Investigation on the Effect of High-Frequency Torsional Impacts on the Torsional Vibration of an Oilwell Drill String in Slip Phase

Abstract

The drilling system usually encounters detrimental stick-slip vibration to the drilling process. One of the approaches for controlling the stick-slip vibration is using high-frequency torsional impact (HFTI). However, how the HTFI will affect the vibration of the drill string is still unknown. This study proposed a mechanical model of the drill string by using continuous system to investigate the effect of HFTI on the vibration of a drill string in slip phase, wherein the HFTI is considered in the model. The mechanical model is investigated through using mode superposition method and conducting case studies. Results show that the HFTI is more sensitive to the drill string close to the drill bit and that the HFTI has little effect on the vibration of drill string. During the drilling process, the HFTI aggravates the rock damage and rock failure, which actually improves the drilling efficiency and mitigates the stick-slip vibration.

Appendices

Appendix A

In this appendix, the approach of processing the impact torque is presented. For the HFTI, it acts on a certain cross section of the drill string. The impact torque is assumed to be periodical pulses shown in Fig. A.1, wherein τ is the load period and A is the load amplitude.

Fig. A.1

Impact torque

According to the figure, the expression of the impact torque can be given as

$$p(t) = A \times \left(1 - \frac{t}{\tau }\right),\;0 \le t \le \tau$$

(A.1)

From the figure, the impact torque is piecewise, which leads to difficulties in solving the equations. The Fourier series is used to solve this matter. The p(t) can be given as

$$\begin{aligned} p(t) & = \frac{{a_{0} }}{2} + a_{1} \cos (\omega t) + a_{2} \cos (2\omega t) + \cdots + b_{1} \sin (\omega t) + b_{2} \sin (2\omega t) + \cdots \\ & = \frac{{a_{0} }}{2} + \sum\limits_{n = 1}^{\infty } {\left( {a_{n} \cos (n\omega t) + \cdots + b_{n} \sin (n\omega t)} \right)} \\ \end{aligned}$$

(A.2)

where ω = 2π/τ is the fundamental frequency, and a₁, a₂, …, b₁, b₂, … are given as

$$a_{0} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} dt = \frac{2}{\tau }\int\limits_{0}^{\tau } {p(t)} dt$$

(A.3)

$$a_{n} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} \cos (n\omega t)dt = \frac{2}{\tau }\int\limits_{0}^{\tau } {p(t)} \cos (n\omega t)dt$$

(A.4)

$$b_{n} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} \sin (n\omega t)dt = \frac{2}{\tau }\int\limits_{0}^{\tau } {p(t)} \sin (n\omega t)dt$$

(A.5)

By substituting Eq. (A.1) into Eqs. (A.1)–(A.5), the results become

$$a_{0} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} dt = A$$

(A.6)

$$a_{n} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} \cos (n\omega t)dt = 0,n = 1,2,3 \ldots$$

(A.7)

$$b_{n} = \frac{\omega }{\pi }\int\limits_{0}^{2\pi /\omega } {p(t)} \sin (n\omega t)dt = - \frac{A}{n\pi },n = 1,2,3 \ldots$$

(A.8)

The expression of impact torque can be given as

$$p(t) = \frac{A}{2} + \frac{A}{n\pi }\sum\limits_{n = 1}^{\infty } {\sin (n\omega_{0} t)}$$

(A.9)

where ω₀ is the impact frequency. By using the first three orders, Eq. (A.9) can be rewritten as

$$p(t) = \frac{A}{2} + \frac{A}{\pi }\sin (\omega_{0} t) + \frac{A}{2\pi }\sin (2\omega_{0} t)$$

(A.10)

Appendix B

The orthogonal character is presented. Equation (7) can be rewritten as

$$GI_{p} \phi_{i}^{\prime \prime } (x) = - \omega_{i}^{2} \phi_{i} (x)$$

(B.1)

Multiplying the first term of Eq. (B.1) by ϕ_j(x) and integrating it can obtain

$$\int\limits_{0}^{l} {GI_{p} \phi_{j} (x)} \phi_{i}^{\prime \prime } (x)\rm{d}x = {GI_{p} \phi_{j} (x)} \phi_{i}^{\prime } (x) |_{0}^{l} - \int\limits_{0}^{l} {GI_{p} \phi_{j}^{{\prime }} (x)} \phi_{i}^{{\prime }} (x)\rm{d}x$$

(B.2)

Eq. (B.2) can be rewritten as

$$\int\limits_{0}^{l} {GI_{p} \phi_{j} (x)} \phi_{i}^{\prime \prime } (x)\rm{d}x{ = }\int\limits_{0}^{l} {GI_{p} \phi_{j}^{\prime \prime } (x)} \phi_{i} (x)\rm{d}x$$

(B.3)

If fixed boundary condition or free boundary condition is applied on the drill string end, the second term of Eq. (B.2) equals 0. Equation (B.2) can then be written as

$$\int\limits_{0}^{l} {GI_{p} \phi_{j} (x)} \phi_{i}^{\prime \prime } (x)\rm{d}x{ = } - \int\limits_{0}^{l} {GI_{p} \phi_{j}^{{\prime }} (x)} \phi_{i}^{{\prime }} (x)\rm{d}x$$

(B.4)

Multiplying Eq. (B.1) by ϕ_j(x) and integrating it shall have

$$\int\limits_{0}^{l} {GI_{p} \phi_{j}^{{\prime }} (x)} \phi_{i}^{{\prime }} (x)\rm{d}x = \int\limits_{0}^{l} {\omega_{i}^{2} \phi_{i} (x)} \phi_{j} (x)\rm{d}x$$

(B.5)

Exchanging the corner marks i and j, Eq. (B.4) becomes

$$\int\limits_{0}^{l} {GI_{p} \phi_{i}^{{\prime }} (x)} \phi_{j}^{{\prime }} (x)\rm{d}x = \int\limits_{0}^{l} {\omega_{j}^{2} \phi_{j} (x)} \phi_{i} (x)\rm{d}x$$

(B.6)

Eqs. (B.5) and (B.6) lead to

$$\left( {\omega_{i}^{2} - \omega_{j}^{2} } \right)\int\limits_{0}^{l} {\phi_{i} (x)} \phi_{j} (x)\rm{d}x = 0$$

(B.7)

If i ≠ j, then ω_i ≠ ω_j; we therefore have

$$\int\limits_{0}^{l} {\phi_{i} (x)} \phi_{j} (x)\rm{d}x = 0$$

(B.8)

Equation (B.8) indicates that the mode shapes are orthogonal with respect to the mass and stiffness.

Appendix C

The derivation process of Eq. (23) is to be discussed. As the mode shapes are orthogonal with respect to the inertia (or mass) and stiffness, then we obtain

$$\int\limits_{0}^{l} {J\phi_{i} (x)\phi_{j} (x)} \rm{d}x = 0,i \ne j$$

(C.1)

$$\int\limits_{0}^{l} {J\phi_{i} (x)\phi_{j} } (x)\rm{d}x = 1,i = j$$

(C.2)

$$\int\limits_{0}^{l} {GI_{p} \frac{{d^{2} \phi_{i} (x)}}{{\rm{d}x^{2} }}\phi_{j} } (x)\rm{d}x = 1,i = j$$

(C.3)

Substituting Eq. (22) into Eq. (2), we have

$$\sum\limits_{i = 1}^{\infty } {GI_{p} \frac{{d^{2} \phi_{i} (x)}}{{\rm{d}x^{2} }}Q_{i} (t) - } \sum\limits_{i = 1}^{\infty } {J\phi_{i} (x)\ddot{Q}_{i} (t) - } \sum\limits_{i = 1}^{\infty } {\beta (x)\phi_{i} (x)\dot{Q}(t)} = M(x,t)$$

(C.4)

The corresponding initial conditions are

$$\theta (x,0) = \theta_{0} (x) = \sum\limits_{i = 1}^{\infty } {\phi_{i} (x)Q_{i} } (0)$$

(C.5)

$$\frac{\partial \theta (x,0)}{\partial t} = \dot{\theta }_{0} (x) = \sum\limits_{i = 1}^{\infty } {\phi_{i} (x)\dot{Q}_{i} (0)}$$

(C.6)

Multiplying Eq. (C.4) by ϕ_j(x) and integrating it, combining with Eqs. (C.1)–(C.3), and (B.3), we have

$$\begin{aligned} \int\limits_{0}^{l} {GI_{p} \frac{{d^{2} \phi_{j} (x)}}{{\rm{d}x^{2} }}} \phi_{j} (x)\rm{d}xQ_{j} (t) - \int\limits_{0}^{l} {J\phi_{j}^{2} (x)\rm{d}x\ddot{Q}_{j} (t)} \\ - \sum\limits_{i = 1}^{\infty } {\int\limits_{0}^{l} {\beta (x)\phi_{i} (x)\phi_{j} (x)\rm{d}x} } \dot{Q}_{i} (t) = \int\limits_{0}^{l} {\phi_{j} (x)} M(x,t)\rm{d}x \end{aligned}$$

(C.7)

By conducting similar operations to Eqs. (C.5) and (C.6), we have

$$Q_{j} (0) = \frac{1}{{\overline{J}_{j} }}\int\limits_{0}^{l} {J\theta_{0} (x)\phi_{j} (x)\rm{d}x}$$

(C.8)

$$\dot{Q}_{j} (0) = \frac{1}{{\overline{J}_{j} }}\int\limits_{0}^{l} {J\dot{\theta }_{0} (x)\phi_{j} (x)\rm{d}x}$$

(C.9)

Combining with Eqs. (18), (19), and (B.1), the first term of Eq. (C.7) can be written as

$$\int\limits_{0}^{l} {GI_{p} \frac{{d^{2} \phi_{j} (x)}}{{\rm{d}x^{2} }}\phi_{j} (x)\rm{d}xQ_{j} (t)} = - \omega_{j}^{2} \overline{J}_{j} Q_{j} (t)$$

(C.10)

and Eq. (C.7) becomes

$$\overline{J}_{j} \ddot{Q}_{j} (t) + \omega_{j}^{2} \overline{J}_{j} Q_{j} (t) + \sum\limits_{i = 1}^{\infty } {\int\limits_{0}^{l} {\beta (x)\phi_{i} (x)\phi_{j} (x)\rm{d}x} } \dot{Q}_{j} (t) = \overline{M}_{j}$$

(C.11)

For the third term of Eq. (C.11), the non-diagonal matrix will lead to difficulties in equation decoupling. In order to decouple the third term of Eq. (C.11), the viscous damping is assumed to be

$$\beta (x) = aJ$$

(C.12)

Defining the jth damping ratio ξ_j is

$$\xi_{j} = \frac{a}{{2\omega_{j} }}$$

(C.13)

where a is constant. By combining Eqs. (C.11)–(C.13), Eq. (23), which is a decoupled differential equation of the generalized coordinate, can be obtained.