**2. Introduction of RFPA2D2.0-Flow**

applications, most of the hydraulic fracturing operations in the oil and gas fields are performed through casing, and, the regular or complex bedding structures likely exist in the rock mass formed in the course of rock formation and tectonic movement, so the research of the effect of perforation and bedding angle and modulus contrasts of rock and bedding in heterogeneous rocks under hydraulic fracturing is necessary. It can not only make fracturing decision-making more scientific, but reduce the fracturing cost and improve the fracturing efficiency, and has great theoretical significance and practical value on the perforation parameters optimization

At present, the perforation parameters are controllable which can be realized easily in practice. Since the hydraulic fracturing technology appeared, many experts have made various researches about the influence of perforation parameters on fracture evolutions under hydraulic fracturing. In [1], Daneshy et al. studied the hydraulic fracturing through perforation in 1973 and found that breakdown pressures of hydraulic fractures would decrease as the number of perforations increased, moreover, the existence of the casing and the perforations had little influence on the direction of the created fracture, which is perpendicular to the minimum principal stress. In [2], Weng et al. studied the hydraulic fracture initiation and propagation from deviated wellbores in 1993, investigated the interaction and link-up of the starter fractures initiated from perforations and the turning of the linked fracture and estab‐ lished a criterion that correlates fracture link-up to stresses and wellbore parameters. In [3], Zhang et al. used two-dimensional model to simulate the initiation and growth of hydraulic fractures in 2011 and developed a dimensionless parameter that is shown to characterise nearwellbore reorientation and curving of hydraulic fractures driven by viscous fluid. In [4], Zhang et al. employed three-dimensional finite element model together with the tensile criterion of rock materials in 2004, investigated that perforation density and perforation angle are the most important parameters controlling the formation fracturing pressure, but the influences of perforation diameter and perforation length are much slighter. In [5], Jiang et al. studied the fracture propagation mechanism of hydraulic fracturing through the experiment in 2009, and the results showed that the turning fracture can be generated by using oriented perforation fracturing technology, and with the increase of azimuth of oriented perforating, the breakdown

Few studies have been carried on for fracture evolutions on heterogeneous rocks with different bedding angles under hydraulic fracturing at present stage. In [6], Bruno and Nakagawa studied fracture propagation path in non-uniform pore pressure field by test method in 1991, and proved that the fracture is influenced by both pore pressure magni‐ tude on a local scale around the crack tip and the orientation and distribution of pore pressure gradient on a global scale. In [7], Li et al. simulated the experiment of hydraul‐ ic fracturing in non-uniform pore pressure field in 2005, and the results are well agreea‐ ble to that of Bruno and Nakagawa's experiments. In 2010, Abbass et al.'s study on Brazilian tensile text of sandstone in [8-9] showed that the breakdown pressure and fracture pattern are considerably affected by the bedding orientation and larger fracture length

design and hydraulic fracturing construction of bedding rockmass.

492 Effective and Sustainable Hydraulic Fracturing

pressure and turning distance are both growing.

correlating with higher strength and applied energy.

RFPA is a numerical experiment tool basing on the realistic failure process analysis method, which can simulate the gradual damage of materials. Its calculation method bases on finite element and statistical damage theory. RFPA considers both heterogeneity of materials and randomness of defect distribution, and puts the statistical distribution hypothesis of these material properties into the numerical calculation method (finite element method) to break the elements which satisfy the strength criterion. The material properties of each element follow Weibull distribution and are different from each other, and the element will fail if its stress reaches the failure strength, moreover, the number of fail elements will increase, which will be connected to each other and form fractures, as the load increases, so that the numerical simulation of heterogeneous material failure process can be realized. RFPA transforms the complex macroscopic nonlinear problem into simple mesoscopic linear problem by consider‐ ing the heterogeneous characteristics of material and the complicated non-continuum me‐ chanics problems into simple continuum mechanics problems by introducing the mathematics continuous and physical discontinuous concept, making the calculation results closer to the actual situation.

In mining and civil engineering projects, the re-distribution of the stress field during the excavation of tunnels and underground chambers leads to the formation of new fractures. The flow and transport behaviour within developing fractures are dramatically different from those in rocks with existing fractures under the same loading, therefore, the permeability of rocks changes dramatically in the process of damage evolution in fracture rocks. RFPA2D2.0- Flow is the software considering the effects of the extension of existing fractures, the initiation of new fractures, the coupled effects of flow, stress and damage on the extension of existing/new fractures, and the permeability change due to damage evolution of the rocks, and is based on the theories of fluid-saturated porous media and damage mechanics. Flow-stressdamage (FSD) coupling model for heterogeneous rocks that takes into account the growth of existing fractures and the formation of new fractures is established herein. In [10-16], RFPA2D2.0-Flow bases on the following five basic assumptions:


$$\varphi(a) = \frac{m}{a\_0} (\frac{a}{a\_0})^{m-1} e^{-(\frac{a}{a\_0})^m} \tag{1}$$

in figure 2, there is a well in the centre of model with radius 0.064m, two perforations approx 0.002m wide and 0.03m long cut into the well to provide initial direction for hydraulic fracture. Casing is in the non perforation area of the well with its strength and stiffness higher and permeability less than rock, so the initiation of fracture only occurs on perforation tip, which is in line with the actual engineering situation. As shown in figure 3, the well's radius is 0.032m and the space of two adjacent parallel beddings 0.04m. In figure 2 and figure 3, the confining pressure of the modelσ*<sup>H</sup>* (in horizontal direction) andσ*<sup>h</sup>* (in vertical direction )are 15MPa and 10MPa respectively with the initial pressure of 12MPa applied in the well and an incremental pressure of 0.1MPa maintained. The mechanical parameters of rock and bedding materials adopted in this simulation are listed in table 1 and table 2. The change of the values of elastic modulus and uniaxial compressive strength of bedding material are listed in table 3, and the value in brackets refers to the ratio of bedding and rock material. Moreover, the bedding angle

Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles…

with other mechanical parameters are in

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495

**Casing**

of seven different bedding materials models is 60o

**Figure 2.** Schematic diagram and RFPA model diagram

**Figure 3.** Figure 3 Schematic diagram and RFPA model diagram

line with table 2.

In this formula, *a* represents the mechanical property parameters of material (rock) elements at meso-scale; *a0* represents the statistical average of mechanical property parameters *a*; *m* is called homogeneity index, and higher *m* value means more homogene‐ ity material; *φ*(*a*) defines the statistical distribution density of mechanical property parameters *a*. Weibull's distribution for mechanical properties of materials with different homogeneity indexes *m* is shown in figure 1.

**Figure 1.** Weibull's distribution for mechanical properties of materials with different homogeneity indexes *m*

#### **3. Model set**

Numerical model is divided into three groups and each of them contains seven models, the fracturing process of rock specimens with different perforation angles, different bedding angles and different bedding materials under increasing hydraulic pressure and constant confining pressure are simulated. Perforation angle α is the angle between perforation and the maximum principal stress direction (horizontal direction), bedding angle α is the angle between bedding and the maximum principal stress direction (horizontal direction), and both of perforation and bedding angles chosen in the simulation are 0o , 15o , 30o , 45o , 60o , 75o and 90o respectively. The geometry of 2D rock model is 0.64m×0.64m and has been discretized into a 320×320(6400 elements) mesh, and the model is calculated by using plane strain. As shown in figure 2, there is a well in the centre of model with radius 0.064m, two perforations approx 0.002m wide and 0.03m long cut into the well to provide initial direction for hydraulic fracture. Casing is in the non perforation area of the well with its strength and stiffness higher and permeability less than rock, so the initiation of fracture only occurs on perforation tip, which is in line with the actual engineering situation. As shown in figure 3, the well's radius is 0.032m and the space of two adjacent parallel beddings 0.04m. In figure 2 and figure 3, the confining pressure of the modelσ*<sup>H</sup>* (in horizontal direction) andσ*<sup>h</sup>* (in vertical direction )are 15MPa and 10MPa respectively with the initial pressure of 12MPa applied in the well and an incremental pressure of 0.1MPa maintained. The mechanical parameters of rock and bedding materials adopted in this simulation are listed in table 1 and table 2. The change of the values of elastic modulus and uniaxial compressive strength of bedding material are listed in table 3, and the value in brackets refers to the ratio of bedding and rock material. Moreover, the bedding angle of seven different bedding materials models is 60o with other mechanical parameters are in line with table 2.

**Figure 2.** Schematic diagram and RFPA model diagram

**4.** In elastic state, stress-permeability coefficient relationship of material is described by

**5.** The mechanical parameters (such as uniaxial compressive strength *fc* and elastic modulus *Ec*) of material at meso-scale (elements) are endowed by the following Weibull distribu‐

In this formula, *a* represents the mechanical property parameters of material (rock) elements at meso-scale; *a0* represents the statistical average of mechanical property parameters *a*; *m* is called homogeneity index, and higher *m* value means more homogene‐ ity material; *φ*(*a*) defines the statistical distribution density of mechanical property parameters *a*. Weibull's distribution for mechanical properties of materials with different

0 0 () ( )

j

*m a <sup>m</sup> <sup>a</sup> a e a a*

**Figure 1.** Weibull's distribution for mechanical properties of materials with different homogeneity indexes *m*

of perforation and bedding angles chosen in the simulation are 0o

Numerical model is divided into three groups and each of them contains seven models, the fracturing process of rock specimens with different perforation angles, different bedding angles and different bedding materials under increasing hydraulic pressure and constant confining pressure are simulated. Perforation angle α is the angle between perforation and the maximum principal stress direction (horizontal direction), bedding angle α is the angle between bedding and the maximum principal stress direction (horizontal direction), and both

 respectively. The geometry of 2D rock model is 0.64m×0.64m and has been discretized into a 320×320(6400 elements) mesh, and the model is calculated by using plane strain. As shown

0 ( ) <sup>1</sup>

*a <sup>m</sup>*


, 15o , 30o , 45o , 60o

, 75o and

negative exponential function;

494 Effective and Sustainable Hydraulic Fracturing

homogeneity indexes *m* is shown in figure 1.

tion:

**3. Model set**

90o

**Figure 3.** Figure 3 Schematic diagram and RFPA model diagram


**4. Simulation results**

(15<sup>o</sup> )

(15<sup>o</sup> )

**4.1. The effect of perforation angles**

The initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different perforation angles under constant confining pressure and increasing hydraulic pressure are simulated. The results reflect the damage evolution process of rock specimen, which causes the macroscopic damage by microscopic under hydraulic fracturing and is consistent with the experimental result in [4]. Pore pressure and the minimum principal stress distribution of specimens with different perforation angles which achieved by numerical simulation are shown from figure 4 to figure 10. The comparison of the numerical simulation

Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles…

same ground stress difference (5MPa) is shown in figure 11, and the values of the initiation

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup>

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup>

**Figure 4.** Pore pressure and the minimum principal stress distribution in fracture evolution process (0o)

Figure 5 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 5 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 5.** Pore pressure and the minimum principal stress distribution in fracture evolution process (15o)

) and under the

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497

)

)

result and the experimental result which has the same perforation angle (60o

and the breakdown pressure are shown in figure 12.

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

**Table 1.** Rock material mechanical parameter


**Table 2.** Bedding material mechanical parameter


**Table 3.** Change of elastic modulus and uniaxial compressive strength values of bedding material
