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This book reports the results of exhaustive research work on modeling and control of vertical oil well drilling systems. It is focused on the analysis of the system-dynamic response and the elimination of the most damaging drill string vibration modes affecting overall perforation performance: stick-slip (torsional vibration) and bit-bounce (axial vibration). The text is organized in three parts.

The first part, Modeling, presents lumped- and distributed-parameter models that allow the dynamic behavior of the drill string to be characterized; a comprehensive mathematical model taking into account mechanical and electric components of the overall drilling system is also provided. The distributed nature of the system is accommodated by considering a system of wave equations subject to nonlinear boundary conditions; this model is transformed into a pair of neutral-type time-delay equations which can overcome the complexity involved in the analysis and simulation of the partial differential equation model.

The second part, Analysis, is devoted to the study of the response of the system described by the time-delay model; important properties useful for analyzing system stability are investigated and frequency- and time-domain techniques are reviewed.

Part III, Control, concerns the design of stabilizing control laws aimed at eliminating undesirable drilling vibrations; diverse control techniques based on infinite--dimensional system representations are designed and evaluated. The control proposals are shown to be effective in suppressing stick-slip and bit-bounce so that a considerable improvement of the overall drilling performance can be achieved.

This self-contained book provides operational guidelines to avoid drilling vibrations. Furthermore, since the modeling and control techniques presented here can be generalized to treat diverse engineering problems, it constitutes a useful resource to researchers working on control and its engineering application in oil well drilling.



Chapter 1. Introduction

The presence of drillstring vibrations is the main cause of performance loss in the perforation process for oil and gas. It provokes premature wear and tear of drilling equipment resulting in fatigue and induced failures such as pipe wash-out and twist-off. It also causes a significant wastage of drilling energy and may induce wellbore instabilities reducing the directional control and its overall shape. In the oil industry, the improvement of the drilling performance is a matter of crucial economic interest. This introductory chapter gives an overview of the three main types of vibrations occurring during drilling operations: torsional (stick-slip), axial (bit-bounce) and lateral (whirling). It also presents the basics of a vertical drilling operation and a description of the system components. The scope and organization of the monograph is provided.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu



Chapter 2. An Overview of Drillstring Models

Over the last half-century, extensive research effort has been conducted to mathematically describe the physical phenomena occurring in real wells. This chapter presents a brief compilation of the most popular modeling strategies allowing the analysis and control of a vertical oilwell drilling system. Drilling models can be classified into three general categories: 1. Lumped parameter models which regard the drillstring as a mass-spring-damper system described by an ordinary differential equation. 2. Distributed parameter models which consider the drillstring as a beam subject to axial and/or torsional efforts. A system of partial differential equations provides a characterization of the drilling variables in an infinite-dimensional setting. The price paid for the model accuracy is the complexity involved in its analysis and simulations. 3. Neutral-type time-delay models which are directly derived from the distributed parameter ones. The involved time delays are related to the speed of the oscillatory waves traveling throughout the rod. This type of model provides a good trade-off between system representation accuracy and complexity of the description.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 3. Bit-Rock Frictional Interface

It is usually assumed that the growth of instabilities eventually leading to stick-slip and bit-bounce oscillations arises from the friction model, which empirically captures the interaction between the cutting device and the rock. An appropriate model of friction allows gaining insight into drillstring vibration phenomena thus characterizing the dynamic behavior at bit level and making possible the development of appropriate control strategies to tackle this problem. This chapter pursues two objectives: first, to compile the main classical friction models and second, to describe the main modeling strategies used to approximate the frictional torque due to the contact between the cutting device and the drilling surface. Most of the approaches include the Coulomb and Karnopp friction models and consider the Stribeck effect which occurs due to the irregular geometry of the contacting surfaces and which is generally assumed to be the main cause of the stick-slip phenomenon.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 4. Comprehensive Modeling of a Vertical Oilwell Drilling SystemBottom hole assembly

This chapter provides a comprehensive mathematical description of the system given by a set of partial differential equations which embeds individual submodels corresponding to the different levels of the drillstring. The upper extremity is accurately described by appropriate top boundary conditions including the actuators’ dynamics. Wave equation models involving viscous and viscoelastic Kelvin–Voigt internal damping are chosen to represent the axial and torsional propagating waves traveling along the drillstring; a distinction between the different sections composing the drilling rod, i.e., the series of drillpipes and the Bottom Hole Assembly (BHA), is made. The system trajectories at the bit level are characterized by the bottom boundary conditions involving a nonlinear term to represent the bit-rock contact from which undesirable vibrations arise. This chapter presents an accurate experience-based model of the frictional torque which considers the drilling surface characteristics and the bit geometry.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu



Chapter 5. Neutral-Type Time-Delay Systems: Theoretical Background

Time-delay systems belong to the class of Functional Differential Equations (FDE) which are infinite-dimensional, as opposed to the standard Ordinary Differential Equations (ODE). A neutral-type time-delay equation also known as Neutral Functional Differential Equation (NFDE) describes a system in which the rate of change of the state depends not only on the present and past values of the system state but also on the earlier value of the state rate change. The wave equation subject to nonlinear boundary conditions describing the propagating waves traveling along the rod during the drilling process can be transformed through the D’Alembert method into a more handleable representation: a NFDE. This representation allows taking advantage of the wide range of existing tools for the stability analysis of time-delay systems. This chapter provides the main theoretical elements to address the analysis of the qualitative dynamic response and stability of the drilling system in the framework of NFDE.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 6. Bifurcation Analysis of the Drilling System

Bifurcation theory is concerned with the study of changes in the qualitative structure of the solutions of a given family of differential equations. In dynamical systems, a bifurcation occurs when a small smooth alteration of the parameter values (the bifurcation parameters) causes a sudden qualitative or topological change in its behavior. Bifurcations can be classified into two main categories: Global bifurcations, which occur when larger invariant sets of the system collide with each other, or with the equilibrium point of the system; and local bifurcations, which can be analyzed through variations of the local stability properties of the equilibrium point. In order to characterize the qualitative dynamic response of the rotary drilling system, this chapter addresses its local bifurcation analysis. Based on the center manifold theorem and normal forms theory, a set of neutral-type time-delay equations that models the coupled axial-torsional drilling vibrations is reduced via spectral projections to a finite-dimensional system described by an Ordinary Differential Equation (ODE) which simplifies the analysis task.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 7. Ultimate Boundedness Analysis

Physically-motivated systems are generally subject to nonlinearities and uncertainties; for these systems, classical stability definitions (like asymptotic or exponential stability in the sense of Lyapunov) can be too restrictive. Namely, the state of a system may be mathematically unstable in some classical sense, but the response oscillates close enough to the equilibrium, to be considered as acceptable. In many stabilization problems, the aim is to bring states close to certain sets rather than to a particular state. In these situations, appropriate performance specifications are given by the concept of ultimate boundedness with a fixed bound, also referred to as practical stability which not only provides information on the stability of the system, but also characterizes its transient behavior with estimates of the bounds on the system trajectories. In this chapter, the ultimate boundedness analysis of the drilling system described by the wave equation with nonlinear boundary conditions is investigated. A proposal of Lyapunov functional allows determining (Linear Matrix Inequalities) LMI-type conditions to establish ultimate bounds on the system response. It is proven that under certain conditions, the nongrowth of the system energy is guaranteed.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu



Chapter 8. Field Observations and Empirical Drilling Control

Several destabilizing dynamics occur when the cutting device comes in contact with the drilling surface giving rise to the occurrence of drillstring vibrations. A detailed description of the three main types of drillstring vibrations (longitudinal, axial, and torsional) is presented in this chapter. Failures induced by drillstring vibrations lead to premature wear of the system components, breakage of drilling bits, unscrewing of pipe connections, wastage of energy, reduction of the rate of penetration,..., which clearly increases the operating costs. Detection guidelines and detrimental consequences of the stick-slip, bit-bounce, and whirling are herein described. Helpful guidelines and empirical control methods to reduce undesirable drillstring behaviors are also reviewed. Furthermore, the most popular tools used in oil platforms to dampen drillstring vibrations, the different methods used for acquiring, monitoring, and transmitting data from the downhole to the surface, and the innovative automated systems that improve the perforation process are addressed in this chapter.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 9. Low-Order Controllers

This chapter presents some of the most frequently used low-order controllers to regulate the angular velocity and tackle the stick-slip phenomenon. First, a proportional-integral (PI)-like control law to maintain a constant rotary speed is presented. The controller is constructed under the basis of a two degree-of-freedom lumped parameter model; its gains are adjusted by means of the classic two-time-scales separation method. Next, two classic solutions to counteract the stick-slip phenomenon are discussed: the soft torque and the torsional rectification controllers. The torsional rectification control allows the absorption of the energy at the top extremity to avoid the reflection of torsional waves back down to the drillstring. The soft torque is one of the most popular vibration control methods; it has the form of a standard speed controller but includes a high-pass filtered torque signal. Finally, a novel technique to reduce the stick-slip and bit-bounce is introduced. Based on the bifurcation analysis of the drilling system, a pair of low-order controllers aimed at eliminating axial and torsional coupled vibrations are designed: delayed proportional and delayed proportional-integral-derivative. The performances of the proposed control techniques are highlighted through simulations of a proposed drilling model.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 10. Flatness-Based Control of Drilling Vibrations

In nonlinear systems theory, the flatness property refers to the ability for dynamical systems of being exactly linearized via endogenous feedback. A system satisfying the flatness property is called a differentially flat system. The main attribute of flat systems is that the state and input variables can be directly expressed without integrating any differential equation, in terms of one particular set of variables called a flat output and a finite number of its derivatives. Through the d’Alembert method, the differential flatness property of the drilling system described by a pair of wave equations subject to Newton-like nonlinear boundary conditions is proved in this chapter. It is worthy of mention that the flatness property of a nonlinear dynamical system is useful to deal with trajectory tracking problems. Based on the idea that the elimination of drilling vibrations requires the angular and axial velocities of the drilling bit to follow a constant reference path, a pair of controllers aimed at tackling the steering problem is designed. Simulation results show that the flatness-based feedback controllers, allowing an exponential convergence of the system trajectories, accurately eliminate the stick-slip and bit-bounce drilling vibrations.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 11. Stick-Slip Control: Lyapunov-Based Approach

Based on the neutral-type time-delay model of the drillstring torsional dynamics, this chapter addresses the design of stabilizing controllers aimed at eliminating the stick-slip phenomenon. Within the framework of Lyapunov theory, two control approaches based on different system representations are proposed. First, a switched model-based control is designed. Since the frictional torque on bit is usually modeled by a nonlinear function subject to the sign function, the drilling model can be regarded as a nonlinear autonomous state-dependent switching system and its stabilization is addressed within the framework of switched systems theory. Next, a multimodel representation-based control is reviewed. This strategy is based on a linear approximation of the neutral-type torsional drilling model describing torsional drilling vibrations. Stabilization conditions are derived from the descriptor approach and a proposal of a Lyapunov-Krasovskii functional. Both strategies lead to (Linear Matrix Inequalities) LMI-type conditions guaranteeing an exponential convergence of the system trajectories. Numerical simulations illustrate the effective suppression of stick-slip vibrations.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 12. Practical Stabilization of the Drilling System

When dynamical systems are subject to external perturbations, it is not possible to establish exponential stability; nevertheless, from engineering point of view, the system response may be considered acceptable. This idea gives rise to the notion of ultimate boundedness or practical stability, which allows characterizing the transient behavior of a perturbed system. Under the assumption that the drilling system is subject to external disturbances, and that certain dynamics are frequently disregarded in the models, it is impractical to design control laws aimed at forcing the system response to reach a particular state; instead, we seek to drive the system trajectories into a given domain guaranteeing an acceptable system performance. This chapter concerns the practical stabilization of the drilling system based on two different modeling approaches to describe coupled axial–torsional rod dynamics. The first approach considers a coupled wave–Ordinary Differential Equation (ODE) model. The practical stabilization of the system is addressed via Lyapunov techniques allowing the design of stabilizing controllers to suppress the stick-slip and the bit-bounce. The second approach is based on a coupled neutral-type time-delay equation–ODE model. A pair of feedback controllers is derived from the attractive ellipsoid method which combines Lyapunov strategies and the principle of attractive sets
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu

Chapter 13. Performance Analysis of the Controllers

Throughout the third part of the monograph, various control strategies aimed at eliminating harmful drilling vibrations have been presented. The proposed control solutions stabilize the system, effectively suppressing the stick-slip and the bit-bounce, notwithstanding, it is important to provide a comparative analysis to highlight the advantages and vulnerabilities of each control method. This chapter presents a contrastive study of the system response to eight different control techniques which can be classified into three categories: low-order control, flatness-based control, and Lyapunov-based control. A pair of coupled neutral-type time-delay equations describing the torsional and axial dynamics of a drillstring subject to a frictional torque model that considers certain drilling bit and surface characteristics representing the interaction between the cutting device and the rock is considered for simulation purposes. The controllers are evaluated by considering different values of the intrinsic specific energy, which is a parameter related to the rock strength. Figures illustrating the system trajectories, data tables, and radar charts allow a direct and simple assessment of the control proposals.
Martha Belem Saldivar Márquez, Islam Boussaada, Hugues Mounier, Silviu-Iulian Niculescu


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