METHOD AND SYSTEM FOR CONTROLLING VIBRATIONS IN A DRILLING SYSTEM

PROCEDE ET SYSTEME POUR CONTROLER LES VIBRATIONS DANS UN SYSTEME DE FORAGE

English Abstract

The invention provides a control system and method for limiting vibrations in a drilling system, including a drill string and a drive system for providing a drive torque for rotating the drill string at a reference frequency (Oref). The control system comprises: - a sensor module for determining at least one uphole parameter of the drilling system; a model module provided with a model of the drilling system, the model module being adapted to provide modeled parameters of the drilling system using the drive torque as an input; - a model gain module for providing a model gain vector (L) to the model module in response to one or more of the modeled parameters and the drive torque (Tm), the model gain vector enabling the module module to update the model thereby obtaining an updated model; and a control module for providing a torque correction factor (u) to the drive system depending on the modeled parameters, the uphole parameter, the reference frequency (Oref ), and the drive torque (Tm).

French Abstract

L'invention concerne un système et un procédé de contrôle pour réduire les vibrations dans un système de forage, le système comprenant un train de tiges et un système d'entraînement permettant de générer un couple d'entraînement de façon à entraîner en rotation le train de tiges à une fréquence de référence (Oref). Le système de contrôle comprend : un module capteur pour déterminer au moins un paramètre vertical du système de forage ; un module de modélisation pourvu d'un modèle du système de forage, ce module étant conçu pour produire des paramètres modélisés du système de forage au moyen du couple d'entraînement comme entrée; un module de gain de modèle permettant d'obtenir un vecteur (L) de gain de modèle par rapport au module de modélisation en réponse à au moins l'un des paramètres modélisés et au couple d'entraînement (Tm), le vecteur de gain de modèle permettant au module de mettre à jour le modèle et d'ontemir un modèle mis à jour; et un module de contrôle permettant d'obtenir un facteur de correction de couple (u) par rapport au système de forage en fonction des paramètres modélisés, du paramètre vertical, de la fréquence de référence (Oref ) et du couple d'entraînement (Tm).

Claims

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CLAIMS
1. A control system for controlling vibrations in a
drilling system, the drilling system (1) including an
elongate body extending from surface into a borehole
formed in an earth formation and a drive system (22) for
providing a drive torque (Tm) to the elongate body for
rotating said elongate body at a reference frequency
(.OMEGA.ref), the control system comprising:
- a sensor module (54) for determining at least one
uphole parameter of the drilling system;
- a model module (16) provided with a model of the
drilling system (1), the model module being adapted to
provide modeled parameters of the drilling system using
the drive torque (Tm) as an input;
- a model gain module (18) for providing a model gain
vector (L) to the model module (16) in response to one or
more of the modeled parameters and the drive torque (Tm),
the model gain vector enabling the module module (16) to
update the model thereby obtaining an updated model; and
- a control module (20) for providing a torque
correction factor (u) to the drive system (22) depending
on the modeled parameters, the uphole parameter, the
reference frequency (.OMEGA.ref), and the drive torque (Tm).
2. The system of claim 1, wherein the modeled parameters
include:
- modeled uphole angular position (~u);
- modeled downhole angular position (~l ); and
- modeled downhole rotary velocity (.omega. l,m).
3. The system of claim 1 or 2, wherein the uphole
parameter of the drilling system as determined by the

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sensor module (54) comprises the uphole rotary velocity
(.omega.u .)
4. The system of any of the previous claims, wherein the
control module (20) is adapted to determine a difference
in uphole angular position and downhole angular position
(~u-~l) using the drive torque (Tm).
5. The system of claim 4, wherein the difference in
uphole angular position and downhole angular position
(~u-~l) is determined by formula:
<IMG>
wherein k .theta. is a constant.
6. The system of any of the previous claims, wherein the
control module (20) is provided with the following
formula to calculate the torque correction factor (u):
<IMG>
wherein k1,k2,k3 and k.theta. are constants.
7. A method of controlling vibrations in a drilling
system, the drilling system (1) including an elongate
body extending from surface into a borehole formed in an
earth formation and a drive system (22) for providing a
drive torque (Tm) to the elongate body for rotating said
elongate body at a reference frequency (.OMEGA.ref), the
method comprising the steps of:
- providing the reference frequency (.OMEGA.ref) to the
drive system (22);
- the drive system (22) providing the drive torque
(Tm) to the elongate body of the drilling system (1);
- determining at least one uphole parameter of the
drilling system (1);

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- providing the drive torque (Tm) to a model module
(16) which is provided with a model of the drilling
system (1), the model module providing modeled parameters
of the drilling system;
- providing one or more of the modeled parameters and
the drive torque (Tm) to a model gain module (18) for
providing a model gain vector (L) to the model module
(16) in response thereto;
- obtaining an updated model by the module module
(16) using the model gain vector; and
- providing the modeled parameters, the uphole
parameter, the reference frequency (.OMEGA.ref), and the drive
torque (Tm) to a control module (20);
- the control module (20) providing a torque
correction factor (u) to the drive system (22) to correct
the drive torque (Tm).
8. The method of claim 7, wherein the modeled parameters
include:
- modeled uphole angular position (~u );
- modeled downhole angular position (~l ); and
- modeled downhole rotary velocity (.omega.l,m ) .
9. The method of claim 7 or 8, wherein the uphole
parameter of the drilling system comprises the uphole
rotary velocity (.omega.u).
10. The method of any of claims 7-9, wherein the control
module (20) determines a difference in uphole angular
position and downhole angular position (.theta.u-.theta.l) using the
drive torque (Tm).

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11. The method of claim 10, wherein the difference in
uphole angular position and downhole angular position
(.theta.u-.theta.l) is determined by formula:
(<IMG>
wherein k.theta. is a constant.
12. The method of any of claims 7-11, wherein the control
module (20) calculates the torque correction factor (u)
using formula:
<IMG>
wherein k1,k2,k3 and k.theta. are constants.
13. The method of any of claims 7-12, including the step
of:
Replacing the drive torque (Tm) with a corrected drive
torque (Tc), using formula:
T c=T m -u.
14. The method of any of claims 7-13, including the step
of using only parameters which can be measured or modeled
uphole.
15. A method of controlling vibrations in a drilling
system, the drilling system including an elongate body
extending from surface into a borehole formed in an earth
formation and a drive system for rotating the elongate
body by providing a drive torque to the elongate body,
the method comprising:
a) operating the drive system to provide a drive
torque to the elongate body, and determining a system
parameter that relates to an uphole parameter of the
drilling system;
b) obtaining a model of the drilling system;

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c) applying the model to determine a modeled system
parameter that corresponds to said system parameter;
d) determining a difference between the system
parameter and the modeled system parameter;
e) updating the model in dependence of said
difference, thereby obtaining an updated model;
f) determining from the updated model at least one
modeled parameter of rotational motion, and adjusting the
drive torque in dependence of each modeled parameter of
rotational motion to control vibrations of the elongate
body.
16. The method of claim 15, wherein said uphole parameter
of the drilling system relates to an uphole torque in the
drilling system.
17. The method of claim 15 or 16, wherein said uphole
parameter of the drilling system relates to torque (T) in
the elongate body at or near the earth's surface.
18. The method of claim 16 or 17, wherein said model of
the drilling system includes a modelled torsional
stiffness (k.theta.m) of the elongate body, and wherein said
drilling parameter comprises a ratio of said torque (T)
over said modelled torsional stiffness (k.theta.m) .
19. The method of claim 15, wherein said modeled system
parameter relates to a modeled difference between an
uphole rotational position of the elongate body and a
downhole rotational position of the elongate body.
20. The method of claim 15, wherein said uphole parameter
of the drilling system is a first uphole parameter, and
wherein step (c) comprises applying the model using an
input parameter relating to a second uphole parameter of
the drilling system.
21. The method of claim 20, wherein the drive system
comprises a rotary drive coupled to an uphole end of the

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elongate body, and wherein said second uphole parameter
is or comprises torque (Tm) provided by the rotary drive
to said uphole end of the elongate body.
22. The method of claim 15, wherein the model includes at
least one modeled state parameter and wherein step (e)
comprises adding to each modeled state parameter the
product of said difference and a respective gain factor
pertaining to the modeled state parameter.
23. The method of claim 22, wherein each modeled state
parameter relates to a modeled parameter of rotational
motion of the elongate body.
24. The method of claim 22, wherein said at least one
modeled state parameter is selected from a modeled
difference between an uphole angular velocity and a
downhole angular velocity of the elongate body, a modeled
uphole angular acceleration of the elongate body, and a
modeled downhole angular acceleration of the elongate
body.
25. The method of claim 22, wherein step (b) comprises
obtaining a state observer in which the model is
included, the state observer further including a gain
module for calculating each said gain factor.
26. The method of claim 15, wherein said at least one
modeled parameter of rotational motion includes at least
one of a modeled difference between an uphole rotational
position and a downhole rotational position of the
elongate body, a modeled uphole angular velocity of the
elongate body, and a modeled downhole angular velocity of
the elongate body.

Description

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METHOD AND SYSTEM FOR CONTROLLING VIBRATIONS IN A
DRILLING SYSTEM
The present invention relates to a method and to a
system for controlling vibrations in a drilling system.
Numerous vibrations can occur in an elongate body
extending into a borehole formed in a subsurface
formation, such as in a drill string operated to drill
the borehole for the production of hydrocarbon fluid from
the subsurface formation.
Drilling of an oil or gas wellbore is typically done
by rotary drilling. Herein the wellbore may include
vertical sections and/or sections deviating from
vertical, e.g. horizontal sections. Rotary drilling
generally employs a drill string including a drill bit at
its downhole end. The drill string typically includes
drill pipe sections which are mutually connected by
threaded couplings. In operation, a drive system located
at or near surface may provide torque to the drill string
to rotate the drill string to extend the borehole. The
drive system may include, for example, a top drive or a
rotary table. The drill string transmits the rotational
motion to the drill bit. Generally the drill string also
provides weight on bit and may transmit drilling fluid to
the drill bit.
As a drill string may be several kilometres long,
e.g. exceeding 5 or 10 km, the drill string may have a
very large length to diameter ratio. As a result, the
drill string behaves as a rotational spring and can be
twisted several turns during drilling. Different modes of
vibration may occur during drilling, e.g. rotational,

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lateral and/or longitudinal (axial) vibrations, possibly
causing alternating slip-stick motions of the drill
string or the drill bit relative to the borehole wall.
Such vibrations are due to, for example, fluctuating bit-
rock interactions and pressure pulses in the drilling
fluid generated by the mud pumps.
In a model description, a drill string can often be
regarded as a torsional pendulum wherein the top of the
drill string rotates with a substantially constant
angular velocity, whereas the drill bit performs a
rotation with varying angular velocity. The varying
angular velocity can have a constant part and a
superimposed torsional vibration part. In extreme cases,
the bit periodically comes to a complete standstill.
Maintaining rotation of the drill string at surface
builds up torque and eventually causes the drill bit to
come loose and to suddenly rotate again, typically
leading to a downhole angular velocity being much higher
than the angular velocity at surface, typically more than
twice the speed of the nominal speed of the motor at
surface, e.g. a top drive or rotary table. The downhole
angular velocity is dampened again whereafter the process
is repeated, causing an oscillating behaviour of the
lower part of the drill string. This phenomenon is known
as stick-slip.
It is desirable to reduce or prevent these vibrations
in order to reduce one or more of shock loads to the
drilling equipment, excessive bit wear, premature tool
failures and relatively poor drilling rate. High peak
speeds occurring during the slip phase can lead to
secondary effects like extreme axial and lateral
accelerations and forces. Proper handling of downhole

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vibrations can significantly increase reliability of the
drilling equipment.
To suppress the stick-slip phenomenon, control
methods and systems have been applied in the art to
control the speed of the drive system such that the
rotational speed variations of the drill bit are dampened
or prevented.
One such method and system is disclosed in
EP-B-443689, whereby the energy flow through the drive
system of the drilling assembly is controlled to be
between selected limits, the energy flow being definable
as the product of an across-variable and a through-
variable. The speed fluctuations are reduced by measuring
at least one of the variables and adjusting the other
variable in response to the measurement.
In EP-B-1114240 it is pointed out that the control
system disclosed in EP-B-443689 can be represented by a
combination of a rotational spring and a rotational
damper associated with the drive system. To obtain
optimal damping, the spring constant of the spring and
the damping constant of the damper are to be tuned to
optimal values, whereby the rotational stiffness of the
drill string plays an important role in tuning to such
optimal values. To aid this tuning, EP-B-1114240
discloses a method and system for determining the
rotational stiffness of a drill string for drilling of a
borehole in an earth formation.
WO 2010/063982 discloses a method and system for
dampening stick-slip operations, wherein the rotational
speed is controlled using a PI controller that is tuned
such that the drilling mechanism absorbs torsional energy
at or near the stick-slip frequency. The method can also
comprise the step of estimating a bit speed, which is the

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instantaneous rotational speed of a bottom-hole assembly.
The bit speed is displayed at a driller's graphical
interface and is regarded as a useful optional feature to
help the driller visualize what is happening downhole.
A basic control theory for a non-smooth mechanical
systems is described in A. Doris, Output-feedback design
for non-smooth mechanical systems: Control synthesis and
experiments, Ph.D. thesis, Eindhoven University of
Technology, September 2007 (hereinafter referred to as
the Doris publication).
The Doris publication uses a dynamic rotor system,
including an upper disc connected to the motor (the top
drive) and a lower disc connected to the bit. Inputs to
the model are the angular position (phase) and the speed
(first derivative of the phase) of both the upper disc
and the lower disc. For the model to provide accurate
results, the speed and phase of the lower disc will have
to be measured using a downhole sensor.
Jens Rudat ET AL: "Development of an innovative
model-based stick/slip control system", SPE/IADC Drilling
Conference and Exhibition, 139996, 1 March 2011,
discloses a system using surface measured rotary speed Q.
and hook load H as inputs to both the real drilling
process and to a model thereof. The output of the model
yin, comprising downhole rotary speed a weight-on-bit W
and torque on bit T, is compared to the measured output
vector y of the drilling process. The comparison is used
to adapt parameters of the model to the real drilling
system. The model ifself however remains the same.
Disadvantage of the system is the requirement to
measure downhole. As disclosed in Rudat et al., in the
bottom hole assembly a dynamics measurement tool was
positioned close to the bit enabling the measurement of

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the downhole parameters downhole rotary speed a weight-
on-bit W and torque on bit T. The model parameters
related to the measured values have to be transmitted to
surface, using limited bandwidth telemetry systems.
US-2009/229882 discloses a system for active
vibration damping which relies on downhole sensors to
measure motions of the drill string.
In practice, it is difficult to accurately measure
the angular position and speed of the downhole disc, i.e.
typically the drill bit. For instance, measurement of
angular position and rotational speed may for instance be
measured using a two-dimensional gravity sensor. During a
slip-phase, wherein the bit suddenly accelerates from a
complete standstill to a rotational speed exceeding the
speed of the top drive, the phase accuracy is often lost.
And not knowing the exact angular position will render
further output of the model inaccurate. Herein, please
note that data transmission rates between a downhole
location and surface, using currently available
techniques, are very low, typically less than 1 Hz. These
low transmission rates allow for instance only one sample
per 15 seconds or less.
In addition, the exact initial angular position of
the drill bit is not known, which implies there is always
an uncertainty or error in the measurement of the angular
displacement of the bit. Since the drill string system is
a non-linear system, for instance due to the friction,
and exhibits multiple steady state solutions for the same
excitation input, this error can drive the system in one
or the other solution. A steady state solution, for
instance, is constant rotational velocity at the top
drive and stick-slip behaviour at the bit. Another steady
state solution, for instance, is constant rotational

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velocity at the top drive and at the bit. Again, this is
also due to the low data transmission rate as explained
above.
The known methods and systems assume a specific
frequency of stick-slip oscillations (vibrations), and
tune the control system to that effect. Such control
strategy is inadequate in case the stick-slip vibrations
occur at a different frequency than the expected
frequency, or when there are multiple vibration
frequencies which may change with operating conditions.
There is a need for an improved method of controlling
vibrations in a drilling system, which overcomes the
drawbacks of the prior art.
In accordance with the invention there is provided a
method of controlling vibrations in a drilling system,
the drilling system including an elongate body extending
from surface into a borehole formed in an earth formation
and a drive system for rotating the elongate body by
providing a drive torque to the elongate body, the method
comprising:
a) operating the drive system to provide a drive
torque to the elongate body, and determining a system
parameter that relates to an uphole parameter of the
drilling system;
b) obtaining a model of the drilling system;
c) applying the model to determine a modeled system
parameter that corresponds to said system parameter;
d) determining a difference between the system
parameter and the modeled system parameter;
e) updating the model in dependence of said
difference, thereby obtaining an updated model;
f) determining from the updated model at least one
modeled parameter of rotational motion, and adjusting the

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drive torque in dependence of each modeled parameter of
rotational motion so as to control vibrations of the
elongate body.
The invention also relates to a control system for
controlling vibrations in a drilling system, the drilling
system including an elongate body extending from surface
into a borehole formed in an earth formation and a drive
system for rotating the elongate body by providing a
drive torque to the elongate body, the control system
comprising:
an operating device for operating the drilling
system to provide a drive torque to the elongate body,
and for determining a system parameter that relates to an
uphole parameter of the drilling system;
a model of the drilling system; and
computer means for applying the model to
determine a modeled system parameter that corresponds to
said system parameter, for determining a difference
between the system parameter and the modeled system
parameter, for updating the model in dependence of said
difference thereby obtaining an updated model, for
determining from the updated model at least one modeled
parameter of rotational motion, and for adjusting the
drive torque in dependence of each modeled parameter of
rotational motion so as to control vibrations of the
elongate body.
By using a model that predicts the responses of the
drilling system, e.g. displacement, angular velocity,
acceleration, frictional torque between drill string and
rock formation, it is achieved that stick-slip vibrations
of the drill string are eliminated for a range of angular
velocities from angular velocities close to zero up to
very high angular velocities. Furthermore, by updating the

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mode 1 in dependence of a difference between the system
parameter and the modeled system parameter, it is
achieved that the parameters of the model converge
rapidly to the parameters of the real drilling system so
that the model accurately represents the state of the
real drilling system. Also, with the method and control
system of the invention, a constructive approach is used
to adjust the drive torque delivered to the drill string
so that there is no need for trial-and-error approach that
can be time consuming. The model can include high system
modes as many as required to simulate the drilling system
accurately for the control purposes. It is robust in terms
of model inaccuracies due to changes in the interaction
between the rock formation and the drill bit or the drill
string (frictional changes, damping changes, etc). The
controller provides information to the drive system to
adjust the drive torque in order to avoid undesirable
drill string vibrations. The adjusted drive torque results
in a winding/unwinding of the drill string able to
eliminate the stick-slip vibrations of the bottom-hole
assembly.
Suitable, said uphole parameter of the drilling
system relates to an uphole torque in the drilling
system. An example of a parameter related to uphole
torque can be a torque parameter provided by a rotary
drive coupled to an uphole end of the elongate body, for
example as available in modern top drives. Alternatively
or in addition a parameter related to uphole torque can
be a torque parameter, such as torque, measured at an
uphole position of the elongate body.
For example, said uphole parameter of the drilling
system suitably relates to torque (T) in the elongate
body at or near the earth's surface.

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In one embodiment, said model of the drilling system
includes a modelled torsional stiffness (kern) of the
elongate body, and wherein said drilling parameter
comprises a ratio of said torque (T) over said modelled
torsional stiffness (kern) .
Suitably, said modeled system parameter relates to a
modeled difference between an uphole rotational position
of the elongate body and a downhole rotational position
of the elongate body.
In one embodiment, said uphole parameter of the
drilling system is a first uphole parameter, and wherein
step (c) comprises applying the model using an input
parameter relating to a second uphole parameter of the
drilling system.
Suitably, the drive system comprises a rotary drive
coupled to an uphole end of the elongate body, and
wherein said second uphole parameter is or comprises
torque (Tm) provided by the rotary drive to said uphole
end of the elongate body.
To accurately model the drilling system, the model
suitably includes at least one modeled state parameter
and wherein step (e) comprises adding to each modeled
state parameter the product of said difference and a
respective gain factor pertaining to the modeled state
parameter.
In one embodiment, each modeled state parameter
relates to a modeled parameter of rotational motion of
the elongate body.
Suitably, said at least one modeled state parameter
is selected from a modeled difference between an uphole
angular velocity and a downhole angular velocity of the
elongate body, a modeled uphole angular acceleration of

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the elongate body, and a modeled downhole angular
acceleration of the elongate body.
In one embodiment, step (b) comprises obtaining a
state observer in which the model is included, the state
observer further including a gain module for calculating
each said gain factor.
Suitably, said at least one modeled parameter of
rotational motion includes at least one of a modeled
difference between an uphole rotational position and a
downhole rotational position of the elongate body, a
modeled uphole angular velocity of the elongate body, and
a modeled downhole angular velocity of the elongate body.
The term "uphole" may refer to locations within, for
example, 200 m from the earth surface or from a drilling
rig used in the method of the invention. In case of an
offshore operation, the earth surface is formed by the
seabed. The term "downhole" may refer to locations within,
for example, 200 m from the lower end of the elongate
body. Suitably, the elongate body comprises a drill
string having a drill bit at its downhole end.
The invention will be described hereinafter in more
detail by way of example, with reference to the drawings,
in which:
Figure 1 schematically shows a drilling system to be
controlled by a preferred embodiment of the method and
control system of the invention;
Figure 2A schematically shows an embodiment of the
control system in modular form;
Figure 2B shows a schematic representation of an
embodiment of the control system of the invention;
Figures 3a, 3b, 3c, 4a, 4b and 4c schematically show
various results achieved using the method and control
system of the invention.

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In the description, like reference numerals relate to
like components.
Fig. 1 shows a drilling system 1 including a drill
string 2 extending from surface into a borehole (not
shown) formed in an earth formation. The drill string 2
can be several thousand meters in length, and therefore
behaves as a torsional spring. A drive system 4 is
arranged at surface to rotate the drill string 2 in the
borehole by providing a drive torque to the drill string
2. The drive system generally includes a motor arranged
to drive a rotary table or a top drive (not shown). The
drill string 2 typically includes a downhole end part 6.
Said downhole end part may include a bottom hole assembly
(BHA) 6 including a drill collar having an increased
weight which provides the necessary weight on bit during
drilling. Top drive may imply a drive system which
rotates an upper end of the drill string. Upper end
implies the end at surface, i.e. near the location where
the drill string is suspended from a drilling rig.
Reference sign 7 represents torque resistance Ti of
the upper part of the drill string, e.g. due to
electrostatic forces in the motor, friction in the ball
bearings, etc. Reference sign 8 represents torque
resistance T of the lower part of the drill string due to
interaction of the BHA with the rock formation and the
drilling mud.
The following parameters are used in the discussion
below:
Tin: drive torque provided by the drive system 4 to the
drill string 2;
V: voltage input to a motor (not shown) of the drive
system 4;

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T: torque in the drill string 2 as determined at or
near the earth surface;
u: an update value for controlling drive torque;
0õ,(91: rotational position of the drill string 2 at
respective uphole and downhole locations;
G, di : rotational velocity of the drill string 2 at
respective uphole and downhole locations;
G, di : rotational acceleration of the drill string 2
at respective uphole and downhole locations;
_ . ..
0,,: model estimates of respective parameters
Ou , du , ou ;
observer estimates of respective
parameters Ou, 01, 0/, o/ ;
du,eqr ,eq d = deq = equilibrium values of Ou and 0/ ;
1 =
Ouuu,0/uu: equilibrium values of Ouand 0/.
Furthermore, arrow 9 (Fig. 1) refers to the parameters
arrow 10 refers to the parameter Tõ,, and arrow
12 refers to the parameters 0
1 r di r oi =
Generally, the subscript "u" ("upper") refers to an
uphole position, preferably at or near the surface of the
earth, and the subscript "1" refers to a downhole
position, preferably at or near the downhole end of the
elongate body. A bar above a symbol indicates a modelled
parameter. A dot above a symbol refers to a single time
derivative, i.e. a single dot indicates a velocity, and a
double dot indicates acceleration. A "hat" above a symbol
(such as 't4i) refers to a parameter of a state observer.
The subscript "eq" refers to an equilibrium value, that is
a value for a state in which the system is free of
vibrations. When the system is in equilibrium, the bit

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will generally rotate at the same frequency as the drill
string at the connection to the motor. Angular velocity is
also referred to as rotational velocity.
Uphole parameters of the drilling system 1 are
determined at or near surface for use in the method of
the invention. At or near surface implies that accurate
measurements can be obtained using high-frequency
sensors. High-frequency is for instance exceeding 1 kHz,
i.e. more than 1000 samples per second. One such uphole
parameter relates to uphole torque T in the upper part of
the drill string 2. In the practice of the invention, the
torque Tin applied in a modern drive system or a parameter
directly related to Tin is often available as a digital
parameter. Generally T differs slightly from Tm due to,
for example, friction in the drive system itself and/or
higher-frequency contributions that may not be transmitted
between the drive system and the drill string. In case the
drive system 4 includes a rotary table, such difference
also can be due to transmission losses. In any event,
uphole torque T or a parameter directly related to T can
be determined for example by measuring, e.g. by a torque
sensor at a location at or near the earth surface. Further
uphole parameters can be measured by suitable sensors.
Uphole rotary velocity du or a related parameter may
also be measured by a sensor at or near surface. Such
related parameter is for example a period of one rotation
of the drill string 2 at an uphole position. The period of
rotation is directly related to and representative of
angular velocity.
Fig. 2A shows a block diagram of a control system for
controlling vibrations in the drilling system 1. The
control system comprises a state observer 14 for
providing an estimate of the state of the drilling system

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1. The state observer 14 may use measurements of the
input and the output of the drilling system 1. The state
observer 14 includes a mathematical model 16 of the
drilling system and a gain module 18 for updating the
model 16. The gain module may use input and output
measurements of the drilling system 1. The model 16 may
typically be implemented in a computer system running
software, e.g. written in Matlab. It is known in the art
how to build a model for a given drill string, and for
the drill string in the borehole. The model 16 can be a
simple two degree-of-freedom (DOF) model, e.g. similar to
the one used in section 6.2.2. of the Doris publication.
The model can also be a more complex multi degree-of-
freedom model. It is also possible to derive a two degree-
of-freedom model from a multi degree-of-freedom model
using model reduction techniques known per se. The skilled
person knows how to build a model that describes the
dynamics of a specific drilling system accurately enough
for the controller needs, by including sufficient eigen-
modes of the drilling system. The control system further
comprises a controller 20 arranged to control a motor 22
that drives the drill string 2.
The state observer 14 may receive an input signal 24
representing Tm. In practice, said motor torque Tm is
available to the driller, as it may be derived from the
current drawn by the top drive. Input signal 24 may also
include T. The model 16 provides output signals 28, 30, 32
representing respective parameters du, di, 6 . d, di are
supplied to both the gain module 18 and the controller 20.
di is supplied to the controller 20. The controller 20
also receives an input signal 34 representing parameter du
and an input signal 36 representing parameters

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¨ 15 ¨
eq = 8u,eq 81,eq = Furthermore, reference sign 38 represents
voltage V supplied to motor 22, reference sign 40
represents drive torque Tin supplied by motor 22 to drill
string 2, and reference sign 42 represents a gain vector L
supplied by gain module 18 to model 16.
During normal use of the drilling system 1, voltage V
is supplied to the motor 22 and as a result the motor
produces torque Tui that drives the drill string 2 in
rotation. T and dare measured at surface. T may be input
to the observer. The observer output comprises du, di andoi .
These parameters together with 19,4 and eq 8u,eq¨ 81,eq are input
to the controller where they are multiplied by a
controller gain. The controller output is -u which is
input back into the motor. The motor adapts Tui by -u and
supplies the adjusted torque to the drill string 2.
A more detailed description of the way in which the
observer 14, the model 16 and the controller 20 are used,
is presented below.
The equations of motions of the drilling system 1 are
governed by two inertias Ju,J/ , a spring flexibility 1(0,
two frictional torques Tfu,Tfl, and torque input from the
motor T.. Ju is rotational inertia of the top drive and
part of the drill string, J1 is rotational inertia of the
Bottom-Hole-Assembly (BHA) and the remaining part of the
drill-pipe. 1(0 is rotational stiffness of the drill
string. Tfu represents torque resistance in torsional
motion of the upper part of the drill string (e.g.
electrostatic forces in the motor, friction in the ball
bearings, etc.) and Tfl represents frictional interaction
of the BHA with the formation and the drilling mud. The

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differential equations that describe the torsional
dynamics of this system are given in formulas (1)-(8):
.1 u = du + 1(0 = (du - 91) + T fu(du) - T. = 0 (1)
.1 i = di - 1(0 = (du - 90+ T fi(di) = 0 (2)
T.(du) = sgn( du ) for du #O (3)
Tfu(Ou)E
[-T. + AT. ,T. + AT.] for du = 0 (4)
T.(du) = T. + AT. = sgn( Ou ) + bõ =Idul+ Abu .O (5)
Tfl (di ) E { Ta ( di ) = sgn( Oi) for di # 0 (6)
[-Tsi,Tsi] for di = 0
(7)
8,
--su
T1(91) =T1 +(T - Ta) = e `D' +b1.191" (8)
wherein:
77,õ , AT , bõ , Abu, Ta , Tsi , bi are constant parameters
governed by frictional torque in the upper part of the
drill string, such as in the drive system, and in the
Bottom-Hole-Assembly. Example values of these parameters
are presented in Table 1 at the end of this section. The
torque T in the drill string 2 at or near surface is:
T = 1(0 = (du - 91) .
The torque T can be derived from the current in the
motor 22, and in practice it is always available to the
drill-string operator.
Equations (9), (10) below are a copy of equations (1),
(2) however with some disturbances included in the
parameters k8,Ju,.11,Tfrand Tfl. These disturbances are used
to simulate modelling inaccuracies when deriving a model
for an oil-field drilling system. This set of equations
(called: model of the drilling system) will be used to
build the observer shown in the cascade of Figure 2A.

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=
jumun(9u9Tjm(9u)Tm00 (9)
J im = W - k oin = (Wu - W) + T Jim (W) = 0 (10)
where kom LIM 1irn,Tjrnajid T Jim are model values of
respective parameters 1(0, Ju,.//,Tfu,Tfy . Tfum and Tfym have the
same structure as Tfu and Tfy respectively. The parameters
of the frictional forces of the model that are different
from those of the drilling system are T. (instead of
Tm), 4. (instead of bu Tam (instead of Td) and
km (instead of I)/ ) .
In state-space form the dynamics of the drilling
system can be written as:
= x2 - x3
1
X2 =r ko Tfu (X2 ) Tin
J (11)
r
X3 =1 [ko - Tfy (X3 )]
where x1 = Ou - , x2 = 19t, and x3 = .
In state-space form the dynamics of the model of the
drilling system can be written as:
-
= 1
[-ken, - Tfun, Tml
len (12)
= 1
= Rom T jim(73)]
Jim
where T1 =8-6, = cou and x3 = .
In state-space form the observer is represented by the
following expression:

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T
11
kat,
1
*µ2. = kOm Tfum (1C2 Tm +12(-- *µ1)1
(13)
urn
em
17/ T
= -L/ComAj / fin, /3
J1111 kan
where /1,/2and /3 are observer gains and
:i2=kand =41; . The
gain vector L = [/1, 4, 131.
By applying the observer design techniques disclosed
in Chapters 5 and 6 of the Doris publication, the values
4, 4, 13 can be computed. The goal of the observer is to
derive estimates of the drilling system states that are as
close as possible to the real drill-string system's
states. To derive the observer gains 4, 4, 4, a set of
linear matrix inequalities (LMIs) is to be solved, for
example using the software Matlab and in particular the
Matlab toolbox LMI-tool. The procedure to derive these
LMIs is described in the Doris publication.
In a further step, an adjustment to the drive torque
is applied for torque control so as to control
vibrations. The adjustment takes the following form in
this example:
u = -k1 [u - - 601,,q)1- k2 - d),,,,q)1-
k3=01- di,eq] (14)
where the subscript eq refers herein to equilibrium values
of the model and the drill-string system. The adjustment u
to the drive torque is calculated using modelled downhole
parameters of rotational motion. keq and di,gare equal to
each other, as these are the desired values of the
drilling system while drilling because no stick-slip
vibrations occur when they are equal. Hence the bit will

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rotate at the same rotational speed as the drill string at
the connection to the drive system.
In order to calculate 0g,eq,01,eq r the acceleration
component in equations (9) , (10) is nullified (i.e.
. .
keg = 0, 4,eg = 0 ) r substitutions e, = k,eg = Oeg,O = 01,eg = kg are
performed, and equations (9), (10) are solved:
kom = (Wu eq - 4 eq ) + Tfum (deg )- Tm = 0
-k = (Wq eq - 4 eq)+T flm(deq) = 0
so that:
kom = (tTou eq - 4 eq) + Tfum (deq)-Tm = -kom = (Wu eq - 4 eq)+Tflm(deq)
and:
= Tflm ¨ Tfum(deq) Tin
u eq ul eq )
2kom
Herein, (keg ¨ 4,N) is constant. Formulas (9) and (10)
may be used to derive either (keg ¨ 4,N) r i.e. the output of
^ ^
the model 16 or (149
, ti eq ¨ 01 eq) r i.e. the output of the
observer 14. Also, (Og eg ¨ O eg) may be derived. Regarding the
latter, optionally a measured value for k eg may be
included, for instance provided by sensor 54. In formula
(14), k1,k2,k3 are constants calculated according to the
control theory of the Doris publication using the model
(9) , (10) . Example values of these parameters are
presented in Table 1.
Formula (14) provides a correction factor to the
torque. The corrected torque Tc applied to the drill
string, after the adjustment, is for instance:
T, =Tm - u

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This torque is then substituted in equation (1) replacing
Tm.
Fig. 2B shows a schematic flow scheme, representing
the above described control system of the invention in an
alternative form.
A driller 50 operates a drilling rig (not shown)
comprising the drive system 22. The driller 50 sets the
voltage input V by signal 38 to the drive system 22. In
response to the voltage signal 38, the drive system 22
will try to rotate the drill string 2 of the drilling
system 1 at a reference rotation nref.
Thus, the reference rotation nref is set by the
driller, by signal 38. When the drill string system would
rotate in equilibrium, the equilibrium rotary speed at
surface k, and downhole Oieg would be equal to the set
reference rotation flref:
keq = I eq = Klref
These values are therefore readily available, and may
be provided for instance by the drive system 22, see
signal 52.
To rotate the drill string, the drive system 22
provides a motor torque Tm to the drilling system 1. In
response to the received motor torque Tm, the drill string
and drill bit of the drilling system 1 will rotate. A
resulting output vector y may include rotary position and
rotary speed both at surface and downhole. However, in the
system of the invention, downhole components of the output
vector y of the drilling system may be disregarded. Only
one or more uphole components, which can be accurately
measured, are required. Downhole measurements are
obviated.

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It is for instance sufficient to measure the rotary
speed co, = k at the connection between the drive system
and the drilling system, for instance using sensor 54.
Sensor 54 may be a separate module, or may be included in
the drive system 22. Said surface rotary speed co, = k may
be provided to the controller 20. See signal 34.
A measured torque value, for instance the motor torque
Tm, may be provided to the model 16 and to the model gain
module 18. See signals 24.
In response to receiving the motor torque Tm, the
model provides a model output vector yin. Said model output
vector yin may comprise angular position and rotary speed
both at surface and downhole respectively.
The signal 52 may also comprise the value of (9-9),
which is available due to the relation thereof to the
torque T in the drill string 2 at or near surface:
T = 1(0 = (9õ - 91) or
T
(Ou
Ice,
The torque T can be derived from the current in the drive
system 22. In practice the value T is available to the
operator 50. Otherwise, T can be measured accurately at or
near the connection between the drive system 22 and the
drill string 2.
When the drilling system rotates in equilibrium or
steady state, the above also provides:
in _ 44 \ = Teq
( i u ,eq u 1 ,eq )
k 0
The value of (0u,eq-01,eq) may thus be derived from the
torque value when the drilling system operates at
equilibrium.

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The value (04-0) may be provided to the control module
20 via signal 52. Alternatively, the contol module 20 may
calculate the value (04-0) using the torque T as provided
by the drive system.
The model module 16 may be provided with any suitable
model of the drilling system 1. Using the input signal 24,
which comprises the motor torque Tm, the model module
provides the model output vector ym.
The model output vector ym comprises for instance the
A A
modelled rotary positions 0.,0/ at surface and downhole
respectively. Also, ym may comprise modelled downhole
: I
rotary speed 0/=00/õ. Herein, Mon and 0/ are both
representations of the modelled downhole rotary speed.
The modelled rotary positions 0.,0/ at surface and
downhole respectively may be provided to a model gain
module 18. See signal 56. The model gain module 18
calculates the gain vector L = [4, 4, 1.3], as described
above relating to Formula (13). The model gain module
provides the gain vector L to the model module 16. See
signal 58. The model module 16 uses the gain vector L to
improve the parameters of the model, and consequently to
improve the output vector ym.
A A
The values of 0.,0/,(0/, as provided by the model module
16, which preferably have been adjusted and improved using
the input of the gain module 18, are provided to control
module 20. See signal 60.
The control module 20 uses the available inputs
(included in signals 34, 52, and 60) in formula (14) to
provide a torque correction factor u:
u =-1(1.0,,-19"1¨(6,,,,,,-91,,,)]-1(2=A-9,,,,,)1¨k3=[19"1¨d1,,,1

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Until the drilling system is in equilibrium, and given
the available inputs, said formula can also be written as:
u = -1(1.[E, - 0, -(1)1-k2=Vou -S1õf)]-k3=[coljn-Slref]
The torque correction factor u is provided to the
drive system 22, which substracts said factor u from the
motor torque Tin, to arrive at a corrected torque value Tc:
T, =Tin -u
The corrected torque Tc is then substituted in equation
(1) replacing Tin.
Herein, please note that the reference frequency Q.ref
as set by the driller 50 is not affected by the above
correction. Rather, the correction is applied to adjust
the torque that the drive system applies to the drill
string to arrive at said Qiref. .
Reference is further made to Figs. 3a-c showing
results for the drill string 2, the model 16 and the
observer 14, whereby the controller 20 is de-activated in
order to illustrate convergence of the drill string states
as determined by the model 16 and the observer 14 to the
real drill string states. Fig. 3a shows a=0õ-0, as a
function of time t. Fig. 3b shows cou = ot, as a function of
time t. Fig. 3c shows co/ = di as a function of time t.
These figures indicate that the drilling system's state as
determined by the observer 16 rapidly converges to the
drilling system's real state.
Reference is further made to Figs. 4a-c showing
results whereby the controller 20 is activated. Herein the
control system operates in closed-loop with the drilling
system so as to dampen stick-slip behaviour of the drill
string 2. Fig. 4a shows a=0-0, as a function of time t,
Fig. 4b shows cou =du as a function of time t, and Fig. 4c

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shows w/ = di as a function of time t. These figures
demonstrate that the control loop is able to rapidly
eliminate the stick-slip behaviour of the drilling system.
Rapidly herein implies for instance in less than a minute.
Example values of the various parameters discussed
hereinbefore are presented in Table 1 below.
The present invention is not limited by the above-
described embodiments thereof, wherein many modifications
are conceivable within the scope of the appended claims.
Features of respective embodiments may for instance be
combined.

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Table 1
Parameter Value Unit
J u 0.4765 kg = m2
J, 0.0414 kg = m2
Tsu 0.37975 N = m
ATsu ¨0.00575 N = m
(1)
_p
cl) bu 2.4245 kg = m2 I rad = s
>,
cl)
o-1 Abu -0 . 008 4 kg=m2/rad=s
-H
,¨I ko 0.0775 N = m I rad
¨1
--1
0.2781 N = m
121 T
si
I'd 0.0473 N=m
0), 1.4302 rad I sec
6, 2.0575 [¨]
b1 0.0105 kg = m2 I rad = s
Icon 0.0787 N = m I rad
J. 0.5003 kg = m2
¨1
Tsum 0.3987 N = m
a)
-0
o bum 2.5457 kg = m2 I rad = s
X
Jun 0.0455 kg = m2
Tam 0.052 N = m
bun 0.0116 kg = m2 I rad = s
o kl 14.5 N = m I rad
_p
= (1)
k 1.5 N = m = sect rad
o ¨1 2
k3 30 N = m = sec/ rad
> II 13.5751
U)
m
12 -4.458
,Q
0 CD
/3 -152.204

Representative Drawing