Keywords

1 Introduction

In rock mechanics laboratories, for various reasons, fracture resistance tests are performed on rock samples with a hole in the center. For example, in hydraulic fracturing tests, to determine the fracture pressure, rock samples with a circular cavity are subjected to triaxial compressive stresses. But it has been observed that the fracture pressure in rock samples, although it depends on the mechanical properties of the rock and the fluid injection rate, but it also depends on the hole radius. In hydraulic fracturing tests, the fracture pressure is also dependent on the length of the initial crack in the cavity wall. The dependence of the fracture pressure on the size of the borehole is also very important in the field, where the breakdown pressure of the hydraulic fracture is supposed to be used in the estimation of the in situ stresses.

In rock and rock masses, failure is mainly caused by shear and tensile stresses caused by compressive stresses [1,2,3,4,5,6,7], in other words, when the tensile stresses exceed the tensile strength of the rock, tensile failure occurs [8,9,10,11,12,13]. This is the traditional classic view of tensile failure, against which there is the view of energy instability. The theory of energy instability, which today has led to the linear elastic fracture mechanics (LEFM) method, states that when the stress intensity factor at the crack tip exceeds the fracture toughness, a crack starts to propagate [14, 15]. On the other hand Leguillon (2002) showed that both the energy criterion and the stress criterion are necessary conditions for failure, but neither of them alone is sufficient [15]. This is while the energy criterion provides the lower bound for crack length and the stress criterion leads to the upper bound for admissible crack length.

If there is a hole in the center of the sample, due to the concentration of tensile stress, tensile failure occurs in the cavity wall and along the maximum in situ stress. Although, due to the use of linear elastic theory, the hole size is not included in the classical criterion based on tensile strength, but the fracture mechanics criterion is able to consider the effect of the initial crack length on the fracture pressure.

Some laboratory studies showed that the fracture pressure is dependent on the tensile strength and the fracture toughness of the material [7, 15], in such conditions, the effect of the size of the hole in the center of the sample can be seen in the fracture pressure. Recently, the authors obtained the breakdown pressure of hydraulic fracture by combining the tensile strength criterion and the fracture mechanics criterion and investigated the effect of borehole radius and initial crack length on the breakdown pressure [16].

This study investigates fracture pressure of brittle rock samples with a circular cavity using a plane strain two-dimensional finite element model. The mixed failure criterion is based on a combination of tensile strength (stress criterion) and fracture toughness (energetic criterion) of the material. The superiority of this model over the previous models is that it examines both the tensile strength of the material and the fracture mechanics at the same time, and by using it, the effect of the radius of the hole in the center of the sample can be seen on the fracture resistance. Basically, the mixed criterion in this paper covers the gap between the tensile strength criterion and the toughness criterion. This model, while being simple, is very effective in determining the fracture stress of brittle materials. In the next section, the geometry of the problem and the solution method are explained, and in the Sect. 3, the results of the numerical analysis are given.

2 Finite Element Numerical Model, Mixed Criterion

As seen in Fig. 1, a sample with a circular cavity of radius Ro = 5 cm is considered. This sample is under compressive stress po. Here, the dimensions of the model are considered to be \(2 \times 2\;m^{2}\). The purpose of this article is to determine the pressure po so that tensile failure occurs in the wall of the cavity and along po. In this article, the mixed criterion is used to find the fracture pressure, which means that material tensile strength criterion and the fracture mechanics criterion must both be satisfied, i.e.:

$$ \left\{ \begin{gathered} K_{I} \left( {p_{0} ,c} \right) = K_{IC} \hfill \\ \sigma_{\theta } \left( {p_{0} ,c} \right) = \sigma_{t} \hfill \\ \end{gathered} \right. \to \left\{ \begin{gathered} p_{f} = p_{0} \hfill \\ c_{f} = c \hfill \\ \end{gathered} \right. $$
(1)

In this relationship, pf is the fracture pressure and cf is the fracture initiation length. \(\sigma_{\theta }\) is the maximum tangential stress along the imaginary crack with length c. \(\sigma_{t}\) is the tensile strength of the rock material and KIc is the fracture toughness. KI is the stress intensity factor at the crack tip with a length of c. To find these values, we draw the graphical form of the first and second criterion in one plot. The point of intersection of two criteria indicates the fracture pressure and the length of fracture initiation. Figure 2 shows an example of a mixed criterion.

After discretizing the geometry of the problem and applying the boundary conditions, the problem is solved by the finite element method and the fracture pressure is obtained. For this, a code is written in the MATLAB program.

Fig. 1.
figure 1

The geometry of 2D numerical model

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Fig. 2.
figure 2

The mixed criterion, \(R_{0} = 50\,mm\,,\,\sigma_{t} \, = \,14\,MPa\,,\,K_{Ic} = \,2.5\,MPa\,\sqrt m\)

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3 Numerical Analysis Results

For the characteristics of the materials presented in Table 1 and using the mixed criteria explained above, various numerical analysis are performed, the results of which are given below.

Table 1. The mechanical properties of model
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Figure 3 shows the fracture pressure (pf) obtained from the mixed criterion against the hole radius for different fracture toughnesses. In this figure, it can be seen that the rate of change of fracture pressure against the hole radius is high at first and then decreases. Therefore, the dependence of the fracture pressure on the cavity radius can be clearly seen. If the fracture pressure is normalized by the tensile strength of the material and plotted against the radius of the hole, Fig. 4 is obtained. The important point that can be obtained from this Fig. 4 is that for Ro > 70 mm, \(\frac{{p_{f} }}{{\sigma_{t} }} \simeq 1\), that is, for holes with a large radius, the tensile strength of the rock determines the fracture pressure, while for holes with a small radius, the fracture mechanics criterion plays an essential role in determining the fracture pressure of the samples. Thus, the mixed criterion captures the size effect well. Figures 3 and 4 show that with increasing toughness, the fracture pressure increases. But for Ro > 70 mm, the fracture pressure for different fracture toughnesses is almost the same, and therefore toughness has no effect on the fracture pressure of the rock samples, this is because for Ro > 70 mm as mentioned, it is the tensile strength of the rock that is effective in the fracture of the model (Fig. 3). As the radius of the hole decreases, the distance between the curves with different toughness increases, because for Ro < 70 mm, the fracture toughness is effective in the fracture of the model.

Figure 5 shows the fracture pressure against the hole radius for different values of tensile strength. In this figure, it can be seen that for Ro < 70 mm, the sensitivity of the model to the tensile strength of the rock is low, but for Ro > 70 mm, the fracture pressure increases slightly with the increase of tensile strength.

Figures 6 and 7 also show the fracture initiation length (cf) versus the hole radius. It can be seen in Fig. 6 that for samples with the same hole radius, the length of fracture initiation increases with the increase of fracture toughness. But Fig. 7 illustrates that, for samples with the same hole radius, as the tensile strength of the sample increases, the fracture pressure decreases. Also, Figs. 6 and 7 illustrate that as the radius of the hole increases, the initiation fracture length increases, the rate of increase of the initiation length of the fracture is high at first, but then it decreases.

Figure 8 shows the comparison between the results of the numerical model presented in this paper and the experiments conducted by Carter et al. (1992) on sandstone [7]. This Fig. Shows that there is a good agreement between the two sets of results.

Fig. 3.
figure 3

Fracture pressure versus cavity radius \(\left( {\sigma_{t} = 14\;MPa} \right)\)

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Fig. 4.
figure 4

Normalized fracture pressure obtained from the mixed criterion versus cavity radius

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Fig. 5.
figure 5

Fracture pressure versus cavity radius \(\left( {K_{Ic} \, = \,2.5\,MPa\sqrt m \,} \right)\)

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Fig. 6.
figure 6

Fracture length versus cavity radius \(\left( {\sigma_{t} \, = \,14\,MPa\,} \right)\)

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Fig. 7.
figure 7

Fracture length versus cavity radius \(\left( {K_{Ic} \, = \,2.5\,MPa\sqrt m \,} \right)\)

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Fig. 8.
figure 8

Comparison between finite element numerical analysis and laboratory results

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4 Conclusion

In this article, using a numerical model based on the finite element method and mixed tensile fracture criterion technique, the fracture pressure of rock samples with a hole in its centre was obtained. Using the technique of mixed fracture criterion, the effect of the size of the hole in the middle of the sample was observed on the fracture pressure. The obtained results show that when Ro > 70 mm, the tensile strength criterion plays a role in determining the fracture pressure of rock samples, while for Ro < 70 mm, the fracture mechanics criterion determines the fracture pressure of the model. Also, as the radius of the hole increases, the fracture pressure decreases, the fracture pressure reduction rate is high at first and then decreases, and the fracture pressure converges to the tensile strength of the rock.