**3. Experimental results**

#### **3.1. Brazilian disc test size dependency**

The size dependency of the absolute size of the Brazilian disc test discs on the tensile strength is shown in Figure 2. Overall 138 Brazilian disc tests are undertaken for up to 6 sizes and four lithologies. The disc diameters, ranging from 30-84 mm, represent the sizes that are mostly tested in laboratories to determine the BDT tensile strength of rock samples. The results of the size dependency tests show no significant relationship between the sizes of the tested disc to

**Figure 1.** Sketches of the three tensile test methods. A: BDT side view, B: MF top view, C: MTT side view cross section

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(upper) and top view (lower).

AE data are sampled by a sampling rate of 10 kHz. The sensors are fixed using hot-melt adhesive to ensure best coupling characteristics. Pencil-break tests (Hsu-Nielsen source) and sensor pulsing runs (active acoustic emission by one sensor) are used to ensure good sensor

Minifrac experiments are carried out mainly on 40 mm cores with a borehole diameter of 4mm. Furthermore some 62 mm cores with a borehole of 6 mm diameter are tested. The samples are loaded axially up to 5 MPa to ensure that the packer mechanism is tight and seals off the borehole openings at the top and at the bottom. The borehole pressure was raised servo controlled with a fixed volume rate of 0.1 ml/s that results in a pressure rate of approximately 0.3 MPa/s. All MF tests are monitored by Acoustic Emissions with four sensors glued directly

All Brazilian disc tests are carried out following the ISRM suggested method [4] at a load rate of 200 N/s. Disc diameters used are 30, 40, 50, 62, 75 and 84 mm, whereas the length to diameter ratio (L/D) was constant at 0.5. All tests are monitored by one AE-sensor glued directly in the middle of the disc specimen. The size dependency is tested with discs from Ruhrsandstone,

The MTT tests are driven load controlled at a rate of 200 N/s that corresponds to a stress rate of 0.02 MPa/s. The axial force is applied from the top (Figure 1). MTT test samples are observed by up to 6 AE-Sensors glued directly to the specimen. The samples were overcored with 62 mm and 30 mm diameters where the overlapping height is 1/3 of the total sample height (Figure 1). The centralizing of the drills was achieved by using a former plate to adjust the sample before drilling. Despite assiduously arrangement the eccentricity of the overcoring was in the range of up to 3 mm due to the imprecise vertical guidance of a standard drilling machine. In order to test the influence of eccentricity we also prepared samples with an eccentricity of 14

The size dependency of the absolute size of the Brazilian disc test discs on the tensile strength is shown in Figure 2. Overall 138 Brazilian disc tests are undertaken for up to 6 sizes and four lithologies. The disc diameters, ranging from 30-84 mm, represent the sizes that are mostly tested in laboratories to determine the BDT tensile strength of rock samples. The results of the size dependency tests show no significant relationship between the sizes of the tested disc to

coupling of the sensor on the sample.

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*2.3.2. Brazilian Disc Tests (BDT) procedure*

*2.3.3. Modified Tension Test (MTT) procedure*

marble, rhyolite and limestone.

**3. Experimental results**

**3.1. Brazilian disc test size dependency**

mm.

*2.3.1. Hydraulic fracturing core experiments (MF) procedure*

to the samples and a fifth sensor placed at the incoming hydraulic line.

**Figure 1.** Sketches of the three tensile test methods. A: BDT side view, B: MF top view, C: MTT side view cross section (upper) and top view (lower).

its calculated tensile strength as long as the length to diameter ratio is held constant as suggested by the ISRM suggested method at a value of 0.5 [4]. There is a marginal tendency for the standard deviation of the tensile strength to decrease with increasing disc size.

Where *R* <sup>2</sup>

and *r* <sup>2</sup>

Ruhrsandstone

rhyolite

limestone

marble

andesite

Bebertal sandstone

MTT MTT eccentric

**Table 3.** Comparison of tensile strength out of three test methods.

scattering is the result of the material heterogeneity itself.

are the outer and inner radius, respectively (Figure 1). Mean values, standard

BDT 13.2 2.1 32 MF 19.0 3.0 10 MTT 5.8 1.0 3

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BDT 15.8 3.2 39 MF 20.1 5.5 5 MTT 4.9 1.4 2

BDT 8.2 2.2 36 MF 10.2 1.7 5 MTT 4.8 1.0 2

BDT 6.4 1.5 32 MF 7.8 1.3 4 MTT 4.3 1.2 2

BDT 14.6 4.5 23 MF 14.4 5.1 4 MTT 8.7 4.4 3

BDT 4.1 1.2 39 MF 4.3 2.0 3

> - 5E-3

1 2

2.4 1.0

One of the main observations is the very low tensile strength measured with the Modified Tension Test method. The MTT results mean values are in the range of 66 % down to 31 % of those obtained with the BDT. In addition to the low tensile strengths obtained by the MTT an eccentricity of the overcoring yields to an additional underestimation of the tensile strength values. The BDT and MF results seem to be more similar. The BDT results lie in the range of 70 % to 100 % of the MF tensile strength, so the MF test yields the highest tensile strengths and also to the highest standard deviations. All measurements are visualized in Figure 3. Doenstedt andesite and Flechtingen rhyolite tensile strengths have the highest standard deviations of the tested rock types. This variance is due to the high amount of natural joints that are assumed to have a different tensile strength with respect to the intact parts. Therefore the tensile strength

deviations and total number of tests for all three testmethods can be found in Table 3.

**Lithology Test method Mean [MPa] Std. dev. [MPa] N [-]**

**Figure 2.** Size dependency of the BDT disc size on the tensile strength for four lithologies. Circles represent the mean values, bars stand for the standard deviation.

#### **3.2. MF, BDT and MTT tensile strength results**

Three different methods for the determination of tensile strength are compared regarding their results. 201 Brazilian disc tests, 31 Minifrac tests and 15 Modified tension tests form the basis of the data evaluation, where *σ<sup>t</sup> BDT* , *σ<sup>t</sup> MF* and *σ<sup>t</sup> MTT* are the tensile strengths indexed by the used method. BDT tensile strengths are calculated as follows [4].

$$\text{'BDT: } \sigma\_t^{BDT} = \text{2P} / \pi Dt \tag{1}$$

Where P is the force at failure, D is the disc diameter and t the disc thickness.

For the MF tests, assuming the rocks to be nearly impermeable and therefore neglecting a relevant pore pressure influence the tensile strength is given directly by the breakdown pressure *Pb* [9].

$$\text{MF: } \sigma\_t^{MF} = P\_b \tag{2}$$

MTT tensile strengths are evaluated by the formula given by [5].

$$\text{MTT: } \sigma\_t^{MTT} = \mathbf{F}\_{\text{max}} \;/\; \mathbf{A}\_{\text{TZ}} = \mathbf{F}\_{\text{max}} \;/\left(\mathbf{R}^2 \boldsymbol{\pi} \cdot \mathbf{r} \;/^2 \boldsymbol{\pi}\right) \tag{3}$$


Where *R* <sup>2</sup> and *r* <sup>2</sup> are the outer and inner radius, respectively (Figure 1). Mean values, standard deviations and total number of tests for all three testmethods can be found in Table 3.

**Table 3.** Comparison of tensile strength out of three test methods.

its calculated tensile strength as long as the length to diameter ratio is held constant as suggested by the ISRM suggested method at a value of 0.5 [4]. There is a marginal tendency

**Figure 2.** Size dependency of the BDT disc size on the tensile strength for four lithologies. Circles represent the mean

Three different methods for the determination of tensile strength are compared regarding their results. 201 Brazilian disc tests, 31 Minifrac tests and 15 Modified tension tests form the basis

For the MF tests, assuming the rocks to be nearly impermeable and therefore neglecting a relevant pore pressure influence the tensile strength is given directly by the breakdown

*MTT* are the tensile strengths indexed by the

*BDT* =2*P* / *πDt* (1)

*MF* <sup>=</sup>*Pb* (2)

*π*) (3)

*π* - *r* <sup>2</sup>

*MF* and *σ<sup>t</sup>*

*BDT* , *σ<sup>t</sup>*

BDT: *σ<sup>t</sup>*

Where P is the force at failure, D is the disc diameter and t the disc thickness.

MF: *σ<sup>t</sup>*

*MTT* <sup>=</sup> *Fmax* / *ATZ* <sup>=</sup> *Fmax* / (*<sup>R</sup>* <sup>2</sup>

used method. BDT tensile strengths are calculated as follows [4].

MTT tensile strengths are evaluated by the formula given by [5].

MTT: *σ<sup>t</sup>*

values, bars stand for the standard deviation.

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of the data evaluation, where *σ<sup>t</sup>*

pressure *Pb* [9].

**3.2. MF, BDT and MTT tensile strength results**

for the standard deviation of the tensile strength to decrease with increasing disc size.

One of the main observations is the very low tensile strength measured with the Modified Tension Test method. The MTT results mean values are in the range of 66 % down to 31 % of those obtained with the BDT. In addition to the low tensile strengths obtained by the MTT an eccentricity of the overcoring yields to an additional underestimation of the tensile strength values. The BDT and MF results seem to be more similar. The BDT results lie in the range of 70 % to 100 % of the MF tensile strength, so the MF test yields the highest tensile strengths and also to the highest standard deviations. All measurements are visualized in Figure 3. Doenstedt andesite and Flechtingen rhyolite tensile strengths have the highest standard deviations of the tested rock types. This variance is due to the high amount of natural joints that are assumed to have a different tensile strength with respect to the intact parts. Therefore the tensile strength scattering is the result of the material heterogeneity itself.

a uniform tensile stress distribution in the annulus of the test samples. It is arguably if this model is the right tool for modeling a tensile stress distribution in rock samples prior to failure. A simple linear elastic 3D FEM model reveals tensile stress concentrations at the edges of the rims in the sample (Figure 5). Fractures may be initiate there at relative low axial forces.

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During preparation of the samples it becomes obvious that exact centralization of the inner overcoring is not always given. Two Bebertal sandstone samples were prepared with a eccentricity of 14 mm resulting in a minimum rim width of 2 mm instead of 16 mm for a perfectly centralized sample. The average eccentricity of our samples is in the range of up to 3 mm. Tensile stress redistribution due to eccentricity is modeled as well and can easily double

**Figure 5.** Slice through a linear elastic 3D FEM model of MTT tensile test. Values of axial stress are given in MPa where negative values stand for tensile stress. Left model represents a perfectly centralized sample. Right model shows the

247 tensile strength test results of BDT, MF and MTT tests vary considerable within one lithology (Figure 6). Therefore it is not trivial to give a reliable prediction of the tensile strength parameter. Results of the BDT tests show no significant variation with respect to the specimen size, as long as the aspect ratio is held constant. Nevertheless the tensile strength data scattering is high, so that it may obscure existing trends. Acoustic Emission evaluation shows that during the BDT multiple fracturing mechanisms are present. Before total fracturing of the sample by a tensile rupture there is a high amount of AE activity that is most likely related to compres‐ sional failure at the top and bottom of the disc. Beside this, compressional stress concentrations and the inhomogeneous tensile stress distribution may lead to tensile cracks before peak load.

the tensile stress in the thinner rim of the annulus (Figure 5).

stress distribution for a eccentricity of 6 mm towards the left edge.

**5. Discussion**

**Figure 3.** Results of all tensile strength test. BDT: Brazilian Disc Test, MF: Minifrac, MTT: Modified Tension Test. Hollow circle in Bebertal sandstone MTT tests represents two results of the highly eccentrical MTT tests.

#### **3.3. Acoustic Emissions results**

Acoustic emission data obtained during the tests give rough insights into the failure processes. It is obvious that all tests end with a spalling of the specimens in parts due to a complete tensile failure. Simple AE count analysis show that the BDT is accompanied with an immense hit-rate long before total failure in comparison to the relatively quiet pre-failure phases of the MF and MTT tests. In good agreement with theoretical considerations of the stress distribution in the Brazil disc [1] these events are most likely due to compressional failure at the top and bottom of the disc, accompanied with crack propagation and coalescence before peak load (Figure 4).

**Figure 4.** AE hits per 0.5 sec., BDT left, MF middle and MTT right showing the huge difference in AE hits before total failure of the sample.

#### **4. Numerical model**

We investigate the effect of eccentricity of the overcoring for the MTT samples by a numerical simulation. A finite element study that has been performed by Plinninger et al. [10] that shows a uniform tensile stress distribution in the annulus of the test samples. It is arguably if this model is the right tool for modeling a tensile stress distribution in rock samples prior to failure. A simple linear elastic 3D FEM model reveals tensile stress concentrations at the edges of the rims in the sample (Figure 5). Fractures may be initiate there at relative low axial forces.

During preparation of the samples it becomes obvious that exact centralization of the inner overcoring is not always given. Two Bebertal sandstone samples were prepared with a eccentricity of 14 mm resulting in a minimum rim width of 2 mm instead of 16 mm for a perfectly centralized sample. The average eccentricity of our samples is in the range of up to 3 mm. Tensile stress redistribution due to eccentricity is modeled as well and can easily double the tensile stress in the thinner rim of the annulus (Figure 5).

**Figure 5.** Slice through a linear elastic 3D FEM model of MTT tensile test. Values of axial stress are given in MPa where negative values stand for tensile stress. Left model represents a perfectly centralized sample. Right model shows the stress distribution for a eccentricity of 6 mm towards the left edge.
