An adaptive observer for hyperbolic systems with application to UnderBalanced Drilling

International audienceWe present an adaptive observer design for a first-order hyperbolic system of Partial Differential Equations with uncertain boundary parameters. The design relies on boundary measurements only, and is based on a backstepping approach. Using a Gradient Descent technique, we prove exponential convergence of the distributed system and estimation of the parameter. This method is applied to the estimation of uncertain parameters during the process of oil well drilling

Stick-slip and Torsional Friction Factors in Inclined Wellbores

Stick slip is usually considered a phenomenon of bit-rock interaction, but is also often observed in the field with the bit off bottom. In this paper we present a distributed model of a drill string with an along-string Coulomb stiction to investigate the effect of borehole inclination and borehole friction on the incidence of stick-slip. This model is validated with high frequency surface and downhole data and then used to estimate static and dynamic friction. A derivation of the torsional drill string model is shown and includes the along-string Coulomb stiction of the borehole acting on the string and the ‘velocity weakening’ between static and dynamic friction. The relative effects of these two frictions is investigated and the resulting drillstring behavior is presented. To isolate the effect of the along-string friction from the bit-rock interaction, field data from rotational start-ups after a connection (with bit off bottom) is considered. This high frequency surface and downhole data is then used to validate the surface and downhole behavior predicted by the model. The model is shown to have a good match with the surface and downhole behavior of two deviated wellbores for depths ranging from 1500 to 3000 meters. In particular, the model replicates the amplitude and period of the oscillations, in both the topside torque and the downhole RPM, as caused by the along-string stick slip. It is further shown that by using the surface behavior of the drill-string during rotational startup, an estimate of the static and dynamic friction factors along the wellbore can be obtained, even during stick-slip oscillations, if axial tension in the drillstring is considered. This presents a possible method to estimate friction factors in the field when off-bottom stick slip is encountered, and points in the direction of avoiding stick slip through the design of an appropriate torsional start-up procedure without the need of an explicit friction test

Stick-slip and Torsional Friction Factors in Inclined Wellbores

Stick slip is usually considered a phenomenon of bit-rock interaction, but is also often observed in the field with the bit off bottom. In this paper we present a distributed model of a drill string with an along-string Coulomb stiction to investigate the effect of borehole inclination and borehole friction on the incidence of stick-slip. This model is validated with high frequency surface and downhole data and then used to estimate static and dynamic friction. A derivation of the torsional drill string model is shown and includes the along-string Coulomb stiction of the borehole acting on the string and the ‘velocity weakening’ between static and dynamic friction. The relative effects of these two frictions is investigated and the resulting drillstring behavior is presented. To isolate the effect of the along-string friction from the bit-rock interaction, field data from rotational start-ups after a connection (with bit off bottom) is considered. This high frequency surface and downhole data is then used to validate the surface and downhole behavior predicted by the model. The model is shown to have a good match with the surface and downhole behavior of two deviated wellbores for depths ranging from 1500 to 3000 meters. In particular, the model replicates the amplitude and period of the oscillations, in both the topside torque and the downhole RPM, as caused by the along-string stick slip. It is further shown that by using the surface behavior of the drill-string during rotational startup, an estimate of the static and dynamic friction factors along the wellbore can be obtained, even during stick-slip oscillations, if axial tension in the drillstring is considered. This presents a possible method to estimate friction factors in the field when off-bottom stick slip is encountered, and points in the direction of avoiding stick slip through the design of an appropriate torsional start-up procedure without the need of an explicit friction test

Axial and torsional self-excited vibrations of a distributed drill-string

We consider a distributed axial-torsional drill-string model with a rate-independent bit-rock interaction law to study the occurrence and non-local characteristics of axial and torsional self-excited vibrations as caused by the regenerative effect. A first contribution of the paper is the derivation of a non-dimensional version of the full non-linear distributed drill-string–bit-rock interaction model and showing how it relates to the minimal set of characteristic quantities. Using this model the study shows how multiple axial modes of the drill-string are excited, or attenuated, depending on the bit rotation rate. This indicates that a lumped drill-string model approximation is insufficient for the general case. Then, a comprehensive simulation study is performed to create a stability map for the occurrence of stick-slip oscillations. In particular, the significance of the axial topside boundary condition, i.e., constant velocity vs. constant hook-load, is evaluated. A central finding is that increasing the axial loop gain (determined by the bit-rock parameters) tends to both increase the area of stable torsional dynamics and increase the rate of penetration for a constant imposed weight on bit. This also corresponds to a more severe axial instability

Dynamics of a distributed drill string system:Characteristic parameters and stability maps

\u3cp\u3eThis paper involves the dynamic (stability) analysis of distributed drill-string systems. A minimal set of parameters characterizing the linearized, axial-torsional dynamics of a distributed drill string coupled through the bit-rock interaction is derived. This is found to correspond to five parameters for a simple drill string and eight parameters for a two-sectioned drill-string (e.g., corresponding to the pipe and collar sections of a drilling system). These dynamic characterizations are used to plot the inverse gain margin of the system, parametrized in the non-dimensional parameters, effectively creating a stability map covering the full range of realistic physical parameters. This analysis reveals a complex spectrum of dynamics not evident in stability analysis with lumped models, thus indicating the importance of analysis using distributed models. Moreover, it reveals trends concerning stability properties depending on key system parameters useful in the context of system and control design aiming at the mitigation of vibrations.\u3c/p\u3

Effect of shock subs on self-excited vibrations in drilling systems

\u3cp\u3eThe present paper studies the effect of an axial elastic tool (known as a shock sub), mounted downhole in the drill-string, on the occurrence of axial and torsional self-excited vibrations. In particular, we evaluate the feasibility of stabilizing the axial dynamics, dominated by a bilateral (feedback) coupling between the bit-rock interaction and the drill-string wave-equations, through the insertion of a passive down-hole tool. We consider the problem of unwanted drill-string vibrations and explain how these vibrations relate to the so-called axial instability using a distributed parameter (infinite dimensional) model. The equations describing the feedback system causing this instability are derived and then extended to accommodate for the inclusion of the effect of the shock sub. Conditions for the design parameters of the shock sub needed to avoid axial instability are then derived and their practical feasibility are considered.\u3c/p\u3

Avoiding stick slip vibrations in drilling through startup trajectory design

International audienceA distributed model of a drill string with a collars section is presented with Coulomb friction as a distributed source term. This model is capable of replicating stick slip oscillations as caused by the reduction in friction from static to dynamic. We design a feed-forward startup trajectory for initiating rotation of the drill string that effectively avoids the stick slip limit cycle. The trajectory design is performed using the differential flatness of the bit angular velocity, and by treating the reduction from static to dynamic friction as an estimated disturbance to be canceled, thus conforming to the canonical 3-DOF controller design for tracking and disturbance rejection. A simulation study illustrates the feasibility of the approach

Axial and torsional self-excited vibrations of a distributed drill-string

\u3cp\u3eWe consider a distributed axial-torsional drill-string model with a rate-independent bit-rock interaction law to study the occurrence and non-local characteristics of axial and torsional self-excited vibrations as caused by the regenerative effect. A first contribution of the paper is the derivation of a non-dimensional version of the full non-linear distributed drill-string–bit-rock interaction model and showing how it relates to the minimal set of characteristic quantities. Using this model the study shows how multiple axial modes of the drill-string are excited, or attenuated, depending on the bit rotation rate. This indicates that a lumped drill-string model approximation is insufficient for the general case. Then, a comprehensive simulation study is performed to create a stability map for the occurrence of stick-slip oscillations. In particular, the significance of the axial topside boundary condition, i.e., constant velocity vs. constant hook-load, is evaluated. A central finding is that increasing the axial loop gain (determined by the bit-rock parameters) tends to both increase the area of stable torsional dynamics and increase the rate of penetration for a constant imposed weight on bit. This also corresponds to a more severe axial instability.\u3c/p\u3