# Modeling of BHA dynamic behaviors

- 1.8k Downloads

## Abstract

Lateral and torsional vibrations of drill strings used in oil well operations are considered. This work presents finite-element-based bottom hole assembly model which addresses natural frequencies determination, and torsional and lateral oscillations on the near vertical well. In terms of the excitations of the developed finite-element model, forces are applied on the bit with time history generated using an Auto-Regressive Moving-Average digital filter. Due to the random pattern of the forces at the drill bit, a stochastic dynamic approach is adopted in investigating the problem. The method of Newmark-beta direct integration is used. It is hoped that this study will enhance the interest in using stochastic dynamic techniques in drilling system analysis and design, as they can capture quite appropriately the inherent uncertainty of the bit forces and, potentially, of other sources. The implementation of torsional degree of freedom during the development of MATLAB code has been considered new attribute regarding previous works based on the same research line. The results from the code developed attained a good agreement with the commercial SPA2000 Advanced 14.0.0 Structural Analysis Program software.

## Keywords

Drill string Bottom hole assembly Dynamic behavior Finite-element model Stochastic excitations ARMA algorithm Newmark-beta direct integration Monte Carlo simulation## Introduction

In the type of deep drilling operations that are commonly used for hydrocarbon exploration and recovery, there are a few common designs. These drilling operations typically make use of drill strings, which are pipes that transmit both drilling fluid and torque necessary to rotate a drill bit. The top end of a drilling string typically has the machinery necessary to pump the drilling fluid, and either a top or kelly drive to provide torque for the drill bit. Pipe typically encases the string below, to contain the drilling fluid. The lower end of the string, consisting of the drill bit, collars and other components that are collectively known as the bottom hole assembly (BHA), is the region of many failures (Macdonald and Bjune 2007). Typical problems that are encountered include severe stick–slip, lateral vibration, hole enlargement, which induce poor directional control. Inevitably this problem causes costly drilling equipment failures.

Development of downhole measurement technique with high frequency resolution has helped appreciating the significance of these vibrations and their impact on equipment failure. Providing consistent and reliable time histories, which are used to study random vibration on drill string, is always of great importance. These problems include random loading force and acceleration at the bit and random formation contact properties. A general vibration perspective of the oil and gas drilling process can be found in Spanos et al. (2003).

One approach to reduce the cost associated with the failures is to use BHA dynamic simulation software and preemptively mitigate the anticipated problems. The ability to pinpoint the source and effects of torsional, axial and lateral oscillations enables users to qualify design changes to the drill-string configuration and optimize parameters, prior to drilling the well, thus minimizing the traditional “trial” and “error” approach and operator risk.

More recently, nonlinear and time-domain modeling solutions have become possible (Spanos et al. 2002). This is critical to achieve the required level of accuracy to make meaningful contributions to drilling system behavioral prediction. While the finite-element method has been used for a number of years in the oilfield, it has been limited to sections of the drill string or discrete components. Simulations for the entire drilling system are exceptionally time consuming and complex both in setup and computational time. These factors are considered when planning the length of drilling intervals to be analyzed.

The paper pursues an analysis of the drill-string dynamic nonlinear problem by resorting to finite-element modeling of the string and using the Newmark-beta direct integration and Monte Carlo simulations to capture the salient features of the problem.

## Dynamic modeling of the drill string

A drill string on a drilling rig is primarily composed of drill pipe, drill collars and drill bit. Its function is to convey drilling fluid (via the mud pumps) and torque (via the kelly drive or top drive) to the drill bit. The drill string is hollow so that drilling fluid can be pumped down through it and circulated back up the annulus, creating clearance between the drill string and the casing/open hole (ASME 2011). A typical drill string can extend over several kilometers.

The BHA is made of a drill bit, which is used to break up the rock formations, drill collars, which are heavy, thick-walled tubes used to apply weight to the drill bit and drilling stabilizers, which keep the assembly centered in the hole. The BHA may also contain other components such as a downhole motor and Rotary steerable system, measurement while drilling (MWD) and logging while drilling (LWD) tools. The components are joined together using rugged threaded connections. Short “subs” are used to connect items with dissimilar threads (Chevallier 2000).

A complete description of the dynamic behavior of a BHA requires three displacements and three rotations assigned at each node. This model is more accurate than certain simplified models which describe only specific vibration modes. Nevertheless, a complete finite-element model demands more computational power. In order to reduce the computational effort and facilitate the interpretation of the numerical results, only lateral and torsional vibrations are examined herein (Chevallier 2000).

Geometrical properties of the BHA

Element | Total length (m) | Length ele (m) | Num of ele | OD (m) | ID (m) |
---|---|---|---|---|---|

675DP1 | 7.68 | 0.48 | 16 | 0.17145 | 0.07145 |

8DC1 | 8.64 | 0.48 | 18 | 0.2032 | 0.07145 |

STAB1 | 0.48 | 0.48 | 1 | 0.2032 | 0.07145 |

8DC2 | 13.92 | 0.48 | 29 | 0.2032 | 0.07145 |

STAB2 | 0.48 | 0.48 | 1 | 0.2032 | 0.07145 |

8DC3 | 6.72 | 0.48 | 14 | 0.2032 | 0.07145 |

Bit | 0.48 | 0.48 | 1 | 0.2032 | 0.07145 |

Material properties of the BHA

Element | Density (kgm | Young (Nm | Iy (m | Iz (m | J (m | G (Pa) | Total mass (kg) | Comp forc (N) |
---|---|---|---|---|---|---|---|---|

675DP1 | 7,833.412 | 2.07E+11 | 4.1136E−05 | 4.1136E−05 | 8.23E−05 | 2.07E+11 | 71.73 | 0 |

8DC1 | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

STAB1 | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

8DC2 | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

STAB2 | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

8DC3 | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

Bit | 7,833.412 | 2.07E+11 | 8.2409E−05 | 8.2409E−05 | 1.65E−04 | 2.07E+11 | 106.86 | −10,000 |

**u**, the coefficient ${k}_{2}$ is the rock stiffness coefficient and $N$ is the total number of degrees of freedom of the mathematical model (Spanos et al. 1997; Santos 2011). Incorporating Eq. (2) into (1) yields

The lateral excitation of the BHA depends strongly on the kind of the drill bit applied to drilling operation. The RC bit and PDC bit differ in their frequency content, as the excitations induced by RC bits tend to possess significant levels of energy over a determined range, whereas PDC bits produce excitations of frequency content centered at the rotational driving frequency (Chevallier 2000).

The time step $\mathrm{\Delta}t$ must be selected so that both the smallest period of the excitation and the natural period of system may be represented adequately. A proper choice of the time interval yields a good approximation of the exact solution.

The random excitations are a set of time histories generated by means of an ARMA algorithm, each of which possesses a determined number of points (Spanos 1983; Spanos et al. 1990; Spanos and Zeldin 1998). For each excitation, the governing dynamic equation of motion provides a different response, in this regard, the simulated responses are submitted to a statistical treatment. The statistical properties of the response are computed as ensemble averages over the simulated responses. This procedure is a simple application of Monte Carlo simulations technique (Chevallier 2000). The time-domain solution with Monte Carlo simulations is based on solving a set of deterministic problems. The statistical characteristics of the excitation, such as its power spectral density, permit generating a set of compatible time histories. These are deterministic since each one represents a particular realization of the random excitation function. The response of the finite-element BHA to each realization can therefore be computed.

Upon having derived the solution to each deterministic time histories, ensemble characteristics of the group of responses can be computed, yielding the statistical properties of the response. In this regard, the major phase in this procedure is to obtain the time-domain response of a nonlinear system to a deterministic excitation.

It is important to mention that the proposed mathematical model herein developed for this study does not take into account the hydraulic effects, friction between the BHA components and the formation, reactive torque from the bit and BHA components with formation, drill string torque, well tortuosity, cutting effect, drag forces among other well complexities all of which may most likely degrade the model accuracy once the numerical results are compared against the real field data. For this study, the input data were from previous published academic articles so as to illustrate and validate the methodology.

## Numerical results

As an illustration, for this numerical application, the stochastic downhole force, i.e., lateral excitations, applied at the bit node and only assigned at the lateral degree of freedom will be associated for a roller cone bit. To this end, time histories compatible with the power spectrum corresponding to the roller cone bit are generated by means of an ARMA algorithm. A set of 100 time histories, each of which possesses 500 data points, is generated. The generated time histories are then used as excitations for the system, being applied at the node which represents the drill bit solely.

Figures 15 and 16 represent the numerical result generated from commercial software aforementioned adopted on this work as a reference according to aforementioned. The *x* and *y* axes in these figures denote the time in seconds and amplitude magnitude in meters, respectively.

## Summary and conclusions

A finite-element model has been used to study the dynamic behavior of the BHA subjected to stochastic excitations. The dynamic response of the BHA has been attained by means of a Newmark-beta direct interaction for two different types of drill bit.

In order to obtain the probability density function of the displacement, Monte Carlo simulation was used. The mathematical model herein developed is able to predict the critical speed range in which the rotary BHA for a near vertical well can potentially experience natural frequency-induced vibrations, and foresee the dynamic rotary BHA response for either PDC or RC bit in terms of lateral and torsional vibrations for a specific applied WOB and rock stiffness coefficient within a drilling scenario.

In this sense, the user is able to filter out the drilling parameters, i.e., RPM and WOB, prone to resonance effects, and detrimental lateral and torsional vibrations based on the material and geometrical properties, applied WOB and rock stiffness coefficient. It is worth beholding the effect of torsional degree of freedom on the BHA response once it has been implemented to the model, the torsional degree of freedom added more inertia to the system, yielding a slight reduction on the magnitude of output results.

By beholding Figs. 14, 15, 16, 17, wherein an ordinary BHA has been simulated subjected to a deterministic excitation applied at the bit node, it is clearly noticeable that the numerical results from finite-element model in conjunction with the Signal Processing algorithm developed on MATLAB for this academic study had a good agreement with the commercial software, SPA2000, with minimal error, validating therefore the mathematical model herein presented.

## Notes

### Acknowledgments

The authors would like to thank the anonymous reviewers for their kind and constructive comments which helped enhance this paper. We also would like to thank North Fluminense State University Darcy Ribeiro, Brazil, and Rice University, USA, for support of this research.

## References

- Aadnoy B, Cooper I, Miska S et al (2009) Advanced drilling and well technology. Society of Petroleum Engineers, RichardsonGoogle Scholar
- ASME Shale Shaker Commitee (2011) Drilling fluids processing handbook, ElsevierGoogle Scholar
- Chevallier A (2000) Nonlinear Stochastic Drilling Vibrations. PhD thesis, Rice University, Department of Mechanical Engineering and Material Science, HoustonGoogle Scholar
- Macdonald KA, Bjune JV (2007) Failure analysis of drillstrings. Eng Fail Anal 14:1641–1666CrossRefGoogle Scholar
- Santos JF (2011) Modeling of BHA dynamic behaviors. MSc thesis, North Fluminense State University Darcy Ribeiro, Department of Petroleum Engineering and Exploration, MacaéGoogle Scholar
- Spanos PD (1983) ARMA algorithms for ocean wave modeling. J Energy Resour Technol 105:300–309CrossRefGoogle Scholar
- Spanos PD, Zeldin BA (1998) Monte Carlo treatment of random fields: a broad perspective. Appl Mech Rev 51(3):219–237CrossRefGoogle Scholar
- Spanos PD, Mushung LJ, Nelson DA, Hamilton DA (1990) Low-frequency spectral representation of space shuttle flight data. J Aerosp Eng 3(2):137–154CrossRefGoogle Scholar
- Spanos PD, Payne ML, Secora SC (1997) Bottom-hole assembly modeling and dynamic response determination. J Energy Resour Technol 119(3):153–158CrossRefGoogle Scholar
- Spanos PD, Chevallier AM, Politis NP (2002) Nonlinear stochastic drill-string vibrations. J Vib Acoust 124(4):512–518CrossRefGoogle Scholar
- Spanos PD, Chevallier AM, Politis NP, Payne ML (2003) Oil and gas well drilling: a vibrations perspective. Shock Vibration Digest 35(2):85–103CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.