- Email: [email protected]
Upper bound analytic mechanics model for rock cutting and its application in field testing
Tunnelling and Underground Space Technology 73 (2018) 287–294
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Upper bound analytic mechanics model for rock cutting and its application in field testing
T
⁎
Qi Wanga,b,c, , Song Gaoa,b,c, Shucai Lia, Manchao Heb,c, Hongke Gaoa,b,c, Bei Jiangd,a,b, Yujing Jiangc a
Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan 250061, China State Key Laboratory for Geo-mechanics and Deep Underground Engineering, China University of Mining & Technology, Beijing 100083, China c State Key Laboratory of Mining Disaster Prevention and Control, Shandong University of Science and Technology, Qingdao, Shandong 266590, China d School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Digital drilling rig Drilling parameter C-φ parameter Upper bound method Rock cutting mechanical model Field test
The rock cohesion c and internal friction angle φ are essential parameters (combined as the c-φ parameter) used to characterise rock strength. Accurate measurement of these parameters is necessary for surrounding rock stability analysis and supporting scheme design in underground engineering. The currently used indoor test procedure is time-consuming and difficult to quantitatively evaluate the mechanical properties of fragmented rocks because these rocks cannot be effectively cored. Most field test methods can measure rock tensile or compressive strength parameters; however, it is difficult to determine the c-φ parameter. “Digital drilling rig” test technology provides a new way to solve the aforementioned problem. The key to implement this technology is to create a quantitative relation between the drilling parameters and the rock c-φ parameter. In this study, based on the characteristics of the rock cutting failure, an upper bound analytic mechanics model is developed for rock cutting. In this model, the ultimate rock cutting force is derived and the relation between the drilling parameters and the c-φ parameter is obtained. A comparative analysis of indoor tests and theoretical calculations shows that the average difference between the drilling parameters from the digital drilling test versus parameters from the theoretical calculation for the limestone tests is 9.33%; for the sandstone tests, the average difference is 5.85%. This validates the rock cutting mechanical model and the formula for the relation between the drilling parameters and the c-φ parameter. Based on these results, a digital drilling measurement method for the surrounding rock c-φ parameter in the field is proposed. The feasibility and effectiveness of the proposed method is verified via indoor testing. This method is convenient to implement in the field and can effectively measure the cφ parameter of both intact and relatively fragmented rock mass in the field.
1. Introduction The rock cohesion c and the internal friction angle φ are the most essential parameters (hereinafter referred to as one combined parameter, the c-φ parameter) used to characterise rock strength. Accurate measurement of the c-φ parameter is the foundation of surrounding rock stability analysis and supporting scheme design in underground engineering. Conventional c-φ parameter measurement methods include indoor testing and field testing. Laboratory triaxial and compression shear test methods can produce accurate results. However, these test methods require the in situ acquisition of rock samples and the transportation of the rock samples to a laboratory for testing; thus, they are time consuming. In particular, the mechanical parameters of the surrounding rock will change when they are disturbed by the
⁎
excavation activity, and the variation of the mechanical parameters of the surrounding rock will significantly affect its deformation and the mechanical conditions of the support structure. However, laboratory test methods cannot provide a timely evaluation of the variations in the mechanical parameters of the surrounding rock. In addition, it is difficult to quantitatively evaluate the mechanical properties of fragmented surrounding rocks via indoor testing because these rocks cannot be effectively cored and protected during transportation. Therefore, field testing methods have been extensively studied to rapidly determine the c-φ parameter. Currently, the main field testing methods are displacement-based back analysis methods (Gioda and Locatelli, 1999; Swoboda et al., 1999; Sakurai, 2003; Feng et al., 2004; Zhang et al., 2006; Yazdani et al., 2012). These techniques determine the rock c-φ parameter through back analysis by using an intelligent algorithm
Corresponding author at: Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan 250061, China. E-mail address: [email protected] (Q. Wang).
https://doi.org/10.1016/j.tust.2017.12.023 Received 26 August 2017; Received in revised form 30 November 2017; Accepted 23 December 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
cutting test (Chaput, 1992; Richard et al., 1998; Richard, 1999) and the numerical simulation (Huang et al., 2013) show that when the rockcutting depth H of the drill bit single-row cutting edge in one rotation increases, the rock failure mode will shift from ductile failure (illustrated in Fig. 2(a)) to brittle failure (illustrated in Fig. 2(b)). Usually, the width of each row of the cutting edge in the drill bit is more than 10 times H, i.e., H and the rock cutting range L are far smaller than the width of each row of the cutting edge. Therefore, in each cycle of rock cutting, the cutting edge follows an approximately linear path. Because H is usually small, the rock in front of the cutting edge demonstrates mostly ductile failure. The rock cutting problem basically satisfies the conditions of a plane strain problem. Based on the aforementioned rock cutting failure characteristics and findings from other researchers (Nishimatsu, 1972, 1993; Gerbaud et al., 2006; Song et al., 2010), the following assumptions are made when developing the rock cutting mechanical model:
or numerical simulation based on displacement data on cross-sections that are obtained after underground engineering excavation. Based on previous researches, a method for determining the c-φ parameters of rock mass by using the “digital drilling rig” test technology is presented in this paper. A “digital drilling rig” (Suzuki et al., 1995; Gui et al., 2002; Kahraman et al., 2003; Yue et al., 2004; Tan et al., 2005; Yang et al., 2012; Chen and Yue, 2015) is a field survey device that provides accurate control and monitoring of the drilling parameters during drilling. The drilling parameters include the drilling rate, the rotating speed, the torque, the thrust and the specific energy. Research shows that the drilling parameters and the rock mechanical parameters are closely related (Kahraman, 1999; Tan et al., 2007; Yaşar et al., 2011; Aalizad and Rashidinejad, 2012); therefore, finding a quantitative relation between the drilling parameters and the rock mechanical parameters is the foundation of using the “digital drilling rig” to obtain the mechanical parameters of the rock. Numerous researchers have established the relationships of the drilling parameters with the uniaxial compressive strength of rocks (Huang and Wang, 1997; Karasawa et al., 2002; Mostofi et al., 2011; Yaşar et al., 2011) and the structural plane parameters of the rock mass (Schunnesson, 1996, 1998; Akin and Karpuz, 2008; Tan et al., 2009) using statistics, intelligent algorithms and energy analysis methods. However, few studies have investigated the relationships between the drilling parameters and the rock c-φ parameter, which has limited further development of the digital-drilling-rig-based drilling test technology. Indoor tests (Chaput, 1992; Richard et al., 1998; Richard, 1999) and numerical simulations (Huang et al., 2013) show that, when the rockcutting depth of the drill bit cutting edge in a rotation is relatively shallow, the rock fails in ductile mode, i.e., the rock c-φ parameter controls the rock cutting failure process. Therefore, it is feasible to establish a relation between the drilling parameters and the rock c-φ parameter based on a mechanical analysis of the rock cutting failure process. Some mechanical models for rock cutting (Nishimatsu, 1972, 1993; Gerbaud et al., 2006; Song et al., 2010) have been developed to investigate the cutting mechanism and optimise drilling tool design. The existing mechanical models for cutting require the assumption of a cutting failure surface shape and the stress distribution. In this study, based on rock cutting failure characteristics, a rock cutting upperbound analytic mechanics model is proposed. In the proposed model, no assumption for the cutting failure surface stress distribution is required. It provides a more accurate match for the actual failure mode and is applicable in a wider range of scenarios. In this study, the ultimate rock cutting force is derived and the formula for the relation between the drilling parameters and the c-φ parameter (hereinafter referred to as the DP-CΦ Formula) is developed based on the proposed rock cutting upper bound analytic mechanics model. Indoor testing is performed, using a multi-function true triaxial rock drilling test system developed in-house, to validate the model. Based on the results, the field surrounding rock c-φ parameter digital drilling measurement method is proposed. The feasibility and effectiveness of the proposed method is verified via indoor testing. This method is convenient to implement in the field and can measure the c-φ parameter of both intact and relatively fragmented rock mass. The proposed method supports surrounding rock stability analysis and support parameter optimisation in underground engineering.
(1) Because the cutting width far exceeds the cutting depth H, the rock cutting problem is simplified as a plane strain problem. (2) Because the weight of the rock (gravitational force) in the cutting area is far smaller than the cutting force, gravitational force is not considered in this model. (3) The rock is treated as an ideal rigid plastic material, i.e., the strain strengthening and softening effects are not considered. (4) The rock follows the Coulomb yield criteria and satisfies the associated flow rule; and (5) Rock cutting failure is caused by shear slip along a plane, the rock above the slip surface is treated as a rigid body and deformation is only in a thin failure layer between the virgin rock and the rock chips. Based on the above assumptions and the rock cutting failure characteristics, a rock cutting mechanical model is developed, as shown in Fig. 3. In the diagram, OA is the cut failure surface, whose angle with respect to the horizontal plane is β; Fc is the cutting force exerted by the cutting edge on the rock in the front; Ff is the force exerted by the cutting edge on the bottom rock; γ is the angle between the cutting force Fc and the normal direction of the surface of the cutting edge; κ is the inclination angle of the cutting edge; δ is the angle between the force Ff and the vertical direction, which is also the friction angle between the cutting edge and the rock; φ is the internal friction angle of the rock; and V is the velocity of the rock chips along the slip surface. Based on the associated flow rule and the Coulomb yield criterion, the angle between the direction of V and the surface of the cut failure is φ (Chen, 1975).
2.2. Analysis of the ultimate rock cutting force Rock cutting failure is caused by the slip of a plane whose angle with respect to the horizontal plane is β. Based on the ultimate analysis upper bound theory, when the rock fracture power W generated by the cutting force Fc and the cut failure surface energy dissipation rate D are equal, the ultimate condition is met. The rock fracture power W is the product of the component of the velocity V along the Fc direction and Fc as follows:
W = Fc V cos(β + φ + κ + γ )
2. Analysis of the theory of rock cutting
(1)
The energy dissipation rate D on the cut failure surface OA is as follows:
2.1. Mechanism of rock cutting and the basic theoretical assumptions The rock cutting is performed using conventional polycrystalline diamond compact (PDC) drill bits, wherein the PDCs are embedded in a matrix to form cutting edges that crushes rock, as shown in Fig. 1. The rock is crushed by the cutting edge under the combined effects of the vertical force F1 and the horizontal force F2, as shown in Fig. 2. The rock
D = c (V cosφ)
H sinβ
(2)
When the rock reaches the ultimate state, the rock fracture power W and cut failure surface energy dissipation rate D are equal, as follows: 288
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 1. Three-dimensional schematic diagram of the PDC drill bit.
Fig. 3. Rock cutting upper bound analytic mechanics model.
(a) Rock ductile failure
Fc =
2cH cosφ 1−sin(φ + γ + κ )
(5)
3. Relation between the drilling parameters and the rock c-φ parameter 3.1. Solution for the drill bit torque M The drill bit torque M consists of two components, the cut torque Mc and the friction torque Mf, as shown in Fig. 3. The cut torque Mc is the drilling rig moment for rock cutting, which is obtained by calculating the moment of the horizontal component Fc with respect to the drill bit centre; Mf is the moment exerted by the drilling rig to overcome the friction between the drill bit and the rock below, which is obtained by calculating the moment of the horizontal component of Ff with respect to the drill bit centre. As shown in Fig. 4, the drill bit radius is R; there are 3 rows of cutting edges; the length of the i-th row of the cutting edge is Li; and the distance from a section of a row of the cutting edge to the drill bit centre O is r. In any segment dr, the torque on the drill bit is as follows:
(b) Rock brittle failure Fig. 2. Rock cutting process: (a) Rock ductile failure, (b) Rock brittle failure.
Fc V cos(β + φ + κ + γ ) = c (V cosφ)
H sinβ
(3)
dM = dMc + dMf = Fc r cos(κ + γ ) dr + Ff r sinδdr
After simplification, Fc is as follows:
Fc =
cH cosφ sin(φ + 2β + κ + γ )−sin(φ + κ + γ )
(6)
The integral of the moment dM along the cutting edge length is calculated and the moments at all row cutting edges are summed to obtain the overall drilling rig torque M as follows:
(4)
3
According to the minimum energy principle, when ∂Fc / ∂β = 0 , there is a minimum upper bound for Fc and β = (π /2−φ−κ−γ )/2 . The above expression is substituted into Eq. (4) to obtain the ultimate cutting force as follows:
M=
R ∑ ∫R−Li (Fc cos(κ + γ ) + Ff sinδ ) rdr
i=1 1
= 2 [Fc cos(κ + γ ) + Ff sinδ ][2R (L1 + L2 + L3)−(L12 + L22 + L32)] 289
(7)
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 4. Schematic diagram of the drill bit and one row of the cutting edge.
3.2. Solution for the thrust F The thrust F is the pressure exerted by the drilling rig to overcome the vertical component of Fc and Ff. For the drill bit shown in Fig. 3, F is as follows: (8)
Fig. 5. Multi-function true triaxial rock drilling test system and the PDC exploration drill bit developed for this study.
3.3. Formula for the relation between the drilling parameters and the rock cφ parameter
square PDCs. During drilling, the shape and the stress characteristics of the PDC will not change even if there is wear, which reduces the impact of PDC wear on the test data. In addition, a square PDC is a more accurate match for the theoretical assumptions used in this study. The geometrical parameters of the drill bit are listed in Table 1. In the tests, the same batch of limestone and sandstone specimens are used. The specimen dimension is 150 mm × 150 mm × 200 mm, as shown in Fig. 6. And the powdery debris is flushed by water and the water pressure is 0.15 MPa. Triaxial and uniaxial tests are performed for two types of rock specimens to measure their mechanical parameters. The results are listed in Table 2.
F = [Fc sin(κ + γ ) + Ff cosδ ](L1 + L2 + L3)
Eqs. (7) and (8) are combined to eliminate the unknown force Ff as follows:
M=
1 Fc [cos(κ + γ )−sin(κ + γ )tanδ ][2R (L1 + L2 + L3) 2 L 2 + L22 + L32 ⎞ 1 −(L12 + L22 + L32)] + F tanδ ⎛⎜2R− 1 ⎟ L1 + L2 + L3 ⎠ 2 ⎝
(9)
When the drilling rig drills into the rock, sensors measure the drilling parameters, including the rotating speed N, the drilling rate Vd, the torque M and the thrust F. The rock-cutting depth H of a single row of the drill bit cutting edge in one rotation is as follows:
V H= d mN
4.2. Test plan
(10)
The tests are based on the mode that controls the drilling rate Vd, the drill rotating speed N, the collection torque M and the thrust F. A series of five tests are designed for each type of rock. The test series for limestone and sandstone are designated Ai and Bi (i = 1, 2, 3, 4, 5). The series for limestone includes a combination of different N and Vd; the test series for sandstone includes identical N with different Vd. In this way, the relation formula for the drilling parameters and the rock c-φ parameter is thoroughly verified. The detailed test plans are listed in Table 3:
where m is the number of rows of cutting edges in the drill bit. Eqs. (5) and (10) are substituted into Eq. (9) to obtain the DP-CΦ Formula as follows:
M=
cVd cosφ [cos(κ + γ )−sin(κ + γ )tanδ ] [2R (L1 + L2 + L3) mN [1−sin(φ + κ + γ )] −(L12 + L22 + L32)] +
L 2 + L22 + L32 ⎞ 1 F tanδ ⎜⎛2R− 1 ⎟ 2 L1 + L2 + L3 ⎠ ⎝
(11)
4.3. Comparative analysis of the test results versus the theoretical calculation
4. Indoor test verification 4.1. Test equipment and test material
4.3.1. Test results and the torque The variation curves of M versus Dh (Figs. 7 and 9) and F versus Dh (Figs. 8 and 10) for different drilling tests are created based on one set of typical data from each test series for limestone and for sandstone, where Dh is the rock-drilling depth of the drilling rig. The data in Figs. 7 and 8 show that in the different tests for limestone, the variation rules of M and F versus Dh are similar, consisting of
To investigate the relation of the drilling parameters to the rock mechanical parameters and the rock mass characteristic parameters, we have developed a multi-function true triaxial rock drilling test system, as shown in Fig. 5. When performing rock specimen drilling tests, this system can provide a maximum torque of 400 N·m, a maximum rotating speed of 400 r/min and a maximum thrust of 50 kN. The system can control and monitor any of the four drilling parameters, including the drilling rate, the rotating speed, the thrust and the torque; record the drilling parameters during drilling; and display the variation curve versus time automatically. The drill bit used in the test is an exploration PDC drill bit designed in-house, as shown in Fig. 5. The cutting edge of this drill bit is based on
Table 1 Geometrical parameters of the exploration PDC drill bit.
290
Bit parameters
R (mm)
L1 (mm)
L2 (mm)
L3 (mm)
κ (°)
Sizes
30
18
18
27
15
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 6. Limestone and sandstone specimens before drilling and after drilling.
14
Table 2 Physical and mechanical parameters of the limestone and the sandstone samples used in the tests.
Limestone Sandstone
Uniaxial compressive strength, UCS (MPa)
Elastic modulus, E (GPa)
Cohesion, c (MPa)
Internal friction angle, φ (°)
69.2 49.8
23.31 19.82
14.27 11.56
43.86 39.52
10
8
F (kN)
Rock type
12
6 4
A1
A2
A4
A5
A3
2 Table 3 Detailed test plans for the limestone and the sandstone specimen drilling tests. Rock type
Test scheme label
Vd (mm/ min)
N (r/min)
Number of specimens
Limestone
A1 A2 A3 A4 A5
60 60 80 80 110
100 200 100 200 250
3 3 3 3 3
B1 B2 B3 B4 B5
30 60 80 110 140
200 200 200 200 200
3 3 3 3 3
0
2
4
6
8
10
12
14
16
Dh (cm) Fig. 8. Variation curves of F versus Dh for different limestone drilling tests.
90 80 70 60
M (N•m)
Sandstone
0
50 40 30 20
B1
B2
10
B4
B5
0
0
2
4
6
8 10 Dh (cm)
12
B3
14
16
Fig. 9. Variation curves of M versus Dh for different sandstone drilling tests.
8 7 6
F (kN)
5
4 3
Fig. 7. Variation curves of M versus Dh for different limestone drilling tests.
the following two stages:
2
B1
B2
1
B4
B5
0
①Before the drill bit enters the rock, M and F are small; after it drills into the rock, M and F increase sharply in a small Dh. This is the rising stage. ②After the drill bit reaches a shallow drilling depth, M and F reach a stable stage. During this stable stage, when Dh increases, M fluctuates slightly around a stable value. This behaviour may be related to rock heterogeneity and the alternating stages of cutting rock.
0
2
4
6
8 10 Dh (cm)
12
B3
14
16
Fig. 10. Variation curves of F versus Dh for different sandstone drilling tests.
versus Dh in different tests for sandstone is similar to that for limestone. However, the behaviour of the sandstone is more stable overall than that of the limestone. The reason is that some limestone specimens have few fractures, while sandstone specimens are relatively intact.
The data in Figs. 9 and 10 show that the variation rule of M and F 291
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
The test calculation methods for M and F for the limestone and sandstone specimens are identical. The calculation for M is used as an example to illustrate the process. Before the rock specimen is drilled, the initial torque is MI, the average torque in the stable stage is MS and the specimen torque test value is M = MS−MI . Using the data for the limestone specimen in the A1 test in Fig. 7 as an example, MI = 26.25 N m, MS = 125.37 N·m and M = MS−MI = 99.12 N·m . In each test, the theoretical value is compared with the average M from the three specimens tested.
4.3.2. Selection of the parameters for the theoretical analysis The geometrical parameters of the drill bit designed for this study are listed in Table 1. The values of c and φ are based on measurements of the specimens in the triaxial tests and are listed in Table 2. γ is based on the research findings by Huang et al. (2013); and κ has a significant impact on γ, i.e., as κ increases, γ decreases. For the drill bit used in this study, κ = 15°and γ = 12°. Based on the research findings by Li et al. (2015), δ is 12°. The values defined for Vd and N in each test series are listed in Table 3. The aforementioned parameters and the thrust F are substituted into Eq. (11) to obtain the theoretical torque M.
Fig. 12. Comparative analysis diagram of the torque indoor tests versus the theoretical calculations for the sandstone tests.
5. Field surrounding rock c-φ parameter digital drilling measurement method 5.1. The proposed method Based on the DP-CΦ Formula developed in this study, a field surrounding rock c-φ parameter digital drilling measurement method is proposed (hereinafter referred to as the digital drilling measurement method). Implementation of this method includes three steps: Step A: A field rock drilling test is performed via the digital drilling measurement method and the rock cutting strength Sc is calculated based on the acquired drilling parameters. The cutting strength Sc, as defined in this paper, is calculated via Eq. (11), resulting in the calculation formula as follows:
4.3.3. Comparison and evaluation of the theoretical analysis with respect to the indoor test results The comparative analysis diagrams of the torque measured in the indoor tests versus the theoretical calculations for the limestone tests and the sandstone tests are shown in Figs. 11 and 12. The data show that, in both the limestone and the sandstone tests, the measured torque from the indoor tests and the theoretical calculations are consistent; however, the sandstone tests exhibit better conformity. To quantitatively evaluate the difference between the results from the theoretical method proposed in this study and the results from conventional experiments, a difference rate index ξ is defined as follows:
ξ=
|thR−exR| × 100% exR
Sc =
2c cosφ 1−sin(φ + κ + γ ) L2 + L 2 + L 2
mN ⎡2M −F tanδ ⎛2R− L1 + L2 + L 3 ⎞ ⎤ 1 2 3 ⎠ ⎝ ⎣ ⎦ = Vd [cos(κ + γ )−sin(κ + γ )tanδ ][2R (L1 + L2 + L3)−(L12 + L22 + L32)]
(12)
(13) where thR is the result from the theory or method proposed in this study and exR is the result from the conventional experiment. This index is also used in other sections of this paper, with the same definition. The analysis shows that the average ξ for the limestone tests is 9.33% and the average ξ for the sandstone tests is 5.85%. This also indicates that the model exhibits better conformity for sandstone. This is because some limestone specimens contain fracture, which increase the discreteness of the test data. For both types of rock, the difference rates ξ for the test versus theory are less than 10%. This proves the feasibility and the effectiveness of the rock cutting upper bound analytic mechanics model and DPCΦ Formula proposed in this study.
This formula, in tandem with Eq. (5), shows that the physical meaning of Sc is the ultimate cutting force per unit depth of rock. Step B: Based on the drilling parameters of the digital drilling rig (Huang and Wang, 1997; Karasawa et al., 2002; Mostofi et al., 2011) or a point load test (Bieniawski, 1974; Panek and Fannon, 1992; Al-Derbi and Freitas de, 1999; Basu and Aydin, 2006), the uniaxial compressive strength Rc is obtained. A large number of comparative test results show that the rock uniaxial compressive strength can be calculated based on the drilling parameters of the digital drilling rig or the point load strength. According to the Mohr-Coulomb yield criterion, the relation between the rock uniaxial compressive strength Rc and the c-φ parameter is as follows:
Rc =
2c cosφ 1−sinφ
(14)
Step C: The rock internal friction angle φ and the cohesion c are calculated based on Sc and Rc obtained from the previous two steps. The equation for the internal friction angle calculation is obtained by combining Eqs. (13) and (14) as follows:
1−
Rc R + c sinφ−sin(κ + γ + φ) = 0 Sc Sc
(15)
Subsequently, φ from Eq. (15) is substituted into Eqs. (13) or (14) to obtain c.
Fig. 11. Comparative analysis diagram of the torque indoor tests versus the theoretical calculations for the limestone tests.
292
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Table 4 Comparative analysis of the c-φ parameter from the digital drilling measurement method versus that from the triaxial method in the limestone and sandstone tests. Rock type
Test scheme
φd (°)
ξφ
cd (MPa)
ξc
Rock type
Test scheme
φd (°)
ξφ
cd (MPa)
ξc
Limestone
A1 A2 A3 A4 A5 Average value
41.26 46.03 39.14 40.77 42.59 41.96
5.93% 4.96% 10.77% 7.05% 2.91% 6.32%
15.67 13.97 16.45 15.85 15.19 15.43
9.84% 2.12% 15.30% 11.10% 6.47% 8.97%
Sandstone
B1 B2 B3 B4 B5 Average value
31.72 38.35 39.83 40.29 36.88 37.41
19.74% 2.95% 0.79% 1.94% 6.67% 6.42%
13.88 12.05 11.66 11.54 12.45 12.31
20.09% 4.24% 0.83% 0.22% 7.67% 6.61%
5.2. Method verification
Acknowledgments
The proposed digital drilling measurement method is verified with drilling parameter data obtained from limestone and sandstone specimen drilling tests using the multi-function true triaxial rock drilling test system (Figs. 7–10), described in Section 4, and the uniaxial compressive strength data for the two types of rock (Table 2). The internal friction angle φd and the cohesion cd measured by the digital drilling measurement method are calculated using Eqs. (13) and (15) in the limestone and sandstone tests. These results, combined with φ and c for the two types of rock from the triaxial test (Table 2) and Eq. (12), are used to calculate the internal friction angle difference rates ξφ and the cohesion difference rates ξc for the two methods. The results are shown in Table 4. The data in Table 4 shows that in the limestone tests, the average difference rate of the internal friction angle from the digital drilling measurement method versus that from the triaxial method is ξφ = 6.32%; for cohesion, it is ξc = 8.97%. In the sandstone tests, the average difference rate of the internal friction angle is ξφ = 6.42%; for cohesion, it is ξc = 6.61%. In all the tests, for both types of rock, the average difference rates of the cohesion and the internal friction angle are always less than 10%. This proves the effectiveness and the feasibility of the field surrounding rock c-φ parameter digital drilling measurement method proposed in this paper.
This work was supported by the Natural Science Foundation of China (grant numbers 51674154, 51704125); Open Fund for State Key Laboratory for Geo-mechanics and Deep Underground Engineering, China University of Mining & Technology (grant numbers SKLGDUEK1519, SKLGDUEK1717); and the China Postdoctoral Science Foundation (grant numbers 2016M590150, 2016M602144). The authors of this article are so grateful to the anonymous reviewers for their valuable comments and modification suggestions on the improvement of this paper. References Aalizad, S.A., Rashidinejad, F., 2012. Prediction of penetration rate of rotary-percussive drilling using artificial neural networks-a case study. Arch. Min. Sci. 57 (3), 715–728. Akin, S., Karpuz, C., 2008. Estimating drilling parameters for diamond bit drilling operations using artificial neural networks. Int. J. Geomech. 8 (1), 68–73. Al-Derbi, M.S., Freitas de, M.H., 1999. Use of the boussinesq equation for determining the distribution of stress within a diametrical point load test. Rock Mech. Rock Eng. 32 (4), 257–265. Basu, A., Aydin, A., 2006. Predicting uniaxial compressive strength by point load test: significance of cone penetration. Rock Mech. Rock Eng. 39 (5), 483–490. Bieniawski, Z.T., 1974. Estimating the strength of rock materials. J. South African Institute Mining Metall. 74 (8), 312–320. Chaput, E.J., 1992. Observations and analysis of hard rocks cutting failure mechanisms using pdc cutters. Technical Report, Project at Imperial College, London, Report published by Elf-aquitaine, France. Chen, J., Yue, Z.Q., 2015. Ground characterization using breaking-action-based zoning analysis of rotary-percussive instrumented drilling. Int. J. Rock Mech. Min. Sci. 75, 33–43. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier. Feng, X.T., Zhao, H., Li, S., 2004. A new displacement back analysis to identify mechanical geo-material parameters based on hybrid intelligent methodology. Int. J. Numer. Anal. Meth. Geomech. 28, 1141–1165. Gerbaud, L., Menand, S., Sellami, H., 2006. PDC Bits: all comes from the cutter rock interaction. In: IADC/SPE Drilling Conference. U.S.A, Miami, Florida, pp. 1–9. Gioda, G., Locatelli, L., 1999. Back analysis of the measurements performed during the excavation of a shallow tunnel in sand. Int. J. Numer. Anal. Meth. Geomech. 23, 1407–1425. Gui, M.W., Soga, K., Bolton, M.D., Hamelin, J.P., 2002. Instrumented borehole drilling for subsurface investigation. J. Geotech. Geoenviron. Eng. 128 (4), 283–291. Huang, H., Lecampion, B., Detournay, E., 2013. Discrete element modeling of tool-rock interaction I: rock cutting. Int. J. Numer. Anal. Meth. Geomech. 37 (13), 1913–1929. Huang, S.L., Wang, Z.W., 1997. The mechanics of diamond core drilling of rocks. Int. J. Rock Mech. Min. Sci. 34 (3/4), 612. Kahraman, S., 1999. Rotary and percussive drilling prediction using regression analysis. Int. J. Rock Mech. Min. Sci. 36 (7), 981–989. Kahraman, S., Bilgin, N., Feridunoglu, C., 2003. Dominant rock properties affecting the penetration rate of percussive drills. Int. J. Rock Mech. Min. Sci. 40 (5), 711–723. Karasawa, H., Ohno, T., Kosugi, M., Rowley, J.C., 2002. Methods to estimate the rock strength and tooth wear while drilling with roller-bits. J. Energy Resour. Technol. 124 (3), 125–132. Li, J.Y., Tan, Z.Y., Li, W., Yue, P.J., 2015. Experimental simulation of dynamic friction characteristics of interface between diamond drill and rock under impact-rotational loading. J. Vibr. Shock 34 (22), 210–214 (in Chinese). Mostofi, M., Rasouli, V., Mawuli, E., 2011. An estimation of rock strength using a drilling performance model: a case study in Blacktip field, Australia. Rock Mech. Rock Eng. 44 (3), 305–316. Nishimatsu, Y., 1972. The mechanics of rock cutting. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 9 (2), 261–270. Nishimatsu, Y., 1993. Theories of rock cutting. In: Hudson, J.A. (Ed.), Comprehensive Rock Engineering: Principles Practice and Projects. Pergamon Press, Oxford, New York. Panek, L.A., Fannon, T.A., 1992. Size and shape effects in point load tests of irregular rock fragments. Rock Mech. Min. Sci. 25 (1), 109–140. Richard, T., 1999. Determination of rock strength from cutting tests. Dissertation,
6. Conclusions (1) In this study, based on rock cutting failure characteristics, a rock cutting upper bound analytic mechanics model is proposed. This model avoids assuming a stress distribution on the rock cutting failure surface and can be applied to a wide range of scenarios. Based on the principle of virtual work, the ultimate rock cutting force is derived and the DP-CΦ Formula is developed. (2) The drilling tests for the limestone and the sandstone specimens are performed using a multi-function true triaxial rock drilling test system. The drilling parameters from the tests and the parameters from the theoretical calculation are compared and analysed. The results show that in the limestone tests, the average difference rate ξ is 9.33%, while the average difference rate ξ for the sandstone tests is 5.85%. This validates the rock cutting mechanical model and the DP-CΦ Formula. (3) Based on the relation formula for the drilling parameters and the rock c-φ parameter, the field surrounding rock c-φ parameter digital drilling measurement method is proposed. Results using the proposed method and results from the conventional triaxial method are compared and analysed. The results show that the average difference rates of the cohesion and the internal friction angle are less than 10%. This proves the effectiveness and the feasibility of the digital drilling measurement method proposed in this study. (4) The field surrounding rock c-φ parameter digital drilling measurement method proposed in this study is convenient to implement in the field. The method can measure the c-φ parameter for both intact and relatively fragmented rock mass in the field and provides references for surrounding rock stability analysis and optimisation design of support parameters in underground engineering. 293
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
granite formation. Chin. J. Rock Mech. Eng. 26 (3), 478–483 (in Chinese). Tan, Z.Y., Wang, S.J., Cai, M.F., 2009. Similarity identification method on formational interfaces and application in general granite. Int. J. Miner. Metall. Mater. 16 (2), 135–142. Yang, W.W., Yue, Z.Q., Tham, L.G., 2012. Automatic monitoring of inserting or retrieving SPT sampler in drillhole. Geotech. Test J. 35 (3), 420–436. Yaşar, E., Ranjith, P.G., Viete, D.R., 2011. An experimental investigation into the drilling and physico-mechanical properties of a rock-like brittle material. J. Petrol. Sci. Eng. 76 (3), 185–193. Yazdani, M., Sharifzadeh, M., Kamrani, K., Ghorbani, M., 2012. Displacement-based numerical back analysis for estimation of rock mass parameters in Siah Bisheh powerhouse cavern using continuum and discontinuum approach. Tunn. Undergr. Space Technol. 28, 41–48. Yue, Z.Q., Lee, C.F., Law, K.T., Tham, L.G., 2004. Automatic monitoring of rotary-percussive drilling for ground characterization—illustrated by a case example in Hong Kong. Int. J. Rock Mech. Min. Sci. 41 (4), 573–612. Zhang, L.Q., Yue, Z.Q., Yang, Z.F., Qi, J.X., Liu, F.C., 2006. A displacement-based backanalysis method for rock mass modulus and horizontal in situ stress in tunneling – Illustrated with a case study. Tunn. Undergr. Space Technol. 21, 636–649.
University of Minnesota. Richard, T., Detournay, E., Drescher, A., Nicodème, P., Fourmaintraux, D., 1998. The scratch test as a means to measure strength of sedimentary rocks. In: SPE/ISRM Rock Mechanics in Petroleum Engineering, Trondheim, Norway. Sakurai, S., 2003. Back analysis for tunnel engineering as a modern observational method. Tunn. Undergr. Space Technol. 185, 196. Schunnesson, H., 1996. RQD predictions based on drill performance parameters. Tunn. Undergr. Space Technol. 11 (3), 345–351. Schunnesson, H., 1998. Rock characteristics using percussive drilling. Int. J. Rock Mech. Min. Sci. 35 (6), 711–725. Song, L., Li, N., Liu, F.Y., 2010. Research on the mechanics of rotary penetration test. Chin. J. Rock Mech. Eng. 29 (Supp. 2), 3519–3525 (in Chinese). Suzuki, Y., Sasao, H., Nishi, K., Takesue, K., 1995. Ground exploration system using seismic cone and rotary percussion drill. AIJ J. Technol. Design 1 (1), 180–184. Swoboda, G., Ichikawa, Y., Dong, Q.X., Zaki, M., 1999. Back analysis of large geotechnical models. Int. J. Numer. Anal. Meth. Geomech. 23, 1455–1472. Tan, Z.Y., Cai, M.F., Yue, Z.Q., Tham, L.G., Lee, C.F., 2005. Application and reliability analysis of DPM system in site investigation of HK weathered Granite. J. Univ. Sci. Technol. Beijing 12 (6), 481–488. Tan, Z.Y., Yue, Z.Q., Cai, M.F., 2007. Analysis of energy for rotary drilling in weathered
294
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Upper bound analytic mechanics model for rock cutting and its application in field testing
T
⁎
Qi Wanga,b,c, , Song Gaoa,b,c, Shucai Lia, Manchao Heb,c, Hongke Gaoa,b,c, Bei Jiangd,a,b, Yujing Jiangc a
Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan 250061, China State Key Laboratory for Geo-mechanics and Deep Underground Engineering, China University of Mining & Technology, Beijing 100083, China c State Key Laboratory of Mining Disaster Prevention and Control, Shandong University of Science and Technology, Qingdao, Shandong 266590, China d School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Digital drilling rig Drilling parameter C-φ parameter Upper bound method Rock cutting mechanical model Field test
The rock cohesion c and internal friction angle φ are essential parameters (combined as the c-φ parameter) used to characterise rock strength. Accurate measurement of these parameters is necessary for surrounding rock stability analysis and supporting scheme design in underground engineering. The currently used indoor test procedure is time-consuming and difficult to quantitatively evaluate the mechanical properties of fragmented rocks because these rocks cannot be effectively cored. Most field test methods can measure rock tensile or compressive strength parameters; however, it is difficult to determine the c-φ parameter. “Digital drilling rig” test technology provides a new way to solve the aforementioned problem. The key to implement this technology is to create a quantitative relation between the drilling parameters and the rock c-φ parameter. In this study, based on the characteristics of the rock cutting failure, an upper bound analytic mechanics model is developed for rock cutting. In this model, the ultimate rock cutting force is derived and the relation between the drilling parameters and the c-φ parameter is obtained. A comparative analysis of indoor tests and theoretical calculations shows that the average difference between the drilling parameters from the digital drilling test versus parameters from the theoretical calculation for the limestone tests is 9.33%; for the sandstone tests, the average difference is 5.85%. This validates the rock cutting mechanical model and the formula for the relation between the drilling parameters and the c-φ parameter. Based on these results, a digital drilling measurement method for the surrounding rock c-φ parameter in the field is proposed. The feasibility and effectiveness of the proposed method is verified via indoor testing. This method is convenient to implement in the field and can effectively measure the cφ parameter of both intact and relatively fragmented rock mass in the field.
1. Introduction The rock cohesion c and the internal friction angle φ are the most essential parameters (hereinafter referred to as one combined parameter, the c-φ parameter) used to characterise rock strength. Accurate measurement of the c-φ parameter is the foundation of surrounding rock stability analysis and supporting scheme design in underground engineering. Conventional c-φ parameter measurement methods include indoor testing and field testing. Laboratory triaxial and compression shear test methods can produce accurate results. However, these test methods require the in situ acquisition of rock samples and the transportation of the rock samples to a laboratory for testing; thus, they are time consuming. In particular, the mechanical parameters of the surrounding rock will change when they are disturbed by the
⁎
excavation activity, and the variation of the mechanical parameters of the surrounding rock will significantly affect its deformation and the mechanical conditions of the support structure. However, laboratory test methods cannot provide a timely evaluation of the variations in the mechanical parameters of the surrounding rock. In addition, it is difficult to quantitatively evaluate the mechanical properties of fragmented surrounding rocks via indoor testing because these rocks cannot be effectively cored and protected during transportation. Therefore, field testing methods have been extensively studied to rapidly determine the c-φ parameter. Currently, the main field testing methods are displacement-based back analysis methods (Gioda and Locatelli, 1999; Swoboda et al., 1999; Sakurai, 2003; Feng et al., 2004; Zhang et al., 2006; Yazdani et al., 2012). These techniques determine the rock c-φ parameter through back analysis by using an intelligent algorithm
Corresponding author at: Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan 250061, China. E-mail address: [email protected] (Q. Wang).
https://doi.org/10.1016/j.tust.2017.12.023 Received 26 August 2017; Received in revised form 30 November 2017; Accepted 23 December 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
cutting test (Chaput, 1992; Richard et al., 1998; Richard, 1999) and the numerical simulation (Huang et al., 2013) show that when the rockcutting depth H of the drill bit single-row cutting edge in one rotation increases, the rock failure mode will shift from ductile failure (illustrated in Fig. 2(a)) to brittle failure (illustrated in Fig. 2(b)). Usually, the width of each row of the cutting edge in the drill bit is more than 10 times H, i.e., H and the rock cutting range L are far smaller than the width of each row of the cutting edge. Therefore, in each cycle of rock cutting, the cutting edge follows an approximately linear path. Because H is usually small, the rock in front of the cutting edge demonstrates mostly ductile failure. The rock cutting problem basically satisfies the conditions of a plane strain problem. Based on the aforementioned rock cutting failure characteristics and findings from other researchers (Nishimatsu, 1972, 1993; Gerbaud et al., 2006; Song et al., 2010), the following assumptions are made when developing the rock cutting mechanical model:
or numerical simulation based on displacement data on cross-sections that are obtained after underground engineering excavation. Based on previous researches, a method for determining the c-φ parameters of rock mass by using the “digital drilling rig” test technology is presented in this paper. A “digital drilling rig” (Suzuki et al., 1995; Gui et al., 2002; Kahraman et al., 2003; Yue et al., 2004; Tan et al., 2005; Yang et al., 2012; Chen and Yue, 2015) is a field survey device that provides accurate control and monitoring of the drilling parameters during drilling. The drilling parameters include the drilling rate, the rotating speed, the torque, the thrust and the specific energy. Research shows that the drilling parameters and the rock mechanical parameters are closely related (Kahraman, 1999; Tan et al., 2007; Yaşar et al., 2011; Aalizad and Rashidinejad, 2012); therefore, finding a quantitative relation between the drilling parameters and the rock mechanical parameters is the foundation of using the “digital drilling rig” to obtain the mechanical parameters of the rock. Numerous researchers have established the relationships of the drilling parameters with the uniaxial compressive strength of rocks (Huang and Wang, 1997; Karasawa et al., 2002; Mostofi et al., 2011; Yaşar et al., 2011) and the structural plane parameters of the rock mass (Schunnesson, 1996, 1998; Akin and Karpuz, 2008; Tan et al., 2009) using statistics, intelligent algorithms and energy analysis methods. However, few studies have investigated the relationships between the drilling parameters and the rock c-φ parameter, which has limited further development of the digital-drilling-rig-based drilling test technology. Indoor tests (Chaput, 1992; Richard et al., 1998; Richard, 1999) and numerical simulations (Huang et al., 2013) show that, when the rockcutting depth of the drill bit cutting edge in a rotation is relatively shallow, the rock fails in ductile mode, i.e., the rock c-φ parameter controls the rock cutting failure process. Therefore, it is feasible to establish a relation between the drilling parameters and the rock c-φ parameter based on a mechanical analysis of the rock cutting failure process. Some mechanical models for rock cutting (Nishimatsu, 1972, 1993; Gerbaud et al., 2006; Song et al., 2010) have been developed to investigate the cutting mechanism and optimise drilling tool design. The existing mechanical models for cutting require the assumption of a cutting failure surface shape and the stress distribution. In this study, based on rock cutting failure characteristics, a rock cutting upperbound analytic mechanics model is proposed. In the proposed model, no assumption for the cutting failure surface stress distribution is required. It provides a more accurate match for the actual failure mode and is applicable in a wider range of scenarios. In this study, the ultimate rock cutting force is derived and the formula for the relation between the drilling parameters and the c-φ parameter (hereinafter referred to as the DP-CΦ Formula) is developed based on the proposed rock cutting upper bound analytic mechanics model. Indoor testing is performed, using a multi-function true triaxial rock drilling test system developed in-house, to validate the model. Based on the results, the field surrounding rock c-φ parameter digital drilling measurement method is proposed. The feasibility and effectiveness of the proposed method is verified via indoor testing. This method is convenient to implement in the field and can measure the c-φ parameter of both intact and relatively fragmented rock mass. The proposed method supports surrounding rock stability analysis and support parameter optimisation in underground engineering.
(1) Because the cutting width far exceeds the cutting depth H, the rock cutting problem is simplified as a plane strain problem. (2) Because the weight of the rock (gravitational force) in the cutting area is far smaller than the cutting force, gravitational force is not considered in this model. (3) The rock is treated as an ideal rigid plastic material, i.e., the strain strengthening and softening effects are not considered. (4) The rock follows the Coulomb yield criteria and satisfies the associated flow rule; and (5) Rock cutting failure is caused by shear slip along a plane, the rock above the slip surface is treated as a rigid body and deformation is only in a thin failure layer between the virgin rock and the rock chips. Based on the above assumptions and the rock cutting failure characteristics, a rock cutting mechanical model is developed, as shown in Fig. 3. In the diagram, OA is the cut failure surface, whose angle with respect to the horizontal plane is β; Fc is the cutting force exerted by the cutting edge on the rock in the front; Ff is the force exerted by the cutting edge on the bottom rock; γ is the angle between the cutting force Fc and the normal direction of the surface of the cutting edge; κ is the inclination angle of the cutting edge; δ is the angle between the force Ff and the vertical direction, which is also the friction angle between the cutting edge and the rock; φ is the internal friction angle of the rock; and V is the velocity of the rock chips along the slip surface. Based on the associated flow rule and the Coulomb yield criterion, the angle between the direction of V and the surface of the cut failure is φ (Chen, 1975).
2.2. Analysis of the ultimate rock cutting force Rock cutting failure is caused by the slip of a plane whose angle with respect to the horizontal plane is β. Based on the ultimate analysis upper bound theory, when the rock fracture power W generated by the cutting force Fc and the cut failure surface energy dissipation rate D are equal, the ultimate condition is met. The rock fracture power W is the product of the component of the velocity V along the Fc direction and Fc as follows:
W = Fc V cos(β + φ + κ + γ )
2. Analysis of the theory of rock cutting
(1)
The energy dissipation rate D on the cut failure surface OA is as follows:
2.1. Mechanism of rock cutting and the basic theoretical assumptions The rock cutting is performed using conventional polycrystalline diamond compact (PDC) drill bits, wherein the PDCs are embedded in a matrix to form cutting edges that crushes rock, as shown in Fig. 1. The rock is crushed by the cutting edge under the combined effects of the vertical force F1 and the horizontal force F2, as shown in Fig. 2. The rock
D = c (V cosφ)
H sinβ
(2)
When the rock reaches the ultimate state, the rock fracture power W and cut failure surface energy dissipation rate D are equal, as follows: 288
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 1. Three-dimensional schematic diagram of the PDC drill bit.
Fig. 3. Rock cutting upper bound analytic mechanics model.
(a) Rock ductile failure
Fc =
2cH cosφ 1−sin(φ + γ + κ )
(5)
3. Relation between the drilling parameters and the rock c-φ parameter 3.1. Solution for the drill bit torque M The drill bit torque M consists of two components, the cut torque Mc and the friction torque Mf, as shown in Fig. 3. The cut torque Mc is the drilling rig moment for rock cutting, which is obtained by calculating the moment of the horizontal component Fc with respect to the drill bit centre; Mf is the moment exerted by the drilling rig to overcome the friction between the drill bit and the rock below, which is obtained by calculating the moment of the horizontal component of Ff with respect to the drill bit centre. As shown in Fig. 4, the drill bit radius is R; there are 3 rows of cutting edges; the length of the i-th row of the cutting edge is Li; and the distance from a section of a row of the cutting edge to the drill bit centre O is r. In any segment dr, the torque on the drill bit is as follows:
(b) Rock brittle failure Fig. 2. Rock cutting process: (a) Rock ductile failure, (b) Rock brittle failure.
Fc V cos(β + φ + κ + γ ) = c (V cosφ)
H sinβ
(3)
dM = dMc + dMf = Fc r cos(κ + γ ) dr + Ff r sinδdr
After simplification, Fc is as follows:
Fc =
cH cosφ sin(φ + 2β + κ + γ )−sin(φ + κ + γ )
(6)
The integral of the moment dM along the cutting edge length is calculated and the moments at all row cutting edges are summed to obtain the overall drilling rig torque M as follows:
(4)
3
According to the minimum energy principle, when ∂Fc / ∂β = 0 , there is a minimum upper bound for Fc and β = (π /2−φ−κ−γ )/2 . The above expression is substituted into Eq. (4) to obtain the ultimate cutting force as follows:
M=
R ∑ ∫R−Li (Fc cos(κ + γ ) + Ff sinδ ) rdr
i=1 1
= 2 [Fc cos(κ + γ ) + Ff sinδ ][2R (L1 + L2 + L3)−(L12 + L22 + L32)] 289
(7)
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 4. Schematic diagram of the drill bit and one row of the cutting edge.
3.2. Solution for the thrust F The thrust F is the pressure exerted by the drilling rig to overcome the vertical component of Fc and Ff. For the drill bit shown in Fig. 3, F is as follows: (8)
Fig. 5. Multi-function true triaxial rock drilling test system and the PDC exploration drill bit developed for this study.
3.3. Formula for the relation between the drilling parameters and the rock cφ parameter
square PDCs. During drilling, the shape and the stress characteristics of the PDC will not change even if there is wear, which reduces the impact of PDC wear on the test data. In addition, a square PDC is a more accurate match for the theoretical assumptions used in this study. The geometrical parameters of the drill bit are listed in Table 1. In the tests, the same batch of limestone and sandstone specimens are used. The specimen dimension is 150 mm × 150 mm × 200 mm, as shown in Fig. 6. And the powdery debris is flushed by water and the water pressure is 0.15 MPa. Triaxial and uniaxial tests are performed for two types of rock specimens to measure their mechanical parameters. The results are listed in Table 2.
F = [Fc sin(κ + γ ) + Ff cosδ ](L1 + L2 + L3)
Eqs. (7) and (8) are combined to eliminate the unknown force Ff as follows:
M=
1 Fc [cos(κ + γ )−sin(κ + γ )tanδ ][2R (L1 + L2 + L3) 2 L 2 + L22 + L32 ⎞ 1 −(L12 + L22 + L32)] + F tanδ ⎛⎜2R− 1 ⎟ L1 + L2 + L3 ⎠ 2 ⎝
(9)
When the drilling rig drills into the rock, sensors measure the drilling parameters, including the rotating speed N, the drilling rate Vd, the torque M and the thrust F. The rock-cutting depth H of a single row of the drill bit cutting edge in one rotation is as follows:
V H= d mN
4.2. Test plan
(10)
The tests are based on the mode that controls the drilling rate Vd, the drill rotating speed N, the collection torque M and the thrust F. A series of five tests are designed for each type of rock. The test series for limestone and sandstone are designated Ai and Bi (i = 1, 2, 3, 4, 5). The series for limestone includes a combination of different N and Vd; the test series for sandstone includes identical N with different Vd. In this way, the relation formula for the drilling parameters and the rock c-φ parameter is thoroughly verified. The detailed test plans are listed in Table 3:
where m is the number of rows of cutting edges in the drill bit. Eqs. (5) and (10) are substituted into Eq. (9) to obtain the DP-CΦ Formula as follows:
M=
cVd cosφ [cos(κ + γ )−sin(κ + γ )tanδ ] [2R (L1 + L2 + L3) mN [1−sin(φ + κ + γ )] −(L12 + L22 + L32)] +
L 2 + L22 + L32 ⎞ 1 F tanδ ⎜⎛2R− 1 ⎟ 2 L1 + L2 + L3 ⎠ ⎝
(11)
4.3. Comparative analysis of the test results versus the theoretical calculation
4. Indoor test verification 4.1. Test equipment and test material
4.3.1. Test results and the torque The variation curves of M versus Dh (Figs. 7 and 9) and F versus Dh (Figs. 8 and 10) for different drilling tests are created based on one set of typical data from each test series for limestone and for sandstone, where Dh is the rock-drilling depth of the drilling rig. The data in Figs. 7 and 8 show that in the different tests for limestone, the variation rules of M and F versus Dh are similar, consisting of
To investigate the relation of the drilling parameters to the rock mechanical parameters and the rock mass characteristic parameters, we have developed a multi-function true triaxial rock drilling test system, as shown in Fig. 5. When performing rock specimen drilling tests, this system can provide a maximum torque of 400 N·m, a maximum rotating speed of 400 r/min and a maximum thrust of 50 kN. The system can control and monitor any of the four drilling parameters, including the drilling rate, the rotating speed, the thrust and the torque; record the drilling parameters during drilling; and display the variation curve versus time automatically. The drill bit used in the test is an exploration PDC drill bit designed in-house, as shown in Fig. 5. The cutting edge of this drill bit is based on
Table 1 Geometrical parameters of the exploration PDC drill bit.
290
Bit parameters
R (mm)
L1 (mm)
L2 (mm)
L3 (mm)
κ (°)
Sizes
30
18
18
27
15
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Fig. 6. Limestone and sandstone specimens before drilling and after drilling.
14
Table 2 Physical and mechanical parameters of the limestone and the sandstone samples used in the tests.
Limestone Sandstone
Uniaxial compressive strength, UCS (MPa)
Elastic modulus, E (GPa)
Cohesion, c (MPa)
Internal friction angle, φ (°)
69.2 49.8
23.31 19.82
14.27 11.56
43.86 39.52
10
8
F (kN)
Rock type
12
6 4
A1
A2
A4
A5
A3
2 Table 3 Detailed test plans for the limestone and the sandstone specimen drilling tests. Rock type
Test scheme label
Vd (mm/ min)
N (r/min)
Number of specimens
Limestone
A1 A2 A3 A4 A5
60 60 80 80 110
100 200 100 200 250
3 3 3 3 3
B1 B2 B3 B4 B5
30 60 80 110 140
200 200 200 200 200
3 3 3 3 3
0
2
4
6
8
10
12
14
16
Dh (cm) Fig. 8. Variation curves of F versus Dh for different limestone drilling tests.
90 80 70 60
M (N•m)
Sandstone
0
50 40 30 20
B1
B2
10
B4
B5
0
0
2
4
6
8 10 Dh (cm)
12
B3
14
16
Fig. 9. Variation curves of M versus Dh for different sandstone drilling tests.
8 7 6
F (kN)
5
4 3
Fig. 7. Variation curves of M versus Dh for different limestone drilling tests.
the following two stages:
2
B1
B2
1
B4
B5
0
①Before the drill bit enters the rock, M and F are small; after it drills into the rock, M and F increase sharply in a small Dh. This is the rising stage. ②After the drill bit reaches a shallow drilling depth, M and F reach a stable stage. During this stable stage, when Dh increases, M fluctuates slightly around a stable value. This behaviour may be related to rock heterogeneity and the alternating stages of cutting rock.
0
2
4
6
8 10 Dh (cm)
12
B3
14
16
Fig. 10. Variation curves of F versus Dh for different sandstone drilling tests.
versus Dh in different tests for sandstone is similar to that for limestone. However, the behaviour of the sandstone is more stable overall than that of the limestone. The reason is that some limestone specimens have few fractures, while sandstone specimens are relatively intact.
The data in Figs. 9 and 10 show that the variation rule of M and F 291
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
The test calculation methods for M and F for the limestone and sandstone specimens are identical. The calculation for M is used as an example to illustrate the process. Before the rock specimen is drilled, the initial torque is MI, the average torque in the stable stage is MS and the specimen torque test value is M = MS−MI . Using the data for the limestone specimen in the A1 test in Fig. 7 as an example, MI = 26.25 N m, MS = 125.37 N·m and M = MS−MI = 99.12 N·m . In each test, the theoretical value is compared with the average M from the three specimens tested.
4.3.2. Selection of the parameters for the theoretical analysis The geometrical parameters of the drill bit designed for this study are listed in Table 1. The values of c and φ are based on measurements of the specimens in the triaxial tests and are listed in Table 2. γ is based on the research findings by Huang et al. (2013); and κ has a significant impact on γ, i.e., as κ increases, γ decreases. For the drill bit used in this study, κ = 15°and γ = 12°. Based on the research findings by Li et al. (2015), δ is 12°. The values defined for Vd and N in each test series are listed in Table 3. The aforementioned parameters and the thrust F are substituted into Eq. (11) to obtain the theoretical torque M.
Fig. 12. Comparative analysis diagram of the torque indoor tests versus the theoretical calculations for the sandstone tests.
5. Field surrounding rock c-φ parameter digital drilling measurement method 5.1. The proposed method Based on the DP-CΦ Formula developed in this study, a field surrounding rock c-φ parameter digital drilling measurement method is proposed (hereinafter referred to as the digital drilling measurement method). Implementation of this method includes three steps: Step A: A field rock drilling test is performed via the digital drilling measurement method and the rock cutting strength Sc is calculated based on the acquired drilling parameters. The cutting strength Sc, as defined in this paper, is calculated via Eq. (11), resulting in the calculation formula as follows:
4.3.3. Comparison and evaluation of the theoretical analysis with respect to the indoor test results The comparative analysis diagrams of the torque measured in the indoor tests versus the theoretical calculations for the limestone tests and the sandstone tests are shown in Figs. 11 and 12. The data show that, in both the limestone and the sandstone tests, the measured torque from the indoor tests and the theoretical calculations are consistent; however, the sandstone tests exhibit better conformity. To quantitatively evaluate the difference between the results from the theoretical method proposed in this study and the results from conventional experiments, a difference rate index ξ is defined as follows:
ξ=
|thR−exR| × 100% exR
Sc =
2c cosφ 1−sin(φ + κ + γ ) L2 + L 2 + L 2
mN ⎡2M −F tanδ ⎛2R− L1 + L2 + L 3 ⎞ ⎤ 1 2 3 ⎠ ⎝ ⎣ ⎦ = Vd [cos(κ + γ )−sin(κ + γ )tanδ ][2R (L1 + L2 + L3)−(L12 + L22 + L32)]
(12)
(13) where thR is the result from the theory or method proposed in this study and exR is the result from the conventional experiment. This index is also used in other sections of this paper, with the same definition. The analysis shows that the average ξ for the limestone tests is 9.33% and the average ξ for the sandstone tests is 5.85%. This also indicates that the model exhibits better conformity for sandstone. This is because some limestone specimens contain fracture, which increase the discreteness of the test data. For both types of rock, the difference rates ξ for the test versus theory are less than 10%. This proves the feasibility and the effectiveness of the rock cutting upper bound analytic mechanics model and DPCΦ Formula proposed in this study.
This formula, in tandem with Eq. (5), shows that the physical meaning of Sc is the ultimate cutting force per unit depth of rock. Step B: Based on the drilling parameters of the digital drilling rig (Huang and Wang, 1997; Karasawa et al., 2002; Mostofi et al., 2011) or a point load test (Bieniawski, 1974; Panek and Fannon, 1992; Al-Derbi and Freitas de, 1999; Basu and Aydin, 2006), the uniaxial compressive strength Rc is obtained. A large number of comparative test results show that the rock uniaxial compressive strength can be calculated based on the drilling parameters of the digital drilling rig or the point load strength. According to the Mohr-Coulomb yield criterion, the relation between the rock uniaxial compressive strength Rc and the c-φ parameter is as follows:
Rc =
2c cosφ 1−sinφ
(14)
Step C: The rock internal friction angle φ and the cohesion c are calculated based on Sc and Rc obtained from the previous two steps. The equation for the internal friction angle calculation is obtained by combining Eqs. (13) and (14) as follows:
1−
Rc R + c sinφ−sin(κ + γ + φ) = 0 Sc Sc
(15)
Subsequently, φ from Eq. (15) is substituted into Eqs. (13) or (14) to obtain c.
Fig. 11. Comparative analysis diagram of the torque indoor tests versus the theoretical calculations for the limestone tests.
292
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
Table 4 Comparative analysis of the c-φ parameter from the digital drilling measurement method versus that from the triaxial method in the limestone and sandstone tests. Rock type
Test scheme
φd (°)
ξφ
cd (MPa)
ξc
Rock type
Test scheme
φd (°)
ξφ
cd (MPa)
ξc
Limestone
A1 A2 A3 A4 A5 Average value
41.26 46.03 39.14 40.77 42.59 41.96
5.93% 4.96% 10.77% 7.05% 2.91% 6.32%
15.67 13.97 16.45 15.85 15.19 15.43
9.84% 2.12% 15.30% 11.10% 6.47% 8.97%
Sandstone
B1 B2 B3 B4 B5 Average value
31.72 38.35 39.83 40.29 36.88 37.41
19.74% 2.95% 0.79% 1.94% 6.67% 6.42%
13.88 12.05 11.66 11.54 12.45 12.31
20.09% 4.24% 0.83% 0.22% 7.67% 6.61%
5.2. Method verification
Acknowledgments
The proposed digital drilling measurement method is verified with drilling parameter data obtained from limestone and sandstone specimen drilling tests using the multi-function true triaxial rock drilling test system (Figs. 7–10), described in Section 4, and the uniaxial compressive strength data for the two types of rock (Table 2). The internal friction angle φd and the cohesion cd measured by the digital drilling measurement method are calculated using Eqs. (13) and (15) in the limestone and sandstone tests. These results, combined with φ and c for the two types of rock from the triaxial test (Table 2) and Eq. (12), are used to calculate the internal friction angle difference rates ξφ and the cohesion difference rates ξc for the two methods. The results are shown in Table 4. The data in Table 4 shows that in the limestone tests, the average difference rate of the internal friction angle from the digital drilling measurement method versus that from the triaxial method is ξφ = 6.32%; for cohesion, it is ξc = 8.97%. In the sandstone tests, the average difference rate of the internal friction angle is ξφ = 6.42%; for cohesion, it is ξc = 6.61%. In all the tests, for both types of rock, the average difference rates of the cohesion and the internal friction angle are always less than 10%. This proves the effectiveness and the feasibility of the field surrounding rock c-φ parameter digital drilling measurement method proposed in this paper.
This work was supported by the Natural Science Foundation of China (grant numbers 51674154, 51704125); Open Fund for State Key Laboratory for Geo-mechanics and Deep Underground Engineering, China University of Mining & Technology (grant numbers SKLGDUEK1519, SKLGDUEK1717); and the China Postdoctoral Science Foundation (grant numbers 2016M590150, 2016M602144). The authors of this article are so grateful to the anonymous reviewers for their valuable comments and modification suggestions on the improvement of this paper. References Aalizad, S.A., Rashidinejad, F., 2012. Prediction of penetration rate of rotary-percussive drilling using artificial neural networks-a case study. Arch. Min. Sci. 57 (3), 715–728. Akin, S., Karpuz, C., 2008. Estimating drilling parameters for diamond bit drilling operations using artificial neural networks. Int. J. Geomech. 8 (1), 68–73. Al-Derbi, M.S., Freitas de, M.H., 1999. Use of the boussinesq equation for determining the distribution of stress within a diametrical point load test. Rock Mech. Rock Eng. 32 (4), 257–265. Basu, A., Aydin, A., 2006. Predicting uniaxial compressive strength by point load test: significance of cone penetration. Rock Mech. Rock Eng. 39 (5), 483–490. Bieniawski, Z.T., 1974. Estimating the strength of rock materials. J. South African Institute Mining Metall. 74 (8), 312–320. Chaput, E.J., 1992. Observations and analysis of hard rocks cutting failure mechanisms using pdc cutters. Technical Report, Project at Imperial College, London, Report published by Elf-aquitaine, France. Chen, J., Yue, Z.Q., 2015. Ground characterization using breaking-action-based zoning analysis of rotary-percussive instrumented drilling. Int. J. Rock Mech. Min. Sci. 75, 33–43. Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier. Feng, X.T., Zhao, H., Li, S., 2004. A new displacement back analysis to identify mechanical geo-material parameters based on hybrid intelligent methodology. Int. J. Numer. Anal. Meth. Geomech. 28, 1141–1165. Gerbaud, L., Menand, S., Sellami, H., 2006. PDC Bits: all comes from the cutter rock interaction. In: IADC/SPE Drilling Conference. U.S.A, Miami, Florida, pp. 1–9. Gioda, G., Locatelli, L., 1999. Back analysis of the measurements performed during the excavation of a shallow tunnel in sand. Int. J. Numer. Anal. Meth. Geomech. 23, 1407–1425. Gui, M.W., Soga, K., Bolton, M.D., Hamelin, J.P., 2002. Instrumented borehole drilling for subsurface investigation. J. Geotech. Geoenviron. Eng. 128 (4), 283–291. Huang, H., Lecampion, B., Detournay, E., 2013. Discrete element modeling of tool-rock interaction I: rock cutting. Int. J. Numer. Anal. Meth. Geomech. 37 (13), 1913–1929. Huang, S.L., Wang, Z.W., 1997. The mechanics of diamond core drilling of rocks. Int. J. Rock Mech. Min. Sci. 34 (3/4), 612. Kahraman, S., 1999. Rotary and percussive drilling prediction using regression analysis. Int. J. Rock Mech. Min. Sci. 36 (7), 981–989. Kahraman, S., Bilgin, N., Feridunoglu, C., 2003. Dominant rock properties affecting the penetration rate of percussive drills. Int. J. Rock Mech. Min. Sci. 40 (5), 711–723. Karasawa, H., Ohno, T., Kosugi, M., Rowley, J.C., 2002. Methods to estimate the rock strength and tooth wear while drilling with roller-bits. J. Energy Resour. Technol. 124 (3), 125–132. Li, J.Y., Tan, Z.Y., Li, W., Yue, P.J., 2015. Experimental simulation of dynamic friction characteristics of interface between diamond drill and rock under impact-rotational loading. J. Vibr. Shock 34 (22), 210–214 (in Chinese). Mostofi, M., Rasouli, V., Mawuli, E., 2011. An estimation of rock strength using a drilling performance model: a case study in Blacktip field, Australia. Rock Mech. Rock Eng. 44 (3), 305–316. Nishimatsu, Y., 1972. The mechanics of rock cutting. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 9 (2), 261–270. Nishimatsu, Y., 1993. Theories of rock cutting. In: Hudson, J.A. (Ed.), Comprehensive Rock Engineering: Principles Practice and Projects. Pergamon Press, Oxford, New York. Panek, L.A., Fannon, T.A., 1992. Size and shape effects in point load tests of irregular rock fragments. Rock Mech. Min. Sci. 25 (1), 109–140. Richard, T., 1999. Determination of rock strength from cutting tests. Dissertation,
6. Conclusions (1) In this study, based on rock cutting failure characteristics, a rock cutting upper bound analytic mechanics model is proposed. This model avoids assuming a stress distribution on the rock cutting failure surface and can be applied to a wide range of scenarios. Based on the principle of virtual work, the ultimate rock cutting force is derived and the DP-CΦ Formula is developed. (2) The drilling tests for the limestone and the sandstone specimens are performed using a multi-function true triaxial rock drilling test system. The drilling parameters from the tests and the parameters from the theoretical calculation are compared and analysed. The results show that in the limestone tests, the average difference rate ξ is 9.33%, while the average difference rate ξ for the sandstone tests is 5.85%. This validates the rock cutting mechanical model and the DP-CΦ Formula. (3) Based on the relation formula for the drilling parameters and the rock c-φ parameter, the field surrounding rock c-φ parameter digital drilling measurement method is proposed. Results using the proposed method and results from the conventional triaxial method are compared and analysed. The results show that the average difference rates of the cohesion and the internal friction angle are less than 10%. This proves the effectiveness and the feasibility of the digital drilling measurement method proposed in this study. (4) The field surrounding rock c-φ parameter digital drilling measurement method proposed in this study is convenient to implement in the field. The method can measure the c-φ parameter for both intact and relatively fragmented rock mass in the field and provides references for surrounding rock stability analysis and optimisation design of support parameters in underground engineering. 293
Tunnelling and Underground Space Technology 73 (2018) 287–294
Q. Wang et al.
granite formation. Chin. J. Rock Mech. Eng. 26 (3), 478–483 (in Chinese). Tan, Z.Y., Wang, S.J., Cai, M.F., 2009. Similarity identification method on formational interfaces and application in general granite. Int. J. Miner. Metall. Mater. 16 (2), 135–142. Yang, W.W., Yue, Z.Q., Tham, L.G., 2012. Automatic monitoring of inserting or retrieving SPT sampler in drillhole. Geotech. Test J. 35 (3), 420–436. Yaşar, E., Ranjith, P.G., Viete, D.R., 2011. An experimental investigation into the drilling and physico-mechanical properties of a rock-like brittle material. J. Petrol. Sci. Eng. 76 (3), 185–193. Yazdani, M., Sharifzadeh, M., Kamrani, K., Ghorbani, M., 2012. Displacement-based numerical back analysis for estimation of rock mass parameters in Siah Bisheh powerhouse cavern using continuum and discontinuum approach. Tunn. Undergr. Space Technol. 28, 41–48. Yue, Z.Q., Lee, C.F., Law, K.T., Tham, L.G., 2004. Automatic monitoring of rotary-percussive drilling for ground characterization—illustrated by a case example in Hong Kong. Int. J. Rock Mech. Min. Sci. 41 (4), 573–612. Zhang, L.Q., Yue, Z.Q., Yang, Z.F., Qi, J.X., Liu, F.C., 2006. A displacement-based backanalysis method for rock mass modulus and horizontal in situ stress in tunneling – Illustrated with a case study. Tunn. Undergr. Space Technol. 21, 636–649.
University of Minnesota. Richard, T., Detournay, E., Drescher, A., Nicodème, P., Fourmaintraux, D., 1998. The scratch test as a means to measure strength of sedimentary rocks. In: SPE/ISRM Rock Mechanics in Petroleum Engineering, Trondheim, Norway. Sakurai, S., 2003. Back analysis for tunnel engineering as a modern observational method. Tunn. Undergr. Space Technol. 185, 196. Schunnesson, H., 1996. RQD predictions based on drill performance parameters. Tunn. Undergr. Space Technol. 11 (3), 345–351. Schunnesson, H., 1998. Rock characteristics using percussive drilling. Int. J. Rock Mech. Min. Sci. 35 (6), 711–725. Song, L., Li, N., Liu, F.Y., 2010. Research on the mechanics of rotary penetration test. Chin. J. Rock Mech. Eng. 29 (Supp. 2), 3519–3525 (in Chinese). Suzuki, Y., Sasao, H., Nishi, K., Takesue, K., 1995. Ground exploration system using seismic cone and rotary percussion drill. AIJ J. Technol. Design 1 (1), 180–184. Swoboda, G., Ichikawa, Y., Dong, Q.X., Zaki, M., 1999. Back analysis of large geotechnical models. Int. J. Numer. Anal. Meth. Geomech. 23, 1455–1472. Tan, Z.Y., Cai, M.F., Yue, Z.Q., Tham, L.G., Lee, C.F., 2005. Application and reliability analysis of DPM system in site investigation of HK weathered Granite. J. Univ. Sci. Technol. Beijing 12 (6), 481–488. Tan, Z.Y., Yue, Z.Q., Cai, M.F., 2007. Analysis of energy for rotary drilling in weathered
294