# A New Method for Predicting Residual Strength of Rock in Water Diversion Tunnel Using Drilling Process Monitoring

Authors: Wanfei Gao
Journal: Shock and Vibration
 (2022)

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## Abstract

The peak and residual strengths of rock materials are important parameters for the stability evaluation of rock engineering, especially water diversion tunnels. A friction-strengthening model based on Hoek–Brown model is proposed to determine peak and residual strength of rock. The proposed model parameter, contact friction coefficient of rock, controls the nonlinearity of the peak strength and residual strength. With a great amount of data from publications examined, the relationship between the friction coefficient and the parameters in the Hoek–Brown model is studied. A new field method using drilling process monitoring was proposed to predict the peak strength and residual strength. The results show that the predicted strength for four types of rock is in good agreement with those of the standard tests in laboratory. Their errors are within the error range of 15% compared to the results from the tested results in laboratory. Although the proposed model is empirical, the parameters in the proposed model have clear physical meaning, and it can successfully predict the peak strength and residual strength of hard rock, medium rock, and weak rock using drilling process monitoring in the field. This practical method should have a great potential for field application in rock engineering.


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## Body

## 1. Introduction

The peak and residual strengths of rock materials are important parameters for the stability evaluation of rock engineering, especially for water diversion tunnels [1]. Peak strength has been well investigated, and the peak strength envelopes have been characterized using many models [2–9]. The residual strength is also related to the assessment of the stability of large-scale excavations in water diversion tunnels [1, 10], and the collapse properties of rock in deep tunnels for mining [11].Many researchers have established some models for the estimation of peak and residual strengths, such as the M-C model [12], J-B model [13], GSI-softening model [14], cohesion loss model [15], and H–B failure criterion, which was proposed by Hoek and Brown [2].(1)σ1=σ3+UCS⋅mσ3UCS+sa,Where σ1 is the major principal stress, σ3 is the minor principal stress, UCS is the unconfined compressive strength of intact rock, and s, a, and m are material constants. For the intact rock, m is equal to mi, a = 0.5, and s = 1. These models have been used in a variety of rock engineering. However, the M−C model can be used for the peak and residual strengths at the confining stress of <60 MPa due to its clear physical meaning, and it cannot describe the nonlinear strength properties of rock [15]. Due to the extent dependence of the confinement and unclear physical meaning, the J-B model cannot describe the residual strength properties so well, that it cannot be widely applied in engineering [13]. The GSI-softening model for estimating the residual strength is largely dependent on the determination of a residual GSI [14]. It is difficult to estimate the residual strength of the most brittle rock at zero confinement. The cohesion loss model [15], only parameter and simple form, can well present the residual strength nonlinearity. The parameter lacks physical meanings and should be determined by using a great amount of test data [15]. Eberhardt [16] has presented the advantages and limitations of the H–B model. The H–B model has been widely used in most types of rock engineering due to the nonlinear capturing of the peak strength envelopes [14, 17]. Since the H–B model can characterize the nonlinear properties over a wide range of confining stresses, it is also used to determine the residual strength of rock materials [18]. However, due to the physically incorrect reduction of σc to a residual parameter σr, the H–B model is not widely used to estimate the residual properties in rock engineering applications [15]. From the literature review, a reasonable model of strength envelope of rock should obey nonlinearity and the clear physical meanings of model parameters.In this work, the friction-strengthening model is established to estimate the peak and residual strengths of rock materials. The model parameter, contact friction coefficient, has the physical meaning of rock. The relationship between the friction coefficient and the model parameters is studied by using a great amount of data collected from the literature. A new method using drilling process monitoring was proposed to predict the peak strength and residual strength. Although the proposed model is empirical, it can successfully predict the peak strength and residual strength of hard rock, medium rock, and weak rock using drilling process monitoring in the field.


## 2. Data Collection

In this study, a great amount of the peak and residual strength data were collected from 28 various rock types. Data for these rocks were obtained from the literature as shown in Table1. Due to the cohesion lost largely in the deformation process of rock, the frictional properties of rock primarily control the residual strength [19, 20]. The friction of rock is only dependent on rock type at high confining pressures [21]. However, frictional properties vary significantly under low confining stresses, which is relevant to most rock engineering applications [21, 22]. Therefore, the data is collected at low confining pressures, which are lower than 60 MPa.Table 1
Fitted parameters of the peak and residual strength using the H–B model and the proposed model.
No.Rock typesPeak strengthResidual strengthφCReferencesσcmiR2ηmiηR2(°)(MPa)1Amarelo Pais granite76.5940.960.995.350.1350.9757.5912.42Arzúa et al. [35]2Blanco Mera granite13137.820.9825.120.1310.8059.5218.79Arzúa et al. [35]3Vilachan granite116.2928.230.9942.980.1090.8254.9119.39Arzúa et al. [35]4Blanco Mera granite122.7341.610.99144.880.1170.9560.3716.76Arzúa et al. [37]5Stanstead granite13625.90.9823.040.1170.7753.2024.98Walton [38]6Bunt sandstone6710.60.9678.560.7790.9933.6316.88Gowd [39]7Indiana limestone buff743.30.993.300.7510.9823.0924.45Walton et al. [40]8Red sandstone936.50.992.390.4240.9832.9125.29Yang et al. [41]9Gosford sandstone6113.30.993.300.2760.9340.2414.14Masoumi [42]10Carrara marble945.50.993.340.5070.9638.7522.55Walton et al. [43]11Westerly granite13848.30.997.020.1450.9664.2215.79Walton et al. [29]12Rhyolite1033.40.991.920.7000.9814.2040.09Walton et al. [29]13Indiana limestone gray497.30.997.040.7790.9738.219.95Walton et al. [29]14Porphyry10111.20.984.120.3680.9637.6924.80Walton et al. [29]15Rajnagar marble84.86.80.933.330.4900.8635.6920.92Kumar et al. [44]16Biotite chlorite schist67.35.40.975.400.3950.6231.7518.60Kumar et al. [44]17Purulia gneiss153.624.30.973.170.1310.9151.2828.76Kumar et al. [44]18Kirthai granite gneiss143.332.20.992.940.1090.9353.1032.80Kumar et al. [44]19Tennessee sandstone229.23.20.992.400.7510.9317.62103.08Cuss et al. [45]20Coal38.96.310.560.4900.9930.0011.23Yumlu and ozbay [46]21Sandstone81.322.70.983.420.1510.9952.0014.00Yumlu and ozbay [46]22Norite220.122.50.994.670.2080.9955.0034.70Yumlu and ozbay [46]23Quartzite264.854.6110.190.0910.9968.0025.74Yumlu and ozbay [46]24Hejiata coal G137.77.70.893.510.4560.8937.809.40Gao and kang [47]25Hejiata coal G237.212.60.962.810.2230.8844.408.00Gao and kang [47]26Coarse marble71.25.10.992.590.5070.9628.0022.14Yang et al. [48]27Medium marble112.97.217.200.5650.9934.4031.70Yang et al. [48]28Sandstone93.37.70.995.200.6750.9735.4028.36Yang et al. [48]In addition, the four rock types (limestone, slate, granite, and red sandstone) were obtained by a drilling process by monitoring apparatus (DPM) [23]. The DPM can continuously measure and record drilling operational data such as the thrust force, torque, penetration rate, rotation speed, drilling depth, and penetration depth per rotation. The data for the four rock types (limestone, granite, slate, and sandstone; see Figure 1) were performed using a WDT Rock Mechanics Testing System [24, 25], and testing protocols in the laboratory. The samples were obtained from a water conveyance tunnel in the Weihe River Basin, Shaanxi Province, China. The rock samples are also made with a standard size (height 100 mm × diameter 50 mm) for the compression test. To prevent eccentric compression, the two end faces of a cylindrical sample must remain flat and coincident in the axial direction. Specifically, the end surface deviation is controlled at ±0.05 mm, and the diameter and length error cannot exceed ±0.02 mm. The peak strength and residual strength of rock specimens were obtained by triaxial compression test with the confining pressure of 5, 10, 15, 20, and 25 MPa.Figure 1
Rock types tested in the work (slate, granite, red sandstone, and limestone).


## 3. A New Method for Peak and Residual Strength Predictions Using Drilling Process Monitoring

### 3.1. A Frictional Strengthening Model for Determining Peak and Residual Strengths of Rock

In general, there are two main components: frictional strength and cohesive strength for rock strength [15, 26]. The rock deformation also referred normally to the cohesion degradation and friction mobilization [20], which have been widely used to investigate the deformability and failure of rock [27, 28]. The parameters mb and s of the H–B model are analogous to the cohesion and internal friction in the M-C model, respectively. The parameter s is related to the rock fragmentation [29]. The maximum and minimum value of s is 1.0 and 0, which are related to intact rock and heavily jointed or damaged rock, respectively [29]. For intact rock s = 1, the parameter mi depends mainly on the lithology, crystallization degree, mineral composition, and grains of rock. The physical properties of rock have a great effect on the resistance caused by dislocation and slipping movement, namely, “friction” [29]. Due to the deformation generation, the fraction is mobilized gradually in stages I, II, and III (see Figure 2). The fractional strength gradually occupies a leading position for the rock strength. A frictional model is proposed to describe the peak strength properties of rock and can be expressed as(2)σ1=σ3+UCS⋅λtanθ+bσ3UCS+10.5,where σ1 is the major principle stress, σ3 is minor principal stress, UCS is the unconfined compressive strength of intact rock, λ and b are the model parameters, and tanθ is the contact friction coefficient between the fracture faces of the rock, which can be determined by drilling process monitoring.Figure 2
The frictional strength in the deformation process [29].In consideration of the frictional strength mobilization in the deformation stage of rock, the damage of macroscopic fracture significantly leads to a large loss of cohesion. Figure2 shows that the friction strength keeps almost a constant value in stage IV, and even after. The reduction of strength is due to the loss of cohesion strength. Considering its physical nature, the residual strength is purely “frictional” in nature in most cases [29]. In stage III, the fraction is related to the sliding friction between fracture surfaces. The frictional strength results from sliding, which occupies fully a leading position for the residual strength of rock. The parameter mi for the residual strength in the H–B model is taken as η(λtanθ+b). The parameter s is taken as zero [15]. The frictional strength will reach the limit and be close to a constant in the residual deformation stage (see Figure 2). For intact rock a = 0.5, a new model is proposed to capture the residual strength envelopes of rock, and it can be expressed as(3)σ1=σ3+UCS⋅ηλtanθ+bσ3UCS0.5,where η is a model parameter, which is mainly dependent on the fraction coefficient of rock.


### 3.2. Contact Fraction Coefficient Response to Internal Friction Coefficient of Rock

Generally, all of these drilling parameters are also related to the types of rock [4] and their mechanical properties [30]. Many researchers also conduct the drilling tests, the obtained results of which indicate the linear relation between the penetration and torque force Ft [31]. The aspect ratio of the Ft ∼ Fn linear curve is mainly dependent on the friction angle of the rock. The aspect ratio of Ft ∼ Fn linear curve is a constant as follows [30]:(4)FtFn=1tana+θ=H,where θ is the friction angle between the fracture faces of the rock, H is the slope of Ft ∼ Fn linear curve and is constant, and a is the rake angle of drilling bit cutter. After some algebraic manipulation, we have(5)tana=1−HtanθH+tanθ.During the rock drilling process, the friction coefficient can be calculated as [30](6)tanφ′=tana+θ−tana1−tanatana+θ,where φ′ is the friction angle between the intact rock and compressed crushed zone. By estimating the friction angle φ′ using (6), the internal friction angle φ of rock is calculated as [32, 33](7)tanφ′=π2tanφ,where φ is the internal friction angle of the rock. According to (6) and (7), the contact friction coefficient tanθ can be calculated as(8)tanθ=tanarc tanπ/2tanφ+tana1+π/2tanφtana−a.Using (5) and (8), the variation of the contact friction coefficient tanθ with an internal friction angle of rock (φ) is also expressed as(9)tanθ=πH2tanφ−πtanφ2−2πHtanφ+2H2.The variation of the contact friction coefficient tanθ with the internal friction angle of rock (φ) can be well described. Given the aspect ratio of Ft ∼ Fn linear curve and the rake angle of bit cutter, the internal friction angle of rock and friction coefficient is also determined using (8) and (9).


### 3.3. Determination of Model Parameters

The proposed model has two dimensionless parameters (λ and b) in (2) and (3). The function (λtanθ+b) is related to the parameter mi in the H–B model [34]. The model parameters (λ, b, and η) are determined by a great amount of the tested data from the published literature. The relationship between the contact friction coefficient and the parameter mi is established.The peak strength, residual strength, cohesion, and internal friction angle of rock are mostly provided, and some are collected by interpreting the M-C model in the published literature. The H–B model is used to fit the peak strength [29], and the proposed model in (3) is used to fit the data of the residual strength. Table 1 presents the fitted parameter mi. Using (8), the contact friction coefficient tanθ is calculated by using the internal friction angle of rocks in the published literature at the rake angle of bit cutter of 5°, 10°, 15°, and 20°, respectively. Table 2 shows the friction angle θ and contact friction coefficient tanθ of the proposed model. A linear relationship between the friction coefficient and the parameter mi is presented in Figure 3, which can be expressed as(10)mi=λtanθ+b.Table 2
Summary of the calculated contact friction coefficient at the different rake angles.
No.Rock typesφ' (°)a = 5°a = 10°a = 15°a = 20°θ (°)tanθθ (°)tanθθ (°)tanθθ (°)tanθ1Amarelo Pais granite67.9863.522.0158.731.6553.641.3648.241.122Blanco Mera granite69.4564.912.1460.071.7454.941.4349.521.173Vilachan granite65.8961.541.8556.851.5351.821.2746.451.054Blanco Mera granite70.0965.512.2060.651.7855.511.4650.081.195Stanstead granite64.5260.251.7555.621.4650.641.2245.301.016Bunt sandstone46.2441.400.8838.050.7834.130.6829.690.577Indiana limestone buff33.7938.170.7935.080.7031.410.6127.170.518Red sandstone45.4643.690.9640.150.8436.070.7331.480.619Gosford sandstone53.0449.571.1745.591.0241.130.8736.220.7310Carrara marble51.5646.941.0743.150.9438.860.8134.090.6811Westerly granite72.9068.212.5063.261.9858.061.6052.611.3112Rhyolite21.6639.170.8136.000.7332.240.6327.940.5313Indiana limestone gray51.0341.680.8938.300.7934.370.6829.900.5814Porphyry50.5047.231.0843.420.9539.110.8134.320.6815Rajnagar marble48.4445.331.0141.670.8937.480.7732.800.6416Biotite chlorite schist44.1745.631.0241.940.9037.730.7733.030.6517Purulia gneiss62.9558.781.6554.221.3949.291.1644.000.9718Kirthai granite gneiss64.4460.181.7455.551.4650.571.2245.241.0119Tennessee sandstone26.5142.250.9138.820.8034.850.7030.350.5920Coal42.1943.270.9439.770.8335.720.7231.160.6021Sandstone63.5459.441.6954.851.4249.901.1944.590.9922Norite65.9660.091.7455.461.4550.481.2145.161.0123Quartzite75.5770.772.8765.762.2260.531.7755.081.4324Hejiata coal G150.6147.331.0843.510.9539.190.8234.400.6825Hejiata coal G256.9653.201.3448.971.1544.300.9839.220.8226Coarse marble39.8542.970.9339.490.8235.460.7130.920.6027Medium marble47.0744.080.9740.510.8536.410.7431.800.6228Sandstone48.1342.640.9239.190.8235.180.7030.660.59Figure 3
A linear relationship between the contact friction coefficient and the parametermi of the H–B model.
(a)(b)(c)(d)The correlation coefficients (R2 > 0.95) show that the parameters λ and b can be well obtained using equation (10). The value of λ in the proposed model is 26.296, 35.143, 45.272, and 56.498, and the value of b is -19.602, -23.287, -25.7, and -28.362 at the rake angle of bit cutter of 5°, 10°, 15°, and 20°, respectively. A linear relationship between the parameters λ and b of the proposed model and the rake angle of bit cutter is presented in Figure 4, which can be expressed as(11)λ=2.0147a+15.619,b=−0.5739a−17.065.Figure 4
A linear relationship between the parametersλ and b of the proposed model and the rake angle of the bit cutter.The squared correlation coefficientR2 is larger than 0.99. Equation (11) is used to determine the parameters λ and b. The frictional model for the determination of the peak strength of rock can be rewritten as(12)σ1=σ3+UCS⋅2.0147a+15.619tanθ−0.5739a−17.065σ3UCS+10.5.Given the friction angle and the rake angle of the bit, the peak strength can be estimated using (12).The proposed model (3) is used to fit the data of the rock residual strength. The model parameters (λ, b, and tanθ) are obtained by (10) and (11). The model parameter η is determined using the data collected from literature as shown in Table 1. This parameter η is presented as the exponential function of the friction coefficient as shown in Figure 5, which can be expressed as(13)η=2.1287mi−0.7826.Figure 5
Correlation of the parametersη and mi of the H–B model.The linear correlation coefficientR2 is larger than 0.85. Since mi is accepted as a fitting parameter, whose range is from 0.001 to 25, the range of the parameter η is 0.17 to 474 according to (13). In addition, there are several definitions of mi according to different perspectives, such as the lithology, crystallization degree, and mineral composition and grains of rock. Some studies also show that parameter mi is linked to the tensile strength and compressive strength of rock [4, 34]. This implies that the parameter η is dependent on the physical and mechanical properties of rock. Equation (13) is used to determine the model parameter η. The frictional model (3) can be written as(14)σ1=σ3+2.1287σ3UCS2.0147a+15.619tanθ−0.5739a−17.0650.21740.5.Given the friction angle and rake angle of the bit, the residual strength can be estimated using (14).


## 3.1. A Frictional Strengthening Model for Determining Peak and Residual Strengths of Rock

In general, there are two main components: frictional strength and cohesive strength for rock strength [15, 26]. The rock deformation also referred normally to the cohesion degradation and friction mobilization [20], which have been widely used to investigate the deformability and failure of rock [27, 28]. The parameters mb and s of the H–B model are analogous to the cohesion and internal friction in the M-C model, respectively. The parameter s is related to the rock fragmentation [29]. The maximum and minimum value of s is 1.0 and 0, which are related to intact rock and heavily jointed or damaged rock, respectively [29]. For intact rock s = 1, the parameter mi depends mainly on the lithology, crystallization degree, mineral composition, and grains of rock. The physical properties of rock have a great effect on the resistance caused by dislocation and slipping movement, namely, “friction” [29]. Due to the deformation generation, the fraction is mobilized gradually in stages I, II, and III (see Figure 2). The fractional strength gradually occupies a leading position for the rock strength. A frictional model is proposed to describe the peak strength properties of rock and can be expressed as(2)σ1=σ3+UCS⋅λtanθ+bσ3UCS+10.5,where σ1 is the major principle stress, σ3 is minor principal stress, UCS is the unconfined compressive strength of intact rock, λ and b are the model parameters, and tanθ is the contact friction coefficient between the fracture faces of the rock, which can be determined by drilling process monitoring.Figure 2
The frictional strength in the deformation process [29].In consideration of the frictional strength mobilization in the deformation stage of rock, the damage of macroscopic fracture significantly leads to a large loss of cohesion. Figure2 shows that the friction strength keeps almost a constant value in stage IV, and even after. The reduction of strength is due to the loss of cohesion strength. Considering its physical nature, the residual strength is purely “frictional” in nature in most cases [29]. In stage III, the fraction is related to the sliding friction between fracture surfaces. The frictional strength results from sliding, which occupies fully a leading position for the residual strength of rock. The parameter mi for the residual strength in the H–B model is taken as η(λtanθ+b). The parameter s is taken as zero [15]. The frictional strength will reach the limit and be close to a constant in the residual deformation stage (see Figure 2). For intact rock a = 0.5, a new model is proposed to capture the residual strength envelopes of rock, and it can be expressed as(3)σ1=σ3+UCS⋅ηλtanθ+bσ3UCS0.5,where η is a model parameter, which is mainly dependent on the fraction coefficient of rock.


## 3.2. Contact Fraction Coefficient Response to Internal Friction Coefficient of Rock

Generally, all of these drilling parameters are also related to the types of rock [4] and their mechanical properties [30]. Many researchers also conduct the drilling tests, the obtained results of which indicate the linear relation between the penetration and torque force Ft [31]. The aspect ratio of the Ft ∼ Fn linear curve is mainly dependent on the friction angle of the rock. The aspect ratio of Ft ∼ Fn linear curve is a constant as follows [30]:(4)FtFn=1tana+θ=H,where θ is the friction angle between the fracture faces of the rock, H is the slope of Ft ∼ Fn linear curve and is constant, and a is the rake angle of drilling bit cutter. After some algebraic manipulation, we have(5)tana=1−HtanθH+tanθ.During the rock drilling process, the friction coefficient can be calculated as [30](6)tanφ′=tana+θ−tana1−tanatana+θ,where φ′ is the friction angle between the intact rock and compressed crushed zone. By estimating the friction angle φ′ using (6), the internal friction angle φ of rock is calculated as [32, 33](7)tanφ′=π2tanφ,where φ is the internal friction angle of the rock. According to (6) and (7), the contact friction coefficient tanθ can be calculated as(8)tanθ=tanarc tanπ/2tanφ+tana1+π/2tanφtana−a.Using (5) and (8), the variation of the contact friction coefficient tanθ with an internal friction angle of rock (φ) is also expressed as(9)tanθ=πH2tanφ−πtanφ2−2πHtanφ+2H2.The variation of the contact friction coefficient tanθ with the internal friction angle of rock (φ) can be well described. Given the aspect ratio of Ft ∼ Fn linear curve and the rake angle of bit cutter, the internal friction angle of rock and friction coefficient is also determined using (8) and (9).


## 3.3. Determination of Model Parameters

The proposed model has two dimensionless parameters (λ and b) in (2) and (3). The function (λtanθ+b) is related to the parameter mi in the H–B model [34]. The model parameters (λ, b, and η) are determined by a great amount of the tested data from the published literature. The relationship between the contact friction coefficient and the parameter mi is established.The peak strength, residual strength, cohesion, and internal friction angle of rock are mostly provided, and some are collected by interpreting the M-C model in the published literature. The H–B model is used to fit the peak strength [29], and the proposed model in (3) is used to fit the data of the residual strength. Table 1 presents the fitted parameter mi. Using (8), the contact friction coefficient tanθ is calculated by using the internal friction angle of rocks in the published literature at the rake angle of bit cutter of 5°, 10°, 15°, and 20°, respectively. Table 2 shows the friction angle θ and contact friction coefficient tanθ of the proposed model. A linear relationship between the friction coefficient and the parameter mi is presented in Figure 3, which can be expressed as(10)mi=λtanθ+b.Table 2
Summary of the calculated contact friction coefficient at the different rake angles.
No.Rock typesφ' (°)a = 5°a = 10°a = 15°a = 20°θ (°)tanθθ (°)tanθθ (°)tanθθ (°)tanθ1Amarelo Pais granite67.9863.522.0158.731.6553.641.3648.241.122Blanco Mera granite69.4564.912.1460.071.7454.941.4349.521.173Vilachan granite65.8961.541.8556.851.5351.821.2746.451.054Blanco Mera granite70.0965.512.2060.651.7855.511.4650.081.195Stanstead granite64.5260.251.7555.621.4650.641.2245.301.016Bunt sandstone46.2441.400.8838.050.7834.130.6829.690.577Indiana limestone buff33.7938.170.7935.080.7031.410.6127.170.518Red sandstone45.4643.690.9640.150.8436.070.7331.480.619Gosford sandstone53.0449.571.1745.591.0241.130.8736.220.7310Carrara marble51.5646.941.0743.150.9438.860.8134.090.6811Westerly granite72.9068.212.5063.261.9858.061.6052.611.3112Rhyolite21.6639.170.8136.000.7332.240.6327.940.5313Indiana limestone gray51.0341.680.8938.300.7934.370.6829.900.5814Porphyry50.5047.231.0843.420.9539.110.8134.320.6815Rajnagar marble48.4445.331.0141.670.8937.480.7732.800.6416Biotite chlorite schist44.1745.631.0241.940.9037.730.7733.030.6517Purulia gneiss62.9558.781.6554.221.3949.291.1644.000.9718Kirthai granite gneiss64.4460.181.7455.551.4650.571.2245.241.0119Tennessee sandstone26.5142.250.9138.820.8034.850.7030.350.5920Coal42.1943.270.9439.770.8335.720.7231.160.6021Sandstone63.5459.441.6954.851.4249.901.1944.590.9922Norite65.9660.091.7455.461.4550.481.2145.161.0123Quartzite75.5770.772.8765.762.2260.531.7755.081.4324Hejiata coal G150.6147.331.0843.510.9539.190.8234.400.6825Hejiata coal G256.9653.201.3448.971.1544.300.9839.220.8226Coarse marble39.8542.970.9339.490.8235.460.7130.920.6027Medium marble47.0744.080.9740.510.8536.410.7431.800.6228Sandstone48.1342.640.9239.190.8235.180.7030.660.59Figure 3
A linear relationship between the contact friction coefficient and the parametermi of the H–B model.
(a)(b)(c)(d)The correlation coefficients (R2 > 0.95) show that the parameters λ and b can be well obtained using equation (10). The value of λ in the proposed model is 26.296, 35.143, 45.272, and 56.498, and the value of b is -19.602, -23.287, -25.7, and -28.362 at the rake angle of bit cutter of 5°, 10°, 15°, and 20°, respectively. A linear relationship between the parameters λ and b of the proposed model and the rake angle of bit cutter is presented in Figure 4, which can be expressed as(11)λ=2.0147a+15.619,b=−0.5739a−17.065.Figure 4
A linear relationship between the parametersλ and b of the proposed model and the rake angle of the bit cutter.The squared correlation coefficientR2 is larger than 0.99. Equation (11) is used to determine the parameters λ and b. The frictional model for the determination of the peak strength of rock can be rewritten as(12)σ1=σ3+UCS⋅2.0147a+15.619tanθ−0.5739a−17.065σ3UCS+10.5.Given the friction angle and the rake angle of the bit, the peak strength can be estimated using (12).The proposed model (3) is used to fit the data of the rock residual strength. The model parameters (λ, b, and tanθ) are obtained by (10) and (11). The model parameter η is determined using the data collected from literature as shown in Table 1. This parameter η is presented as the exponential function of the friction coefficient as shown in Figure 5, which can be expressed as(13)η=2.1287mi−0.7826.Figure 5
Correlation of the parametersη and mi of the H–B model.The linear correlation coefficientR2 is larger than 0.85. Since mi is accepted as a fitting parameter, whose range is from 0.001 to 25, the range of the parameter η is 0.17 to 474 according to (13). In addition, there are several definitions of mi according to different perspectives, such as the lithology, crystallization degree, and mineral composition and grains of rock. Some studies also show that parameter mi is linked to the tensile strength and compressive strength of rock [4, 34]. This implies that the parameter η is dependent on the physical and mechanical properties of rock. Equation (13) is used to determine the model parameter η. The frictional model (3) can be written as(14)σ1=σ3+2.1287σ3UCS2.0147a+15.619tanθ−0.5739a−17.0650.21740.5.Given the friction angle and rake angle of the bit, the residual strength can be estimated using (14).


## 4. Verification

A drilling process monitoring apparatus is used to measure the thrust force, torque force, penetration rate, rotation speed, and penetration depth per rotation of hard rock, medium rock, and weak rock such as granite, limestone, slate, and red sandstone. The rotation speeds of drilling are applied at 200–600 rpm for each rock type. The penetration rates of drilling are set as 0.1–1.2 mm/min. The relationships between the thrust force and torque force of each rock are obtained as presented in Figure6. The triaxial compressive strength, triaxial residual strength, cohesion, and internal friction angle of each type of rock are obtained in the laboratory. We use these test data to verify the proposed model in predicting the peak and residual strength using drilling process monitoring in the field as presented in Figure 7.Figure 6
The thrust force-torque force curves of slate, granite, red sandstone, and limestone.
(a)(b)(c)(d)Figure 7
The peak and residual strength data of slate, granite, red sandstone, and limestone.
(a)(b)(c)(d)Many drilling experimental results show a strong linear relationship between the thrust force and torque force [30]. The slope of the thrust force and torque force linear curve is mainly dependent on the internal friction angle of rock [35]. For limestone, granite, slate, and sandstone, the slopes of the thrust force and torque force linear curve are 0.6717, 0.5553, 0.6592, and 0.758 at a value of 10°, respectively. Using (4) or (9), the contact friction coefficient tanθ is 1.04, 1.23, 1.06, and 0.93, respectively. Based on the proposed model, the peak strength and residual strength of the four types of rock are calculated using equations (12) and (14) as shown in Figure 7. The results show that the predicted peak strength is slightly higher than those from the ISRM standard test especially for weak rocks such as limestone and sandstone. This highness is caused by the high plastic deformity in drilling especially in soft rocks [36]. The error ranges of the predicted peak strength and residual strength of rocks are −13.7%–11.8% and −13.2%–14.6%. Their errors are within the error range of 15% compared to the results from the standard test. Therefore, despite the slight overestimation (or underestimation) of rock strength owing to the existence of a boundary layer of failed rock in the cutting face and the more plastic deformity caused by the high plastic deformities during drilling, this result suggests that the method can predict the peak and residual strength.The proposed friction-strengthening model for the peak strength has a nonlinear form and has three parameters (λ, b, and tanθ). The parameters (λ and b) are related to the drilling parameter. The parameter tanθ is the contact friction coefficient. They control the nonlinearity of the peak and residual strength. The value of λtanθ+b can be correlated with parameter mi [34]. The nonlinearity of the peak strength is generally weaker than that of the residual strength [29]. The parameter η in the proposed friction-strengthening model can also control the strength nonlinearity. Moreover, in rock engineering applications, the new method can provide a continuous prediction for rock residual strength. Besides, it can predict the residual strength of the rock at a speed several orders of magnitude faster than the standard test in the laboratory. Finally, owing to the minimum requirement for test preparations, this method should have a great potential for field application in rock engineering.


## 5. Conclusions

A new model is established to determine the peak strength and residual strength of rock materials. Although it is empirical, the parameter of the proposed model, the contact friction coefficient of rock, can control the nonlinearity of the peak strength and residual strength. The variation of the contact friction coefficient with the internal friction angle of rock can be well described. Given the aspect ratio ofFt ∼ Fn linear curve, the internal friction angle and friction coefficient of rock are also determined using the proposed model. With a great amount of data from publications examined, the linear relationship between the friction coefficient and the parameter mi is established. The frictional model was proposed to predict the peak strength and residual strength using drilling process monitoring. The predicted results are in good agreement with those of the standard laboratory test. The predicted peak strength is slightly higher than those from the ISRM standard test especially for weak rock such as limestone and sandstone. The error ranges of the predicted peak strength and residual strength of rocks are -13.7%–11.8% and -13.2%–14.6%, respectively. It can successfully predict the peak strength and residual strength of hard rock, medium rock, and weak rock using drilling process monitoring in the field. This method should have a great potential for field application in rock engineering.



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*Source: 8054163-2022-01-21.xml*