A new mixed-mode fracture criterion of anisotropic rock
Introduction
As a new type of clean energy, shale gas has become one of the fastest growing and the greatest potential energy industries in the world, owning to its large reserve, stable production capacity and low cost. It is exploited mainly by hydraulic fracturing technology [1], [2], in which the fracturing medium with high pressure (including the water-based fracturing medium of resistance-reducing water and glue and the anthydrous-based fracturing medium of liquefied petroleum gas and supercritical carbon dioxide gas) is injected into the artificial cracks to connect with the natural cracks in the shale gas reservoir in order to form the fracture networks. The formation of fracture networks depends on the fracture mechanism revealed by fracture criterion. Therefore, it is of very importance to study the mixed-mode fracture criterion of anisotropic rock for predicting crack initiation and propagation and improving productivity by optimal design of the artificial cracks.
Currently, there are some mixed-mode fracture criteria available for isotropic rock based on stress, strain or energy: maximum circumferential stress criterion (σ-criterion) [3], maximum tangential principal stress (σ1-criterion) [4], maximum strain criterion (ε-criterion) [5], maximum energy release rate criterion (G-criterion) [6], [7], minimum strain energy density criterion (S-criterion) [8] and tensile-shear energy release rate ratio criterion (F-criterion) [9]. Although the σ-criterion [10], [11], [12], [13], ε-criterion [13], G-criterion [12], [14], [15], S-criterion [12], [13] and F-criterion [16] have been modified to predict the crack initiation angles of anisotropic rock under different loading conditions, by considering the fracture toughness on different plane, the former two criteria can only predict Mode I (tensile) fracture while the latter three criteria have difficulty in judging whether the fracture mechanism is Mode I or Mode II (shear). Therefore, there is imperative need to establish a suitable mixed-mode criterion to predict fracture mechanism for anisotropic rock. Our research group has established a tensile-shear stress intensity factor (SIF) ratio criterion (K-ratio criterion, [17]) to successfully predict both Mode I and Mode II fracture for isotropic rock. The criterion has the potential to be extended for anisotropic rock.
In this study, based on K-ratio of isotropic rock, a new mixed-mode fracture criterion (modified K-ratio criterion) of anisotropic rock was established by taking the ratio of SIFs and fracture toughness on arbitrary plane θ with respect to original crack plane, KI(θ)/KIC(θ) and KII(θ)/KIIC(θ) (the subscripts I and II mean Mode I and Mode II fracture respectively), as basic fracture parameters to predict crack initiation angle and fracture mechanism (Mode I or Mode II fracture). In order to determine KI(θ) and KII(θ) (on the arbitrary plane) related to KI(0) and KII(0) (on the original crack plane), new physical factors (YI and YII) were proposed to describe effects of both geometry and material parameters on KI(0) and KII(0), which could be calculated by finite element method. The predicted results of crack initiation angle and fracture mode were obtained by the modified K-ratio criterion and could be verified by the experimental results of semi-circular bend (SCB) pre-cracked shale specimens under different loads applied to the original crack plane (including pure tensile, pure shear, tension-shear and compression-shear loads).
Section snippets
New mixed-mode criterion of anisotropic rock
Since the brittle rock fracture is not always certainly to occur along its original plane,
it is necessary to firstly calculate Mode I and Mode II SIFs of arbitrary plane (KI(θ) and KII(θ)) which are related to SIFs of original crack plane, KI(0) and KII(0)) and then establish a new mixed-mode criterion of anisotropic rock.
Calculation of new physical factors
In nature, anisotropic rock usually has a geometrical micro-structural axis of symmetry caused by presence of foliation or bedding and exhibits transverse isotropy, such as shale used in this study. Semi-circular bend (SCB) shale specimen under three-point load was selected to verify the new mixed-mode K-ratio criterion owing to its simple geometry, convenient preparation of the original crack and loading set-up. New physical factors of SCB shale specimen must be given firstly to calculate KI
Prediction results of SCB shale specimens under different loads
In this study, the shale is exploited from outcrops of the Longmaxi in Sichuan province, southwest of China (Fig. 2) and a typical transverse isotropy rock. Table 3 lists its main mechanical parameters, which were measured by our uniaxial compression tests (E, ν, E’, ν’), SCB three-point bending test at ψ = 0° , ), and shear-box tests , ) , and calculated (G, G’) by means of E, ν, E’ and ν’. It is found that the measured fracture toughness are only different on the
Three-point bending test of SCB shale specimens
Three-point bending tests of SCB shale specimens (Fig. 22a) were adopted to verify the predicted results obtained by the new K-ratio criterion, where the original crack was prepared by a water-jet cutter of thickness t = 0.5 mm. The detail specimen sizes (a, s, r) were listed in Table 6 according to Table 4 (a/r, s/r). Each SCB specimen was placed onto a three-point bending fixture (Fig. 22a), where the three linear distributed loads were applied by an upper round bar and two below round bars
Conclusions
The following conclusions are obtained based on the theoretical and experimental analysis:
- (1)
A new mixed-model fracture criterion (modified K-ratio criterion) is established to predict both Mode I and Mode II fracture of anisotropic rock based on the ratio of Mode I or Mode II SIF to its fracture toughness (KI(θ)/KIC(θ) or KII(θ)/KIIC(θ)) on the arbitrary plane (θ), rather than the original crack plane (KI(0), KII(0)). It can be verified by test results of the SCB cracked shale specimens.
- (2)
Physical
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (No. 51874351, No.51474251), Center South University Innovation Foundation For Postgraduate (2019zzts874).
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