Distributed parameter model

The oscillation of a physical system can be reproduced by using the wave

equation. Bailey and Finnie (1960) and Finnie and Bailey (1960) were one of

the earliest researchers to develop dynamic models using the classical wave

equation to describe the stick-slip oscillation behaviour of a drillstring supported

by experimental validation.

The general wave equation which is used to describe a drillstring of length , subjected to a purely torsional excitations can be written as (Challamel 2000;

Boussaada et al. 2012):

𝑏

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Where is the drillstring twist angle which depends on the drillstring length and time . The parameters 𝑏 and are the shear modulus, the geometrical moment of inertia, damping and inertia mass moment respectively.

The boundary condition, which is used to solve the general wave equation of a

drillstring (equation 2.1), depends on the dynamics of the drillstring at the upper

and lower parts. Challamel (2000) used the following boundary condition to

solve equation 2.1. 2.2 2.3

The boundary equations in equation 2.2 and 2.3 assume that the speed at the

top of the drillstring is restricted to a constant value which represents the speed of the motor, while the bottom of the drillstring is represented by a

lumped inertia, , of the BHA and the bit subjected to torque which is a function of speed at (total drillstring length). These boundary conditions constrain the dynamic behaviour at the BHA, however the velocity of the motor

the does not match the rotational speed at the top of drillstring and this slight difference results in the local torsion at the top of the drillstring. In order to

overcome this limitation Saldivar et al. (2011) and Saldivar and Mondié (2013)

presented the following boundary condition:

𝑏

2.4

The analysis and simulations of a distributed parameter model are very complex

tasks especially when it subjected to nonlinearities and uncertainties. To solve

this problem, the distributed model was simplified by ignoring the minor

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the dynamic behaviour. A neutral-type time delay equation was used to simplify

the distributed model by transforming the partial differential equation of the

model to that of a delay system of a neutral type.

This is a suitable method to simplify the model of the drillstring, by ignoring the

damping along the drillstring, because most of the energy dissipation in drilling

systems takes place at the bit-rock interaction (Saldivar et al. 2011; Boussaada

et al. 2012; Boussaada et al. 2013). The distributed parameter model of the

drillstring (equation 2.1) can be reduced to the unidimensional wave equation

(Saldivar et al. 2011).

𝑝 _2.5

Where 𝑝 is a constant and √ .

The modelling of stick-slip oscillation using a lumped approach based on the

assumption that the mass, damping and stiffness of the system can be

represented at a certain discrete points, results in a model that is fast and

efficient for analysis and can be used for control when compared with finite

element method (FEM). However, in real systems, these parameters are

distributed, and therefore most accurate way to determine the nature and the

magnitude of influence of these parameters on the system behaviour is by

representing them as a distributed.

Therefore, Finite Element Methods (FEM) focussed on the distributed approach,

have been used to study the drillstring vibration by considering the main

parameters of system (mass, inertia, and damping) to be distributed along the

drillstring. Millheim et al.(1978) is one of the earliest published articles in which

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Apostal et al.(1990) used FEM to investigate the harmonic response of the

bottom hole assembly (BHA). Included in this study were the effects of damping

due to mud viscosity and structural damping along the BHA. However, the

damping along the drillpipe was neglected which is considered a significant

factor when the length of drillsring is increased.

Khulief and Al-Naser (2005) used a Lagrangian approach to formulating the FE

model to describe a rotating vertical drillstring that included drillpipe and

drillcollar. The coupling between torsional and bending vibrations, gyroscopic

effect and axial stiffness were considered in this study. This model was able to

predict the more simplistic response of the drilling operation, but cannot predict

correctly the dynamical response of a real system due to the fact that the model

is too simple compared to a real system and uncertainties are not taken into

account.

Khulief and Al-Sulaiman(2007) calculated the time-response of the drillstring

system in the presence of stick-slip excitations by developing a dynamic model

of the drillstring which included the drillpipe and drillcollar and used the

Lagrangian approach in conjunction with the finite element method to derive the governing equations of motion. Whilst the model accounted for the stick-slip

interaction forces the hydrodynamic damping due to the presence of drilling

mud are were not modelled or investigated which was a limitation of the study.

Models built using the finite element method are computationally expensive and

inefficient when compared to lumped parameter models. Whilst they do provide

additional detail or fidelity, it is with both great computational and time expense

and it can be seen that the models are not compatible with real-time

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