Determining Method of Tensile Strength of Rock Based on Friction Characteristics in the Drilling Process

Abstract

Measuring-while-drilling has been considered as an effective method to determine the mechanical properties of rock. In this paper, drilling models are improved for predicting the rock strength on the basis of nonlinear M-C criterion. The analytical relationship between the ratio of unconfined compressive strength to tensile strength and internal friction angle was derived, and their correlation is studied. A field method of strength prediction is proposed based on the drilling parameters. The proposed models are verified with results from standard test for sandstone, shale, marble and diorite. The experimental results indicate that the prediction error off rock strength within the range of 10%. The correlation between the tensile and compressive strength ratio and the internal friction angle is obtained based on drilling parameters. The parabolic failure criterion is suitable for sandstone and shale, and the hyperbolic failure criterion is suitable for marble and diorite. The strength ratio based on the drilling parameters are compared with the measured values of standard test, including compressive test and Brazilian split test, the point load test and the indentation test to verify the proposed model. The proposed prediction model based on drilling parameters has potential in practical engineering applications.

Highlights

• The drilling models are established for predicting the rock strength on the basis of the nonlinear M-C criterion.

• The analytical relationship between the ratio of UCS to TS and internal friction angle of rock is derived.

• A new method of UCS, TS and TCS prediction is proposed based on the drilling parameters.

Appendices

Appendix

Supplementary Derivation of Equations in Sect. 2

2.1 Derivation of Eq. (1)

The tangential force F_t and thrust force F_n are decomposed into cutting (c) component and wear (w) component:

$$\left\{ \begin{gathered} F_{n} \, = \,F_{n}^{c} \, + \,F_{n}^{w} \hfill \\ F_{t} \, = \,F_{t}^{c} \, + \,F_{t}^{w} \hfill \\ \end{gathered} \right..$$

(23)

Assuming that the stress state in the crushed zone are homogeneous, the mechanical equilibrium requires:

$$\left\{ \begin{gathered} F_{{\text{t}}}^{c} = A\sigma_{{0}} + A\tau_{{0}} \tan \alpha \hfill \\ F_{{\text{n}}}^{c} = A\sigma_{{0}} \tan \alpha + A\tau_{{0}} \hfill \\ \end{gathered} \right.$$

(24)

From Fig. 

Fig. 12

Mechanical equilibrium of: a Crushed zone and b Rock fragment (chip)

12, the mechanical relationship between rock chip and crushed zone is considered as,

$$\left\{ \begin{gathered} \sigma {\text{A/sin}}\psi = \sigma_{0} {\text{Asin}}\psi + \tau_{0} {\text{Acos}}\psi \hfill \\ \tau {\text{A/sin}}\psi = \sigma_{0} {\text{Acos}}\psi - \tau_{0} {\text{Asin}}\psi \hfill \\ \end{gathered} \right.$$

(25)

By considering the condition that τ₀ = σ₀·tanφ', Eq. (25) becomes,

$$\left\{ \begin{gathered} \sigma = \sigma_{0} \left( {\sin {}^{2}\psi { + }\tan \varphi ^{\prime}\cos \psi \sin \psi } \right) \hfill \\ \tau = \sigma_{0} \left( {\cos \psi \sin \psi - \tan \varphi ^{\prime}\sin {}^{2}\psi } \right) \hfill \\ \end{gathered} \right.$$

(26)

Assuming the M-C criterion τ = c + σtanφ on the sliding plane, then we obtain:

$$\sigma_{0} \left( {\frac{1}{2}\sin 2\psi - \tan \varphi ^{\prime}\sin {}^{2}\psi - \sin {}^{2}\psi \tan \varphi - \frac{1}{2}\tan \varphi \tan \varphi ^{\prime}\sin 2\psi } \right) = c$$

(27)

Given the condition for minimum σ₀ and substituting into Eq. (27), the corresponding ψ is calculated as,

$$\left\{ \begin{gathered} \frac{{\partial \sigma_{0} }}{\partial \psi }\,{ = }\,0 \hfill \\ \psi \,{ = }\,\frac{\pi }{4} - \frac{\varphi + \varphi ^{\prime}}{2} \hfill \\ \end{gathered} \right.$$

(28)

By substituting ψ, Eq. (26) becomes,

$$\left\{ \begin{gathered} \sigma = \frac{1}{2}\sigma_{0} \left[ {1 - \sin \left( {\varphi + \varphi ^{\prime}} \right) + \tan \varphi ^{\prime}\cos \left( {\varphi + \varphi ^{\prime}} \right)} \right] \hfill \\ \tau = \frac{1}{2}\sigma_{0} \left[ {\cos \left( {\varphi + \varphi ^{\prime}} \right) + \tan \varphi ^{\prime}\sin \left( {\varphi + \varphi ^{\prime}} \right) - \tan \varphi ^{\prime}} \right] \hfill \\ \end{gathered} \right.$$

(29)

2.2 Derivation of Eqs. (10) and (16)

As shown in Fig. 

Fig. 13

Envelopes of two types of nonlinear M-C criteria

13, the correlation between shear stress and principal stress for parabolic M-C criterion is obtained as:

$$\left\{ \begin{gathered} \left( {\sigma_{1} { + }\sigma_{3} } \right)/2 = \sigma + \tau \cot 2\alpha \hfill \\ \left( {\sigma_{1} - \sigma_{3} } \right)/2 = \tau /\sin 2\alpha \hfill \\ \end{gathered} \right.$$

(30)

where σ₁ and σ₃ are the maximum and minimum components of the principal stress, respectively, and 2α denotes the angle between shear stress τ and normal stress σ, whose values is approximately equal to ψ. The principal stress form of parabolic M-C criterion is obtained as

$$\sigma_{1} = \sigma_{3} + \tau /\sin \psi$$

(31)

Then Eq. (6) is substituted into the expression of shear stress τ in Eq. (1) to eliminate τ, ψ and introduce σ_t.

$$\tau = \sigma_{0} \left[ {\tan \varphi ^{\prime}\cos \left( {\varphi ^{\prime} + \varphi } \right) + \sin \left( {\varphi ^{\prime} + \varphi } \right)} \right] = \frac{{4\sigma_{t} \left[ {\tan \varphi ^{\prime}\cos \left( {\varphi ^{\prime} + \varphi } \right) + \sin \left( {\varphi ^{\prime} + \varphi } \right)} \right]}}{{\tan \varphi ^{\prime}\sin \varphi ^{\prime}\left( {\sin \varphi { + }\cos \varphi } \right) - \cos \varphi ^{\prime}\cos \varphi }}$$

(32)

Equations (28) and (31) are considered in Eq. (30) to result in Eq. (32). As shown in Fig. 13b, the hyperbolic type of nonlinear M-C criterion has similar derivation with that of parabolic type of nonlinear M-C criterion. There is a difference of σ₀ expression in derivation.