Rock Mech Rock Eng DOI 10.1007/s00603-015-0762-6

ORIGINAL PAPER

Simulation on Vibration Characteristics of Fractured Rock Siqi Li1 • Tie Yan1 • Wei Li1 • Fuqing Bi2

Received: 26 March 2014 / Accepted: 1 May 2015  Springer-Verlag Wien 2015

Abstract Modal analysis theory of rock is proposed and the modeling of vibration characteristics of fractured rock is undertaken in this study. The modeling includes two aspects, namely, the natural frequency of rock with a single fracture and the crack expansion energy of rock with multiple fractures. Also, the results of numerical analysis are presented. Four main control parameters are considered, including the material properties, crack size, crack trend and the number of cracks. It is confirmed that the natural frequency of rock will be reduced by the cracks in it. The expansion energy of a single crack is inversely proportional to the number of such cracks in the rock. Namely, the more the similar cracks are, the less the energy required for a single crack expansion is and the smaller the excitation frequency needed for rock resonance is. The vibration characteristics of fractured rock are validated by numerical analysis. The natural frequency of rock increases with the increase of elastic modulus, and decreases with the increase of the angle, the length, the width and the number of cracks. Keywords Natural frequency of rock  Resonant frequency of rock  Crack size  Crack trend  The number of cracks

List of symbols M Mass matrix of rock K Stiffness matrix of rock x Displacement array of rock € x Acceleration array of rock FðtÞ Excitation force array on the rock U Free response amplitude array of rock xi The i order natural frequency k Stiffness of rock m Mass of rock L Length of rock E Elastic modulus of rock A Cross-sectional area of rock r Fracture strength a Half length of fracture cs Density of surface energy pi The number of rock unit which contains i class crack Eio Expansion energy of crack x_ Vibration velocity of rock c Damping coefficient jHðxi Þj Amplitude–frequency characteristics of rock n The number of times of vibration T Harmonic force on the surface of rock f ðtÞ Excitation force

& Wei Li [email protected]

1 Introduction

1

Institute of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, Heilongjiang, China

2

The Fifth Oil Production Plant of Daqing Oilfield, Daqing 163513, Heilongjiang, China

The idea of utilizing high-frequency vibration impact energy to drill rock formation has been proposed in recent years. It is a new technology, still at laboratory stage, which has great potential to improve the rock breaking

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efficiency. The main idea of this technology is that the bit, while rotating, applies a dynamic impacting force with an adjustable high frequency to the rock so as to create resonance conditions for the rock drilled. In fact, the realization of high-frequency vibration impact drilling technology is closely related to the dynamic characteristics of rock structure, such as natural frequency, resonant frequency and vibration response. However, the cracks in the rock will cause dynamic characteristics of rock change. Therefore, the study on the vibration characteristics of fractured rock is a necessary foundation to implement the high-frequency vibration impact drilling technology (Liu et al. 2013; Li et al. 2011). A series of studies on vibration frequency of beam structure have been done by a number of researchers (Chinchalkar 2001; Hu et al. 2007; Kevin and Zhang 2000). Yang et al. (2013a) used the theoretical method to determine the vibrating frequency of skewed girder bridges considering shear deformation effect. Yang et al. (2013b) proposed an approximate calculation method of natural frequency of cracked beam based on the nonlinear contact theory. Wang and Jiang (2003) calculated and analyzed the relationship between the change of the first three natural frequencies of three typical beams and the location and depth of cracks. In addition, some scholars have studied on the vibration frequency of other structures (Jia et al. 2005; Adhikari 2006). Xiang et al. (2012) calculated the natural frequency of rotating functionally graded cylindrical shells by Love’s first approximation theory. Zhou et al. (2012) analyzed the natural frequency of nonlinear transverse vibrations for pipes conveying fluid by multiple scale method. Based on the method of ultrasonic normal incidence, Zou et al. (2012) investigated the relationship between reflection coefficients and resonance frequencies under the boundary condition of an adhesive layer with a spring model. This paper is focused on the vibration characteristics of fractured rock, presents the results of the numerical analysis. The aim is to investigate the influence of the parameters, such as material properties, crack size, crack trend and the number of cracks, on the vibration frequency. The concept of rock modal is proposed in Sect. 2 and the modeling of vibration characteristics of fractured rock is presented in Sect. 3. In addition, the influence of parameters on the vibration frequency is discussed in Sect. 4.

The basic equation of undamped modal analysis of rock structure is a classical solution of eigenvalue. The differential equation of vibration is given as M€ x þ Kx ¼ FðtÞ

ð1Þ

Equation (1) is a vector equation. For the free vibration system, FðtÞ ¼ 0, it can be rewritten as, M€ x þ Kx ¼ 0

ð2Þ

Then, the particular solution is obtained, x ¼ Ueixt   K  x2 M U ¼ 0

ð3Þ ð4Þ

The necessary and sufficient condition of the equation with non-zero solution is that its coefficient matrix determinant is equal to zero.   K  x 2 M  ¼ 0 ð5Þ Equation (5) is the characteristic equation, and it is n times algebraic equation of x2 . There are n different positive roots of the equation, and make them in an ascending sort order as follows: 0\x1 \x2 \    \xn

ð6Þ

Take xi ði ¼ 1; 2; . . .; nÞ into Eq. (4), the corresponding natural mode of rock vibration will be got, Ui ¼ ½/1i ; /2i ; . . .; /ni T

ð7Þ

Each order modal of rock in a certain range of incentive frequency can be obtained through the modal analysis and the actual vibration response can be determined considering the effect of external or internal various vibration sources.

3 Models of Vibration Characteristics of Fractured Rock There are a lot of cracks in the rock which affect the vibration characteristics of rock. They are the preconditions for rock breaking. To simplify the analysis, it is considered as the scalar case to analyze the models of vibration characteristics of fractured rock. 3.1 Mathematical Modeling of Natural Frequency of Rock with a Single Fracture

2 Modal Analysis of Rock Modal analysis is a process which the vibration response of structure is described by its dynamics properties, such as frequency, damping and mode of vibration (Zhang et al. 2009).

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With the increase in the number of cracks or the expansion of cracks, rock strength will be reduced and the natural frequency of rock will also be changed. Assume that rock contains one crack unit, as shown in Fig. 1, and calculate its natural frequency xi .

Simulation on Vibration Characteristics of Fractured Rock

the resonance frequency of xi , and expansion energy of crack of Eio . Apply a periodic vibration excitation F ðtÞ ¼ kA cos xi t on the rock, the rock unit which contains i class crack will resonate and the expansion energy of cracks is of Ei ¼ pi Eio . The excitation energy (Shi 2004) applied by system is, DE1 ¼

Z

¼

_ ¼ F ðtÞxdt Z

Z

T

F ðtÞx_ dt 0

2p xi

kA cos xi t  AjHðxi Þjxi sinðxi t  uÞdt

0

Fig. 1 An analysis unit of rock

¼ KA2 jHðxi Þjxi

Z

2p xi

cos xi t sinðxi t  uÞdt

0

The natural frequency of rock vibration system can be expressed as (Zhang and Li 2010), pffiffiffiffiffiffiffiffiffi xi ¼ k=m ð8Þ Assuming that the rock is an ideal model, its stiffness can be obtained by stiffness formula of elastic material, k ¼ EA=L

ð9Þ

Based on the Griffith fracture criterion, elastic modulus of rock can be derived as, E ¼ r2 pa=2cs Take Eqs. (9) and (10) into Eq. (8), thus, sffiffiffiffiffiffiffiffiffiffiffiffiffi r2 paA xi ¼ 2cs Lm

ð10Þ

ð13Þ Due to the damping in the process of vibration, the energy consumed is, Z Z T Z T DE2 ¼ cx_ dx ¼ cx_x_ dt ¼ c x_2 dx 0

¼ cA2 jHðxi Þj2 x2i

Z

0 2p xi

ð14Þ

sin2 ðxi t  uÞdt

0

¼ cA2 jHðxi Þj2 xi p Based on the Eqs. (13) and (14), the increased energy of vibration system can be obtained, DE ¼ DE1  DE2 ¼ A2 jHðxi Þjp½k sin u  cxi jHðxi Þj

ð11Þ

ð15Þ

qffiffiffiffiffiffiffiffiffi r2 pA Only considering the effect of the size of crack, 2c s Lm is expressed as n, and Eq. (11) can be rewritten by,

The increased energy per cycle is used for the i class cracks expansion,

1

xi ¼ na2 ða\1Þ

ð12Þ

Equation (12) is the theoretical calculation equation of natural frequency of fractured rock. The equation indicates that the natural frequency of rock decreases with the increase of crack size. Therefore, when the cracks of rock expand from small to large, the excitation frequency should be changed from large to small accordingly. In this way, the resonance of down hole rock can be realized furthest in high-frequency vibration impact drilling.

Ei ¼ nDE ¼ nA2 jHðxi Þjp½k sin u  cxi jHðxi Þj  Eio ¼ nA2 jHðxi Þjp½k sin u  cxi jHðxi Þj pi :

ð16Þ ð17Þ

Equation (17) is the modeling of expansion energy of rock crack. The model shows that the expansion energy of a single crack is inversely proportional to the number of such cracks in the rock. Namely, the more the i class cracks are, the less the energy required for a single crack expansion is and the smaller the excitation frequency needed for rock resonance is.

3.2 Mathematical Modeling of Crack Expansion Energy of Rock with Multiple Fractures

4 Numerical Simulations

In the high-frequency vibration impact drilling, the vibration of down hole rock is dependent on its natural frequency. However, the natural frequency is closely related to the distribution of its internal cracks. Divide the rock into n classes according to the crack size and each class corresponds to a resonant frequency. The rock unit which contains i class crack has a number of pi ,

To solve the natural frequency and the resonant frequency of rock, simulations are conducted using modal analysis module and harmony analysis module of ANSYS. In the simulation, the element type is selected as Solid65. The rock is simulated as a cube with a size of 300 mm 9 300 mm 9 300 mm and the crack is simulated as a cube with a size of 0.1 mm 9 2 mm 9 300 mm.

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S. Li et al. 800 700 600 f Hz

500 400

Sandstone

300

Limestone

200

Granite

100

Oil Shale

0

0

2

4

6

8

10

j

Fig. 3 The modals of rock in different orders Fig. 2 The numerical model of ANSYS 800

Assume that the impact forces acting on the surface of rock is 0–1000 N with excitation frequency of 0–5000 Hz. Besides, the fixed transverse constraints are imposed on both sides of rock which make rocks vibrate in the Z direction only. Four types of rock are chosen in the simulation, namely sandstone, limestone, granite and oil shale. Figure 2 shows the numerical model of ANSYS and Table 1 shows the basic physical parameters of the rock. •

600

f Hz

500

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Granite

100

Oil Shale 2

4

6

8

10

j

ð18Þ

ð19Þ

Figure 3 shows that different rocks have different natural frequencies. It can be seen that the larger the elastic modulus is, the larger the natural frequency is. Due to the fact that the larger the elastic modulus of rock is, the harder it can be deformed, more energy and higher excitation frequency are required to make the rock vibrate. In addition, as can be seen by the modals, the average increase of the first five order modals is up to 86 %, while it is only 30 % after five order modals. It implies that the first five order vibrations of rock play a major role. Lithology

Limestone

200

0

4.1 The Influence of Crack on Rock Modal

Table 1 Basic physical parameters of rocks

Sandstone

0

The boundary conditions of displacement are, Xjx¼0 ¼ Xjx¼300 ¼ Xjy¼0 ¼ Xjy¼300 ¼ 0

400 300

The boundary conditions of pressure are, Tjx¼0 ¼ Tjx¼300 ¼ Tjy¼0 ¼ Tjy¼300 ¼ 0; Tjz¼0 ¼ Tjz¼300 ¼ 1000

•

700

Fig. 4 The modals of fractured rock in different orders

As can be seen from Figs. 3 and 4, the modals of fractured rock are lower than those with no cracks in the same conditions, which illustrates that cracks can reduce the natural frequency of rock. Therefore, the relationship among crack, elastic modulus and natural frequency in 3.1 Section is verified by the above results. 4.2 The Influence of Crack Trend on Resonant Frequency of Rock The crack trend was considered first. Take granite as example. The angle between crack and horizontal plane is of h. When h ¼ 0 , the crack is horizontal and when h ¼ 90 , the crack is vertical. To study the effect of a change in h on resonant frequency, the model was simulated for varying h giving the results shown in Fig. 5.

Elastic modulus, E (Pa)

Poisson ratio, l

Density, q (Kg/m3)

Sandstone

4 9 1010

0.34

2560

Limestone

3.7 9 1010

0.31

2660

Granite

2.6 9 10

10

0.26

2790

Oil shale

1.5 9 1010

0.15

2103

Simulation on Vibration Characteristics of Fractured Rock 4500 4000

f Hz

3500 3000 2500 2000

0

20

40

θ

60

80

100

Fig. 5 The relation curve between resonant frequency and crack trend Fig. 6 The resonance frequency of horizontal short crack

As can be seen from this figure, when h ¼ 0 ; 30 ; 45 ; 60 ; 90 , the corresponding resonance frequencies of granite are 2200, 2400, 3300, 3700 and 4300 Hz, respectively. It is obvious that with the decreases of h, the resonance frequency of granite decreases. As we can see the harmonic force acting on the crack is F  sin h, the smaller the h is, the smaller the F  sin h is. The smaller the force suppressing crack propagation is, the more easily the crack can be expanded. Consequently, the larger the angle between crack trend and harmonic force is, the more easily the rock can be vibrated and the smaller its resonant frequency is. 4.3 The Influence of Crack Size on Resonant Frequency of Rock Crack size is another major parameter that affects the resonant frequency of rock. In the simulation, the short crack is simulated as a cube with a size of 0.1 mm 9 2 mm 9 280 mm and the wide crack is simulated as a cube with a size of 1 mm 9 2 mm 9 300 mm. Besides, the other simulation conditions are the same. Figures 6, 7, 8, 9, 10 and 11 are the graphs of resonance frequencies of rock with horizontal and vertical cracks in different length and width, respectively. Simulations show that the resonance frequencies of horizontal crack, horizontal short crack and horizontal wide crack are 3600, 3900 and 3200 Hz, respectively, and the resonance frequencies of vertical crack, vertical short crack and vertical wide crack are 4000, 4250 and 2000 Hz, respectively. By the comparison of graphs, it is intuitive that the longer or the wider the crack is, the smaller the harmonic frequency of rock is. It is caused by the decrease of rock stiffness. The above conclusions have verified the relationship between crack size and natural frequency is correct in Sect. 3.1.

Fig. 7 The resonance frequency of horizontal wide crack

Fig. 8 The resonance frequency of horizontal crack

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S. Li et al. 4500

f Hz

4000 3500 3000 2500 2000

1

2 3 The number of cracks

4

Fig. 12 The relationship between resonant frequency and the number of cracks

Fig. 9 The resonance frequency of vertical short crack

4.4 The Influence of the Number of Cracks on Resonant Frequency of Rock To analyze the influence of the number of cracks on resonant frequency of rock, the simulation is conducted with the excitation frequency of 0–10,000 Hz. The relationship between them is shown in Fig. 12. It can be seen that the resonant frequencies of rocks with 1, 2, 3 and 4 cracks are 4100, 3200, 3000 and 2200 Hz, respectively. With the increase of the number of cracks in the rock, the resonant frequency will decrease. It is consistent with the conclusion in Sect. 3.2.

5 Conclusions

Fig. 10 The resonance frequency of vertical wide crack

Fig. 11 The resonance frequency of vertical crack

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This paper presents the modal analysis theory of rock and the modeling of vibration characteristics of fractured rock, showing the results of the numerical analysis. The modeling includes two aspects, namely, the natural frequency of rock with a single fracture and the crack expansion energy of rock with multiple fractures. Four main control parameters are considered, including the material properties, crack size, crack trend and the number of cracks. Our investigations confirm that the natural frequency of rock is closely related to its material properties. Also, the natural frequency of rock will be reduced by the cracks in it. The expansion energy of a single crack is inversely proportional to the number of such cracks in the rock. Namely, the more the similar cracks are, the less the energy required for a single crack expansion is and the smaller the excitation frequency needed for rock resonance is. The numerical simulation indicates that the natural frequency of rock increases with the increase of elastic modulus, and decreases with the increase of the angle, the length, the width and the number of cracks.

Simulation on Vibration Characteristics of Fractured Rock

Based on the analysis undertaken, it can be concluded that the vibration characteristics of fractured rock are verified by the numerical analysis. High-frequency vibration impact drilling as a new technology is of great significance to enrich drilling methods and promotes the development of oil and gas well engineering. The study on the influence of crack on vibration frequency of rock can provide a theoretical basis for realizing high-frequency vibration impact drilling technology. Acknowledgments The support of National Natural Science Foundation Major Project of China (No. 51490650) and Scientific Research and Technology Development Project of CNPC (No. 2014A-4211) are gratefully acknowledged. The authors would also like to thank the support of Graduate Student Innovation Research Projects of Northeast Petroleum University (No. YJSCX2014-011NEPU).

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