Three-dimensional nonlinear coupling vibration of drill string in deepwater riserless drilling and its influence on wellbore pressure field

Abstract

During deepwater riserless drilling operations, the vibration behavior of drill string may have a significant impact on wellbore pressure, which leads to serious drilling accidents such as well leakage and collapse in shallow risk areas. In order to solve this problem, based on Hamilton's principle, a three-dimensional nonlinear coupling dynamics model of drill string in deepwater riserless drilling is established, taking into account the following factors: heave and offset motion of offshore platform, ocean current load, drill string-borehole contact and bit-rock interaction. The Newmark-β method is used to solve the nonlinear discrete equations of the system. The effectiveness of the model and calculation program is verified by the field test data. Meanwhile, a model of wellbore pressure field is established under the influence of axial-lateral-torsional coupling vibration of drill string. The 3D nonlinear coupling vibration characteristic of drill string and its influence on wellbore pressure fluctuation are investigated. The results indicate that the collision between the drill string system and the borehole usually occurs in the middle of the formation and at the bit, which is easy to cause drilling tools failure. The maximum pressure fluctuation along the wellbore is mainly affected by lateral vibration. The mean value of wellbore pressure fluctuation is mainly determined by torsional vibration. The results may be useful to predict the risks of well leakage and collapse in deepwater riserless drilling operations.

Appendices

Appendix 1

As shown in Fig. 2a, when the turntable rotates at speed $\Omega_{0}$, it causes a complex three-dimensional displacement of the drill string system, including translational displacement $\left( {u,v,w} \right)$ and torsional angle $\phi$. In Fig. 2b, point ${\text{A}}(y,z)$ on the cross section goes to point ${\text{A}}^{\prime }$ through displacement $\left( {v,w} \right)$, and then goes to point ${\text{B}}$ through torsional angle $\phi$. Displacement $(\Delta y,\Delta z)$ is the change in the coordinate from $A^{\prime }$ to ${\text{B}}$.

$$ \left\{ \begin{gathered} \Delta y = r\cos \left( {\alpha + \phi } \right) - r\cos \alpha \\ = r\left( {\cos \alpha \cos \phi - \sin \phi \sin \alpha - \cos \alpha } \right) \\ \Delta z = r\sin \left( {\alpha + \phi } \right) - r\sin \alpha \\ = r\left( {\sin \alpha \cos \phi + \sin \phi \cos \alpha - \sin \alpha } \right) \\ \end{gathered} \right. $$

The coordinates of point ${\text{A}}(y,z)$ are $y = r\cos \alpha$, $z = r\sin \alpha$. The above equation can be written as:

$$ \left\{ \begin{gathered} \Delta y = y\cos \phi - z\sin \phi - y \\ \Delta z = z\cos \phi + y\sin \phi - z \\ \end{gathered} \right. $$

Therefore, the total displacement $\left( {\hat{v},\hat{w}} \right)$ of point ${\text{A}}(y,z)$ is the sum of translational displacement $\left( {v,w} \right)$ and torsional displacement $(\Delta y,\Delta z)$.

$$ \left\{ \begin{gathered} \hat{v} = v + y\cos \phi - z\sin \phi - y \hfill \\ \hat{w} = w + z\cos \phi + y\sin \phi - z \hfill \\ \end{gathered} \right. $$

Appendix 2

Mass matrix ${\mathbf{m}}_{1}$ in Eq. (27) contains only axial interpolation function ${\varvec{N}}_{1} {\varvec{N}}_{1}^{T}$, indicating that ${\mathbf{m}}_{1}$ is a axial one-dimensional mass matrix. ${\mathbf{m}}_{2}$ contains only lateral interpolation functions ${\varvec{H}}_{1} {\varvec{H}}_{1}^{T}$ and ${\varvec{H}}_{2} {\varvec{H}}_{2}^{T}$, indicating that ${\mathbf{m}}_{2}$ is a lateral one-dimensional mass matrix. ${\mathbf{m}}_{3}$ contains only lateral interpolation function ${\varvec{N}}_{2} {\varvec{N}}_{2}^{T}$, indicating that ${\mathbf{m}}_{3}$ is a torsional one-dimensional mass matrix. Similarly, ${\varvec{k}}_{4}$, ${\varvec{k}}_{7}$ and ${\varvec{k}}_{8}$ in Eq. (28) are axial, lateral and torsional one-dimensional mass matrixes, respectively. According to the Rayleigh damping model formula in Sect. 2.2, the axial, lateral and torsional one-dimensional damping matrixes can be expressed as:

$$ \begin{aligned} {\mathbf{c}}_{A} & = \alpha_{A} {\mathbf{m}}_{1} + \beta_{A} {\varvec{k}}_{4} \\ {\mathbf{c}}_{L} & = \alpha_{L} {\mathbf{m}}_{2} + \beta_{L} {\varvec{k}}_{7} \\ {\mathbf{c}}_{T} & = \alpha_{T} {\mathbf{m}}_{3} + \beta_{T} {\varvec{k}}_{8} \\ \end{aligned} $$

where $\alpha_{x}$ and $\beta_{x}$ $\left( {x = A,L,T} \right)$ are mass proportional coefficient and stiffness proportional coefficient, respectively.

Matrix ${\mathbf{m}}_{4}$ contains the lateral-torsional coupling (LTC) interpolation functions ${\varvec{N}}_{2} {\varvec{H}}_{2}^{T}$ and ${\varvec{N}}_{2} {\varvec{H}}_{1}^{T}$, and is the LTC mass matrix. ${\varvec{k}}_{5}$ contains the LTC interpolation functions $N_{2}^{\prime } N_{2}^{\prime T} H_{1}^{\prime } H_{1}^{\prime T}$ and $N_{2}^{\prime } N_{2}^{\prime T} H_{2}^{\prime } H_{2}^{\prime T}$. ${\varvec{k}}_{62}$ contains the LTC interpolation functions $N_{2}^{\prime } H_{1}^{\prime } H_{2}^{\prime \prime T}$ and $N_{2}^{\prime } H_{2}^{\prime } H_{1}^{\prime \prime T}$. They are the LTC stiffness matrixes. Therefore, following the form of the viscous damping formula, the LTC damping matrix ${\mathbf{c}}_{LT}$ can be written as follows:

$$ {\mathbf{c}}_{LT} { = }\sqrt {\alpha_{L} \alpha_{T} } \left( {{\mathbf{m}}_{4} + {\mathbf{m}}_{4}^{{\text{T}}} } \right) + \sqrt {\beta_{T} } \beta_{L} \left( {{\varvec{k}}_{62} + {\varvec{k}}_{62}^{{\text{T}}} } \right) + \beta_{T} \beta_{L} \left( {{\varvec{k}}_{5} + {\varvec{k}}_{5}^{{\text{T}}} } \right) $$

In order to facilitate calculation, the above equation is simplified as:

$$ {\mathbf{c}}_{LT} { = }\sqrt {\alpha_{L} \alpha_{T} } \left( {{\mathbf{m}}_{4} + {\mathbf{m}}_{4}^{{\text{T}}} } \right) + \sqrt {\beta_{T} } \beta_{L} \left( {{\varvec{k}}_{62} + {\varvec{k}}_{62}^{{\text{T}}} + {\varvec{k}}_{5} + {\varvec{k}}_{5}^{{\text{T}}} } \right) $$

Matrix ${\varvec{k}}_{3}$ contains the axial-lateral coupling (ALC) interpolation functions ${\varvec{N}}_{1}^{\prime } {\varvec{H}}_{1}^{\prime } {\varvec{H}}_{1}^{\prime T}$ and ${\varvec{N}}_{1}^{\prime } {\varvec{H}}_{2}^{\prime } {\varvec{H}}_{2}^{\prime T}$, and is the ALC stiffness matrix. ${\varvec{k}}_{1}$ and ${\varvec{k}}_{2}$ contain the lateral nonlinear term of higher order ${\varvec{H}}_{1}^{\prime } {\varvec{H}}_{1}^{\prime T} {\varvec{H}}_{1} ^{\prime}{\varvec{H}}_{1}^{\prime T}$, ${\varvec{H}}_{2}^{\prime } {\varvec{H}}_{2}^{\prime T} {\varvec{H}}_{2}^{\prime } {\varvec{H}}_{2}^{\prime T}$ and ${\varvec{H}}_{1}^{\prime } {\varvec{H}}_{1}^{\prime T} {\varvec{H}}_{2}^{\prime } {\varvec{H}}_{2}^{\prime T}$. In order to facilitate classification and calculation, they are directly incorporated into ALC stiffness matrixes. The ALC damping matrix ${\mathbf{c}}_{AL}$ can be written as follows:

$$ {\mathbf{c}}_{AL} = \sqrt {\beta_{A} } \beta_{L} \left( {{\varvec{k}}_{3} + {\varvec{k}}_{3}^{{\text{T}}} \user2{ + k}_{1} + {\varvec{k}}_{2} + {\varvec{k}}_{2}^{{\text{T}}} } \right) $$

Matrix ${\varvec{k}}_{61}$ contains the axial–torsional coupling (ATC) interpolation function ${\varvec{N}}_{1}^{\prime } {\varvec{N}}_{2}^{\prime } {\varvec{N}}_{2}^{\prime T}$, and is the ATC stiffness matrix. ${\varvec{k}}_{9}$ contains the torsional nonlinear term of higher order ${\varvec{N}}_{2}^{\prime } {\varvec{N}}_{2}^{\prime T} {\varvec{N}}_{2}^{\prime } {\varvec{N}}_{2}^{\prime T}$. In order to facilitate classification and calculation, they are directly incorporated into ATC stiffness matrixes. The ATC damping matrix ${\mathbf{c}}_{AT}$ can be written as follows:

$$ {\mathbf{c}}_{AT} = \sqrt {\beta_{A} } \beta_{T} \left( {{\varvec{k}}_{61} + {\varvec{k}}_{61}^{{\text{T}}} + {\varvec{k}}_{9} } \right) $$

Therefore, the 3D nonlinear coupling damping matrix of drill string element (${\mathbf{c}}_{e}$) can be written as:

$$ {\mathbf{c}}_{e} = {\mathbf{c}}_{A} + {\mathbf{c}}_{L} + {\mathbf{c}}_{T} + {\mathbf{c}}_{AL} + {\mathbf{c}}_{LT} + {\mathbf{c}}_{AT} $$