The research reported in this thesis builds on a novel model developed at the University of Minnesota to analyze the self-excited vibrations that occur when drilling with polycrystalline diamond cutter bits. The lumped parameter model of the drilling system takes into consideration the axial and the torsional vibrations of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process combined with the quasihelical motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The response of this complex system is characterized by a fast axial dynamics superposed to the slow torsional dynamics. A two time scales analysis that uses a combination of averaging methods and a singular perturbation approach is proposed to study the dynamical response of the system. An approximate model of the decoupled axial dynamics permits to derive a pseudo analytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier works. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. In this thesis, we also analyze the axial and torsional vibrations by basing the model on a continuum representation of the drillstring rather than on the low dimensional lumped parameter model. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurrence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring are detected. In particular, stick-slip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters. Finally, we describe the experimental strategy chosen for the validation of the model and discuss results of tests conducted with DIVA, an analog experimental set-up of the lumped parameter model. Some results of the experiments conducted in an artificial rock seem to validate the model studied here although the same experiments obtained with natural rocks were unsuccessful. Different problems with the design of the experimental setup were identified. By using the outcome of the analysis of the uncoupled dynamics, we could provide critical recommendation to elaborate and to design a simpler and stiffer analog experiment (TAZ) used to study the self excitation of the axial dynamics that ultimately lead to the excitation of the torsional dynamics.

Este trabalho apresenta uma análise paramétrica da reposta dinâmica de colunas de perfuração de poços de petróleo com controle proporcional-integral de velocidade não colocalizado. A operação de perfuração de poços de petróleo e gás em águas profundas consiste na abertura de poços em solo rochoso através de uma broca cuja rotação é controlada por uma mesa rotativa na superfície. O torque imposto pela mesa é transmitido à broca por meio de uma coluna de perfuração. Particularmente no caso de perfuração em águas profundas, as colunas de perfuração podem ser muito extensas e, portanto, bastante flexíveis. As vibrações ocasionadas pela grande flexibilidade das colunas de perfuração são as principais responsáveis por falhas no processo de perfuração. Em particular, o fenômeno não-linear conhecido como stick-slip e relacionado às vibrações torcionais da coluna de perfuração, faz com que um sistema de controle projetado para manter a velocidade da mesa constante dê origem a grandes oscilações na velocidade da broca. Na prática, este fenômeno é amplificado pela inerente não-linearidade do contato entre broca e formação rochosa e pela forte não colocalização entre mesa rotativa e broca. Este trabalho tem por principal objetivo realizar uma análise paramétrica da dinâmica do processo de perfuração, usando um modelo de dois graus de liberdade para representar o conjunto mesa rotativa, coluna de perfuração e broca, para identificar condições nas quais uma lei de controle simples do tipo linear proporcional-integral pode fornecer um desempenho de perfuração estável e satisfatório. / This paper presents a parametric analysis of the dynamics of oilwell drillstrings with non-collocated proportional-integral velocity control. The drilling operation for oil and gas in deep waters consists of opening wells in rocky ground formation by a drill, whose angular speed is controlled by a rotary table at the surface. The torque applied by the table is transmitted to the drill-bit through the drillstring. Particularly in the deepwater drilling case, the drillstring can be very long and therefore very flexible. The vibrations caused by the great flexibility of drilling columns are mainly responsible for the failures in the drilling process. In particular, the nonlinear phenomenon known as stick-slip and related to the torsional vibration of the drillstring, makes that a control system designed to maintain a constant angular velocity at the table yield large variations at the drill-bit angular velocity. In practice, this phenomenon is amplified by the inherent nonlinearity of the contact between drill bit and rock formation and by the strong non-colocalization between rotary table and drill-bit. The main objective of this work is to perform a parametric analysis of the dynamics of the drilling process, using a two degrees of freedom model in order to represent the rotary table assembly, the drilling column and drill-bit, to identify conditions in which a simple control law, such as a linear proportional-integral velocity control, can provide a stable and satisfactory drilling performance.

Prospection de pétrole utilise un équipement appelé tube de forage pour forer le sol jusqu'au le niveau du réservoir. Cet équipement est une longue colonne rotative, composée par une série de tiges de forage interconnectées et les équipements auxiliaires. La dynamique de cette colonne est très complexe parce que dans des conditions opérationnelles normales, elle est soumise à des vibrations longitudinales, latérales et de torsion, qui présentent un couplage non linéaire. En outre, cette structure est soumise à effets de frottement et à des chocs dûs aux contacts mécaniques entre les paires tête de forage/sol et tube de forage/sol. Ce travail présente un modèle mécanique-mathématique pour analyser un tube de forage en configuration horizontale. Ce modèle utilise la théorie des poutres qui utilise l'inertie de rotation, la déformation de cisaillement et le couplage non linéaire entre les trois mécanismes de vibration. Les équations du modèle sont discrétisées par la méthode des éléments finis. Les incertitudes des paramètres du modèle d'interaction tête de forage/sol sont prises en compte par l'approche probabiliste paramétrique, et les distributions de probabilité des paramètres aléatoires sont construits par le principe du maximum d'entropie. Des simulations numériques sont réalisées afin de caractériser le comportement dynamique non linéaire de la structure, et en particulier, de l'outil de forage. Des phénomènes dynamiques non linéaires par nature, comme le slick-slip et le bit-bounce, sont observés dans les simulations, ainsi que les chocs. Une analyse spectrale montre étonnamment que les phénomènes slick-slip et bit-bounce résultent du mécanisme de vibration latérale, et ce phénomène de choc vient de la vibration de torsion. Cherchant à améliorer l'efficacité de l'opération de forage, un problème d'optimisation qui cherche à maximiser la vitesse de pénétration de la colonne dans le sol, sur ses limites structurelles, est proposé et résolu / Oil prospecting uses an equipment called drillstring to drill the soil until the reservoir level. This equipment is a long column under rotation, composed by a sequence of connected drill-pipes and auxiliary equipment. The dynamics of this column is very complex because, under normal operational conditions, it is subjected to longitudinal, lateral, and torsional vibrations, which presents a nonlinear coupling. Also, this structure is subjected to friction and shocks effects due to the mechanical contacts between the pairs drill-bit/soil and drill-pipes/borehole. This work presents a mechanical-mathematical model to analyze a drillstring in horizontal configuration. This model uses a beam theory which accounts rotatory inertia, shear deformation, and the nonlinear coupling between three mechanisms of vibration. The model equations are discretized using the finite element method. The uncertainties in bit-rock interaction model parameters are taken into account through a parametric probabilistic approach, and the random parameters probability distributions are constructed by means of maximum entropy principle. Numerical simulations are conducted in order to characterize the nonlinear dynamic behavior of the structure, specially, the drill-bit. Dynamical phenomena inherently nonlinear, such as slick-slip and bit-bounce, are observed in the simulations, as well as shocks. A spectral analysis shows, surprisingly, that slick-slip and bit-bounce phenomena result from the lateral vibration mechanism, and that shock phenomena comes from the torsional vibration. Seeking to increase the efficiency of the drilling process, an optimization problem that aims to maximize the rate of penetration of the column into the soil, respecting its structural limits, is proposed and solved

Este trabalho apresenta uma análise paramétrica da reposta dinâmica de colunas de perfuração de poços de petróleo com controle proporcional-integral de velocidade não colocalizado. A operação de perfuração de poços de petróleo e gás em águas profundas consiste na abertura de poços em solo rochoso através de uma broca cuja rotação é controlada por uma mesa rotativa na superfície. O torque imposto pela mesa é transmitido à broca por meio de uma coluna de perfuração. Particularmente no caso de perfuração em águas profundas, as colunas de perfuração podem ser muito extensas e, portanto, bastante flexíveis. As vibrações ocasionadas pela grande flexibilidade das colunas de perfuração são as principais responsáveis por falhas no processo de perfuração. Em particular, o fenômeno não-linear conhecido como stick-slip e relacionado às vibrações torcionais da coluna de perfuração, faz com que um sistema de controle projetado para manter a velocidade da mesa constante dê origem a grandes oscilações na velocidade da broca. Na prática, este fenômeno é amplificado pela inerente não-linearidade do contato entre broca e formação rochosa e pela forte não colocalização entre mesa rotativa e broca. Este trabalho tem por principal objetivo realizar uma análise paramétrica da dinâmica do processo de perfuração, usando um modelo de dois graus de liberdade para representar o conjunto mesa rotativa, coluna de perfuração e broca, para identificar condições nas quais uma lei de controle simples do tipo linear proporcional-integral pode fornecer um desempenho de perfuração estável e satisfatório. / This paper presents a parametric analysis of the dynamics of oilwell drillstrings with non-collocated proportional-integral velocity control. The drilling operation for oil and gas in deep waters consists of opening wells in rocky ground formation by a drill, whose angular speed is controlled by a rotary table at the surface. The torque applied by the table is transmitted to the drill-bit through the drillstring. Particularly in the deepwater drilling case, the drillstring can be very long and therefore very flexible. The vibrations caused by the great flexibility of drilling columns are mainly responsible for the failures in the drilling process. In particular, the nonlinear phenomenon known as stick-slip and related to the torsional vibration of the drillstring, makes that a control system designed to maintain a constant angular velocity at the table yield large variations at the drill-bit angular velocity. In practice, this phenomenon is amplified by the inherent nonlinearity of the contact between drill bit and rock formation and by the strong non-colocalization between rotary table and drill-bit. The main objective of this work is to perform a parametric analysis of the dynamics of the drilling process, using a two degrees of freedom model in order to represent the rotary table assembly, the drilling column and drill-bit, to identify conditions in which a simple control law, such as a linear proportional-integral velocity control, can provide a stable and satisfactory drilling performance.

Prospection de pétrole utilise un équipement appelé tube de forage pour forer le sol jusqu'au le niveau du réservoir. Cet équipement est une longue colonne rotative, composée par une série de tiges de forage interconnectées et les équipements auxiliaires. La dynamique de cette colonne est très complexe parce que dans des conditions opérationnelles normales, elle est soumise à des vibrations longitudinales, latérales et de torsion, qui présentent un couplage non linéaire. En outre, cette structure est soumise à effets de frottement et à des chocs dûs aux contacts mécaniques entre les paires tête de forage/sol et tube de forage/sol. Ce travail présente un modèle mécanique-mathématique pour analyser un tube de forage en configuration horizontale. Ce modèle utilise la théorie des poutres qui utilise l'inertie de rotation, la déformation de cisaillement et le couplage non linéaire entre les trois mécanismes de vibration. Les équations du modèle sont discrétisées par la méthode des éléments finis. Les incertitudes des paramètres du modèle d'interaction tête de forage/sol sont prises en compte par l'approche probabiliste paramétrique, et les distributions de probabilité des paramètres aléatoires sont construits par le principe du maximum d'entropie. Des simulations numériques sont réalisées afin de caractériser le comportement dynamique non linéaire de la structure, et en particulier, de l'outil de forage. Des phénomènes dynamiques non linéaires par nature, comme le slick-slip et le bit-bounce, sont observés dans les simulations, ainsi que les chocs. Une analyse spectrale montre étonnamment que les phénomènes slick-slip et bit-bounce résultent du mécanisme de vibration latérale, et ce phénomène de choc vient de la vibration de torsion. Cherchant à améliorer l'efficacité de l'opération de forage, un problème d'optimisation qui cherche à maximiser la vitesse de pénétration de la colonne dans le sol, sur ses limites structurelles, est proposé et résolu / Oil prospecting uses an equipment called drillstring to drill the soil until the reservoir level. This equipment is a long column under rotation, composed by a sequence of connected drill-pipes and auxiliary equipment. The dynamics of this column is very complex because, under normal operational conditions, it is subjected to longitudinal, lateral, and torsional vibrations, which presents a nonlinear coupling. Also, this structure is subjected to friction and shocks effects due to the mechanical contacts between the pairs drill-bit/soil and drill-pipes/borehole. This work presents a mechanical-mathematical model to analyze a drillstring in horizontal configuration. This model uses a beam theory which accounts rotatory inertia, shear deformation, and the nonlinear coupling between three mechanisms of vibration. The model equations are discretized using the finite element method. The uncertainties in bit-rock interaction model parameters are taken into account through a parametric probabilistic approach, and the random parameters probability distributions are constructed by means of maximum entropy principle. Numerical simulations are conducted in order to characterize the nonlinear dynamic behavior of the structure, specially, the drill-bit. Dynamical phenomena inherently nonlinear, such as slick-slip and bit-bounce, are observed in the simulations, as well as shocks. A spectral analysis shows, surprisingly, that slick-slip and bit-bounce phenomena result from the lateral vibration mechanism, and that shock phenomena comes from the torsional vibration. Seeking to increase the efficiency of the drilling process, an optimization problem that aims to maximize the rate of penetration of the column into the soil, respecting its structural limits, is proposed and solved