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Scanning Electron Microscopy (SEM) test on 30 pieces of fractured granite has been researched by using S250MK III SEM under triaxial compression of different temperature (25~1000°C) and confining pressure (0~40 MPa). Research results show that (

In 1982, French mathematician Mandelbrot [

In the aspect of studying the variation of FD of rock fracture with the change of confining pressure, Ni et al. [

In the aspect of FD of rock fracture with temperature, Xu et al. [

Based on above research results, the FD of rock fracture mostly reflected the state of uniaxial compression under high temperature and triaxial compression at normal temperature; it is rarely reported in the condition of triaxial compression under high temperature. But in the development of geothermal resources, nuclear waste treatment, underground energy reserves, and other projects, rock is always in the coupling environment of temperature and confining pressure, so the study of deformation and failure mechanism of rock under different temperature and confining pressures has become a hot and frontier issue in the study of rock mechanics [

S250MKIII was used for SEM test, which is in the Analysis and Testing Center of China University of Mining and Technology. Its technical indicators are as follows: resolution is 60 Å, amplification factor is 200 thousand times, and accelerating voltage is 40 KV. AN10000 energy spectrometer and WDX-2A spectrometer are the main accessories. The application is quantitative micro analysis of solid material, surface structure, and element distribution.

This test is mainly on the SEM analysis of granite under triaxial compression after high temperature. Firstly, cut the fracture of sample into appropriate size; clean the sample with acetone or alcohol to prevent illusion or with ultrasonic cleaning oscillator if necessary. Secondly, because rock minerals are nonconducting samples, charge accumulation can be generated under the action of electron beam; this will affect the trajectory of two electron motions of incident electron beam spot and sample emission; then the image quality will decline. Therefore, conductive layer should be sprayed before observation; usually using gold or silver or carbon film with the secondary electron emission coefficient is higher; the film thickness is controlled at about 20

The test has six temperature points of 25 (normal temperature), 200, 400, 600, 800, and 1000°C and five confining pressures of 0 (uniaxial compression), 10, 20, 30, and 40 MPa; it consists of 30 samples; each rock sample scans 8–10 pieces of images.

In fractal theory, the common FD has capacity dimension, information dimension, correlation dimension, general dimension, self-similar dimension, and so on. FD of Cantor set, Koch curve, Sierpinski set, and Menger sponge can be obtained by using self-similar dimension, because they have strict self-similarity, which can be estimated according to the generating element and structural rules. Fractal in nature is statistical self-similarity and unable to determine the generation of a natural fractal; therefore, in the study of rock mechanics, the box-counting dimension is widely used by many researchers, mainly because the box-counting dimension can be a good description and characterization of crack morphology of irregular characteristics, and the mathematical calculation and estimation are relatively easy. In this paper, FD of rock fracture is calculated by box-counting method. The box-counting dimension is calculated using cover method. Because the crack of rock has the characteristics of fractal distribution, it can be regarded as a fractal set of

Because the fractal set of

Based on above basic theory, FD of rock fracture can be obtained by SEM images. Calculation step includes importing image, processing image, calculating area, covering number of statistics box, calculating box-counting dimension, and deriving calculation results. Figure

Calculation software of fractal dimension.

FD of 8 pieces of SEM images for each sample is calculated by box-counting method; the average value is taken as the final value in each case. Select 30 images closest to the average FD in 200 pieces of SEM images, as shown in Figure

SEM image of rock fracture under different temperature and confining pressure.

0 MPa (uniaxial compression)

10 MPa

20 MPa

30 MPa

40 MPa

Through calculation, the average value of FD under various temperature and confining pressure is shown in Table

Average value of fractal dimension of rock fracture under different temperature and confining pressure.

Confining pressure MPa | Temperature °C | |||||
---|---|---|---|---|---|---|

25 | 200 | 400 | 600 | 800 | 1000 | |

Fractal dimension | ||||||

0 | 1.905 | 1.894 | 1.965 | 1.973 | 1.954 | 1.772 |

10 | 1.937 | 1.947 | 1.968 | 1.974 | 1.943 | 1.853 |

20 | 1.915 | 1.957 | 1.955 | 1.974 | 1.949 | 1.940 |

30 | 1.846 | 1.979 | 1.932 | 1.953 | 1.953 | 1.939 |

40 | 1.935 | 1.976 | 1.975 | 1.915 | 1.941 | 1.946 |

According to the data in Table

Relationship between fractal dimension and temperature.

0 MPa (uniaxial compression)

10 MPa

20 MPa

30 MPa

40 MPa

The change of FD with temperature is discussed, respectively, under various confining pressure.

The change of FD with temperature can be divided into two stages under 0 MPa (uniaxial compression). In the first stage, FD of 25~600°C increases with the rising temperature, which is increased from 1.905 of 25°C to 1.973 of 600°C, with an increase of 3.57%; in the second stage, FD of 600~1000°C is rapidly decreased, which is decreased from 1.973 of 600°C to 1.775 of 1000°C, with a decline of 10.19%, reaching the maximum value at 600°C. FD is in accordance with cubic polynomial fitting curve with the increase of temperature. Fitting formula is

When confining pressure is 10 MPa, the variation of FD with temperature is the same as uniaxial compression. It is also in line with cubic polynomial fitting curve. Fitting formula is

When confining pressure is 20 MPa, FD can also be divided into two stages as the temperature increases. The first stage, FD of 25~600°C, rises from 1.937 of 25°C to 1.974 of 600°C, with an increase of 1.91%; the second stage, FD of 600~1000°C drops from 1.974 of 600°C to 1.853 of 1000°C, with a decline of 6.13%, reaching maximum value at 600°C. Fitting formula is

When confining pressure is 30 MPa, FD is fluctuating with temperature. The change of FD can still be divided into two stages except 200°C. It increases from 1.846 of 25°C to 1.953 of 600°C, with 5.79% increase, and drops from 1.953 of 600°C to 1.939 of 1000°C, compared to 0~20 MPa with a smaller decline, only 0.72%. There is no better fitting relationship between FD and temperature. It can be seen that when confining pressure is 30 MPa, the influence of temperature on FD is reduced.

When confining pressure is 40 MPa, FD has no better fitting curve with temperature. It is worth noting that it drops from 1.975 of 400°C to 1.915 of 600°C and then increased to 1.946 of 1000°C, reaching the minimum at 600°C. The change of FD in 400~1000°C is just opposite to the same range of temperature in 0~30 MPa, the increase of confining pressure weakens the influence of temperature on FD, and the confining pressure plays a key role in this case.

At the same time, FD of rock fracture under various confining pressures is plotted in Figure

Relationship between fractal dimension and temperature under confining pressure at all levels.

It can be educed that, in the test range of temperature (25~1000°C) and confining pressure (0~40 MPa), the variation of FD of SEM images as temperature rises is closely related to confining pressure; it can be divided into two categories generally.

In the first category, when confining pressure is in 0~30 MPa, the change of FD with temperature can be divided into two stages. FD first increases and then decreases with the rise of temperature, reaching maximum value at 600°C (except 30 MPa). It can be inferred that 600°C is a critical temperature. FD is in accordance with cubic polynomial fitting curve with temperature, but with the increase of confining pressure, the correlation coefficient of fitting curve decreases; there is no better fitting curve at 30 MPa.

In the second category, when confining pressure is in 30~40 MPa, the change of FD of 400~1000°C is just contrary to the law of 0~30 MPa; the influence of confining pressure plays a key role on FD in this case.

According to the data in Table

Relationship between fractal dimension and confining pressure.

25°C (normal temperature)

200°C

400°C

600°C

800°C

1000°C

The change of FD of rock fracture with confining pressure is discussed, respectively, under different temperature.

FD fluctuates with confining pressure at 25°C; it can be divided into three stages. In the first stage, it increases from 1.905 of 0 MPa to 1.937 of 10 MPa, with an increase of 1.68%; in the second stage, it decreases from 1.937 of 10 MPa to 1.846 of 30 MPa, with a decline of 4.69%; since then, as the confining pressure increases to 40 MPa, it increases to 1.935, with an increase of 4.82.

When temperature is 200°C, FD increases from 1.894 of 0 MPa to 1.979 of 30 MPa, with an increase of 4.49%; then it decreases slightly, down to 1.976 of 40 MPa. At this temperature, the least square method is used to fit the test curve; the FD is better to satisfy exponential growth with the increase of confining pressure. Fitting formula is

When temperature is 400°C, the variation of FD with confining pressure is consistent with 25°C, which can also be divided into three stages. From 0 to 10 MPa, it increases from 1.965 to 1.968, with an increase of 0.15%; from 10 to 30 MPa, it decreases from 1.968 to 1.932, with a decline of 1.83%; thereafter, it increases to 1.975 of 40 MPa, with an increase of 2.23%.

When temperature is 600°C, FD can be divided into two stages with the rise of confining pressure. In the first stage, there is no variation of FD from 0 to 20 MPa; in the second stage, it decreases from 1.974 of 20 MPa to 1.915 of 40 MPa, with a decline of 2.99%. FD decreases with exponential function with confining pressure, fitting formula is

When temperature is 800°C, FD can be divided into three stages with confining pressure, but the change trend is just opposite to 25°C and 400°C. From 0 to 10 MPa, it decreases from 1.954 to 1.943, with a decline of 0.56%, and then increases to 1.953 of 30 MPa, with an increase of 0.52%; thereafter, it is down to 1.941 of 40 MPa, with a decline of 0.61%. As can be seen, FD changes with fluctuation of confining pressure at 800°C; the overall change is smaller. It can be approximated that FD of 800°C is mainly controlled by temperature, and the confining pressure has no obvious effect on it.

When temperature is 1000°C, FD can be divided into two stages. From 0 to 20 MPa, it increases from 1.772 to 1.940, with a decline of 9.48%; it basically has no change from 20 to 40 MPa. FD increases with exponential curve with confining pressure, fitting formula is

In order to carry on the whole analysis, draw the change of FD of rock fracture with confining pressure at all kinds of temperature in the same figure, as shown in Figure

Relationship between fractal dimension and confining pressure under temperature at all levels.

It can be seen that, in the test range of temperature (25~1000°C) and confining pressure (0~40 MPa), the variation of FD of SEM images with confining pressure is closely related to temperature, which can be divided into the following three categories.

In the first category, FD has volatility with the change of confining pressure at 25°C, 400°C, and 800°C; it can be divided into three stages.

In the second category, FD increases exponentially with confining pressure at 200°C and 1000°C. Fitting curve of 200°C is

In the third category, FD decreases exponentially with confining pressure at 600°C. Fitting curve is

It can also be seen by the overall figure that the influence of confining pressure on FD is weakened in 400~800°C; FD is mainly controlled by temperature in this section.

Through this research, when confining pressure is in the range of 0~30 MPa, FD first increases and then decreases with the rise of temperature, reaching the maximum value at 600°C (except 30 MPa); it can be considered that 600°C is the critical temperature of granite. The main reasons are as follows. When temperature is below 600°C, the pore water, interlayer water, and structure water of rock is gradually evaporated; some of the original crack is closured due to the thermal expansion induced by the increase of temperature, and the friction between molecules is enhanced; all of these factors lead to the enhancement of mechanical properties of rock, resulting in more complex cracks when the rock is destroyed, so FD increases with the rise of temperature below 600°C. When temperature exceeds 600°C, thermal stress generated by temperature leads to thermal cracking of rock. By diffraction information of granite [

The study found that when discussing the influence of temperature on FD under various confining pressure, 30 MPa is a watershed. With the increase of confining pressure, rock occurs from brittle to plastic transition. From this research it can be initially considered that 30 MPa is the critical confining pressure of granite from brittle to plastic transition; this result is consistent with Huang et al. [

At the same time, this study also obtained that FD of rock fracture has fluctuation with the increase of confining pressure at 25°C (normal temperature), which can be divided into three stages, reaching the minimum value at 30 MPa. This result is inconsistent with previous research results, such as Chen et al. [

In the test range of temperature (25~1000°C) and confining pressure (0~40 MPa), the variation of FD of SEM images with temperature is closely related to confining pressure; it can be generally divided into two categories. In the first category, when confining pressure is in 0~30 MPa, FD increases with cubic polynomial fitting curve with temperature, but correlation coefficient of fitting curve decreases with the rise of confining pressure; there is no better fitting curve at 30 MPa. In the second category, when confining pressure is in 30~40 MPa, FD has volatility with the change of temperature; the change law of 400~1000°C is just opposite to the same range of temperature of 0~30 MP; confining pressure plays a leading role in this case.

In the test range, the variation of FD with confining pressure is closely related to temperature, which can be divided into three categories. In the first category, FD has volatility with the change of confining pressure at 25°C, 400°C, and 800°C. In the second category, it increases exponentially with confining pressure at 200°C and 1000°C. In the third category, it decreases exponentially with confining pressure at 600°C.

Through this study, it is obtained that 600°C and 30 MPa are the critical temperature and confining pressure of granite. When temperature exceeds 600°C and confining pressure exceeds 30 MPa, rock occurs from brittle to ductile transition; the influence of temperature on FD is weakened by the increase of confining pressure.

It is worth pointing out that this paper considers triaxial compression test of rock after high temperature cooling, while the mechanical properties of rock differ from real time high temperature and high temperature cooling. The variation of FD of rock fracture with the change of temperature and confining pressure under real time high temperature also has a good value of scientific research; this part will be further studied through the follow-up experiment.

Scanning Electron Microscopy

Fractal dimension.

The authors declare that they have no competing interests.

This work had been financially supported by the National Natural Science Foundation of China (no. 11202108, 51604260), the Opening Fund of State Key Laboratory for Geomechanics and Deep Underground Engineering of China University of Mining and Technology (no. SKLGDUEK1204), the Natural Science Foundation of Jiangsu Province of China (no. BK20160416), the China Postdoctoral Science Foundation (no. 2016T90519), and the Jiangsu Postdoctoral Sustentation Fund of China (no. 1501085B).