

13. The following observations are made:

• BTS increases with material property homogenization, consistent with the previous analysis. Homogenization reduces the local tensile stresses developing within a BBM (Dey and Wang 1981; Kranz 1983; Sinha and Walton 2020) and thereby delays fracture development and failure.

• With respect to the calibrated case, the average BTS for grain property homogenized models increased by 1.1–1.9 MPa and the variability is similar across all model types (Fig. 11a). It is not clear why the VO-HGS model exhibited ~ 1 MPa lower average strength than the other three types. In any case, the simple VO-UGS model demonstrated similar strength distribution as the deterministic and semi-deterministic models, consistent with the observations in Sect. 3.2.

• For the complete homogenized models (Fig. 11b), the average strengths are similar across the four model types. It is noted here that 5 realizations were run per face for both VO-HGS and VO-UGS models, but no/minimal deviation was observed (as a result, there is no scatter; see Fig. 11b). This is because the grain structure in the 5 realizations for a face was the same (refer to Sect. 2.2 and Table 6) and homogenizing both the contact and grain properties essentially led to very similar realizations (same grain structure, same grain properties, and very similar contact properties).

• Overall, the average strengths are similar across the four grain structures considered, meaning that the simplest UGS model can be used instead of the more complex deterministic or semi-deterministic model for practical purposes, whether homogeneous or heterogeneous micro-properties are used. The similar performance obtained using the different grain structure assumptions suggests that the conclusions of this study regarding the validity of the Voronoi approach could perhaps be extended to the simulation of monomineralic rocks.